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Automated Variance Reduction Technique for 3-D Monte Carlo Coupled Electron-Photon-Positron Simulation Using Determinist...

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Title: Automated Variance Reduction Technique for 3-D Monte Carlo Coupled Electron-Photon-Positron Simulation Using Deterministic Importance Functions
Physical Description: 1 online resource (185 p.)
Language: english
Creator: Dionne, Benoit
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Three-dimensional Monte Carlo coupled electron-photon-positron transport calculations are often performed to determine characteristics such as energy or charge deposition in a wide range of systems exposed to radiation field such as electronic circuitry in a space-environment, tissues exposed to radiotherapy linear accelerator beams, or radiation detectors. Modeling these systems constitute a challenging problem for the available computational methods and resources because they can involve i) very large attenuation, ii) a large number of secondary particles due to the electron-photon-positron cascade, and iii) large and highly forward-peaked scattering. This work presents a new automated variance reduction technique, referred to as ADEIS (Angular adjoint-Driven Electron-photon-positron Importance Sampling), that takes advantage of the capability of deterministic methods to rapidly provide approximate information about the complete phase-space in order to automatically evaluate variance reduction parameters. More specifically, this work focuses on the use of discrete ordinates importance functions to evaluate angular transport and collision biasing parameters, and use them through a modified implementation of the weight-window technique. The application of this new method to complex Monte Carlo simulations has resulted in speedups as high as five orders of magnitude. Due to numerical difficulties in obtaining physical importance functions devoid of numerical artifacts, a limited form of smoothing was implemented to complement a scheme for automatic discretization parameters selection. This scheme improves the robustness, efficiency and statistical reliability of the methodology by optimizing the accuracy of the importance functions with respect to the additional computational cost from generating and using these functions. It was shown that it is essential to bias different species of particles with their specific importance functions. In the case of electrons and positrons, even though the physical scattering and energy-loss models are similar, the importance of positrons can be many orders of magnitudes larger than electron importance. More specifically, not explicitly biasing the positrons with their own set of importance functions results in an undersampling of the annihilation photons and, consequently, introduces a bias in the photon energy spectra. It was also shown that the implementation of the weight-window technique within the condensed-history algorithm of a Monte Carlo code requires that the biasing be performed at the end of each major energy step. Applying the weight-window earlier into the step (i.e., before the last substep) will result in a biased electron energy spectrum. This bias is a consequence of systematic errors introduced in the energy-loss prediction due to an inappropriate application of the weight-window technique where the actual path-length differs from the pre-determined path-length used for evaluating the energy-loss straggling distribution.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Benoit Dionne.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Haghighat, Alireza.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021718:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021718/00001

Material Information

Title: Automated Variance Reduction Technique for 3-D Monte Carlo Coupled Electron-Photon-Positron Simulation Using Deterministic Importance Functions
Physical Description: 1 online resource (185 p.)
Language: english
Creator: Dionne, Benoit
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Three-dimensional Monte Carlo coupled electron-photon-positron transport calculations are often performed to determine characteristics such as energy or charge deposition in a wide range of systems exposed to radiation field such as electronic circuitry in a space-environment, tissues exposed to radiotherapy linear accelerator beams, or radiation detectors. Modeling these systems constitute a challenging problem for the available computational methods and resources because they can involve i) very large attenuation, ii) a large number of secondary particles due to the electron-photon-positron cascade, and iii) large and highly forward-peaked scattering. This work presents a new automated variance reduction technique, referred to as ADEIS (Angular adjoint-Driven Electron-photon-positron Importance Sampling), that takes advantage of the capability of deterministic methods to rapidly provide approximate information about the complete phase-space in order to automatically evaluate variance reduction parameters. More specifically, this work focuses on the use of discrete ordinates importance functions to evaluate angular transport and collision biasing parameters, and use them through a modified implementation of the weight-window technique. The application of this new method to complex Monte Carlo simulations has resulted in speedups as high as five orders of magnitude. Due to numerical difficulties in obtaining physical importance functions devoid of numerical artifacts, a limited form of smoothing was implemented to complement a scheme for automatic discretization parameters selection. This scheme improves the robustness, efficiency and statistical reliability of the methodology by optimizing the accuracy of the importance functions with respect to the additional computational cost from generating and using these functions. It was shown that it is essential to bias different species of particles with their specific importance functions. In the case of electrons and positrons, even though the physical scattering and energy-loss models are similar, the importance of positrons can be many orders of magnitudes larger than electron importance. More specifically, not explicitly biasing the positrons with their own set of importance functions results in an undersampling of the annihilation photons and, consequently, introduces a bias in the photon energy spectra. It was also shown that the implementation of the weight-window technique within the condensed-history algorithm of a Monte Carlo code requires that the biasing be performed at the end of each major energy step. Applying the weight-window earlier into the step (i.e., before the last substep) will result in a biased electron energy spectrum. This bias is a consequence of systematic errors introduced in the energy-loss prediction due to an inappropriate application of the weight-window technique where the actual path-length differs from the pre-determined path-length used for evaluating the energy-loss straggling distribution.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Benoit Dionne.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Haghighat, Alireza.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021718:00001


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AUTOMATED VARIANCE REDUCTION TECHNIQUE FOR 3-D MONTE CARLO
COUPLED ELECTRON-PHOTON-POSITRON SIMULATION
USING DETERMINISTIC IMPORTANCE FUNCTIONS




















By

BENOIT DIONNE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































2007 Benoit Dionne
































To my parents, Normand and Mireille, without whom this long road to a boyhood dream would
not have been possible.









ACKNOWLEDGMENTS

I would like first to express my thanks to Dr. Haghighat for his support and guidance over

the years as well as to the committee members. I would also like to gratefully acknowledge the

many members of the UFTTG next to whom I spend these many years. In particular, Mike and

Colleen, whose discussions, friendships and willingness to listen to my rants gave me the support

I needed.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IS T O F T A B L E S ............. ..... ............ ................. ............................ ............... 8

LIST OF FIGURES .................................. .. .... ..... ................. 10

A B S T R A C T ................................ ............................................................ 14

CHAPTER

1 INTRODUCTION ............... .......................................................... 16

O b j e ctiv e ................... ...................1...................7..........
L iteratu re R ev iew .............................................................................18

2 T H E O R Y ................... ...................2...................8..........

M onte Carlo Transport Theory: General ...................................... ............... 29
Deterministic Transport Theory: Forward Transport ......................................32
Deterministic Transport Theory: Backward Transport .................................................... 34
Electron, Photon and Positron Interactions ................................ ...............36
N um erical C considerations ...............................................................40

3 ADEIS METHODOLOGY CONCEPTS AND FORMULATIONS ........... ............... 49

Im portance Sam pling ...............................................49..........
ADEIS Angular Transport Biasing ................................. ......................... .. ......51
A D E IS S ou rce B iasin g ..................................................................................................... 5 5
A D E IS C ollision B iasin g .................................................................................................. 56
Criteria for Applying Weight-Window .................................................56
Selection of the Adjoint Source ....... ......... ........ .......................................57
Comparison with Methodologies in Literature Review ..................................... .................. 60

4 ADEIS METHODOLOGY IMPLEMENTATION....................................65

M onte Carlo Code: M CN P5 ............................................ ................... ............... 65
Deterministic Codes: ONELD, PARTISN and PENTRAN ................ ...............66
A utom action : U D R ....................................................................................... 67
M modifications to M C N P 5 ..................................................................... ...........................67
Generation of the Deterministic M odel .................. ..................... ................... ....... 68
Generation of the W eight-W window ......................................................... ............... 71
M CNP5 Parallel Calculations ............................................................................ ..... ..........72









5 IMPACT OF IMPORTANCE QUALITY ........................................ ........................ 74

R reference C ase .............................................................76
Im portance Function Positivity......................................................... .......................... 77
Positrons Treatment and Condensed-History in ADEIS ................................ ...............85
C o n c lu sio n s ............................................................................................................................. 9 8

6 IMPROVING THE QUALITY OF THE IMPORTANCE FUNCTION .............................100

Grid Sensitivity and Automatic Spatial Meshing Schemes ..............................................102
Energy Group and Quadrature Order .................................................................... ...... .111
A ngular B iasing .................. ................ ........ ............. ..... .................. 122
Coupled Electron-Photon-Position Simulation.................................................... 126
A joint Source Selection ..................................... .......... ........ .............. ... 128
C onclu sions.......... .........................................................13 1

7 MULTIDIMENSIONAL IMPORTANCE FUNCTION.... ............................133

Generation of 3-D Importance Function Using PENTRAN ................................................133
Generation of 1-D Importance Functions Using PARTISN.................. .............. 143
Biasing Along the Line-of-Sight Using the MCNP5 Cylindrical Weight-Window.............144
Generation of 2-D (RZ) Importance Functions Using PARTISN ............. ................145
Speedup Comparison between 1-D and 2-D (RZ) Biasing ..............................................146
C o n c lu sio n s............................................................................. .14 8

8 CONCLUSIONS AND FUTURE WORK.......................... .................................150

C o n c lu sio n s............................................................................. .1 5 0
F utu re W ork ..............................................................................152

APPENDIX

A V A R IO U S D ER IV A TIO N S .................................................. ......................................... 154

Selection of an Optimum Sampling Distribution in Importance Sampling........................154
Biased Integral Transport Equation ............... ..................... .................................155
Lower-weight Bounds Formulation and Source Consistency ...........................................156
Determination of the Average Chord-Length for a Given Volume................. ......... 157

B IM PLEM ENTATION DETAILS .............. ............ ....... ..................... ............... 158

U universal D river (U D R ) ....................................................................... ..........................158
Perform ing an AD EIS Sim ulation ................. ............................................ ............... 159
A D E IS M C N P 5 Input C ard ................................................................. .........................160









C ELECTRON SPECTRUM BIAS SIDE STUDIES................................161

Impact of Tally Location ................................................ 161
Impact of the Number of Histories on Convergence........................................................... 163
Impact of Electron Energy and Energy Cutoff ........................................... ...............166
Im pact of Lateral Leakage ................. ................................................. ............... 167
Impact of Knock-On Electron Collision Biasing..... .......... ...................................... 168
Impact of Weight-Window Energy Group Structure................................................169
Im pact of K nock-O n Electrons...... .......................... ............................... ............... 171
Impact of the Energy Indexing Schem e ....................................................... .............. 172
Impact of Russian Roulette Weight Balance.............................................173
Impact of Coupled Electron-Photon-Positron Simulation.................................................175
C onclu sions.......... .............................. ...............................................176

LIST OF REFERENCES ......... ......... ........................... ............... ................... 178

B IO G R A PH IC A L SK E T C H ........................................... ......... ................... .......................... 185









LIST OF TABLES


Table page

3-1 Comparison of other variance reduction methodology with ADEIS ................................61

5-1 M materials and dim tensions of reference case ............................. .....................75

5-2 Test case sim ulation param eters ...................................................................... 75

5-3 Reference case spatial mesh structure producing a positive importance function.............79

5-4 Average FOM and RFOM for all approaches.................. ................... 85

5-5 Impact of biasing on annihilation photons sampling .......................................................87

5-6 Impact of explicit positron biasing on annihilation photons sampling ............................89

5-7 Impact explicit positron biasing on average FOM and RFOM........................................ 91

6-1 V various test cases of the analysis plan ..............................................................................101

6-2 Other reference case simulation parameters of the analysis plan.................................... 102

6-3 CEPX S approxim ated detour factors ........................................... ......................... 105

6-4 Calculated detour factors for each case of the analysis plan............................................106

6-5 M materials and dimensions of new simplified test case ..................................................... 107

6-6 Speedup for multi-layered geometries using automatic mesh criterion............................107

6-7 Speedup gain ratios from boundary layers scheme #1 in Cases 1 and 9.........................109

6-8 Speedup gain ratios from boundary layers scheme #2 in Cases 1 and 9.........................111

6-9 Total flux and relative error in the ROI for all cases of the analysis plan.........................114

6-10 Average ratio of track created to track lost for cases with same energy .........................14

6-11 Speedup with angular biasing for Case 7 with a source using an angular cosine
distribution and redu ced size ................................................. ................. ..................... 12 5

6-12 Speedup as a function of the number of energy groups for Cases 6 and 9........................127

6-13 Electron and photon tally speedup using ADEIS with angular biasing ..........................128

6-14 Electron and photon flux tally speedup using different objective particles ....................130









6-15 Energy deposition tally speedup in the reference case for various objective particles .....130

7-1 PENTRAN and ONELD simulation parameters for solving problem #1 .......................134

7-2 PENTRAN and ONELD simulation parameters for solving problem #2.........................139

7-3 Other sim ulation param eters for problem #3 ........................................ .....................141

7-4 Energy deposition tally speedup for ONELD and PARTISN simulation of Chapter 5
referen ce c a se .......................................................................... 14 3

7-5 M materials and dimensions of reference case................................................................... 147

7-6 Energy deposition tally speedup for 1-D and 2-D biasing ............................. .............147

B -l T he A D E IS keyw words ............................................................................ .....................160









LIST OF FIGURES


Figure page

3-1 Field-of-V iew (FO V ) concept............................................................................. .... .. 54

4-1 A utom ated A D E IS flow chart......................................... .............................................65

4-2 L ine-of-sight approach ............................................................................. ....................69

4-3 Two-dimensional model generation using line-of-sight....... .......................................69

5-1 R reference case geom etry ......................................................................... ....................75

5-3 Relative error and variance of variance in ADEIS with importance function
sm nothing ................ ..... .......... ...........................................78

5-4 Relative error and variance of variance in ADEIS with "optimum" mesh structure.........80

5-5 Relative difference between importance with 1st and 2nd order CSD differencing ............81

5-6 Relative error and variance of variance for ADEIS photon tally with 1st order CSD
differencing scheme and 75 energy groups. ........................................... ............... 82

5-7 Relative error and variance of variance in ADEIS with 75 energy groups ....................82

5-8 Relative error and variance of variance in ADEIS with CEPXS-GS ..............................83

5-9 Impact of large variation in importance between positron and photon ...........................87

5-10 Electron and annihilation photon importance function in tungsten target.........................88

5-11 Positron and annihilation photon importance function in tungsten target.........................88

5-12 Surface Photon Flux Spectra at Tungsten-Air Interface................................ ............89

5-13 Relative error and VOV in ADEIS with CEPXS and explicit positron biasing ...............90

5-14 Regions of interest considered in simplified test case .....................................................91

5-15 Relative differences between the standard MCNP5 and ADEIS total fluxes for three
energy-loss approaches ...................... .................... .. .. .......................92

5-16 Relative differences between the standard MCNP5 and ADEIS electron energy
spectrum for Case 3 in the five regions of interest ........................ ................93

5-17 Relative differences in electron spectra for undivided and divided models ....................94









5-18 Impact of the modification of condensed-history algorithm with weight-window on
the relative differences in total flux between standard MCNP5 and ADEIS ....................95

5-19 ADEIS normalized energy spectrum and relative differences with the standard
MCNP5 at 70% of 2MeV electron range with the CH algorithm modification ................96

5-20 Relative differences between the standard MCNP5 and ADEIS at various fraction of
the range ................ .... ......... ............... ................................97

6-1 Sim plified reference m odel............................ ...................................... ............... 101

6-2 Speedup as a function of FOD for Cases 1, 4 and 7 ........................................103

6-3 Speedup as a function of FOD for Cases 2, 5 and 8 ........................................103

6-4 Speedup as a function of FOD for Cases 3, 6 and 9 ........................................104

6-5 M ulti-layered geom entries. ............................................................................. ...............107

6-6 Automatic boundary layer meshing scheme #1 .................................... ............... 108

6-7 Automatic boundary layer meshing scheme #2 ............................ ..... ...........110

6-8 Speedup as a function of the number of energy groups for Cases 1, 2 and 3 ................12

6-9 Speedup as a function of the number of energy groups for Cases 4, 5 and 6................12

6-10 Speedup as a function of the number of energy groups for Cases 7, 8 and 9................13

6-11 Importance functions for source and knock-on electrons at a few energies for Case 7...115

6-12 FOM as a function of the number of energy groups for modified Case 9................116

6-13 Number of knock-on electrons and their total statistical as function of the number of
energy groups for Case 3 ........ .......................... .............. ........ .. 117

6-14 Splitted electron energy as a function position for a 1000 source particles in Case 3.....118

6-15 Rouletted electron energy as a function position for a 1000 source particles in
C ase 3 ......... ... ........................... ............. .................................. 1 18

6-16 Splitted electron weight as a function of position for a 1000 source particles in
C ase 3 ............ .......................................................................... 119

6-17 Importance functions for 15 and 75 energy groups at 3.06 cm for Case 3 .......... ......119

6-18 Impact of discrete ordinates quadrature set order on speedup for all cases of the
an aly sis p lan ............................................................................................... 12 1









6-19 Electron tracks for a 20 MeV electron pencil beam impinging on water (Case 7)..........123

6-20 Electron tracks for a 20 MeV electron cosine beam impinging on tungsten (Case 7
with a source using an angular cosine distribution)....................................................... 124

6-21 Electron tracks for a 0.2 MeV electron pencil beam impinging on tungsten (Case 3)....124

7-1 Problem #1 geom etry ................................... .............. ............... ............ 135

7-2 Importance function for fastest energy group (0.9874 MeV to 1.0125 MeV) in
p rob lem # 1 ...............................................................................13 5

7-3 Ratio of ONELD importance over PENTRAN importance for energy groups in
p rob lem # 1 ...............................................................................13 6

7-4 Mesh refinement to resolve boundary layers at the edges of model for problem #1 .......137

7-5 Ratio of ONELD importance over PENTRAN importance for four energy groups in
problem #1 with mesh refinem ent ............................................................................137

7-6 Ratio of the ONELD and PENTRAN importance functions for group 1 obtained
using S16 and S32 quadrature order with mesh refinement in problem #1 .....................138

7-7 P problem #2 geom etry ...................... .. .. ......... .. ........................... ............................. 139

7-8 M esh structure for problem #2............................................... .............................. 139

7-9 Impact of differencing scheme on importance function for group 20 in problem #2......140

7-10 P problem #3 geom etry .................................................................................... .......... 14 1

7-11 Ratio of ONELD importance over PENTRAN importance for four photon energy
group in problem #3 ..................................... ................. .......... ....... ..... 142

7-12 One-dimensional (R) and two-dimensional (RZ) weight-window mesh along the
line-of-sight in a 3-D geom etry............................................... ............................. 144

7-13 Modified reference case geometry............ ... .... ......... ............... 147

B Exam ple of U D R input file syntax......... ................. .............................. ............... 159

B-2 Exam ples of calls to adeisrun. ............................................... .............................. 160

B-3 N ew simulation sequence in M CNP5 ........................................ ........................ 160

C-l ADEIS normalized spectrum and relative difference with standard MCNP5 for tally
located at 70% of 2 MeV electron range ..... ................................................162









C-2 ADEIS normalized spectrum and relative difference with standard MCNP5 for tally
located at 2 M eV electron range ...................................................... ............... 162

C-3 Relative differences between standard MCNP5 and ADEIS for tally located at 70%
of the 2 MeV electron range at various number of histories.................. ..................... 164

C-4 Norm of relative differences between standard MCNP5 and ADEIS for tally located
at 70% of the 2 MeV electron range at various number of histories ..............................164

C-5 Norm of relative differences between standard MCNP5 and ADEIS for tally located
at the 2 MeV electron range at various number of histories ................. .....................165

C-6 Relative differences between the tally electron spectra from ADEIS and standard
MCNP5 for a 2 MeV electron beam at two energy cutoff...............................166

C-7 Relative differences between the tally electron spectra from ADEIS and standard
MCNP5 for a 13 MeV electron beam at two energy cutoff................... ..................... 167

C-8 Relative differences in spectrum between the standard MCNP5 and ADEIS for
various model sizes and an energy cutoff of 0.01 M eV ............................................... 168

C-9 Relative differences in spectrum between the standard MCNP5 and ADEIS for the
reference case with and without collision biasing for knock-on electrons ................169

C-10 Norm of relative differences between standard MCNP5 and ADEIS for tally located
at the 2 MeV electron range at various number of histories with condensed-history
group structure ........................................................................ ......... 170

C-11 Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with various energy groups................. ........... ............... 171

C-12 Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with and without knock-on electron production..........................172

C-13 Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with the MCNP and ITS energy indexing scheme....................... 173

C-14 Ratios of weight creation over weight loss for 5x105 to 2x106 histories......................... 174

C-15 Ratios of weight creation over weight loss for 4x106 to 1.6x107 histories .................... 174

C-16 Ratios of weight creation over weight loss for 3.2x107 to 1.28x108 histories .................175

C-17 Relative differences between standard MCNP5 and ADEIS for tally located at the
2 M eV electron range in coupled electron-photon model ............................................ 176









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AUTOMATED VARIANCE REDUCTION TECHNIQUE FOR 3-D MONTE CARLO
COUPLED ELECTRON-PHOTON-POSITRON SIMULATION
USING DETERMINISTIC IMPORTANCE FUNCTIONS


By

Benoit Dionne

December 2007

Chair: Alireza Haghighat
Major: Nuclear Engineering Sciences

Three-dimensional Monte Carlo coupled electron-photon-positron transport calculations

are often performed to determine characteristics such as energy or charge deposition in a wide

range of systems exposed to radiation field such as electronic circuitry in a space-environment,

tissues exposed to radiotherapy linear accelerator beams, or radiation detectors. Modeling these

systems constitute a challenging problem for the available computational methods and resources

because they can involve; i) very large attenuation, ii) large number of secondary particles due to

the electron-photon-positron cascade, and iii) large and highly forward-peaked scattering.

This work presents a new automated variance reduction technique, referred to as ADEIS

(Angular adjoint-Driven Electron-photon-positron Importance Sampling), that takes advantage

of the capability of deterministic methods to rapidly provide approximate information about the

complete phase-space in order to automatically evaluate variance reduction parameters. More

specifically, this work focuses on the use of discrete ordinates importance functions to evaluate

angular transport and collision biasing parameters, and use them through a modified

implementation of the weight-window technique. The application of this new method to complex

Monte Carlo simulations has resulted in speedups as high as five orders of magnitude.









Due to numerical difficulties in obtaining physical importance functions devoid of

numerical artifacts, a limited form of smoothing was implemented to complement a scheme for

automatic discretization parameters selection. This scheme improves the robustness, efficiency

and statistical reliability of the methodology by optimizing the accuracy of the importance

functions with respect to the additional computational cost from generating and using these

functions.

It was shown that it is essential to bias different species of particles with their specific

importance functions. In the case of electrons and positrons, even though the physical scattering

and energy-loss models are similar, the importance of positrons can be many orders of

magnitudes larger than electron importance. More specifically, not explicitly biasing the

positrons with their own set of importance functions results in an undersampling of the

annihilation photons and, consequently, introduces a bias in the photon energy spectra.

It was also shown that the implementation of the weight-window technique within the

condensed-history algorithm of a Monte Carlo code requires that the biasing be performed at the

end of each major energy step. Applying the weight-window earlier into the step, i.e., before the

last substep, will result in a biased electron energy spectrum. This bias is a consequence of

systematic errors introduced in the energy-loss prediction due to an inappropriate application of

the weight-window technique where the actual path-length differs from the pre-determined path-

length used for evaluating the energy-loss straggling distribution.









CHAPTER 1
INTRODUCTION

Three-dimensional Monte Carlo coupled electron-photon-positron transport calculations

are often performed to determine characteristics such as energy or charge deposition in a wide

range of systems exposed to radiation fields, such as electronic circuitry in a space-environment,

tissues exposed to radiotherapy linear accelerator beams, or radiation detector. Modeling these

systems constitutes a challenging problem for the available computational methods and resources

because they can involve: i) very large attenuation (often referred to as deep-penetration

problem), ii) large number of secondary particles due to the electron-photon-positron cascade

and iii) large and highly forward-peaked scattering cross sections.

Monte Carlo methods are generally considered more accurate since very complex 3-D

geometries can be simulated without introducing systematic errors related to the phase-space

discretization. However, even for problems where source particles have a reasonable probability

of reaching the region of interest, tracking large number of secondary particles may result in

unreasonable simulation times. Therefore, selecting the right particles through the use of

variance reduction (VR) techniques can be highly beneficial when such performing coupled

electron-photon-positron simulations for complex 3-D geometries. Note that hereafter, the

expressions variance reduction and biasing will refer to fair-game techniques that achieve

precise unbiased estimates with a reduced computation time unless implied otherwise by the

context. Typical biasing methodologies often require a lot of experience and time from the user

to achieve significant speedup while maintaining accurate and precise results. It is therefore

useful for a VR methodology to minimize the amount user involvement in order to achieve high

efficiency.









Deterministic approaches solving a discretized form of an equation representing the

particle balance in phase-space have also been used successfully to model systems involving

coupled electron-photon-positron transport. The current deterministic methods presented in the

publicly available literature are limited in scope since they either accurately describe the physical

processes only for 1-D or 2-D geometries, or by obtaining approximate solutions for 3-D

geometries (e.g. fast semi-analytic deterministic dose calculations1). The main advantage of these

deterministic approaches is that they provide information about the whole phase-space relatively

quickly compared to Monte Carlo (MC) methods. However, such methods require even more

experience and time from the user to obtain accurate and precise results in reasonable amount of

time. Therefore, despite the large increase in computer performance, a methodology to perform

more efficient and accurate coupled electron-photon-positron is needed.

Objective

Therefore, the objective of this work is to develop a new automated variance reduction

methodology for 3-D coupled electron-photon-positron MC calculations that takes advantage of

the capability of deterministic methods to rapidly provide approximate information about the

complete phase-space in order to automatically evaluate the VR parameters. Such methodology

significantly reduces the computation time (as shown previously in similar work performed for

neutral particles2) and the engineering time if a sufficient amount of automation is implemented.

Furthermore, it reduces the chance of unstable statistical behavior associated with VR techniques

requiring users to manually select VR parameters.

More specifically, this work focuses on the use of discrete ordinates (SN) importance

functions to evaluate angular transport and collision biasing parameters and accelerate MC

calculations through a modified implementation of the weight-window technique. This

methodology is referred to as Angular adjoint-Driven Electron-photon-positron Importance









Sampling (ADEIS) from hereon. To maximize the increase efficiency and reduce the amount of

engineering time spent on evaluating ADEIS VR parameters, a high level of automation is

implemented. As presented in the literature review, the idea of using importance functions to

accelerate MC calculations is not new, however as far as surveyed, no work has been done to

perform angular transport and collision biasing using deterministic importance functions in

coupled electron-photon-positron problems.

Literature Review

This section presents a summary of previous work performed over the past few decades on

coupled electron-photon-positron MC simulation and their associated variance reduction

techniques, and on coupled electron-photon deterministic methods. Note that some of the work

presented in this section may address only electron transport simulation since the major

difficulties in performing such calculations arise from modeling electron interaction with matter.

Monte Carlo

Monte Carlo (MC) methods were developed in the 1940s by scientists involved in nuclear

weapon research. Based on their work, one of the first accounts of the method was written by

Metropolis and Ulam3 in 1949. Interestingly enough the authors suggested that the method is

inherently parallel and should be applied to many computers working in parallel which seems to

be becoming the standard approach. Nowadays, the term Monte Carlo refers to numerical

methods based on the use of random numbers to solve physical and mathematical problems.

Kalos and Whitlock4 provide a good general survey of various MC techniques with applications

to different fields.

Radiation transport MC calculations simulate a finite number of particle histories by using

pseudo-random numbers to sample from probability density functions (PDF) associated with the

various kinds of physical processes. Statistical averages and their associated variances are then









estimated using the central limit theorem. Note that an overview of the different aspects of the

MC method in the context of radiation transport is presented in Lewis and Miller5 as well as in

Shultis and Faw6.

Over the years many MC production codes have been developed to solve various problems

involving coupled electron-photon-positron processes. Some of the major codes still in use in

various fields are:

* MCNP57, developed at Los Alamos national Laboratory,

* ITS8, developed at Sandia National Laboratory,

* PENELOPE9, developed by university research groups in Spain and Argentina,

* MARS10, developed at Fermi National Accelerator Laboratory,

* EGS11, evolution of EGS3 developed by SLAC and other international agencies,

* GEANT12, developed as a worldwide effort and managed by CERN,

* DPM13, developed at the University of Michigan,

* FLUKA14, developed within INFN (National Institute of Nuclear Physics),

* TRIPOLI-415, developed at CEA (Commisariat de l'Energie Atomique) in France

All these codes use a technique referred to as condensed history (CH) Monte Carlo in order

to circumvent some of the difficulties created by the large interaction rate of electrons (e.g., an

electron slowing down from 0.5 MeV to 0.05 MeV will undergo between 105 to 106 collisions).

This method developed by Berger16 condensed a large number of collisions in a single electron

step. To predict the change of energy and direction of motion at the end of each step, cumulative

effects of individual interactions are taken into account by analytical theories such as Bethe

energy-loss theory1, Goudsmith-Saunderson18' 19 or Moliere scattering theory20, and Landau21

energy-loss straggling theory. Larsen22 has proven that in the limit of an infinitesimal step size,









the CH approach is a solution of the Boltzmann transport equation. A good review and analysis

of CH techniques is presented by Kawrakow and Bielajew23

Variance Reduction

Many techniques have been developed to increase the efficiency of MC calculations by

reducing either the variance or the computation time per history. As discussed more at length in

Chapter 2, these techniques modify the physical laws of radiation transport in an attempt to

transport the particle toward a region of interest without introducing a bias in the statistical

estimates. A complete summary of all the VR techniques is outside the scope of this review;

therefore only techniques particle importance will be presented.

Kalos24 described the importance sampling technique and its relation to the hypothetical

zero-variance solution. Coveyou et a125 developed formulations involving the use of importance

functions to reduce the variance through source and transport biasing. Many studies26'27' 28, 29,30,

32, 33, 34, 36, 37, 38 were also carried out, with various degrees of success, to use a deterministic

approximate importance functions to accelerate neutral particles MC transport calculations.

Tang and Hoffman26 used 1-D importance function along the axial and radial direction of a

shipping cask to bias the neutron reaction rates at two detectors (axial and radial detectors) using

a combination of source energy biasing, energy biasing at collision site, splitting and Russian

roulette, and path-length stretching. This approach suggests that different importance functions

should be used when the biasing objectives are significantly different.

Mickael27 uses an adjoint diffusion solver embedded into MCNP to generate a

deterministic neutron importance map. In this methodology, the group constants are generated by

using an estimate of the flux from a short analog MC calculation. However, this approach cannot

be applied to electron and positron simulation since a diffusion treatment is inadequate for highly

angular phenomena such as high energy electron scattering.









Turner and Larsen28'29 developed the LIFT (Local Importance Function Importance) VR

technique for neutral particle transport. This method uses an analytic formulation that

approximates the solution within each energy group and spatial cell of a MC model. This

formulation is based on an approximate importance solution which can be estimated using

various techniques such as diffusion, SN or SPN. This methodology biases the source distribution,

the distance to collision, the selection of postcollision energy groups, and trajectories. Note that

this approach approximates a theoretical zero-variance method. However, in this version of the

methodology, only linear anisotropic scattering is considered, which makes it unsuitable for

high-energy beam-like problems and most electron/positron simulations.

Van Riper et al.30 developed the AVATAR (Automatic Variance and Time of Analysis

Reduction) methodology for neutral particles. This methodology uses a 3-D importance function

calculated from THREEDANT31 in order to evaluate lower-weight bounds of the MCNP weight-

window using the basic inverse relation between statistical weight and importance. An angular

dependency was introduced into the MCNP weight-window by using an approximation to the

angular importance function. Note that in Ref 30, no mention was made of preserving the

consistency between the source and weight-window definitions.

Wagner and Haghighat32 33 developed the CADIS (Consistent Adjoint-Driven Importance

Sampling) for neutral particles. CADIS uses the concept of importance sampling to derive

consistent relations for source biasing parameters and weight-window bounds. This methodology

implemented within a patch to MCNP is referred to as A3MCNP (Automated Adjoint

Accelerated MCNP) and uses TORT34 to estimate the three-dimensional importance functions.

A3MCNP automatically generates the input for the 3-D transport code by using a transparent









mesh and a back thinning technique. Cross-section mixing and various other tasks are handled

through various scripts.

Shuttleworth et a135 developed an inbuilt importance generator which uses the adjoint

multigroup neutron diffusion equation to estimate the importance function. Note that even if

special diffusion constants are used to provide a closer approximation to transport theory, this is

still inadequate for electron/positron transport.

As reported by Both et a136 and Vergnaud et a137, a fairly automatic importance-generation

technique was implemented in the MC code TRIPOLI. This technique uses exponential biasing,

a form of splitting call quota sampling, and collision biasing. Their importance function is the

product of three factors depending on space, energy and angle but essentially assumes a spatial

exponential behavior. This method requires that the user provides input values of apriori

parameters.

Giffard et al.38 developed a different variance reduction for TRIPOLI which does not

require much apriori expertise. As many others, this methodology uses a deterministic code to

generate an approximate importance function. However, they use that information to create a

continuous-energy importance function using an interpolation scheme. This methodology uses

these importance functions for source biasing, transport kernel biasing, Russian roulette and

splitting.

Wagner39 developed the ADVANTG (Automated Deterministic VAriaNce reducTion

Generator) code based on the previously developed and proven CADIS methodology. The main

difference between ADVANTAG and A3MCNP is that ADVANTG uses the standard MCNP

interface file named wwinp.









The following methodologies do not use deterministic importance functions but are still

importance-based. Goldstein and Greenspan40 developed a recursive MC (RMC) method where

the importance is estimated by solving the forward problem for extensively subdivided geometric

regions. The methodology sprinkles the problem with test particles and tracks them until they

score or die. Scoring particles make a contribution to the importance at their birth location.

Booth41 developed an importance estimation technique known as the weight-window

generator since it generates importance functions to be used in MCNP weight-window. In this

methodology, it is considered that the cell (or mesh) from which a particle emerges after an event

may be considered as the starting point for the remainder of the history. A contribution to the

importance estimate is thus made in every cell (or mesh) that the particle passes through.

Two others methods35 have been implemented in the MCBEND code. In the first method,

the test particles are initially generated in a cell of the importance mesh that contains the detector

and soon generates a reliable value for the importance. These particles are then generated in cells

adjacent to the target and then tracked. If they cross into the target, their expected score is known

from the value of the importance that has already been calculated. Processing then moves on to

next layer of cells surrounding the ones already completed and so on. The second method, named

MERGE, uses an approach similar to weight-window generator. However, the partially

completed importance function is then merged with an initial estimate of the importance.

Murata et al.42 methodology is also similar to the weight-window generator with the

exception that the contribution to the detector, and consequently the importance at the location, is

evaluated for each scattering point using the point detector tally. Note that the authors introduce

the idea of using angular-dependent parameters in form of angular meshes, but do not present

any results.









Finally, a multigroup adjoint transport simulation can be performed in MCNP57 to obtain

biasing functions which are then used through an energy-dependent but cell-based weight-

window. However, this capability requires the use of an undocumented feature of the CEPXS

package51 and the use of the unsupported CRSRD code to generate multigroup libraries that are

suitable to be used in a MC code.

Deterministic Methods

A complete review of all deterministic methods is also out of the scope of this work,

therefore only methods that are of interest for estimating deterministic coupled electron-photon-

positron importance are discussed as well as some early work that pioneered the field.

Spencer's approach43 was one of the first methodology performing numerical calculations

of electron spatial distribution taking into accounts both energy-loss and change of direction.

Spencer's method is based on previous work from Lewis44 and considers monoenergetic and

mono-directional sources distributed uniformly over an infinite plane in homogeneous media.

This work recasts the Lewis equation in terms of residual ranges, and expands it into a series of

spatial, angular and residual range moments. He then proceeded to numerically evaluate those

moments and compare his results with experiments.

Based on our survey, Bartine et a145 were the first to use the SN method to calculate the

spatial distribution of low-energy electrons. In their approach, the scattering integral is rewritten

using an asymptotic approach (Taylor series expansion) to obtain a continuous slowing down

term for representing the small-angle (soft) inelastic collisions. Moreover, to reduce the number

of moments required to represent the highly forward-peaked elastic scattering cross-sections still

represented by an integral kernel, the extended transport correction46 is applied. These

modifications where implemented in the standard code ANISN47. Nowadays, this form of the









transport equation is referred to as the Boltzmann-CSD and constitutes the basis, or is related to,

most of the current practical work on deterministic electron transport.

Melhom and Duderstadt48 modified the TIMEX code49 to provide time-dependent Fokker-

Planck (FP) solutions for one-dimensional slab and spherical geometries assuming that the

scattering can be decomposed in a continuous energy-loss term and a continuous angular

diffusion term. Note that the pure FP solutions are considered inadequate for electrons since they

lack the ability to properly represent the hard collisions.

Morel5o developed a method for performing Boltzmann-FP calculations using a standard

SN production code. In this approach, Morel defines the scattering multigroup Legendre cross-

section in terms of the FP functions. The SN quadrature set must be defined such that the

Boltzmann solution converges to the Boltzmann-FP solution as the SN space-angle-energy mesh

is refined. If the continuous angular diffusion term is neglected, this methodology therefore solves

the Boltzmann-CSD transport equation as in the previous work from Bartine. Note that the

Boltzmann-CSD equation is more amenable for electron transport simulation than either the

Boltzmann and FP equations alone. Lorence and Morel used this approach to develop the

CEPXS/ONELD package5

Przybylski and Ligou52 investigated two numerical approaches to solve the Boltzmann-FP

equation using a discrete ordinates approach for the angular dependency of the angular flux and

the angular diffusion term of the Fokker-Planck scattering kernel. They compared a multigroup

approach to a method which uses a linear-diamond scheme on space and energy. The goal was to

mitigate numerical instabilities usually resulting from the finite-difference approximations of the

derivatives in the Fokker-Planck scattering kernel. Since the linear-diamond scheme is not









guaranteed positive, they also developed a fix-up scheme that preserves the energy balance over

the cell.

Filippone53 developed a methodology, named SMART matrix, to generate more accurate

scattering matrices. This methodology allows the definition of a scattering matrix that guarantees

nearly exact calculations of electron angular distribution at each discrete direction. Using the

Goudsmit-Saunderson theory18, 19, it is possible to find an integral expression for the scalar flux

(as a function of the electron path-length) that can be integrated using a quadrature set. By

comparing the scalar flux expression with the explicit expression obtained from the discretized

SN transport equation (using a characteristic differencing scheme); it is possible to derive an

expression of the scattering kernel for which both formulations are identical. Fillipone also

shows that smart matrices reducing energy and angular discretization errors can be derived for

any discretization scheme.

Drumm54 improved a methodology developed by Morel5o by eliminating the need to

exactly integrate a scattering kernel containing delta functions so that general quadrature sets can

be used instead of the Gauss or the Galerkin sets. This methodology combined Morel and

Filiponne53 approaches, resulting in better scattering cross-sections since they are more often

positive, tend to exhibit smaller values than the true interaction cross-sections, and are not tied to

any specific quadrature set. Note that this work is based on the Boltzmann-FP equation instead of

the Spencer-Lewis equation43'44 used by Filiponne. The author also mentions that the use of

these cross-sections eliminates some of the well-known numerical oscillations present in

CEPXS/ONELD results, and that the convergence rate of the source iteration technique is also

generally much faster.









For completeness, it must be mentioned that besides these SN-like methods, there are other

methodologies which have been used to perform deterministic electron transport. For instance,

Corwan et a155 derived a multigroup diffusion formulation from the Fokker-Planck equation,

Haldy and Ligou56 developed a code to calculate angular and spatial moments of the electron

distribution based on the work by Spencer43, Honrubia and Aragones57 developed a methodology

using a finite element discretization of the Boltzmann-FP equation where either the space-energy

or space-energy-angle variables are treated in a coupled way, Prinja and Pomraning58 developed

a generalized Fokker-Planck methodology which introduced higher order terms in the asymptotic

expansion of the scattering kernel, and finally, Franke and Larsen59 developed a methodology to

calculate exact multidimensional information concerning the spreading of 3-D beams by solving

a coupled set of 1-D transverse radial moment equations. This method is related to the method of

moments developed by Lewis44

The methodology developed in this work, ADEIS, uses a modified version of the MCNP57

weight-window algorithm to implement an angular extension (similar to Ref 42) of the

CADIS32 33 methodology to coupled electron-photon-positron simulations using beam sources.

ADEIS uses the CEPXS/ONELD51, CEPXS-GS54 and PARTISN60 packages to calculate the

deterministic importance functions required to evaluate the VR parameters. Before presenting

this methodology in more detail (see Chapter 3), it is useful to review the major theoretical

concepts (see Chapter 2).














CHAPTER 2
THEORY

While the literature review presented in the previous chapter provided a general

background to the current effort to develop a VR methodology, this chapter provides a

more in-depth theoretical review of some of the aforementioned concepts and

formulations. A quick overview of the MC and deterministic approaches to radiation

transport are presented followed by an introduction to the electron, photon and positron

interaction physics. Finally, the remainder of this chapter discusses various numerical

considerations.

To fully characterize all the particles, the positions and velocities of each particle

before and after each collision must be known. To accomplish this, it is necessary to

describe these collisions in a six-dimensional space (three dimensions for position and

three dimensions for the velocities) called the phase-space. In a general way, the transport

of radiation through matter can be represented by Eq. 2-1.

E(P) = JdP'Q(P')T(P P') (2-1)

This equation describes a source particle (Q(P')) located at P' in phase-space being

transported (T(P' -> P)) to another location in the phase-space P, and contributing to an

average quantity of interest (E(P)) at that location. In nuclear engineering, it is

customary to represent a phase-space element P as d-rdEdQ where di represents the

position component of phase-space and dEdQ represents the velocity component of

phase-space in terms of energy (dE) and direction (dQ).









Monte Carlo Transport Theory: General

The MC method could solve Eq. 2-1 by sampling the integrand using random

numbers. However, the exact probability density function (PDF) of a complex process

such as transporting particles through a 3-D geometry is never known, thus T(P' -> P) is

implicitly sampled by tracking all the microscopic events in the histories of a large

number of particles. In MC calculations, it is possible to estimate the expected value of

the quantities of interest by calculating average properties from a set of particle histories

using laws of large number, e.g., the Strong Law of Large Number and the Central Limit

Theorem.

Using a simplified notation, a PDF, f(x) can be used to describe a particle being

transported and contributing to the quantity of interest. Consequently, the expected value

E(x) of that quantity would be calculated by Eq. 2-2.

E(x) = xf(x)dx = true mean (2-2)

The true mean can then be estimated by the sample mean x calculated using Eq. 2-3.

1 N
x Yx, (2-3)


In this equation, x, is the value of x selected from f(x) for the ith history and N is the

total number of histories. This mean is equivalent to the expected value since the Strong

Law of Large Numbers states that if E(x) is finite, x will tend toward E(x) as N

approaches infinity61. Note that the numerical operator used to estimates the mean is

often referred as an estimator.

Since in practical simulation N will be smaller then infinity, it is necessary to

evaluate the statistical uncertainty associated with using x The variance of the









population x values is a measure of their spread around the expected value and can be

evaluated by Eq. 2-4.

U2 = [x -E(x)]2 f(x)dx= E(x2) [E(x)]2 (2-4)

As with the true mean, the bias-corrected variance of the population can only be

estimated based on the distribution of the sampled scores using Eq. 2-5.

S2 Y -(x Y)2 (2-5)
N-1

However, it is more useful to know the variance associated with the average value (x)

being calculated. If the Central Limit Theorem is valid, the sample variance of x should

be given by Eq. 2-6.

S2
S= (2-6)
N

It is possible to define an estimated relative error to represent the statistical precision at

lo-level (i.e., x is within the interval x + Sx 68% of the time) as Eq. 2-7.


R = S S (2-7)
x xJN

It must be noted that there is an important difference between precision and accuracy of a

MC simulation. The precision is a measure of the uncertainty associated with x due to

statistical fluctuations, while the accuracy is related to the fidelity of the model in

representing the actual system and physics.

In addition to the variance associated with each mean, it is important to verify that

the tally is statistically well-behaved otherwise erroneous results could be obtained.

Another useful quantity is the relative variance of variance (VOV) which is the estimated

relative variance of the estimated R and therefore is much more sensitive to large score









fluctuations than R. Eq. 2-7 highlights a drawback of the MC method, i.e., the reduction

of the statistical uncertainty requires a large number of histories. For example, to

decrease the relative error associated with a converged result by a factor 2, the total

number of histories must be increased by a factor of 4.

When evaluating the efficiency of any MC simulation, three factors are important:

i) the history scoring efficiency, ii) the dispersion of non-zero scores, and iii) the

computer time per history. The first factor is essentially the fraction of source particles

that contribute to a given tally, the second factor is related to the spread of the particle

weights scores (and therefore the variance of a tally), and the third is related to the

number of histories that can be simulated in a given unit of time. Therefore, the scoring

efficiency, the ratio of the largest tally score to the average tally score and the number of

simulated histories per minute can be used to take a detailed look at the performance of

the simulation. However, these three factors are generally folded into a single metric to

get a figure-of-merit (FOM) describing the performance of the simulation. This FOM is

defined by Eq. 2-8.

1
FOM = (2-8)
R2T

This metric takes into account the competing effects of the decreasing variance (as

measured by the square of the relative error R2) and the increasing computation time (T)

as a function of the number of histories. It is possible to get the speedup obtained from a

variance reduction (VR) technique by comparing the FOM from two simulations. By

assuming that both simulations reach the same precision (relative error) in their

respective time, an estimate of the speedup can simply be obtained by Eq. 2-9.









FOM2 T,
=FM2- = speedup (2-9)
FOM1 T2

It is important to note that if the relative errors are different, Eq. 2-9 still provide a good

estimation of the speedup. Since the VOV is more sensitive to the statistical fluctuations

of an estimator, another useful metric used to measure the statistical reliability is the

Figure-of-Reliability (FOR) defined by Eq. 2-10.

1
FOR = (2-10)
VOV2T

Deterministic Transport Theory: Forward Transport

By considering particles and the target atoms as two component of a gas, it is

possible to write an equation, either in integral or integro-differential form, to

characterize the behavior of particles. In that sense, the deterministic approach to

radiation transport differs significantly from the MC approach since the average

quantities of interest are calculated from the solution of that equation.

Boltzmann resolved the task of writing an equation representing all the particles,

with their respective positions and velocities, by assuming that it was only necessary to

know precisely the state of motion within an infinitesimal volume element of the phase-

space when you are interested in average macroscopic properties. Consequently, the

macroscopic states of the gas are not represented by point-wise functions; rather, they are

represented by density functions.

This equation constitutes a balance of the various mechanisms by which particles

can be gained and lost from a phase-space element drdEdQ 62. It is possible to write a

linear form of the Boltzmann equation, sometimes called the forward transport equation,

by assuming; i) that one component of the gas (particle) is considered having a very









small density so that collisions of that component with itself can be neglected in

comparison with the collisions with the other component (target atoms), and ii) the

properties of the target material do not depend on the behavior of the particle type of

interest. The resulting time-independent integro-differential form of the linear Boltzmann

equation (LBE) for a non-multiplying media is expressed by Eq. 2-11

Q V rEQ) + a,(fE) (,E,Q) =
(2-11)
SJdQ' dE'a,(r,E' -> E,Q' -> Q) q(F,E',') +Q(F,E,Q)
47 0

Where y(r,E,Q) drdEdQ is the angular flux with energy E within the energy range dE,

at position r within the volume element ctr, and in direction Q within the solid angle

dQ. Similarly, S(r,E,Q) dFdEdQ is the angular external source, i.e., the rate at which

particles are introduced into the system in a given phase-space element. The double

integral term, referred to as the scattering source, represents the sum of the particles

scattered into dEdQ from all the dE'dQ' after a scattering collision represented by the

double differential cross-section a,(F,E' E,Q' -> Q). For simplicity, the transport

equation (Equation 2-11) can be written in operator form.

H (,E, ) =Q(r,E, ) (2-12)

The operator H in Eq. 2-12 is defined by Eq. 2-13.


H = +a,(rF,E) dQ' dE'a(,E' -> E, Q' -> Q) (2-13)
47 0

Note that an analytical solution is possible only for very limited simple cases. It is

therefore necessary to use numerical methods for solving this equation. Such discussions

are reserved for a later section of this chapter.









Deterministic Transport Theory: Backward Transport

At this point, it is interesting to discuss the concept of backward (adjoint) transport

as it relates to the importance functions. Using the operator notation introduced at the end

of the previous section, we can define the mathematical adjoint of the LBE by using the

adjoint property62

< y'Hy >=< pH'H1' > (2-14)

In the adjoint property shown in Eq. 2-14, t denotes the adjoint and the Dirac brackets

< > signify integration over all independent variables. For the adjoint property to be

valid, vacuum boundary conditions for the angular flux and its adjoint (as shown in Eqs

2-15a and 2-15b) are required.

S(F, Q,E)= 0, Fe ,Q. < 0 (2-15 a)

y'(FQ,E) 0, r r, h > 0 (2-15 b)

In these equations r is the boundary surface and h is an outward unit vector normal to

the surface. Using this adjoint property, it can be proven that the adjoint to the LBE can

be written in operator form.

H' /'(E, ) = Q'(FE,E) (2-16)

Where, the operator Hf is defined by Eq. 2-17.

co 4a
H' = -Q V + a JdE' dfd'o,(E -> E',P -> ') (2-17)
0 0

By using a physical interpretation, it is possible to derive a balance equation for

particle importance62, which is equivalent to the adjoint transport equation (Eq. 2-16 and

Eq. 2-17) without the vacuum boundary restriction. Hence, the solution to the adjoint









LBE relates to the physical importance of a particle toward a given objective represented

by the adjoint source Qt.

Adjoint Source and Objective

Let's consider a transport problem represented by Eq. 2-12 and Eq. 2-16 and look

at an example of the relationship between the source of the adjoint problem and the

objective of the calculation. In this case, the objective of the simulation is to calculate the

response of a detector in term of counts (related to the reaction rate in the detector). From

theforward transport simulation, the response can be calculated according to Eq. 2-18

where R represents the detector response and a7d (cm-) is the detector cross-section.

R =< /i (rF,i,E)ad (F,E) > (2-18)

It is possible to define a commutation relation by multiplying Eq. 2-12 and

Eq. 2-16 by the adjoint function and angular flux respectively. Doing so, we obtain a

commutation relation.

< V'Hy >- < pH'H/f' >=< 'eQ > < VQw > (2-19)

Using the adjoint property shown in Eq. 2-14 (i.e., assuming vacuum boundary

conditions), we can rewrite Eq. 2-19:

< V'eQ >=< VQw > (2-20)

Using an adjoint source equal to the detector cross-section (Q = cr), Eq. 2-21 provides

an alternative way to evaluate the reaction rate.

R =< y'Q > (2-21)

This shows that by using the detector cross-section as the adjoint source, the

resulting solution to the adjoint problem gives the importance toward producing a count









in the detector. Note that for this case, an analysis of the units of Eq. 2-16 would show

that the resulting adjoint function is unitless (or per count).

Electron, Photon and Positron Interactions

This section will mainly be descriptive since a complete review of the electron,

photon and positron interactions is out of the scope of this work. Note that for the details

of the mechanisms the reader is referred to Evans63. An excellent review (even though it

is relatively old) of electron interactions and transport theory is given by Zerby and

Keller64. Note that in the context of particle interactions, the incident particle can often be

referred to as a source particle or a primary particle while particles resulting from the

interactions are referred to as secondary particles.

Electrons and Positrons

The most important interactions of electrons and positrons below 10 MeV are

elastic scattering, inelastic scattering from atomic electrons, bremsstrahlung, and

annihilation for positrons. Typically, electron collisions are characterized either as soft or

hard. This classification relates to the magnitude of the energy loss after the collision;

interactions with a small energy loss are referred to as soft while interactions with large

energy loss are referred as hard. However, the energy-loss threshold that distinguishing a

soft from a hard collision is arbitrary.

Elastic scattering

For a wide energy range (-100 eV to -1 GeV), elastic collisions (sometimes

referred to as Coulomb scattering) can be described as the scattering of an

electron/positron by the electrostatic field of the atom where the initial and final quantum

states of the target atom stay the same. This type of interaction causes most of the angular

deflections experienced by the electrons/positrons as they penetrate matter. Note that









there is a certain energy transfer from the projectile to the target, but because of the large

target to projectile mass ratio, it is usually neglected.

The electron/positron elastic scattering cross sections are large and concentrated in

the forward directions resulting mostly in small deflections with an occasional large-

angle scattering. Electron elastic scattering interactions are usually represented by the

Mott65 cross-section with a screening correction from Moliere66. The positron elastic

scattering cross-section is often approximated by the electron cross-section. This

approximation is most accurate in low-Z materials and for small angles of deflection9. In

high-Z material and for larger angles of deflection, the two cross sections can differ up to

an order of magnitude. However the differential cross-section for such large angle

deflections is at least several orders of magnitude lower than for the smaller angles.

Inelastic scattering

Passing through matter, electrons and positrons lose small amount of energy due to

their interactions with the electric fields of the atomic electrons. However, an electron

colliding with another electron can exchange nearly all its energy in a single collision and

produce knock-on electrons (delta-rays). For energies below a few MeV, these processes

are responsible for most of the energy losses. The fraction of these interactions resulting

in hard events is often modeled through the Moller64 cross-section for the electrons and

the Bhabba67 cross-section for the positrons. However, as it will be seen later, many of

these collisions produce small energy loses and are often represented by a continuous

energy loss without angular deflection. This approach uses collisional stopping powers

and related ranges. By comparing the values of stopping powers for electron and

positron9 for various element and energies, it is possible to observe that the largest

differences occurs (e.g., -30% at 10 eV in gold) below 1 keV while above 100 keV the









two quantities seem, from a practical standpoint, identical. Since most coupled electron-

photon-positron simulations do not include particles below 1 keV, using the same

collisional stopping power values for electron and positron seem a valid approximation.

This continuous slowing down (CSD) approach needs to be supplemented by an energy-

loss straggling model to correct for the fact that the CSD approximation forces a one-to-

one relationship between depth of penetration in the target material and the energy loss,

while in reality the energy-loss is a stochastic variable following a distribution, such as

the Landau distribution21

Bremsstrahlung

The sudden change in the speed of a charged particle (in this case electron or

positron) as it passes through the field of the atomic nuclei, or the atomic electrons field,

produces bremsstrahlung (braking) photons. At very high energies, most of the energy is

lost through this process. This process is often represented through the use of radiative

stopping powers. According to Ref 9, even though the radiative stopping power of

electrons differs significantly from positron at energies below 1 MeV (almost an order of

magnitude at 10 keV), the differences in the total range of the electron and positron is

minimal. This can be explained by the fact that below 1 MeV, the collisional energy-loss

dominates. It is therefore a valid approximation to use electron radiative stopping range

for positron.

Positron annihilation

A typical way to model the positron annihilation is to assume that it occurs only

when the energy of the positron falls below the energy cutoff of the simulation. Upon

annihilation, two photons are produced of equal energy are produced. This implicitly









assumes that a positron annihilates only when "absorbed" and therefore neglecting the

small fraction of annihilation that occurs in flight.

Photons

For the photon energies considered in this work, the main interaction mechanisms

are the photoelectric effect, Compton scattering (incoherent scattering) and pair

production. Note that the fluorescent photons and coherent scattering can also be

important mechanisms in some cases.

Photoelectric effect (and fluorescence)

In the photoelectric effect, the photon interacts with the atom as a whole, gets

"absorbed", and a photoelectron is emitted (usually from the K shell). As the vacancy left

by the photoelectron is filled by an electron from an outer shell, either a fluorescence

x-ray, or Auger electron may be emitted.

Coherent scattering

The coherent scattering (sometime referred to as Rayleigh scattering) results from

an interaction of the incident photon with the electrons of an atom collectively. Since the

recoil momentum is taken up by the atom as a whole, the energy loss (and consequently

the change of direction) is really small and usually neglected.

Incoherent scattering

Incoherent scattering refers to a scattering event where the photon interacts with a

single atomic electron rather then with all the electrons of an atom (coherent). The

incident photon loses energy by transferring it to this single electron (referred to as recoil

electron) which gets ejected from the atom. This phenomenon is represented by the

double differential Klein-Nishina cross-section. However, this cross-section was derived

assuming scattering off a free electron, which is invalid when the kinetic energy of the









recoil electron is comparable to its binding energy. Therefore, a correction for the

electron binding energy is usually applied using a scattering form factor. Qualitatively, its

effect is to decrease the Klein-Nishina cross-section (per electron) in the forward

direction, for low E and for high Z, independently.

Pair production

In pair production, the incident photon is completely absorbed and an electron-

positron pair is created. This interaction has a threshold energy of 2mec2 (1.022 MeV)

when it is a result of an interaction with the nucleus electric filed or 4mec2 when a result

of an interaction with an electron electric field (also called triplet production). Note that

the triplet production process is relatively not important, and therefore generally ignored.

Numerical Considerations

Some MC simulations cannot reach a certain statistical precision with a reasonable

amount of time and therefore, it is necessary to use VR technique. A MC simulation

using VR techniques is usually referred to as non-analog since it uses unnatural

probabilities or sampling distributions as opposed to the analog MC which uses the

natural correct probabilities and distributions. For completeness, the discussion on the

importance-based VR presented in the literature review is extended to include non

importance-based VR techniques that are often used in coupled electron-photon-positron

calculations to provide more theory about some of the other techniques used in this work.

Ref 68 provides a review of the variance reduction methods implemented in MCNP.

Bielajew and Rogers69 as well as Kawrakow and Fippel70 present discussions of different

techniques including electron-specific and photon-specific techniques. Note that the

McGrath's report71 provides a comprehensive list of variance reduction techniques.









In the second part of this section, a general overview of the SN method is presented

since it is the numerical method used for solving the integro-differential form of the

linear Boltzmann equation. Finally, specific numerical techniques used to resolve issues

arising when performing SN calculation for electron/positron will be discussed. Note that

a comprehensive review of the deterministic methods for neutral particles is given by

Sanchez and McCormick72

Monte Carlo Method: Variance Reduction

In general, variance reduction techniques can be divided into four classes68

truncation methods, population control methods, modified sampling methods, and

partially-deterministic methods. The following list provides a short introduction to most

of the VR techniques used in coupled electron-photon-positron transport.

1. Truncation methods (note that truncation methods usually introduce a degree of
approximation and may reduce the accuracy of the calculation)

a. Energy cutoff: an increase of the overall efficiency of the simulation can be
achieved by increasing the energy cutoff because particles will then be tracked
over a smaller energy range and less secondary particles will be produced.
However, it is difficult to derive limits on acceptable energy cutoff based on
theoretical consideration, so it must be used carefully because improper cutoffs
can result in the termination of important particles before they reach the region
of interest.

b. Discard within a zone (electron trapping): in this technique, improvement in
the efficiency is obtained because, if the electron ranges are smaller than the
closest boundary of the zone, they are not transported and their energy is
deposited locally. Note that this approach neglects the creation and transport of
bremsstahlung photons (or other secondary particles) which may have otherwise
been created.

c. Range rejection: this approach is similar to the "discard within a zone" method
with the exception that the electron is discarded if it cannot reach some region
of interest instead of the closest boundary of the current zone.

d. Sectioned problem: it is often possible to separate a problem in various
sections where different parts are modeled with different levels of accuracy. For
example, one could model separately a complex geometry and store the phase-









space parameters (energy, direction and position) at the surface of the geometry.
It is then possible to use this information repeatedly at no extra computational
cost. Note that this is usually done at the price of some accuracy.

2. Population control methods

a. Geometry splitting: if the particles are in an important region of the problem, it
is often advantageous to increase their number (and decrease their weight
accordingly) by splitting them. Commonly, this splitting is performed at
boundary crossings. This increases the amount of time per history but reduces
the variance by having more particles scoring in the tally.

b. Russian roulette: if the particles are in an unimportant region of the problem,
or they are unimportant themselves (because of a small weight resulting from
the use of another VR technique), the particles can be terminated. Rather than
simple termination, Russian roulette must be played to avoid introducing a bias
in the estimators by not conserving the total number of particles. In this
technique, a particle with a weight below a given threshold has a set probability
of being terminated and has its weight increased by the inverse of this
probability if it survives.

c. Weight-window: this technique provides a utility to administer splitting and
Russian roulette within the same framework. This technique can be implement
with various useful features; i) biasing surface crossings and/or collisions, ii)
controlling the severity of splitting or Russian roulette, and iii) turning off
biasing in selected space or energy regions. The typical weight-window
technique allows the use of space- and energy-dependent weight bounds to
control particle weights and population.

3. Modified sampling methods

a. Implicit capture: This is probably the most universally used VR technique. In
this technique, the weight of the particle is reduced by a factor corresponding to
its survival probability. Note that one must then provide a criterion for history
termination based on weight.

b. Source biasing: In this technique, the source distribution is modified so more
source particles are started in phase-space locations contributing more to the
estimator. As always, to preserve unbiased estimators, the weight of the
particles must be adjusted by the ratio of unbiased and biased source
probabilities.

c. Secondary particle enhancement: To enhance the number of certain
secondary particles considered important for a given problem, it is possible to
generate multiple secondary particles once a creation event has taken place. The
secondary particles energy and direction are sampled to produce many









secondary particles emanating from a single interaction point. Note again that
the weight must be adjusted to preserve unbiased estimators.

d. Electron history repetition: This technique increases the efficiency of electron
dose calculation by re-using a pre-calculated history in water. The starting
positions and directions of the recycled electrons are different when they are
applied to the patient geometry.

4. Partially-deterministic methods

a. Condensed-history: This procedure uses analytical formulations to represent
the global effect of multiple collisions as a single virtual collision, therefore
reducing the amount of time required to sample the excessively large number of
single event collisions.

It is useful to discuss in more details the CH method since it constitutes one of the

major differences between neutral particle and coupled electron-photon-positron MC

simulations. As mentioned before, this approach uses an analytical theory to sum the

effect of many small momentum transfers from elastic and inelastic collisions into a

single pseudo-collision event often refer to as a CH step. The various flavors of

implementations of this technique, developed by Berger16, can fall into the following two

categories. From Refs. 73 and 74, these two classes can be described as:

* Class I: in this scheme, the particles move on a predetermined energy loss grid.
This approach provides a more accurate treatment of the multiple elastic scattering
but have disadvantages related to, i) the lack of correlation between energy loss and
secondary particle production, and ii) interpolation difficulties when CH step does
not conform to the pre-determined energy grid because of interfaces and/or energy-
loss straggling. This scheme is implemented into ITS and MCNP5.

* Class II: in this scheme, the hard (or catastrophic) events, e.g., bremsstrahlung
photon and Moller knock-on electrons, created above a certain energy threshold are
treated discretely, while sub-threshold (referred to as soft events) processes are
accounted for by a continuous slowing down approximation. This class of scheme
is implemented in EGS, DPM and PENELOPPE.

Deterministic Discrete Ordinates (SN) Method

One of the most widely used numerical methods to solve the integro-differential

form of the transport equation is the SN method. In the nuclear community, the current









method evolved from the early work of Carlson75. The method solves the transport

equation along a set of discrete ordinates (directions) typically selected such that physical

symmetries and moments of direction cosines are preserved. This set of directions and

associated weights are referred to as a quadrature set76. It is possible to use biased

quadrature sets which are useful for specific applications requiring highly directional

information. However, as it will be discuss later, the selection of quadrature sets to

perform electron can be limited. The energy variable is generally discretized into a finite

number of energy groups and cross sections are averaged over these intervals. A large

variety of approaches are used to discretize the spatial variable (including finite

difference and finite element methods) resulting in different representations of the

streaming term. In order to numerically represent the behavior of particles in a spatial

mesh, auxiliary equations, referred to as differencing schemes are needed. A good review

of the main differencing schemes is provided by Sjoden77'78

From a theoretical point of view, the linear Boltzmann equation is valid for charged

particles transport79 but the usual numerical approaches fail for various reasons: i) the

elastic scattering cross-section is so forwarded-peaked that a Legendre polynomial (or

spherical harmonics) expansion would lead to an excessively large number of moments54

(-200), ii) the required quadrature order (>200) to accurately represent the large number

of scattering kernel moments, and iii) the number of energy groups required to properly

represent the small energy changes resulting from soft inelastic collision51 (>160).

Therefore, various other numerical and mathematical treatments have been studied.








Fokker-Planck equation

One approach consists of replacing the integral scattering operator of the LBE by a

differential operator. This results in the Fokker-Planck (FP) equation, which can be

written for a homogeneous and isotropic medium as Eq.2-22.

Q Vq/(E, Q) + a,(E) y(E, Q) =

T(E)L y( y2) )2 W(E, Q)+ (2-22)

Q[S(E) y(E, )] 02 [R(E) V(E, )]
+ + Q(E, >)
aE 8E2

where

T(E) = r dE' d (l ),(E,E',4) (2-23)


S(E) = 2r dE' J d (E E') a (E,E',) (2-24)


R(E) = 2n dE' J1 d(E-E')2 (E,E', ) (2-25)

The left-hand terms of Eq. 2-22 represent the streaming of particles and their

absorption. The first term on the right-hand side represents the angular diffusion where

T(E) can be considered as some sort of diffusion coefficient. This term causes the

particles to redistribute in direction without change in energy. The second and third terms

(S(E) and R(E)) represent the energy-loss as a convective and diffusive process,

respectively. Note that these last two terms cause the particles to redistribute in energy

without directional change. Pomranning8o showed that this equation is an asymptotic

limit of the Boltzmann equation that is valid when the deviation of the scattering angle

from unity, the fractional energy change after a single scattering, and the scattering mean-

free-path (mfp) are all vanishingly small. This asymptotic analysis also shows that one









could set R(E) to zero and still get the leading order behavior in energy transfer.

Therefore, the FP equation, by definition, does not include any large energy transfers and

is inappropriate for a large number of problems involving lower energy electrons.

Boltzmann-CSD equation

To take into account the large deflection events, a scattering kernel combining the

Boltzmann and FP formalisms was introduced and referred to as the Boltzmann-FP

equation. This equation combines the advantages of the usual transport equation (large

energy transfer) with the FP formalism, which is very accurate for highly anisotropic

collisions. However, a simplified form of the Boltzmann-FP equation, referred to as the

Boltzmann-CSD (Continuous Slowing Down), is generally used. This form can be

obtained by neglecting the diffusive terms in angle and energy and is given by Eq. 2-26.

Q. [-(r,E,Q)+S ,(r,E) y(,, E,] ()

SdQ' dE'ar,E' -> E, ) V (,E', Q') + +a[s ) E Q(,E,) (226)
8E

Note that integral limits of the scattering kernel and the stopping power (S(E)) must

reflect the energy boundary between the hard and soft collisions, and the tilde indicates

that soft collisions have been excluded from the integral scattering kernel.

Goudsmith-Saunderson equation

The Goudsmit-Saunderson equation8 19, as shown in Eq. 2-27, solves for the

electron angular flux in an infinite homogeneous media.


a(E) W(E,up)= d c,,(E,) y(E,')+ S(E ( )] (2-27)
OE

Eq. 2-27 takes into account the following physical phenomena: i) the elastic

scattering for directional change without energy-loss, and ii) the soft inelastic scattering









part for energy loss without significant directional change. Note that this equation

neglects the hard inelastic scattering. The major advantage of this formulation is that it

can be solved exactly for a source free media as shown in Eq. 2-28.


S(E) (E,p) = (,2n +1Jd u'P(p)e_ ,EOd r.'-')]P,,(,u')S(E,) y(E,,p') (2-28)
n=1 2

CEPXS methodology

To perform deterministic electron transport calculations, CEPXS generates effective

macroscopic multigroup Legendre scattering cross sections which, when used in a

standard SN code, effectively solves the Boltzmann-CSD equation. To achieve this,

CEPXS uses the following treatments:

* A continuous slowing-down (CSD) approximation is used for soft electron inelastic
scattering interactions and radiative events resulting in small-energy changes, i.e.,
restricted stopping powers are used.

* The extended transport correction46 is applied to the forward-peaked elastic
scattering cross-section.

* Interactions resulting in hard events are treated through the use of differential cross
sections.

* First or second-order energy differencing scheme is applied to the restricted CSD
operator.

These pseudo cross-sections are unphysical since they do not posses associated

microscopic cross sections and require the use of specific quadrature sets, e.g. Gauss-

Legendre, not available in multi-dimension. The efficiency and accuracy of this

technique highly depends on the proper selection of the discretization parameters.

CEPXS-GS methodology

The CEPXS- GS (Goudsmit-Saunderson) approach combines the elastic scattering

and CSD in a single downscatter operator that is less anisotropic than the scattering cross

section. From the Goudsmit-Saunderson equation (Eq. 2-28), it is possible to define









multigroup Legendre scattering cross sections by doing the following; i) specify Eo as the

upper bound of energy group g, ii) relate the angular flux at the energy group boundary to

the group average through an energy-differencing scheme, iii) divide the resulting

discretized equation by the energy group width, iv) truncate the Legendre expansion, and

v) compare it with the standard multigroup-Legendre expansion to obtain the various

terms. It is then possible to use these cross sections in a standard SN code to perform

coupled electron-photon-positron in multi-dimensional geometries since they do not

depend, as with the CEPXS cross-sections, on the quadrature set. These cross sections are

an improvement on the CEPXS cross sections since they result in a faster convergence,

eliminates some well known numerical oscillations, require smaller expansion orders, and

can be guaranteed positive under certain conditions.














CHAPTER 3
ADEIS METHODOLOGY CONCEPTS AND FORMULATIONS

The ADEIS (Angular adjoint-Driven Electron-photon-positron Importance

Sampling) methodology is based on the same principles as the CADIS (Consistent

Adjoint Driven Importance Sampling) methodology32. In both these methodologies,

importance sampling is used to performed transport and collision biasing through the

weight-window technique using deterministic importance functions to determine variance

reduction parameters. However, in order to address issues related to coupled electron-

photon-positron transport, many specific features had to be developed and implemented

in ADEIS. Before discussing these features, it is useful to present in more detail the

concept of importance sampling, and the different mathematical formulations used in

ADEIS.

Importance Sampling

The general idea of the importance sampling technique is to take into account that

certain values of a random variable contribute more to a given quantity being estimated

and consequently, sampling them more frequently will yield an estimator with less

variation. Therefore, the basic approach is to select a biased sampling distribution (PDF)

which encourages the sampling of these important values, while weighting these

contributions in order to preserve the correct estimator. Using a simplified notation, this

concept can be represented mathematically by introducing a biased PDF in the

formulation of the unbiased expected value shown in Eq. 3-1.









b
(g) = g(x)f(x)dx (3-1)
a

It is also necessary to introduce a biased contribution function to preserve the expected

value as shown in Eq. 3-2.

b
(g)= Jw(x)g(x)f(x)dx, (3-2)
a

In this equation, (g) is the estimated quantity, g(x) is a function of the random variable


x defined over the range [a,b], f(x) represents the sampling PDF, f(x) represents the

biased sampling PDF, and w(x) = f(x)/f(x) represents the weight of each contribution.

As shown in Appendix A, choosing an optimum biased sampling PDF with the

same shape as f(x)g(x) will yield a zero-variance solution. However, this implies an a

priori knowledge of the solution defying the purpose of performing the simulation.

However, this suggests that an approximation to that optimum biased sampling PDF can

be used to reduce the variance with a minimal increase in computation time per history. It

also suggests that the closer that approximated integrand is to the real integrand; the more

the variance should be reduced. To apply this methodology to a particle transport

problem it is useful to use a more detailed form of the equation representing the transport

process (Eq. 2-1) as shown in Eq. 3-3.

W(P) = f( (P")C(P"-P)dP" + Q(P') ) T(P'P)dP' (3-3)

In this equation, I(P) represents the integral quantity being estimated, C(P"->P')

represents the collision kernel, T(P'->P) represents the transport kernel, and Q(P')

represents the external source of primary particles. The collision kernel describes the

particles emerging from a phase-space element after either a scattering or the creation of









a secondary particle. The transport kernel simply reflects the change in phase-space

location due to streaming and collision. As shown in Appendix A, it is possible to use the

concept of importance sampling to write the equation representing the biased transport

process as shown in Eq. 3-4. Note that Eq. 3-4 and its derivation (as presented in

Appendix A) are slightly different the previous work23' 33, and show more clearly the

separation between collision and transport biasing.

S(P) = Jf J(P")C(P"-P')dP"T(P'--P)dP' + JQ(P')}T(P'-P)dP' (3-4)

In this equation, '(P) represents the biased estimator, Q(P') represents the biased

source, T(P'-P) represent the biased transport kernel, and C(P"-P') represents the

biased collision kernel. Eq. 3-4 shows that performing importance sampling on the

integral transport equation is equivalent to performing transport, source and collision

biasing in a consistent manner. The following three sections present ADEIS approach to

these three type of biasing.

ADEIS Angular Transport Biasing

From Appendix A, it can be seen that the biased transport kernel is described by

Eq. 3-5.


T(P'-P) = T(P P) P) (3-5)
(P')

From a physical point of view, this biased transport kernel can be seen as an

adjustment of the number of particles emerging from a phase-space element according to

the ratio of importance of the original and final phase-space elements. Since no explicit

PDF of T(P'-P) is available to be modified, it is possible to achieve a modification of

the transfers by creating extra particles when the original particle transfers from a less to









a more important region of phase-space or by destroying a particle when it transfers from

a more to a less important region of phase-space. This creation/destruction can be

performed using the standard splitting/rouletting VR techniques and following the rules

given in Eqs 3-5 and 3-6.

xy-(P)
-- > 1 particles are created (splitting) (3-5)
v (P)

<1 particles are destroyed (Russian roulette) (3-6)
W't(P')

As discussed in the previous section and in Appendix A, there is an inverse

relationship between the biased sampling distribution (in ADEIS case the importance

functions) and the resulting weight of the contribution to a given estimator.

Consequently, by associating the importance of a given phase-space element to a

corresponding weight, it is possible to force the statistical weight of a particle to

corresponds to the importance of the phase-space region by following the new set of rules

presented in Eqs 3-7 and 3-8.

w(P)
w- < 1 particles are created (splitting) (3-7)
w(P')

w(P)
S> 1 particles are destroyed (Russian roulette). (3-8)
w (P ')

However, the cost of performing splitting and rouletting every time a particle

changes phase-space location could offset the benefit gained by this technique. Therefore,

it is generally useful to define a range of weights that are acceptable in a given phase-

space element. Consequently, the weight-window technique allows for particles with

weights within a given window (range) to be left untouched, while others are splitted









and/or rouletted to be forced back into the window. The standard approach is to define the

lower-weight bounds of this window and set the upper bound as multiple of the lower-

weights. Note that more statistically reliable results are possible because a better control

over the weights scored by individual particles is achieved.

It is possible to write the formulation for the lower-weight bounds by using the

expression for the biased angular flux and the conservation law shown in Eq. 3-9.

S(P)= w (P) (3-9)

The resulting expression for the lower-weight bounds is given in Eq. 3-10.

R
w1(P) =- (3-10)
Y (P) C,

Where w,(P) is the lower-weight bound, R is the approximated estimator, Yf(P) is the

importance function value, and C, is the constant multiplier linking the lower and upper

bound of the weight-window. Even though Eq. 3-10 states that the lower-weight depends

on all phase-space variables (r, E, Q), it typically depends only on space and energy.

Note that this formulation is slightly different then the formulation used in previous
work32, 33
work

However, for most problems considered in this work, the flux distributions can be

highly angular-dependent because: i) the source characteristics (e.g. high-energy electron

beam); ii) the geometry of the problem (e.g. duct-like geometry or large region without

source); and iii) the scattering properties of high-energy electrons and photons.

Therefore, to achieve a higher efficiency, it is expected that the weight-window bounds

should be also angular-dependent. However, in the context of a deterministic importance-

based VR technique, it is important to be able to introduce this angular dependency









without using a complete set of angular fluxes which requires an unreasonable amount of

memory. To address this issue, ADEIS uses the concept of field-of-view (FOV) where

the angular importance is integrated within a field-of-view subtending the region of

interest. Figure 3-1 illustrates simplified space-dependent FOVs in 1-D and 2-D

geometries.



1I







A) B)

Figure 3-1. Field-of-View (FOV) concept. A) in 1-D geometry B) in 2-D geometry

It is therefore necessary to calculate two sets of lower-weight bounds for directions

inside and outside the FOV as shown in Eqs 3-11 and 3-12. Note that the FOV,,

represents the field-of-view associated with a given particle type n since it may be useful

to bias differently various particle species. Note that in principle, the FOV could be

dependent on energy; however, it is not considered for this version of ADEIS.

Corresponding lower-weights for positive and negative directions on the FOV are defined

by:

R
w ( ,,E)= (3-11)
((FE,E) =d (,E,) + FO and FO (3-12)C

P+(p-,E) = dQ T (r-, E, Q) + : Q e FOV, and -: Q V FOV,. (3-12)









ADEIS Source Biasing

A formulation for a better sampling of the source probability distribution can be

developed by using the relative contribution of the sampled source phase-space element

to the estimated quantity. From Appendix A, it can be seen that this biased source is

described by Eq. 3-12.,

St(P ')
Q(P') = Q(P') (3-13)
R

More specifically, in ADEIS, the biased source would be calculated using the

formulation presented in Eq. 3-14.

SP(Fr, E) Q (r, E)
Q(r,E) Q ) (3-14)
R

Where p+(FE) the FOV-integrated angular importance function and R is is the

approximated estimator. Again, to preserve the expected number of particles, the weight

of the biased source particles would be adjusted according to Eq.3-15.

w Q(F,E)
1 w = (f (3-15)
Q(f,E)

However, all examples studied in this work are mono-energetic pencil beam and

therefore, no source biasing was implemented at this time and is only shown for

completeness. In ADEIS, the approximated value R is calculated according to either one

of the two formulations shown by Eqs 3-16 and 3-17.

R = dE dVQ (F,E) p (F,E) + dE dV Q_ (,E) (F,E) (3-16)

R = J ddE J dQ o (r, E)(F, E)
F Ah. (3-17)
+ JddE JdQa h '(r, E) (, E)
F n-








Where Q (F,E) and ',(r,E) are the projections of the volumetric and surface source

over the discretized phase-space of the deterministic calculation, and h is the outward

normal to the surface F. Eq. 3-16 is used to calculate the approximated estimator value

for cases with a volumetric source and vacuum boundaries while Eq. 3-17 is used for

cases with an incoming source at a boundary.

As shown in Appendix A, using Eq. 3-11 to calculate the lower-weight bound also

ensures that the source particles are generated at the upper-weight bounds of the weight-

window if a mono-directional, mono-energetic point source is used. This is a useful

characteristic since, for such problems, it is possible to maintain the consistency between

the weight-window and the source without having to perform useless source biasing

operations that would not increase the efficiency of the simulation.

ADEIS Collision Biasing

Collision biasing can be achieved by playing the weight-window game at every

collision, on the primary particles and on most of the secondary particles before they are

stored into the bank. This is the standard approach used in MCNP5. Note that in electron

transport, the term collision can be interpreted as the end of each major energy step in the

CH algorithm.

Criteria for Applying Weight-Window

To minimize the amount of computational overhead associated with comparing the

particle weight to the transparent mesh associated with the weight-window, it is

important to optimize the frequency of these checks. Each check against the transparent

weight-window mesh has a computational cost associated with the binary search

algorithm. This increased cost has to be as small as possible in order to maximize the









increase in FOM. Earlier studies2 33 found that checking the weight every mean-free-path

(mfp) was the near-optimum criterion for neutral particles. For electrons, it is necessary

to check the weight against the weight-window at most after each major energy step of

the electron condensed-history (i.e., at every pseudo-collision) in order not to introduce a

bias in the electron spectrum.

Selection of the Adjoint Source

As discussed in Chapter 2, the importance function is related to the objective for

which the user wishes to bias the simulation. In a similar approach developed for neutral

particle transport81, the objectives were typically reaction rates in a small detector (e.g.

multigroup detector response cross-section from the BUGLE-96 library82). Consequently,

the objectives of these calculations were defined by the library containing the detector

response cross sections. For coupled electron-photon-positron, MC calculations are often

performed to determine energy deposition (dose) profile and accurate (i.e. adjusted to fit

experimental data) coefficients such as flux-to-dose conversion factors may not be

readily available for all materials. To circumvent this difficulty, ADEIS allows the use of

two automatically determined adjoint sources; i) a local energy deposition response

function to approximate dose in the ROI, and ii) a uniform spectrum to maximize the

total flux in the ROI. Note that the ADEIS methodology uses a spatially uniform adjoint

source over the whole ROI.

Local Energy Deposition Response Function

A local energy deposition response function can be use as an adjoint source for

problems where the objective is related to the energy deposited (MeV) in the ROI. From

the forward transport simulation, the energy deposition can be defined by Eq. 3-22.









R =< i (r,,E)Eo, (r,E) >


Where ox (F,E) is some sort of energy deposition coefficient.

By comparing Eq. 2-14 and Eq. 3-22, it is possible to deduce that if Eo- (F,E)

(MeV cm-) is used as an adjoint source, the importance function units would be MeV per

count. Consequently, the solution of the adjoint problem represents the importance of a

particle toward energy deposition. To evaluate these coefficients for photons, different

assumptions can me made which then results in different coefficients6 as listed below.

* Linear absorption: assumes that when a photoelectric or pair production event
occurs, all the energy is deposited locally, i.e., no energy is "re-emitted" in the form
of fluorescence x-rays, bremsstrahlung annihilation photon or other secondary
particles,

* Linear pseudo-energy-transfer: similar to the linear absorption with the
exception that the energy "re-emitted" in the form of annihilation photon,

* Linear energy-transfer: similar to the linear pseudo-energy-transfer with the
exception that energy is also "re-emitted" in the form of fluorescence x-rays,

* Linear energy absorption: similar to the linear energy-transfer with the exception
that energy is "re-emitted" from bremsstrahlung through radiation.

It may be argued that by a phenomenon of error compensation, these various

approximations result in almost the same dose6 when multiplied with the appropriate

fluence. However, in the context of a VR technique, an approximate objective can be

used since only an approximate importance function is needed. Therefore, at this point,

ADEIS uses the absorption cross-section multiplied by the energy of the group as an

adjoint source as shown in Eq. 3-23.

Qe(E) = Eoa (E) = Ea, (E)- Eo, (E) (3-23)

In this equation, oa, the total collision cross-section, and o7, is the total scattering cross-

section. It must be noted that using the absorption cross-section as an energy deposition


(3-22)









coefficient contains the additional assumption that no energy is deposited when a

Compton scattering event occurs. This adjoint source spectrum was chosen because these

cross sections are readily available from CEPXS.

For electrons, the situation is slightly more problematic since such coefficients are

not readily available because of the use of the continuous slowing-down approximation.

However, it is possible to conclude from the previous discussion on photons that any

quantity that represents the deposited energy per unit path-length (MeV cm1) would

constitute a sufficient approximation in the context of a VR technique. Therefore, it is

possible to define such a quantity as the energy imparted in a volume divided by the

mean chord length of the volume. By energy imparted, it is usually meant the sum of the

energies of all charged and neutral ionizing particles entering the volume minus the sum

of the energies of all charged and neutral ionizing particles leaving the volume. The

adjoint source in ADEIS approximates this quantity based on the energy deposited by an

electron in the ROI divided by the average chord length.

For each electron energy group, a surface source, with an angular distribution

proportional to the cosine of the angle and with an energy corresponding to the midpoint

of the group, is assumed, so an average chord length can be calculated (see Appendix A).

Using a CSD approximation, the energy deposited can be approximated by:

1. Subtract the average chord-length (r) from the range of the electron in the energy
group (R ) being considered, i.e., R' = Rg -r.

2. IfR' < 0, an amount of energy corresponding to the middle point of the energy
group g is assumed deposited (Ed = Eg).

3. IfR' > 0, the energy group g corresponding to that residual range if found and an
interpolation if performed to find the energy (E' ) corresponding to that range. The









difference between the midpoint of the original energy group and that remaining
energy is assumed deposited (Ed = Eg E).

4. Finally, the adjoint source is defined as Qt(E) = -d (MeV cm-1).
r

Uniform Spectrum

In theory, an adjoint source uniform spectrum could be used for a problem where

the objective is related to the total flux in the ROI. This objective may not be optimal for

problems concerned with energy deposition, but could be sufficient to produce significant

speedups. From theforward transport simulation, the average total flux over the ROI can

be defined as

1 1 1
-< /(r,E)>= JdVJdEf dQ (F,Q,E) dVD(F) D (3-23)
V V V

By comparing Eq. 2-14 and Eq. 3-23, it is possible to deduce that if a uniform spectrum,

equal to the inverse of the volume of the ROI, is used as an adjoint source

(Qe(r, ,E) = V 1 cm-3), the importance function units would be per count per cm2

Consequently, the solution of the adjoint problem represents the importance of a particle

toward the average total flux in the ROI.

Comparison with Methodologies in Literature Review

Most techniques described in Chapter 1 differs significantly from ADEIS either

because they; i) focus exclusively on neutral particle transport, ii) use diffusion or linear

anisotropic scattering approximations, iii) use importance functions generated from the

convolution of various functions for each phase-space variable. It is therefore difficult to

compare the ADEIS methodology with such approaches. However, it is possible to do a

more detailed point-by-point comparison with four approaches that share common

features with ADEIS: i) multigroup adjoint transport in MCNP5 (referred to as MGOPT);









ii) weight-window generator (referred to as WWG); iii) ADVANTAG/A3MCNP/CADIS

(referred to as CADIS); and, iv) AVATAR. Note that ADVANTAG, A3MCNP and

CADIS are grouped together since A3MCNP and ADVANTAG are both rather similar

implementations of the CADIS methodology.

These four methodologies were chosen because they all take advantage of the

weight-window technique implemented in various versions of MCNP (either in an

original or modified form) to perform transport and collision biasing using a transport-

based importance functions. The first two approaches were also chosen because they can

be used to perform coupled electron-photon biasing even though the importance

functions are determined using MC simulations rather then deterministic. Alternatively,

the last two approaches were chosen because the importance functions are deterministic-

based even though they were developed for neutral particle biasing. Table 3-1 presents a

summarized point-by-point comparison of the four methodologies and is followed by a

slightly more in-depth discussion of certain points.

Table 3-1. Comparison of other variance reduction methodology with ADEIS
ADEIS MGOPT WWG CADIS AVATAR
Coupled
electron/photon / / /
biasing
Deterministic
importance
Explicit positron ,
biasing
Angular biasing / /
Source biasing / / / /
3-D importance / / / /
function
Automation / /
Mesh-based
weight-window









Deterministic Importance Function

Despite some difficulties related to discretization, cross-sections, input files

generation and transport code management, the use of deterministic importance functions

constitute a significant advantage since information for the whole phase-space of the

problem can be obtained relatively rapidly. The MC approach to generate importance

functions is limited by the fact that obtaining good statistics for certain region of phase-

space can be extremely difficult, hence the need for VR methodology for the forward

problem. This difficulty is often circumvented to some extent by generating the

importance function recursively, i.e., using the incomplete phase-space information

generated in the prior iteration to help obtaining better statistics for the current iteration

and so on until the user is satisfied with the quality of his importance functions. This

requires a lot of engineering time and expertise to be used efficiently.

Explicit Positron Biasing

Currently, ADEIS in the only methodology that generates a distinctive set of

importance functions for the positron and biases them independently of the electrons.

Further discussion on this topic is provided in Chapters 4 and 5.

Angular Biasing

The major issue with angular biasing is the amount of information that is possibly

required. ADEIS and AVATAR use completely different schemes to circumvent this

issue. While AVATAR uses an approximation to the angular importance function,

ADEIS uses the concept of field-of-view to introduce an angular dependency for each of

weight-window spatial mesh and for each particle species. Further discussion of this topic

is provided in Chapters 4 and 6.









3-D Importance Functions

Since ADEIS is based on a modified version of the mesh-based weight-window

implemented in MCNP5, it is theoretically possible to use 3-D importance functions.

However, because of the computational cost and difficulties (large number of groups,

possible upscattering, high orders for quadrature and scattering expansion, optically thick

spatial meshes) of generating the 3-D coupled electron-photon-positron importance

functions, it was chosen that only 1-D and 2-D (RZ) importance functions would be used.

This can be justified by the following arguments, i) large computational cost of a

generating a 3-D importance function may offset the gain in variance, ii) a large class of

problem of interest in coupled electron-photon-positron can be adequately approximated

by 1-D and 2-D models, and iii) the line-of-sight approach introduces an additional

degree of freedom to better approximate a 3-D geometry. Obviously, highly three-

dimensional problem by nature might not be properly approximated by such treatments

and may require 3-D importance functions.

Automation

Even though a small degree of automation is incorporated into WWG and MGOPT;

the user-defined spatial mesh structure and energy group structure, the necessary

renaming of files and manual iterative process to generate statistically reliable importance

functions still requires too much engineering time and expertise to really qualify as

automated. Note that the AVATAR package was not marked as automated either since

from the available papers, it is difficult to judge the extent of automation implemented in

the code.

The A3MCNP implementation of the CADIS methodology was automated to a

large extent since the deterministic model was automatically generated, the energy group









structure determined from the cross-section library and the data manipulation handle

through scripts. The degree of automation was extended in ADEIS in order to ensure that

all aspects of the VR methodology are transparent to the user; only a simple input card

and command-line option are required to use the VR methodology.

Mesh-Based Weight-Window

The use of the MGOPT option in MCNP5 seems to generate importance function

for the cell-based weight-window. This constitutes a significant disadvantage, since to

generate and use 3-D importance functions, it is necessary to subdivide the geometry in

many sub-cells. In addition to the additional engineering time required to perform this,

the presence of additional surfaces can considerably slow the particle tracking process

and introduce a systematic error due to the use of a class-I CH algorithm in MCNP (see

Chapter 5 for further discussion on this topic).











CHAPTER 4
ADEIS METHODOLOGY IMPLEMENTATION

Implementing a deterministic importance-based VR technique such as ADEIS requires


various processing tasks such as generating the deterministic model and the lower-weight bounds

for the weight-window. All implementations choices described in this chapter are made with


automation in mind since a large degree of automation is required for this technique to be

efficient and practical. Figure 4-1 shows the flow chart of the automated ADEIS.


Start MCNPS Link file / / adeisinp

r - - --- I
New Generate input files
command line
option
SCalls CEPXS/CEPXS-GS as a
Generate xs file shared library
New card: adeis D
.-.----.--. R Generate Calls ONELD/PARTISN as a
Calls ADEIS as a importance I shared library
shared library

call mcrun --- Generate wwinp



End MCNP5 / wwinp /



Figure 4-1. Automated ADEIS flow chart

The following sections are addressing topics related the different parts of the automated ADEIS


flow chart. More details about certain aspects are given in Appendix B

Monte Carlo Code: MCNP5

Since a large number of MC codes are available to the nuclear engineering community, it

was necessary to select a single Monte Carlo code to implement ADEIS. For this work, MCNP5

was selected for the following reasons:

* It is well known and benchmarked.

* The availability of a weight-window algorithm with a transparent mesh capability.









* Uses a standard interface file for the weight-window.

* The experience with previous version of the code using similar techniques30,32

* The availability and clarity of the source code and documentation.

Deterministic Codes: ONELD, PARTISN and PENTRAN

A few deterministic transport code systems are available to the community such as

DANTSYS31, DOORS34, PARTISAN0 and PENTRAN83'84

ONELD is a special version of the 1-D SN code ONEDANT code (part of the DANTSYS

package) that includes a spatial linear-discontinuous differencing scheme that lessens the

constraints on the numerical meshes. This selection was motivated by the fact that

CEPXS/ONELD5, a package developed at Sandia National Laboratory, already made use of this

transport code to perform coupled electron-photon-positron transport simulation. Moreover,

CEPXS/ONELD has a few advantages including:

* It is well known and benchmarked.

* It has some degrees of automation which facilitated testing and verification of the earlier
non-automated versions of ADEIS.

However, it is not possible to generate multi-dimensional importance functions using ONELD;

therefore the PARTISN and PENTRAN codes were considered. PARTISN was selected for the

following reasons:

* It is an evolution of the DANTSYS system so input file and cross-section formats remain
the same as in ONELD.

* Linear discontinuous differencing scheme is also available.

* It contains 1-D, 2-D and 3-D solvers so it may be possible to perform all the required
transport simulations within one framework.

PENTRAN was selected for the following reasons:

* Familiarity and experience with the code to perform 3-D transport calculations.









* The availability of different quadrature sets and an adaptive differencing strategy including
a family of exponential differencing schemes85' 86 which might be useful for electron
transport.

* The availability of pre- and post-processing tools.

* The capability of performing full domain decomposition (space, angle and energy) and
memory partitioning in parallel environments.

Note that the expansion to multidimensional calculations requires the use of the CEPXS-GS

version of cross-section generator.

Automation: UDR

To reach a high degree of automation, a Universal DRiver (UDR) was developed to

manage the different processing tasks required by the implementation of ADEIS within a single

framework. UDR is a library that can be linked (or shared) with any pre-existing computer

program to manage an independent sequence of calculations. In addition to the automation, UDR

allows for more input flexibility through a free-format input file, better error management and a

more consistent structure than a simpler script-based approach. Moreover, UDR has utilities that

allows for general data exchange between the various components of the sequence and the parent

code. In the context of this work, this implies that ADEIS is a sequence of operations managed

by UDR and called by MCNP5. Additional details about UDR are given in Appendix B.

Modifications to MCNP5

A standard MCNP5 simulation involves processing the input, the cross sections, and

performing the transport simulation. However, an ADEIS simulation requires a few other tasks

before performing the actual transport simulation. To address this issue, a new command line

option was implemented into MCNP5. By using this option, MCNP5 performs the following

tasks: i) process input and cross sections; ii) generate the deterministic model; iii) extract

material information and other necessary parameters; iv) run the independent ADEIS sequence;









v) process the modified weight-window information; and, vi) perform the biased transport

simulation. In addition to this new command line option, an adeis input card has also been

implemented in MCNP5. Additional details about the command line option, the new MCNP5

simulation sequence and the adeis card are given in Appendix B.

An important change to the MCNP5 code concerns the weight-window algorithm, which

was modified to take into account various combination of biasing configuration: i) standard

weight-window; ii) angular-dependent weight-window without explicit positron biasing; iii)

explicit positron biasing without angular dependency; and, iv) explicit positron biasing with

angular dependency. The application of the weight-window within the CH algorithm was also

modified to ensure that the electron spectrum would not be biased (see Chapter 5).

Generation of the Deterministic Model

Because of the high computational cost associated with performing exclusively 3-D

deterministic transport simulation for coupled electron-photon-positron problems, ADEIS allows

in principle the use of 1-D, 2-D and 3-D deterministic transport simulation to obtain the

importance functions. Other considerations such as the material compositions and the energy

group structure are also automatically managed by ADEIS before performing the deterministic

transport simulation.

1-D Model (X or R) Generation

In order to automatically generate applicable 1-D (X or R) importance functions, a line-of-

sight approach is used. In this approach, the user defines a line-of-sight between the source origin

and the region of interest (ROI). A model is then generated along that line by tracking through

the geometry and detecting material discontinuities as illustrated in Figure 4-2.










BEAM
MC f----------- --- -- LOS



Deterministic
0.0 0.1 0.25 9.0 9.05 10.15 10.25 100.0 110.0

Figure 4-2. Line-of-sight approach

This approach is better suited for problem types in which the beam is relatively well

collimated and the overall behavior of the solution is 1-D-like.

2-D Model (XY or RZ) Generation

In order to automatically generate an applicable 2-D model (XY or RZ), a perpendicular

direction to the line-of-sight is defined either by default or by the user. While the model is being

generated by tracking along the line-of-sight, the tracking algorithm recursively branch along the

perpendicular direction each time a material discontinuity is encountered. That new direction is

tracked and material discontinuities are recorded until a region of zero importance is

encountered. At this point, the algorithm returns to the branching point, and continues along the

line-of-sight as illustrated in Figure 4-3.

Y 4 ............ .................. +






LOS






xo xI X2

Figure 4-3. Two-dimensional model generation using line-of-sight









These various material regions can then be regrouped into coarse meshes and automatically

meshed as described in the next section. Note that this methodology also allows for the

generation of 3-D (XYZ) models by branching along a third perpendicular direction at each

material discontinuity. Obviously, models generated by such an approach are approximate, but

they are sufficient for the purposes of generating relative importance functions to be used in the

context of a VR methodology.

Automatic Meshing of Material Regions

Earlier studies87' 88, 89 required a significant amount of engineering time to determine an

appropriate spatial mesh structure. Moreover, another study90 showed that an improper meshing

can introduce unphysical oscillations in the importance function (especially with the use of the

CEPXS package) and be partly responsible for statistical fluctuations in the photon tallies

obtained from the coupled electron-photon-positron simulation. Therefore, automating the

selection of a proper mesh density in each material region constitute an important consideration,

both from practical (less engineering time) and technical (reducing possible unphysical

oscillations in the importance functions) perspectives.

Different automated meshing schemes have been implemented and studied (see Chapter 6):

i) uniform mesh size; ii) selective refinement of a boundary layer at material and source

discontinuities; and, iii) material region mesh size based on partial range associated with electron

energy. Moreover, for all these automatic meshing schemes, the approximate rule-of-thumb91

shown in Eq. 4-1 is respected.

SXR(E) G

In this equation, R(E1) is the range associated with electron in the fastest energy group, G is the

total number of groups and v is the ratio of the mean vector range to the CSD range. This rule









ensures that possible fluctuations in the energy domain are not transmitted to the spatial domain

by maintaining a mesh size larger than the partial range associated with the slowing down of

electrons from one group to the next. This is especially useful when using the CEPXS cross

sections.

Material Composition

The composition of each material region is extracted from MCNP5 after the input file has

been processed. A special attention is given to the fact that different MCNP5 cells can have the

same material but different densities and that certain material can be gaseous (important for the

density correction of the stopping power). This information is then used to automatically

generate an input file for either of the CEPXS or CEPXS-GS codes.

Energy Group Structure

ADEIS is highly flexible and allows any group structure to be used since the cross sections

are generated on-the-fly for each problem using CEPXS or CEPXS-GS. Because of the absence

of resonance regions in the cross sections, the accuracy of the results is not as sensitive to the

multigroup structure as in deterministic neutron transport. Therefore, a uniform or logarithmic

distribution of the energy group width is generally sufficient.

Generation of the Weight-Window

The generation of the weight-window requires additional tasks addressing practical

concerns related to the implementation of the methodology described in Chapter 3.

Importance Function Treatment

Because of the numerical difficulties inherent in the deterministic coupled electron-photon-

positron transport calculations, the importance functions may exhibit undesirable characteristics

which make them inappropriate to calculate physical quantities such as the lower-weight bounds.

A few possible problems have been identified: i) the use of CEPXS cross sections may lead to









unphysical and negative values for the importance function; ii) extremely small or large values of

the importance; and, iii) numerical round-off resulting in importance values of zero. To address

these various possible problems, the following steps are taken to eliminate undesirable numerical

artifacts.

First, ADEIS eliminates the negative values by a simple smoothing procedure based on the

knowledge that in most cases, the average of those oscillations is correct91. For each negative

value detected within an energy group, the importance value can either be interpolated or

extrapolated from the closest neighbor points, depending on the locations of the negatives value

within the model. Many smoothing passes may be performed to ensure that no negative values

remain. In order to avoid numerical problems with extremely small and large numbers during the

Monte Carlo simulation, the importance function values are limited to the same values used in

MCNP fur huge (1036) and tiny (10-36) numbers. Finally, importance values that are equal to zero

are set to the minimum value of the importance of that energy group. This is necessary since

weight-windows bounds equal to zero are usually used to indicate a region in phase-space where

no biasing is required.

MCNP5 Parallel Calculations

The MCNP5 code can perform parallel calculations, i.e., distributing the simulation over

many processors. In the case of Monte Carlo simulations, the parallelization of the tasks is quite

natural considering that each particle history can be simulated independently of the others. More

specifically, MCNP5 parallelized the simulation by breaking the total of number of particle

histories over the total number of processors. In this work, the simulations are performed on a

parallel machine (cluster) using essentially a distributed memory architecture where each

processor has access to its own independent memory. In the MCNP5 version used in this work,

the communications between the various processors are handled through the use of message









passing via the MPI library. A complete discussion of the various aspects of parallel computing

and its implementation within MCNP5 are beyond the scope of this work, and the reader is

referred to the MCNP5 user's manual7. It is however important to mention that it was necessary

to implement the following ADEIS feature within the parallel framework of MCNP5 in order to

be able to perform parallel calculations; the possibility of using field-of-view (FOV) depending

on particle type and space required additional message passing at the onset of the simulation to

communicate the FOV to all processors. Note that the addition of new ledgers to tally the amount

of weight created and lost through splitting and Russian roulette over the weight-window

transparent mesh also required a parallel implementation.









CHAPTER 5
IMPACT OF IMPORTANCE QUALITY

In Appendix A, it was shown that a biased sampling distribution with the exact shape of

the integrand would result in a zero-variance solution, therefore it is expected that a biased PDF

that only approximates that shape would still yield a reduction in variance. Consequently, it can

also be expected that the more accurate the importance function, the larger the reduction in

variance. However, obtaining and using more accurate importance functions has a computational

cost that can offset the gain in variance and results in the reduction of the FOM. This implies that

for a given problem, there is a combination of the importance function accuracy and cost that

should result in a maximum increase in FOM. This combination might be difficult to find and,

most of the time, a given accuracy of the importance function is arbitrarily chosen. The accuracy

of the importance function may also affect the statistical reliability of the estimators and

introduce statistical fluctuations that delay or even prevent the convergence of the estimator. It is

therefore possible to refer to the importance function quality, i.e., the desirable characteristics to

produce accurate and statistically well behaved tallies when used for biasing in ADEIS. In

previous work on neutral particle2' 33, it was shown that methodologies similar to ADEIS produce

significant speedup with relatively approximate importance functions. From these studies, it

appears that the quality of the importance function was not a critical issue for neutral particle. It

is, however, important to verify how the quality of coupled electron-photon-positron importance

functions impacts the efficiency and accuracy of the ADEIS methodology.

To study this impact, a reference case with a poor quality importance function was

deliberately chosen. This reference case considers a mono-energetic 6 MeV electron beam

impinging a tungsten target 100 cm away from a region of interest (ROI) composed of water.









The heterogeneous geometry illustrated in Figure 5-1 represents a simplified accelerator head


and patient.


ROI --


Beam


12 4 6 8
Figure 5-1. Reference case geometry

The details associated with each zone (1 to 8) are presented in Table 5-1.

Table 5-1. Materials and dimensions of reference case


Zone Description
1 Target
2 Heat dissipator
3 Vacuum
4 Vacuum window
5 Air
6 Flattening filter
7 Air
8 ROI (tally)


Color
Dark gray
Orange
White
Light gray
White
Dark gray
White
Blue


Material

Tungsten
Copper
Low density air
Beryllium
Air
Tungsten
Air
Water


# of
Size (cm) me
meshes


0.1
0.15
8.75
0.05
1.1
1.0
88.85
0.1


2
3
175
1
22
20
1775
2


The other simulation parameters for this reference case are presented in Table 5-2.


Table 5-2. Test case simulation parameters
Monte Carlo
Energy-loss straggling is not sampled
Mode: Electrons and photons

Energy cutoff at 0.025 MeV
Default value for ESTEP in CH algorithm


Electron-Photon Adjoint Transport
CEPXS cross sections
43 uniform electron groups
30 uniform photon groups
Energy cutoff at 0.025 MeV
S16-P15
Flat adjoint source spectrum
No smoothing


Different factors suspected of influencing the quality of the importance function are then

varied and the statistical behavior of the tallies as a function of the number of histories is


JAt-










investigated using two parameters; i) the relative error of the total flux, and ii) the variance of

variance of the total flux. The energy spectra are also studied to verify that no bias is introduced

by the use of an importance function of poor quality.

Reference Case

The behavior of the relative error and the variance of variance of a surface flux tally at the

air-tungsten interface are studied as a function of the number of histories for a standard MCNP5

simulation without variance reduction. It is possible to see in Figure 5-2 that the tally is

statistically well behaved since it is rapidly converging (FOM of 1798) to a low relative error and

VOV. These values then smoothly decrease as the number of histories increases.


10
SRelative Error
--Variance of Variance




0.1


0.01


0.001


0.0001
0.E+00 5.E+05 1.E+06 2.E+06 2.E+06 3.E+06 3.E+06 4.E+06 4.E+06
# of histories
Figure 5-2. Relative error and variance of variance for a statistically stable photon tally in a
standard MCNP5 simulation

For the ADEIS simulation, a uniform spatial mesh of 0.05 cm (size of the smallest material

region, zone 4, vacuum window) is used throughout the model. The selection of this mesh size

obviously assumes that the user would have no knowledge or experience with deterministic

methods. This exercise is however useful to illustrate the impact of the quality of the importance

function and the need to automate the process and encapsulate within the code the knowledge









about generating importance functions of good quality. It is important to mention that

simulations performed with the parameters given in Table 5-2 result in importance function

values that are negative for large portions of the model and therefore, cannot be used to calculate

any physical quantities such as the weight-window bounds. It is therefore essential to ensure that

the importance function values are positive everywhere in the model.

Importance Function Positivity

In deterministic electron transport, significant numerical constraints can be imposed on the

differencing scheme since most practical mesh size can be considered optically thick because of

the large electron total cross sections. These constraints can produce oscillating and negative

solutions when a lower-order differencing scheme, such as linear-diamond, is used. The spatial

linear-discontinuous scheme used in ONELD reduces these constraints by introducing some

additional degrees of freedom. Moreover, the introduction of a differential operator to represent

part of the scattering allows similar constraints to produce oscillations in the energy domain

which, in certain cases, can propagate into the spatial domain. For all these reasons, the use of a

deterministic method to obtain the electron-photon-positron importance function can result in

solutions of poor quality (negative and oscillating) if proper care is not given to, among other

things, the selection of the discretization parameters. In the context of an automated VR

procedure, the robustness of the methodology is especially important to minimize user's

intervention. As mentioned earlier, it is essential to ensure, as a minimum, the positivity of the

importance function. However, the chosen approach to ensure positivity should not excessively

degrade the importance functions quality or increase significantly the computation time.

Importance Function Smoothing

The first solution, considered to address this issue, was to smooth the importance function

to remove negative and zero values. The importance function values are also limited to prevent










numerical problems with extremely small or large numbers. After applying the smoothing

procedure, it is possible to bias the reference case using the parameters presented in Table 5-2.

However, large statistical fluctuations are observed in the photon tallies as shown in Figure 5-3.


10
Relative Error
Variance of Variance




0.1


0.01


0.001


0.0001
0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
# of histories
Figure 5-3. Relative error and variance of variance in ADEIS with importance function
smoothing

By looking at the relative error and VOV, it is obvious that small values are rapidly

obtained (FOM of 6648 after 3.5x105 histories) but as the simulation progresses, statistical

fluctuations degrade the performance of the tally (FOM of 329 after 3.5x106 histories). The

presence of these fluctuations is, as expected, especially visible in the VOV. Obviously, by

simulating an extremely large number of histories it would be possible to obtain a converged

tally, but this would defy the purpose of using a VR technique.

Before addressing this issue of statistical fluctuation, it is useful to investigate other

methods to obtain positive importance functions without numerical artifacts since it is possible

that an importance function of better quality would resolve this issue. However, the importance

smoothing approach will be kept since it provides more robustness to the methodology and is not

incompatible with other methods.









Selection of Spatial Meshing

Early studies of the CEPXS methodology showed86 that the numerical oscillations in the

energy domain have a wavelength that is equal to twice the energy group width and since the

CSD operator forces a correlation between the path-length and energy-loss, these oscillations

could propagate in the spatial domain. The approximate rule-of-thumb described in Eq. 4-1 was

developed to ensure that the mesh size exceeds the path-length associated with the oscillations in

the energy domain and therefore mitigate these oscillations. It is possible to manually select a

mesh structure meeting that criterion and therefore generate importance functions of higher

quality. Table 5-3 presents a mesh structure for the reference case that produces a positive

importance function throughout the model and for all energy groups.

Table 5-3. Reference case spatial mesh structure producing a positive importance function
Zone # of meshes
1 5
2 25
3 10
4 5
5 5
6a (10.15cm to 11.0cm) 20
6b (11.0cm to 11.15cm) 20
7 10
8 5


In addition to the engineering time, the design of this mesh structure requires a more in-

depth knowledge of deterministic methods and a certain familiarity with the CEPXS/ONELD

package. Moreover, if thin regions are considered, it may not be possible to respect the criterion

for all cases. This definitely highlights the need for an automatic mesh generator and for

complementary techniques to further ensure the robustness of the methodology. Even though

positivity is obtained for this spatial mesh structure, statistical fluctuations are still present in the

photon tallies as shown in Figure 5-4



















0.01



0.001


0.0001
0.OE+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
# of histories
Figure 5-4. Relative error and variance of variance in ADEIS with "optimum" mesh structure

In this case, it is also possible to observe that the relative error and VOV reach small

values rapidly (FOM of 5699 after 3.5x105 histories) but as the simulation progresses, statistical

fluctuations degrade the performance of the simulation (FOM of 1306 after 3.5x106 histories).

However, these statistical fluctuations are slightly smaller then those observed in the previous

section. From this, it can be concluded that in addition to help achieving positivity, the selection

of the mesh structure also affect the statistical behavior of the tallies by influencing the quality of

the importance functions used by ADEIS.

First-Order Differencing of the CSD Scattering Term

By default, CEPXS uses a second-order differencing scheme for the restricted CSD

operator since it provides a more accurate solution by reducing the amount of numerical

straggling. Note that numerical straggling refers to the variation in the electron energy loss due to

the discretization approximation rather than the physical process. However, this differencing

scheme is responsible for the spurious oscillations in the energy domain. These oscillations can

be suppressed by selecting a first-order differencing scheme. However, this criterion is not










sufficient to ensure positivity of the importance function, since negative importance function

values are still obtained when the first-order scheme is selected for the reference case mesh

structure and it was necessary to perform smoothing on the importance functions. Alternatively,

the use of the mesh structure described in Table 5-3 in conjunction with the first-order

differencing scheme for the CSD operator produces an importance functions which is too

inaccurate. As shown in Figure 5-5, the relative difference between the importance functions of

certain energy groups obtained with the first and second-order differencing scheme of the CSD

operator are significant.

0 -










SGmup 36
Grup 34
S---- Group 1
-60




-80
011 11.05 11.1 11.15
Position [cm]

Figure 5-5. Relative difference between importance with 1st and 2nd order CSD differencing

The lower-order differencing scheme is well-known91 to produce large numerical

straggling degrading the accuracy of the transport solution. This translates in poor performance

when the importance function is used in ADEIS. However, increasing the number of energy

groups should improve that solution, since smaller energy group widths are more appropriate for

first-order differencing scheme.















Relative Error
Variance of Variance
1



0.1



0.01



0.001



0.0001
0.OE+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
# of histories

Figure 5-6. Relative error and variance of variance for ADEIS photon tally with 1st order CSD
differencing scheme and 75 energy groups.


Even though statistical fluctuations are still presents, Figure 5-6 shows that they are


significantly smaller. Therefore, using the first-order differencing scheme and a larger number of


group seems to improve the quality of the importance. This is reinforced by the fact that


performing the simulation with 75 energy groups and using the second order differencing


scheme results in a significantly worse statistical behavior of the photon tallies as shown in


Figure 5-7.


10
-*- Relative Error
-- Variance of Variance
1



0.1



0.01



0.001



0.0001
0.OE+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
# of histories

Figure 5-7. Relative error and variance of variance in ADEIS with 75 energy groups











In the context of an automated VR technique where only approximated solutions are

required, the robustness provided by the first order differencing scheme could be a useful

advantage. However, using a larger number of energy groups may prove to be computationally

too expensive in some cases. Therefore, further studies are required to investigate ways to

improve the quality of the importance at a minimum computational cost.

CEPXS-GS Methodology

In addition to the capability to perform multidimensional coupled electron-photon-


positron, the CEPXS-GS methodology eliminates the oscillations in the energy domain54 even

for very small mesh size. However, CEPXS-GS with the reference or improved mesh structures

still results in negative importance functions requiring the use of the smoothing technique. This

is understandable since these negative values can also be the results of the spatial differencing

scheme and/or optically thick regions. To perform a fair comparison between CEPXS-GS and

CEPXS, simulations using a first-order differencing scheme and 75 energy groups were

performed. Figure 5-8 shows that the relative error and the VOV using CEPXS-GS are not

significantly different from the ones obtained with CEPXS (see Figure 5-6).


10
SRelative Error
--Variance of Variance



0.1


0.01


0.001


0.0001
0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
# of histories
Figure 5-8. Relative error and variance of variance in ADEIS with CEPXS-GS









Impact of Importance Quality on Statistical Fluctuations: Preliminary Analysis

In previous sections, the approaches considered to obtain usable importance functions (i.e.,

functions of sufficient quality) ensured the positivity either by themselves or in combination with

smoothing and resulted in various degrees of statistical fluctuations. Since none of these

approaches resolved completely the problem of statistical fluctuations, the comparison presented

in this section must be considered preliminary. Even though it is possible to qualitatively

compare the various approaches by comparing, as previously done, the curves of the relative

error and VOV as a function of histories, it would be interesting to have a more quantitative

criterion.

The figure-of-merit (FOM) is typically used to indicate the efficiency of a simulation tally

and consequently, it can be assumed that the higher the FOM, the better the quality of the

importance functions. However, since all results presented earlier are not fully converged and the

final value of the FOM cannot be used, it becomes interesting to look at how the FOM changes

as a function of the number of histories. To simplify the analysis, it seems pertinent to look at the

average FOM (characterize overall performance) for different number of histories and the

relative variation of these FOMs (characterize statistical fluctuation). Eq. 5-1 shows the

formulation used to calculate the relative variation of the FOM.

SFOM
RFOM -FOM (5-1)
XFOM

SFoM is the standard deviation of the FOM, and it is estimated from the FOMs obtained at

various number of histories during the simulation. xFOM is estimated by calculating the average

of these FOMs.









Table 5-4. Average FOM and RFOM for all approaches
Case Average FOM RFOM
Standard MCNP5 1516 0.139
Optimal Mesh Structure 3907 0.422
1st order differencing scheme for CSD 3776 0.247
+ optimal mesh + 75 energy groups
+ smoothing
CEPXS-GS + 1st order differencing 4711 0.405
scheme for CSD + optimal mesh
+ 75 energy groups + smoothing


Table 5-4 indicates that that the standard MCNP5 results have the smallest efficiency

(smallest average FOM) and the smallest amount of statistical fluctuations (smallest RFOM). The

use of first-order differencing scheme and CEPXS seems to produce the least amount of

statistical fluctuation while the use of CEPXS-GS with first-order differencing scheme produces

the larger increase in FOM. Besides the gain in FOM from ADEIS, it is also possible to observe

that none of those approaches tested completely eliminate the excessive statistical fluctuations. It

is therefore important to further study the root cause of these statistical fluctuations before any

other conclusions can be drawn from this comparison.

Positrons Treatment and Condensed-History in ADEIS

The quality of the importance function, as defined earlier, is related to the characteristics of

the function that results in accurate and statistically reliable tallies. However, the quality of an

importance can appearpoor if improperly used within the MC code because of various

implementation considerations. Therefore, this section presents studies evaluating the impact of

positron treatment during the simulation, and implementation of the ADEIS weight-window

based methodology within the context of the CH algorithm.

Positron Biasing in ADEIS

MCNP5 follows the traditional approach of using the same scattering physics for electrons

and positrons, but onlyflags the particle as a positron for special purposes such as annihilation









photon creation and charge deposition. Note that CEPXS follows a similar approach by using the

same scattering laws and stopping powers for electrons and positrons. Therefore, following the

traditional approach, the ADEIS used the electron importance function to bias both the electrons

and positrons. However, this treatment revealed to be inappropriate within the context of the

ADEIS VR methodology as shown in the following studies.

Generally, the types of statistical fluctuations presented in the previous sections are an

indication that undersampling of an important physical process is occurring. In ADEIS, the large

differences (at certain location in phase-space) between the electron importance function and

photon importance functions are in part responsible for this undersampling and statistical

fluctuations. More precisely, such variations between the importance functions (ratio larger then

5 orders of magnitude) produces statistical fluctuations in the photon tallies because positrons

surviving Russian roulette game see their weight increased significantly because of the low

importance predicted by the electron weight-window bounds. Consequently, annihilation

photons generated by the surviving positrons may result in infrequent high weight scores,

therefore leading to statistical fluctuations in the photon tallies. More specifically, for the

reference case, statistical fluctuations occur because,

1. due to the low importance of the positrons in the flattening filter as predicted by the
electron importance function, most positrons are killed by Russian roulette,

2. but, infrequently, a positron will survive Russian roulette and therefore its weight increased
significantly to balance the total number of positrons in the simulation,

3. however this positron will annihilate quickly and produce high weight annihilation
photons,

4. which, because of the geometry of the problem, are likely to contribute directly to the
tallies at the surface of the flattening filter or in the ROI,

5. and increase the spread of the scores distribution which affect the variance of variance and
possibly, the variance itself.










In the physical process illustrated in Figure 5-9, the thickness of the arrows represents the weight

of the particles.

STungsten

Water

Air

Photon
Annihilation photon
-- Positron
Figure 5-9. Impact of large variation in importance between positron and photon

By comparing the average weight per source particle created as annihilation photons in a

standard MCNP5 and ADEIS simulation, it is possible to observe that this excessive rouletting of

the positrons results in the annihilation photons being undersampled in ADEIS.

Table 5-5. Impact of biasing on annihilation photons sampling
Case Annihilation Photon Weight /
Source Particle
Standard MCNP5 1.646E-02
ADEIS 3.51E-04

However, this artificial effect stems from the use of the electron importance function to

bias the positrons. This phenomenon can be easily understood by comparing the electron

importance function with the importance function of the annihilation photons. By definition, the

importance of a particle should include its own importance toward the objective and the sum of

the importance of all its progenies including secondary particles. However, near the cutoff

energy (i.e., the energy at which positrons annihilate), the electron importance function is

significantly smaller than the annihilation photon importance function as shown in Figure 5-10.

Consequently, the electron importance function cannot be used to represent the positron

importance for which the annihilation photons are progenies.












6.E-02 1.E+00
1.E-02
1.E-04
5.E-02 1.E-06
Annihilation 1.E-08

4.E-02 Photon Importance 1.E-12
1.E-14
1.E-16
S 3.E-02 1.E-18
S1.E-20
1.E-22 t
S2.E-02 1.E-24
Electron 1.E-26
Importance 1.E-28
1.E-02 1.E-30
1.E-32
1.E-34
O.E+00 i**- --*- 1.E-36
10.15 10.16 10.17 10.18 10.19 10.2 10.21 10.22 10.23 10.24 10.25
Position [cm]

Figure 5-10. Electron and annihilation photon importance function in tungsten target


Moreover, the electron importance function greatly underestimate the importance of the


positrons and, as observed, results in excessive rouletting of the positrons and undersampling of


the annihilation photons. A more realistic and physical positron importance function calculated


by CEPXS/ONELD is compared to the annihilation photon importance function in Figure 5-11.


1.2E-01


1.OE-01

Positron Importance
8.0E-02


t 6.0E-02


4.0E-02 Annihilation Photon
Importance

2.0E-02


0.OE+00
10.15 10.16 10.17 10.18 10.19 10.2 10.21 10.22 10.23 10.24 10.25
Position [cm]

Figure 5-11. Positron and annihilation photon importance function in tungsten target










As expected, the positron importance function values are slightly larger than the

annihilation photon. Using a modified version of the MCNP5 weight-window algorithm,

importance sampling is therefore performed using a distinct importance function for the

positrons (explicit positron biasing). Table 5-6 shows that performing such biasing eliminates

the annihilation photons undersampling.

Table 5-6. Impact of explicit positron biasing on annihilation photons sampling
Case Annihilation Photon Weight /
Source Particle
Standard MCNP5 1.646E-02
ADEIS 1.652E-02

It is also interesting to examine the surface photon flux spectrum at the interface between

regions 6 and 7, i.e., at the surface of the flattening filter. By examining the spectrum coming out

of this region, it is possible to better observe the impact of the positron biasing through the

annihilation photons before this effect is smeared by scattering in the rest of the model.

2.0E-05

1.8E-05
ADEIS
1.6E-05 -- ADEIS with positions
-- Standard MCNP5
S1.4E-05

1.2E-05

O1.0E-05
~ 8.0E-06
6.0E-06

4.0E-06 0.5 to 0.525 Me-V.

2.0E-06

0.0E+00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Energy [MeV]
Figure 5-12. Surface Photon Flux Spectra at Tungsten-Air Interface

Figure 5-12 shows that when positrons are not biased explicitly, a single energy bin

presents a bias (-35% smaller and not within the statistical uncertainty). Again, this effect is due










to the undersampling of the annihilation photons since that energy bin (0.5 MeV to 0.525 MeV)

tallies mainly the 0.511 MeV annihilation photons. Note that the 1-o statistical uncertainty on

these results is smaller than the size of the points. Finally, it is interesting to see in Figure 5-13

that all statistical fluctuations in the relative error and the variance of variance disappear when

the positrons are explicitly biased.

10






0.1
-*-Relative Error
-*-Variance of Variance
0.01


0.001


0.0001 -- ------
0.E+00 5.E+05 1.E+06 2.E+06 2.E+06 3.E+06 3.E+06 4.E+06 4.E+06
# of histories
Figure 5-13. Relative error and VOV in ADEIS with CEPXS and explicit positron biasing

Impact of Importance Quality on Statistical Fluctuations: Final Analysis

Since the major statistical fluctuations have been eliminated through explicit positron

biasing, it is to compare again the approaches listed in Table 5-4 by examining the average FOM

and its relative variation as a function of histories. By comparing Tables 5-4 and 5-7, it is

possible to observe that the RFOM are decreased to about the same value as the standard MCNP5

simulation and that the average FOM is increased significantly. It can also be observed that no

significant gains in FOM or statistical stability are obtained from using either the CEPXS or

CEPXS-GS. However, a significant improvement in the average FOM and RFOM is observed

when the 2nd-order CSD operator and a smaller number of energy groups are used.









Table 5-7. Impact explicit positron biasing on average FOM and RFOM
Case Average FOM RFOM
Standard MCNP5 1516 0.139
Optimal Mesh Structure + Smoothing 8754 0.111
1st order differencing scheme for CSD 6842 0.166
+ optimal mesh + 75 energy groups
+ smoothing
CEPXS-GS + 1st order differencing 6930 0.150
scheme for CSD + optimal mesh
+ 75 energy groups + smoothing


This behavior could be attributed to various factors affecting the quality of the importance

function. It is possible that; i) the number of energy groups is too small for the first-order

differencing scheme to have the same accuracy as the second-order scheme, and ii) the larger

number of energy groups increases the computational cost. Further studies on this topic and the

optimization of other discretization parameters are presented in Chapter 6.

Condensed-history algorithm and weight-window in ADEIS

This section presents studies related to the accuracy of electron tallies and the

implementation of the ADEIS weight-window based VR methodology within the context of the

CH algorithm. However, to better understand the impact of the implementation of the weight-

window within the CH algorithm, it is useful to simplify the test case. Therefore, an electron-

only simulation is performed in a simple cube of water with a 13 MeV pencil beam impinging on

the left surface. As illustrated in Figure 5-14, five regions of interest are considered; 0.15 cm

starting at about 70%, 80%, 84%, 92% and 100% of the CSD range of the source electrons.








Figure 5-14. Regions of interest considered in simplified test case











In the MCNP5 implementation of the CH algorithm, all distributions are evaluated on a


predetermined energy-loss grid at the beginning of the simulation. Path-lengths associated with


the major steps are used to model the energy loss using the CSD expected value and the


Landau/Blunck-Leisegang distribution for energy-loss straggling. In CEPXS, the energy-loss is


modeled through the use of a differential cross-section for hard collisions and restricted stopping


powers for the soft collisions (no energy-loss straggling is considered for soft collisions).


Therefore, for each ROI, the electron total flux and spectrum are estimated with three different


energy-loss approximations:


* Case 1: CSD expected value of the energy loss in MC and unrestricted stopping power in
deterministic

* Case 2: CSD expected value of the energy loss in MC and implicitly modeled energy-loss
straggling using differential cross-section for hard collisions in deterministic

* Case 3: CSD expected value of the energy loss and sampling of the Landau/Blunck-
Leisegang energy-loss distribution in MC and implicitly modeled energy loss straggling
using differential cross-section for hard collisions in deterministic.

Figure 5-15 shows the percentage of relative difference between the standard MCNP5 and


ADEIS electron total fluxes in the ROI for the three energy-loss approximations.

1.50
1.00
0.50
0.00
-0.50 -
-1.00 -Case : With CSDA Only
-1.50
-*-Case 2: With Energy-Loss
S-2.00 Straggling in Deterministic Only
-2.50 ase 3: With Energy-Loss
Straggling
3.00
-3.50
0.6 0.7 0.8 0.9 1.0 1.1
Depth [fraction of CSDA range]


Figure 5-15. Relative differences between the standard MCNP5 and ADEIS total fluxes for three
energy-loss approaches











By comparing the relative differences in total fluxes for the Cases 1 and 2, it is possible to


conclude that discrepancies in the energy-loss models are not responsible for the bias observed in


Figure 5-15. This conclusion can be reached since the relative difference behaviors of these two


cases are not significantly different in spite of having significantly different energy-loss models.


Therefore, it can be implied that this bias is somewhat related to the energy-loss straggling


sampling within the CH algorithm and the use of the weight-window in the ADEIS VR


methodology. Consequently, it is interesting to look further at the relative differences between


the electron energy spectra in the ROI from the standard MCNP5 and ADEIS calculations for


Case 3.


4.00
0.69 CSDA Range
"--- 0.84 CSDA Range
2.00 --0.79 CSDA Range

02.00 -4 1.00 CSDA Range
S2.00

-4.00

S-6.00

S-8.00

S -10.00
12.00

-14.00

-16.00

-18.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Energy [Mev]


Figure 5-16. Relative differences between the standard MCNP5 and ADEIS electron energy
spectrum for Case 3 in the five regions of interest

Figure 5-16 indicates that systematic errors are introduced in the spectra from the ADEIS


VR methodology and that those errors seem to increase with increasing penetration depths. This


behavior is analogous to the systematic errors introduced in the energy spectrum when an


electron track is interrupted by cell boundaries92 in the class-I CH algorithm as implemented











within MCNP5. This can be simply shown by dividing the simplified test base into small sub-


regions and calculating the relative difference between the resulting spectrum and the spectrum


obtained from the undivided model. Note that these simulations are performed in an unmodified


standard version of the MCNP5 code with the ROI located at about the range of the source


particles.


10.0

0.0

-10.0

-20.0
-A-With 5 subregions
30. 0 With 9 subregions
---With 18 subregions
-40.0

-50.0

S-60.0

-70.0

-80.0

-90.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Energy [MeV]


Figure 5-17. Relative differences in electron spectra for undivided and divided models

Comparison of Figures 5-16 and 5-17 shows that the relative difference behaviors are


similar. However, it can be observed that these systematic errors introduced by the ADEIS VR


methodology are much smaller. These systematic errors are introduced when a major step is


interrupted by a cell boundary and the real path-length is shorter than the predetermined path-


length used in the pre-determination of the energy-loss straggling distribution in that step.


Obviously, this systematic error increases with the increasing number of surface crossings. In a


similar manner, it is important that the weight-window be applied at the end of major step of a


class-I algorithm otherwise a similar systematic error will be introduced since particles that are











splitted or have survived Russian roulette will have experienced an energy-loss based on the full


length of the step rather then partial length where the weight-window is applied.


A review of the MCNP5 CH algorithm revealed an indexation error resulting in the


weight-window being applied before the last substep rather than after. Because of this error, all


particles created through the weight-window technique have inaccurate energy losses due to the


small truncation of the full path-length of the major step. This error is small but accumulates as


particles penetrate deeper into the target material. Therefore, the MCNP5 CH algorithm was


modified to ensure that the weight-window is applied at the end of each major step. To further


illustrate the impact of applying the weight-window before the end of a major CH step,


Figure 5-18 shows the impact on the total flux at different depth of performing the bias at three


locations within the major step: i) just before the second-to-last substep, ii) just before the last


substep, and iii) just after the last substep.

1.00


0.00


S-1.00oo

-- Weight-window applied just
-2.00 before the second-to-last
S2substep
--Weight-window applied just
before the last substep
3 3.00
S Weight-window applied just
after the last substep
-4.00


-5.00
0.6 0.7 0.8 0.9 1.0 1.1
Depth [fraction of CSDA range]


Figure 5-18. Impact of the modification of condensed-history algorithm with weight-window on
the relative differences in total flux between standard MCNP5 and ADEIS










As expected, Figure 5-18 shows that the systematic error is larger for the case where the

weight-window is applied before the second-to-last substep since the error in the energy-loss

prediction is larger. It can also be seen that no bias is introduced when the weight-window is

applied after the last substep. Figure 5-19 shows that the large systematic bias in the electron

tally spectra shown in Figure 5-16 disappears when the weight-window is applied after the last

substep of each major step.

1.E-01 4.0
-*-Normalized spectrum
-- Relative difference 3.5
1.E-02 -3.0

2.5
1.E-03
2.0

1.5 .
S1.E-04
a- 1.0

-" 0.5 <
1.E-05
___-_- 0.0

1.E-06 -0.5
-1.0

1.E-07 -1.5
0.E+00 1.E+00 2.E+00 3.E+00 4.E+00 5.E+00 6.E+00 7.E+00 8.E+00
Energy [MeV]

Figure 5-19. ADEIS normalized energy spectrum and relative differences with the standard
MCNP5 at 70% of 2MeV electron range with the CH algorithm modification

Above results also indicate that, as expected, the largest relative differences are located at

the tail of the spectrum. For most problems and tally locations, no bias is observed in the

spectrum, as shown in Figure 5-20. Note that the uncertainties in the relative difference were

obtained using a standard error propagation formula.











4.00


2.00


0.00


-2.00


S-4.00


-6.00


-8.00


-10.00


0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Energy [Mev]


Figure 5-20. Relative differences between the standard MCNP5 and ADEIS at various fraction of
the range


Even though the total flux and its spectrum are unbiased for most cases, it is important to


note that a small bias (within 1-o statistical uncertainty) can remain in the tail of the spectrum for


tallies located pass the range of the source particle. At this point, the source of this small bias is


not fully understood but it is possible that difference in straggling models between the


deterministic importance and the MC simulation could be responsible. In previous studies91, it


was also supposed that discrepancies between electron MC and deterministic results might be


caused by the differences in straggling models. Moreover, no bias can be seen with the 99%


confidence interval and this small bias has a negligible impact on the integral quantities of


interest such as energy deposition. No further studies of this aspect will be presented here;


however, for completeness, a series of analyses in search of the specific cause of this behavior


are presented in Appendix C.


I -LaL-t_~


S0.69 CSDA Range
S0.79 CSDA Range
0.84 CSDA Range
-- --0.92 CSDA Range
S1.00 CSDA Range


t


-


-


-


i









Conclusions

First, the analyses presented in chapter showed that the CEPXS methodology can be used

to generate importance functions for coupled electron-photon-positron transport and collision

biasing. However, numerical difficulties in obtaining physical importance functions devoid of

numerical artifacts were encountered. Our studies indicate that a combination of limited

smoothing, a proper selection of the mesh structure and the use of a first-order differencing

scheme for the CSD operator (0 (AE)) circumvents some of the numerical difficulties but large

statistical fluctuations remain in the photon tallies. Note that the need for smoothing and

selecting a proper spatial mesh highlights the fact that automation must be an essential aspect of

this methodology in order to be practical.

Secondly, it was shown that it is essential to bias different species of particles with their

specific importance function. In the specific case of electrons and positrons, even though the

physical scattering and energy-loss models are similar, the importance of positrons can be many

orders of magnitudes larger then the electron importance functions due to the creation of

annihilation photon from positrons. More specifically, it was shown that not explicitly biasing

the positrons with their own importance functions results in an undersampling of the annihilation

photons, and consequently introduces a bias in the photon energy spectra. Therefore, in ADEIS,

the standard MCNP5 weight-window algorithm was modified to perform explicit biasing of the

positrons with a distinctive set of importance functions. It is important to note that the

computational cost of generating coupled electron-photon-positron importance functions may

become noticeable in multidimensional problems due to upscattering.

Thirdly, it was shown that the implementation of the weight-window technique within the

CH algorithm, as implemented with MCNP5, requires that the biasing be performed at the end of









each major step. Applying the weight-window earlier into the step, i.e., before the last substep,

results in a biased electron energy spectrum. This bias is a consequence of systematic errors

introduced in the energy-loss prediction due to an inappropriate implementation of the weight-

window. More specifically, these errors occur if the path-lengths between weight-window events

differ from the pre-determined path-lengths used for evaluating the energy-loss straggling

distribution. Therefore, in ADEIS, the standard MCNP5 CH algorithm was modified to ensure

that the weight-window is applied after the last substep of each major step.

Finally, in general, it can be concluded that improving the quality of the importance

function could improve the statistical reliability of the ADEIS methodology. However, the

analyses in this chapter did not address in detail an important reason of performing non-analog

simulations; i.e., achieving speedups. Therefore, various strategies to further improve the quality

of the importance function are studied in Chapter 6. These strategies are aim at improving and/or

maintaining the statistical reliability (robustness of the methodology) of the tallies as well as

maximizing the speedup.









CHAPTER 6
IMPROVING THE QUALITY OF THE IMPORTANCE FUNCTION

An increase in accuracy of the importance function may result in larger decrease in

variance, but depending on the choice of phase-space discretization, it may also have a

computational cost that may offsets the gain in variance. In theory this implies that for a given

problem, there is a combination of accuracy and cost of the importance function that should

result in a maximum increase in FOM and statistical reliability. Such importance functions could

be referred to as importance functions of good quality. The previous chapter highlighted the need

for an automatic discretization scheme to encapsulate within the code the knowledge necessary

to obtain an importance function of good quality. Moreover, it was concluded that such

automatization schemes increase the robustness and statistical reliability of the methodology

while reducing the amount of engineering time necessary to use ADEIS.

Therefore, the present chapter studies strategies to automatically select discretization

parameters that improve the quality of the importance function. This problem is two-fold: i) the

selection of discretization parameters that generates a positive importance function of sufficient

accuracy, and ii) the maximization of the variance reduction in the MC simulation while

minimizing the computational overhead cost. Note that for most cases using 1-D deterministic

importance functions, the overhead cost associated with performing the deterministic calculation

(a few seconds) is negligible compared to MC simulation time (tens of minutes at the least).

Therefore, most of the conclusions presented in this chapter reflect primarily the impact of the

accuracy of the importance function used in the VR technique.

To simplify the analyses, a reference case representing a cube of a single material (one

layer), with an impinging monoenergetic electron pencil beam and a region of interest (ROI)

located slightly pass the range of the source particle is considered. The ROI has a thickness of









approximately 10% of the range of the source particle. The geometry of this reference case is

illustrated in Figure 6-1.


Y



BEAM






Figure 6-1. Simplified reference model

It is well known that the accuracy and efficiency of a coupled electron-photon-positron

deterministic discretization scheme depends on the energy of the source particles as well on the

atomic number (Z) of the material. These two parameters influence: i) the anisotropy of the

scattering, ii) the total interaction rate, and iii) the yield of secondary particles creation.

Therefore, different combinations of source particle energies and materials are considered as part

of the analysis plan presented in Table 6-1. The other reference case simulation parameters such

as the number of energy groups, the energy cutoff values, the quadrature order or the Legendre

expansion order are presented in Table 6-2.

Table 6-1. Various test cases of the analysis plan
Case Energy (MeV) Material Average Z Thickness (cm)
1 0.2 Water 8 0.0450
2 0.2 Copper 29 0.0075
3 0.2 Tungsten 74 0.0045
4 2.0 Water 8 0.9800
5 2.0 Copper 29 0.1550
6 2.0 Tungsten 74 0.0840
7 20.0 Water 8 9.3000
8 20.0 Copper 29 1.1700
9 20.0 Tungsten 74 0.5000









Table 6-2. Other reference case simulation parameters of the analysis plan
Monte Carlo Electron Adjoint Transport
Energy-loss straggling is sampled CEPXS cross sections
Mode: electrons only 50 equal width electron groups
Energy cutoff at 0.01 MeV Energy cutoff at 0.01 MeV
Default value for ESTEP in CH algorithm Ss-P7
No angular biasing Flat adjoint source spectrum
1st order CSD operator discretization
Smoothing


Grid Sensitivity and Automatic Spatial Meshing Schemes

As discussed in Chapter 5, the automatic selection of a spatial mesh structure is essential

for the robustness and ease of use of the methodology. As described in Chapter 4, the

deterministic models are automatically created by tracking the material discontinuities along a

line-of-sight between the source and the region of interest (ROI). This section presents a series of

studies aimed at identifying the impact of the mesh size within each of those material regions on

ADEIS efficiency. This aspect is essential to the development of an automatic discretization

scheme.

Uniform Mesh Size

The simplest approach is to select a default mesh density to be applied throughout the

model. Even though this approach is not believed to be the most efficient, it provides a better

understanding of the impact of different mesh sizes on the efficiency and accuracy of the ADEIS

methodology. The optimum mesh size should be related to the energy of the source particle and

the average Z of the material, and consequently it should be problem-dependent (i.e., different

for each case of the analysis plan). Since the weight-window is applied at the end of each major

energy step of the CH algorithm, it might be more insightful to study the speedup as a function

of the ratio of the mesh size to the partial range associated with a major energy step (referred to

as DRANGE in MCNP5) as given by Eq. 6-1.












Ax
Fraction of DRANGE (FOD) = (6-1)
DRANGE,


Moreover, by calculating this ratio using the DRANGE of the first major energy step, this


parameter becomes a function of the source particle energy and the average Z, and therefore is


problem-dependent. Figures 6-2 to 6-4 show the ADEIS speedup obtained as a function of this


ratio for all the test cases in the analysis plan.


1.E+05


1.E+04


1.E+03


1.E+02


1.E+01


1.E+00


1.E-01 I '',I
1.0E-02 1.OE-01 1.0E+00 1.0E+01
Fraction of DRANGE


Figure 6-2. Speedup as a function of FOD for Cases 1, 4 and 7


1 .E+05


1.E+04


1.E+03


1.E+02


1.E+01


1.E+00


1.E-01


1.0E-02 1.OE-01 1.0E+00 1.0E+01
Fraction of DRANGE


Figure 6-3. Speedup as a function of FOD for Cases 2, 5 and 8


Average Z = 8 Case 1:0.2 MeV

-C*-Case 4: 2.0 MeV

-U-Case 7: 20.0 MeV







"', *- -- ------ ...^ ~--


Average Z = 29 Case 2: 0.2 MeV

-*-Case 5: 2.0 MeV

Case 8: 20.0 MeV







--- E











1.E+05


Average Z = 74 Case 3: 0.2 MeV

S-*-Case 6: 2.0 MeV

-I- Case 9: 20.0 MeV







-. I. .


1.E+04


1.E+03


1.E+02


1.E+01


1.E+00


L.E-01


1.0E-02 1.OE-01 1.0E+00 1.0E+01
Fraction of DRANGE


Figure 6-4. Speedup as a function of FOD for Cases 3, 6 and 9

In all these cases, it can be observed that the speedup increases as the mesh size decreases


until a plateau is reached at, or below, mesh sizes close to the DRANGE. It is interesting to note


that, in Figures 6-2 to 6-4, the inconsistent ups and downs in speedup are produced by the fact


that the numerical artifacts are not completely removed by the smoothing.


For 1-D deterministic transport, the overhead computational cost associated with


generating and using the importance function is always significantly smaller than the


computational cost associated with the actual MC simulation. This explains why the speedup


plateau covers such a wide range of mesh sizes. However, for multi-dimensional importance, it is


expected that, as the mesh size is further reduced, the speedup would decrease due to the


computational cost of generating and using such detailed importance functions. Therefore, it is


important to select a mesh size as large as possible to reduce the computational cost in future


multi-dimensional deterministic transport simulation. Therefore, the onset of the speedup plateau


is indicative of the criterion that should be used to automatically select the appropriate mesh size


in each material region of the problem. It can be observed in Figures 6-2 to 6-4 that the onset of









the speedup plateau occurs at FOD that are similar to the approximate detour factor91' 93,94 values

presented in Table 6-3, which are defined as the ratio of the projected range to the CSD range.

Table 6-3. CEPXS approximated detour factors
Z number Detour factor
Low (Z<6) 0.5
Medium (6 High(Z>30) 0.1


These results imply that, for a given problem, a near optimal mesh size correspond to about

the average depth-of-penetration along the line-of-sight before the first weight-window event

evaluated by multiplying the detour factor by the CSD range. An alternative scheme using the

average energy of an electron in a coarse mesh (calculated using a CSD approximation and the

distance traveled from the source) to evaluate DRANGE was also implemented. This scheme is

not discussed in details since it resulted in extremely high mesh density near ROIs located deep

within the model and therefore led to significant reduction the efficiency of the methodology.

Note that the onsets of the speedup plateau do not exactly occur as predicted by the detour

factors presented in Table 6-3. In part, this can be explained by the fact that these approximated

detour factor values are mainly valid for energies below 1 MeV where they are independent of

the energy of the particle. A more recent and accurate set of semi-empirical formulas for electron

detour factors95 can be used to determine the detour factors for each case of the analysis plan

presented in Table 6-4. These various detour factors are plotted as vertical lines in the previous

Figures 6-2 to 6-4 and it can be seen that they agree slightly better with the onset of the speedup

plateau, and it is probable that a more accurate estimate of the average depth-of-penetration

along the line-of-sight would provide a better indication of the onset of the speedup plateau.


a The projected range is the projection of the vector distance from the starting point to the end point of a trajectory
along the initial direction of motion









Table 6-4. Calculated detour factors for each case of the analysis plan
Case Detour factor
1 0.49
2 0.28
3 0.18
4 0.54
5 0.32
6 0.20
7 0.78
8 0.56
9 0.40


It appears that, for Cases 7 and 8, the speedups are somewhat insensitive to the selection of

the mesh size, and that for Case 9, the onset of the speedup plateau occurs at FODs much larger

then one. This can be explained by the following facts: i) the gain in efficiency for these cases

results mainly from rouletting low-energy electrons, and ii) Russian roulette is much less

sensitive to the selection of the discretization parameters as it will be shown later. Therefore,

based on all these analyses, ADEIS will use the empirical formulas presented in Ref. 90 and the

DRANGE of the first major energy step to automatically determine the mesh size for each

material region.

Multi-layered geometry

Since most realistic cases are composed of more than one material, it is important to study

the automatic meshing scheme for such problems. Therefore, two new test case with a 2 MeV

electron beam impinging on three material layers are considered as illustrated in Figure 6-5.

Table 6-5 provides more detailed information about these new test case geometries while the

other simulation parameters are the same as given in Table 6-2. Using the spatial mesh criterion

described in the previous section, simulations are performed for these multi-layered geometries

and the results are presented in Table 6-6.













Beam Beam'


A) 1 23 B) 21 3

Figure 6-5. Multi-layered geometries. A) Tungsten-Copper-Water B) Copper-Tungsten-Water

Table 6-5. Materials and dimensions of new simplified test case
Zone Color Material Size (cm)
1 Dark Gray Tungsten 0.035
2 Orange Copper 0.055
3 Blue (ROI) Water 0.295


Table 6-6. Speedup for multi-layered geometries using automatic mesh criterion
Case Code FOM Speedup
Standard MCNP5 7.1x10-3
W-Cu-H20 8732
ADEIS 62
Cu-W-0 Standard MCNP5 3.2x10-3 1
Cu-W-H20 A 310313
ADEIS 33


These results clearly indicate that the automatic meshing criterion is applicable for geometries

with multiple materials and produce significant speedup.

Boundary Layer Meshing

In certain deterministic transport problem involving charge deposition near material

discontinuities or photoemission currents, it is important to select a mesh structure that can

resolve the boundary layer near the material and source discontinuities, i.e., the region near a

discontinuity where rapid changes in the flux occur. In CEPXS, this is achieved by generating a

logarithmic mesh structure where the coarse mesh size decreases as depth increases and

material/source discontinuities are approached. This approach is well suited for problems

involving a source on the left-hand side of the model but may not be adequate for ADEIS needs.









It is therefore useful to study various boundary layer meshing schemes and measure their impact

on the robustness and efficiency of the ADEIS methodology. However, in the ADEIS

methodology, the resolution of the boundary layers may affect both theforwardMC simulation

(i.e., using accurate values of the importance for biasing when approaching the material and

source discontinuities from the source side) and the backward (adjoint) deterministic simulation

(i.e., using appropriate meshing when approaching the material and source discontinuities from

the ROI side to generate accurate importance functions). Therefore, the automatic boundary layer

meshing scheme allows for appropriate meshing on either or both side of each discontinuity.

Automatic scheme #1

This automatic scheme has two steps. The first step is similar to the CEPXS approach,

where the coarse mesh size is decreased as the distance to a material discontinuity is decreasing.

In the second step, the fine mesh density in each coarse mesh is automatically selected based on

the criterion described earlier. This scheme is illustrated in Figure 6-6.

Source side ROI side




MC geometry





Deterministic
coarse meshes



Figure 6-6. Automatic boundary layer meshing scheme #1









This scheme is first used while performing simulations for Cases 1 and 9 in order to

evaluate the impact of properly modeling the boundary layers at the edge of the adjoint source

region (ROI) and source interface. Using this scheme with five and ten coarse meshes, the

changes in speedup with respect to the plateau speedup (see Figures 6-2 to 6-4) are presented in

Table 6-7. It is obvious that the change in speedup obtained by resolving the boundary layers at

the source region interface is rather small and can even results in a slight decrease in efficiency.

Table 6-7. Speedup gain ratios from boundary layers scheme #1 in Cases 1 and 9
Case # of coarse Forward Backward Speedup
Forward Backward
meshes gain ratio
5 Yes No 1.09
No Yes 1.09
10 Yes No 1.15
No Yes 1.14
5 Yes No 0.94
No Yes 0.93
10 Yes No 0.97
No Yes 0.99


Then this scheme was applied to the multi-layer geometry described in Figure 6-5 A) and

Table 6-5. The resulting speedup was reduced by a factor 3 compared to the results shown in

Table 6-6. It is therefore concluded that any gain obtained by refining the meshes at the material

or source discontinuities is lost because of the extra computational cost of searching through the

many additional coarse meshes.

Automatic scheme #2

This second scheme is based on the knowledge that if the selected mesh size can resolve

the lowest energy group flux near the material boundary then all the fluxes for all energies will

be resolved in that region. However, such a refined meshing would be extremely

computationally costly, and therefore should be used only within a short distance of material or

source discontinuities. Even though this distance is somewhat arbitrary, it is possible to make an









informed selection. This distance is chosen as the distance along the line-of-sight between the

boundary and a fraction of the partial range representing the slowing down of the fastest

electrons to the next adjacent energy group. Finer meshes in that region should properly describe

the exponential drop of the higher energy fluxes and the buildup of the lower energy fluxes as

illustrated in Figure 6-7.













Slower flux

Faster flux

Partial Partial
range range

Figure 6-7. Automatic boundary layer meshing scheme #2

As with the automatic scheme #1, this scheme is first applied to the test cases 1 and 9.

Table 6-8 clearly shows that the gain in speedup from resolving the boundary layer at the source

region interface using the automated meshing scheme #2 is minimal, and can even results in a

slight decrease of performance. Application of this scheme also resulted in a decrease of

efficiency for the multi-layered geometry. It is interesting to note that the change in speedups

using the automated scheme #2 is quite similar to the previous automated scheme both in the

case of source region and material discontinuities.









Table 6-8. Speedup gain ratios from boundary layers scheme #2 in Cases 1 and 9
Case Size of S
refined Forward Backward gan rop
gain ratio
region
0.5 R1 Yes No 1.09
No Yes 1.10
R1 Yes No 1.09
No Yes 1.09
0.5 Ri Yes No 0.95
No Yes 0.94
Ri Yes No 0.94
No Yes 0.94


Conclusions

Even though it was shown in previous studies89 that manually adjusting the mesh structure

to resolve the boundary layer at certain material discontinuity had a positive impact of the

efficiency and statistical reliability of the tallies, this section showed that a systematic and

automatic approach to perform such a task does not appear to yield any improvement, and

therefore, will not be used by default in ADEIS.

Energy Group and Quadrature Order

Performing discrete ordinates simulations also require the selection of the number and

structure of the energy groups as well as the quadrature set order. It is therefore important to

study the impact of these parameters on the ADEIS methodology speedup in order to properly

select a criterion for the automatic scheme.

Number of Energy Groups

To study the impact of the number of energy groups, the test cases of the analysis plan

presented in Table 6-1 are used. Using a mesh size equal to the crow-flight distance associated

with the slowing down from the first to second energy group, each test case is simulated with

different number of energy groups ranging from 15 to 85 energy groups of equal width.

Figures 6-8 to 6-10 present the speedup obtained from all cases of the analysis plan.














1.E+05



1.E+04



1.E+03



1.E+02



1.E+01



1.E+00



1.E-01


0 10 20 30 40 50 60 70 80 90
Number of Groups



Figure 6-8. Speedup as a function of the number of energy groups for Cases 1, 2 and 3


1.E+05



1.E+04



1.E+03



1.E+02



1.E+01



1.E+00



1.E-01


0 10 20 30 40 50 60 70 80 90

Number of Groups



Figure 6-9. Speedup as a function of the number of energy groups for Cases 4, 5 and 6


-* Case 1: Z-8
--Case 2: Z-29
- Case 3: Z-74 Energy = 0.2 MeV


" Case 4: Z-8
--Case 5: Z-29
-- Case 6: Z-74 Energy = 2.0 MeV











1.E+05


1.E+04


1.E+03


1.E+02


1.E+01


1.E+00 Case 7: Z-8
-- Case 8: Z-29
-- Case 9: Z-74 Energy = 20.0 MeV
1.E-01 .
0 10 20 30 40 50 60 70 80 90
Number of Groups


Figure 6-10. Speedup as a function of the number of energy groups for Cases 7, 8 and 9

A few observations can be made about Figures 6-8 to 6-10; i) no clear optimal values

seems to apply to all cases, ii) the dependency on the number of energy group is rather weak for

cases with an average low Z number (Cases 1, 4 and 7), iii) the dependency on the number of

energy group is rather weak for cases with high energy source electrons (Cases 7, 8 and 9), iv)

speedup can vary by a few order of magnitudes depending on the number of energy groups,

which clearly illustrates the need for an automatic selection of the discretization parameters, v)

higher maximum speedups are obtained for cases with larger average Z, and vi) the speedup

plateau seems to be reached at about 65 energy groups for Cases 5 and 6.

A clear optimal value for the number of energy groups is difficult to pinpoint in

Figures 6-8 to 6-10, because these test cases are inherently different as demonstrated by the ROI

total fluxes shown in Table 6-9.









Table 6-9. Total flux and relative error in the ROI for all cases of the analysis plan
Case Total flux (cm-2) Relative error
1 1.5405E+00 0.0034
2 9.9798E-02 0.0111
3 1.6629E-04 0.0349
4 2.9529E-03 0.0033
5 5.0038E-05 0.0190
6 1.6312E-08 0.0900
7 2.3149E-03 0.0009
8 4.9834E-02 0.0011
9 4.5018E-02 0.0015


To explain this behavior, it is useful to first remember that the Russian roulette and

splitting games improve the simulation efficiency in completely different ways; Russian roulette

reduces the time per history by killing time consuming unimportant particles but increases the

variance while splitting decreases variance by multiplying important particles but increases the

time per history. Therefore, for a given problem, these two mechanisms compete to produce an

increase in efficiency. This can be seen by looking at Table 6-10 showing the average ratio of

tracks created from splitting to tracks lost from Russian roulette of the cases with same source

electron energy given in Table 6-10.

Table 6-10. Average ratio of track created to track lost for cases with same energy
Electron energy Ratio of track created
(MeV) to track lost
0.2 0.64
2.0 0.17
20.0 0.04
a Tracks are created through splitting and lost through Russian roulette

For example, Figure 6-10, as well as Tables 6-9 and 6-10, indicate that cases with a

20 MeV electron beam have a low ratio of track created to track lost, lower speedups, higher

total fluxes and a rather weak dependency on the number of energy groups. This suggests that,

for these cases, Russian roulette dominates because; i) a significant amount of secondary

electrons will have to be rouletted, and ii) it is relatively easy for a source particle to reach the









ROI as shown by the total flux. Figure 6-11 shows the importance of the source particles and

knock-on electrons at a few energies. Note that in certain cases the ADEIS smoothing algorithm

limits the importance function values.








S10-
S103- 19.3MeVto20.7MeV



0.01 MeV to 1.38 MeV

10-23


2.0 4.0 6.0 8.0 10.0
Position along x-axis [cm]


Figure 6-11. Importance functions for source and knock-on electrons at a few energies for Case 7

Figure 6-11 clearly shows that the importance of the source electrons (in red) will differ

significantly from the importance of the knock-on electrons created in the other energy ranges.

Considering that 99% of the knock-on generated from the source electrons will be created below

1.38 MeV, it is obvious by looking at the importance that the very large majority of them will be

rouletted.

However, for cases (e.g. 5 and 6) where splitting is more important (lower total flux in ROI

and larger ratio of track creation to track loss), the use of a larger number of energy groups

increases the quality of the importance function used in ADIES. As seen previously for the

spatial meshing, an increase in accuracy is accompanied by an increase in speedup until a plateau

is reached. Once again this plateau may not be as wide for multi-dimensional calculations.











To further demonstrate this effect, Case 9 was modified to increase the importance of

splitting by tallying at larger depths within the target material. As expected, Figure 6-12 shows a

smaller total flux in the ROI, a larger ratio of track created to track lost and stronger dependency

on the number of energy groups.


1.E+05


1.E+04
Tracks created / tracks lost = 0.064
Total flux in ROI= 6.1265E-07
1.E+03


S1.E+02


1.E+01


1.E+00


1.E-01
0 10 20 30 40 50 60 70 80 90 100
Position along the x-axis [cm]


Figure 6-12. FOM as a function of the number of energy groups for modified Case 9

It can also be observed in Figure 6-12 that the onset of the speedup plateau for this

modified case is reached at about 65 energy groups as seen previously for Cases 5 and 6. Finally,

it is interesting to study in more details Case 3 for which the efficiency is reduced rather then

increased as the number of energy groups increases. The reduction in efficiency is caused by

additional splitting produced by the increase in the number of energy groups, resulting in a

significant increase in the number of secondary electrons. This can be demonstrated by

examining the changes in the number of knock-on electrons and their total statistical weight as a

function of the number of energy groups as shown in Figure 6-13.











1.E+08 1.6

1.4



1.E+07
-- Number of knock-on electrons 1

6 Total weight of knock-on 0.8 "
electrons

0.6
1.E+06
0.4

0.2

1.E+05 0.0
0 10 20 30 40 50 60 70 80
Number of energy groups


Figure 6-13. Number of knock-on electrons and their total statistical as function of the number of
energy groups for Case 3.

Figure 6-13 clearly shows the large increase in the number of knock-on electrons as the


number of energy group increases, while the total sum of their weights remains almost constant


to conserve the total number of particles. Therefore, part of the decrease in efficiency observed


for Case 3 in Figure 6-8 can be explained by the additional computational cost associated with


simulating these additional secondary electrons.


Another part of the decrease in efficiency comes from the overhead computational cost


associated with performing additional splitting and rouletting. To illustrate this fact, it is


interesting to look at scatter plots of the energy of electrons that have been splitted or rouletted as


a function of their position in the model. Note that in Figures 6-14 to 6-16, the grid represents the


spatial and energy discretization of the weight-window.













.o .5165 *'9 V!

-51R.217
**. f 3.




0.l13 .









Figure 6-14. Splitted electron energy as a function position for a 1000 source particles in Case 3.
A) for a weight-window using 15 energy groups B) for a weight-window using 75
energy groups

o. 13 ,- ... ... .-- ------- -.....
0.11 .. 1.11. ,.



















o .,,,.' .i .
0.052 v 0.002










.-9 -. 5 -..033 :-. .

.) Po lo Al.. .I.a.. I.n I'j. B ) Poo.lion 311g 0.l. [_nx 10,.

Figure 6-14. Splitted electron energy as a function position for a 1000 source particles in Case 3.
A) for a weight-window using 15 energy groups B) for a weight-window using 75






















using 75 energy groups

By comparing Figures 6-14 and 6-15, it can be seen that slightly more biasing is performed

on high-energy electrons near the source for the case with 15 energy groups while significantly

more splitting and rouletting of low-energy electrons is performed near the ROI (deep within the
target material) for the case with 75 energy groups. Figures 6-16 A) and B) examine the











statistical weight of the splitted electrons as a function of the position in the model for a weight-
window with 15 and 75 energy groups.
tae .r f.F





Figure 6-15. Rouletted electron energy as a function position for a 1000 source particles in

using 75 energy groups














window with 15 and 75 energy groups.












1 I.E*O1

-. -; L. i ""., -,

P i ":.. .lo 1 ] n m 1
i L.E-0 io c03
Su i .. p. tL e oon





A) brp eons it Eo is4d [cmexst0gB) to compare the i0 o


Figure 6-16. Splitted electron weight as a function of position for a 1000 source particles in
Case 3. A) for a weight-window using 15 energy groups B) for a weight-window
"."**' + ,,, ..E-l '





Figure 6-16. Splitted electron weight as a function of position for a 1000 source particles in
Case 3. A) for a weight-window using 15 energy groups B) for a weight-window
using 75 energy groups.

There is obviously no physical justification for this change in behavior as a function of the

number of energy groups. It is therefore possible that the quality of the importance function

might be responsible. Consequently, it is interesting to compare the importance functions

spectrum obtained with 15 and 75 energy groups. In Figure 6-17, the importance function for the

case with 75 groups shows a significant amount of unphysical oscillations that are obviously

degrading the quality of the function and the efficiency.






101


101


0 103


10 -


0.05 0.1 0.15
Energy [MeV]


Figure 6-17. Importance functions for 15 and 75 energy groups at 3.06 cm for Case 3.









Conclusions

It can be concluded that for problems where Russian roulette is the dominant factor in the

improvement in efficiency, the speedup is not significantly affected by the number of energy

groups. On the other hand, for most cases where the splitting is the dominant factor, the quality

of the importance function improves as the number of energy group increased. A speedup

plateau is reached around 65 groups, where both the accuracy and efficiency are optimum. The

increased computational cost associated with a larger number of energy groups is not strongly

influencing the efficiency when using 1-D importance functions. It was also shown that a high

number of energy groups can be significantly detrimental to the efficiency of an ADEIS

simulation in certain cases because of the additional computational cost from the unnecessary

splitting and rouletting near the ROI. ADEIS simulation should therefore be performed with a

weight-window using at most 35 energy groups for which the speedup plateau is almost reached

and no degradation in efficiency was noticed. Moreover, it is expected a smaller number of

energy groups will results in significant computation time savings in multi-dimensional

simulations.

Quadrature Order

The quadrature order represents the number of discrete directions used to solve for the

deterministic importance functions. In the non-angular biasing, the angular importance along

these directions is integrated into a scalar importance functions. However, to properly model the

angular behavior of the solution before integration, it is important to have a number of directions

that adequately represents the physics of the problem. Higher anisotropy requires a larger

number of directions. Typically, unbiased quadrature sets are symmetric along [t (over the unit

sphere in the case of 3-D simulations) and have an even number of directions equal to the order

(e.g. S4 correcponds to 4 angles). According to previous studies86 performed using the











CEPXS/ONELD package, an S16 quadrature order is required to properly model the highly


angular behavior of an electron beam. However, these studies also show that an Ss quadrature

order is sufficient to model problems with a distributed volumetric source. These


recommendations can easily be extended to the adjoint calculations performed in the ADEIS


simulations. It is therefore expected that an Ss quadrature set should be sufficient since the


adjoint source is usually distributed over the ROI. For completeness, the various test cases of the


analysis plan are simulated with three different SN order (quadrature order); S4, Ss, and S16


corresponding to 4 angles, 8 angles, and 16 angles. Note that the Gauss-Legendre quadrature set


is used to meet the limitation of the CEPXS methodology.


100000



10000 s

-case 1
Case 2
1000 -case3
Case 4
to case 5
"i case 6
100 case 7
case 8


10




0 2 4 6 8 10 12 14 16 18
SN order


Figure 6-18. Impact of discrete ordinates quadrature set order on speedup for all cases of the
analysis plan.

Figure 6-18 shows that speedups do not strongly depend on the number of directions.


However, the cases exhibiting the larger variations in speedup are the cases where splitting is the


dominating VR game. From these results, it can be concluded that for cases where Russian









roulette dominates, the speedup dependency on quadrature order is rather weak. It can also be

concluded that the quality of the importance functions obtained with 4 directions is not optimal

and would be probably even less adequate for a smaller ROI. As expected, it does not seem

necessary to increase the order to S16. This is a desirable characteristic for the calculations of

multi-dimensional importance functions where the number of directions for a given order is

much larger (e.g., an S16 level symmetric quadrature set have 288 directions in 3-D). Therefore,

by default, ADEIS will use an Ss quadrature set until the impact of the parameter on multi-

dimensional importance function calculations is observed.

Angular Biasing

It is also important to study the angular aspect of the biasing to verify if the field-of-view

(FOV) approach is appropriate for all cases and for all particles. For these studies, Case 1 and

Case 7 were simulated by using various constant and changing FOVs as listed below:

* p e [0, 1]: the FOV for truly 1-D geometries is equivalent to biasing in the forward
direction.

* Cp e [0.78, 1]: Calculating [t subtending the ROI from the location where the beam
impinges on the face of the model gives a FOV of [t e [0.89, 1]. However this direction
falls between two directions of the Ss quadrature set. This FOV integrates all the directions
of the quadrature set that have smaller |ts and the next immediate direction.

* p e [0.95, 1]: Calculating [t subtending the ROI from the location where the beam
impinges on the face of the model gives a FOV of [t e [0.89, 1]. However this direction
falls between two directions of the Ss quadrature set. This FOV integrates all the directions
of the quadrature set that have smaller hts.

* p e [0.98, 1]: For completeness a more forward-peaked biasing is analyzed. Note that the
quadrature order had to be increased to S16 to have a FOV subtending a smaller solid
angle. This highlights one of the limitations of the FOV methodology since the size of ROI
and the quadrature set order are linked. This limitation will be further discussed in Chapter
8.

* Space-dependent p-FOV: As shown in Figure 3-1 A), it is possible to define different [t
subtending the ROI at different depth and use them to calculate space-dependent FOVs.









In Figure 6-19, tracks from a standard MCNP5 and non-angular ADEIS simulations for

Case 7 are shown. Note that the four space-independent FOVs studied in this analysis are

overlaid on Figure 6-19 B). By looking at the tracks, it is obvious that some angles of travel are

already favored in the non-angular ADEIS since the lower energy electrons (less forward-peaked

particles) are already rouletted. Therefore, the out-of-FOV biasing will not be responsible for a

significant of amount of rouletting since few electrons are naturally out of the selected FOV. The

major impact of the angular biasing should therefore be to increase the splitting of the particles

traveling within the FOV.

F-N
FO, 1.1




IO "' 4






A) B)

Figure 6-19. Electron tracks for a 20 MeV electron pencil beam impinging on water (Case 7) A)
standard MCNP5 B) ADEIS.

However, compared to non-angular biasing, a loss in efficiency (between 25% and 50% of

the non-angular speedup) was obtained from additional angular biasing in this case since most

electrons are naturally traveling within the FOV. Therefore, the additional computational cost of

the extra splitting with the FOV provides little reduction in variance. To further study the

usefulness of the angular biasing, the source of Case 7 was slightly modified to introduce an

angular dependency in the form of a cosine distribution along it.














c ,, O \







A) B)

Figure 6-20. Electron tracks for a 20 MeV electron cosine beam impinging on tungsten (Case 7
with a source using an angular cosine distribution). A) standard MCNP5 B) ADEIS.

Figure 6-20 shows that because of the source angular profile, the non-angular ADEIS

biasing does not favor a directional behavior. Therefore the loss in efficiency compared to non-

angular biasing is not as large, between 70% and 85% of the non-angular speedup. However, in

Case 7, the spreading of the beam is similar to the FOV subtending the ROI. It can be therefore

supposed that the increase of efficiency from angular biasing should occur if a significant

amount of particles are traveling outside the FOV. First, angular biasing is performed for Case 3

since the spreading of the beam is significantly smaller then the FOV as shown in Figure 6-21.


FOV: [0 [0,1]
FOV: i [0.78,1]



,. FO%. l1.9N.,






A) B) _


Figure 6-21. Electron tracks for a 0.2 MeV electron pencil beam impinging on tungsten (Case 3)
A) standard MCNP5 B) ADEIS.









Again, as expected, angular biasing reduces the gain in speedup compared to non-angular

biasing by 50% since most electrons biased by the non-angular version of ADEIS reaches the

ROI. It is interesting to note, in Figures 6-21 A) and B), the difference in behavior between the

unbiased and biased electrons. It is clear that in the ADEIS simulation the higher energy

electrons (red) are significantly splitted close to the source so more of them can reach the ROI.

Secondly, Case 7 is further modified to reduce by 75% the size of the ROI along the z-axis and

locate it at larger depth in the target material. This obviously decreases the number of electrons

naturally traveling within the FOV to the ROI. Table 6-11 gives the various speedups obtained

from the different FOVs for this case at two different depths; i) ROI at 10.5cm, ii) ROI at

11.5 cm.

Table 6-11. Speedup with angular biasing for Case 7 with a source using an angular cosine
distribution and reduced size
FOV Speedup: case i) Speedup: case ii)
None
n/a n/a
(standard MCNP5)
None
122 378 (386/362)
(non-angular ADEIS)
t e [0,1] 137 334 (464/490)
[t e [0.78,1] 138 335 (464/495)
et [0.95,1] 137 334 (464/496)
[t e [0.98,1] 152 463
Space-dependent FOV 17 44


Based on these results, it seems that the angular biasing improve the efficiency of non-

angular ADEIS simulations for cases with a significant number of electrons remaining outside

the FOV if only space-energy biasing is performed. It can also be seen that for these two cases,

the largest improvement is obtained from highly forward-peak biasing. It seems that having

space-dependent FOV along the line-of-sight significantly reduces the speedup, and might be

useful only when multi-dimensional problems are studied.









It is important to discuss the fact that there are some issues related to the quality of the

angular information. It is a well known fact that angular fluxes are generally less accurate than

the scalar fluxes because of the errors compensation. A similar error compensation phenomenon

occurs when calculating the partially integrated values of the FOV's importance. However, the

integration is performed over a small fraction of the unit sphere resulting in values less accurate

than the scalar fluxes. Consequently, the ADEIS angular importance functions contain a much

larger fraction of negative values requiring smoothing, which may further decrease the quality.

In Table 6-11, this is obvious by looking at the speedup values in parentheses. These values were

obtained by calculating the importance function with higher quadrature orders, S16 and S32

respectively. As mentioned before, the computational cost associated with calculating the

importance functions is minimal and therefore the increase in speedup essentially reflect the

increase in accuracy.

Coupled Electron-Photon-Position Simulation

Most realistic simulations require the modeling of the complete cascade and therefore

necessitate coupled electron-photon-positron simulations. In such coupled problems, ADEIS

uses weight-window spatial mesh determined for electrons because, i) the same spatial meshing

must be used in ONELD and MCNP5 to bias all particles, and ii) the accuracy of the electron

importance is much more sensitive to the mesh size as discussed previously. However, there is

no need to use the same energy group structure for photon and electrons, therefore the number of

energy groups considered for the weight-window should be optimized. Note that, because of the

CEPXS methodology, the positrons energy group structure must be the same as the electrons.

Moreover, in ADEIS, the positrons cannot be used as the particles of interest since they cannot

be tallied in MCNP5. Therefore, the adjoint source is set equal to zero for the positrons energy

groups if they are present in the simulation.









To study the impact of the photon energy group structure in coupled electron-photon

simulations, Cases 6 and 9 were modified to tally the photon flux in the ROI rather then the

electron flux and accordingly, a flat adjoint spectrum is defined only for the photon energy

groups. These two cases were selected because the medium and high-energy electrons interacting

with tungsten will create photons through bremsstrahlung and therefore create a model where

electrons and photons are tightly coupled. The number of energy groups for the electrons and the

other discretization parameters are kept identical to cases of the analysis plan while the number

of photon energy groups is varied. Two group structure are also studied for these different

number of photon energy groups; linear and logarithmic. Previous studies86 using the CEPXS

package showed that photon groups with a logarithmic structure describe more accurately the

bremsstrahlung by reducing the group width at lower energies. This should be useful in the

context of the ADEIS VR methodology since for the same computational cost the accuracy of

the importance function, and consequently, the speedup could be increased.

Table 6-12. Speedup as a function of the number of energy groups for Cases 6 and 9
# of energy groups Speedup: Case 6 Speedup: Case 9
15 linear 0.97 0.51
15 logarithmic 5.48 8.37
25 linear 0.46 0.38
25 logarithmic 5.85 8.47
35 linear 1.30 0.63
35 logarithmic 5.55 8.01
45 linear 6.13 0.92
45 logarithmic 6.00 8.20


From Table 6-12, two interesting observation can be made; i) the logarithm energy group

structure has a significant impact on the efficiency of the ADEIS methodology, and ii) 15 energy

groups seems to be sufficient if the logarithmic group structure is used. Consequently, by default,

ADEIS will use these parameters.









It is also important to study the impact of the ADEIS angular biasing methodology for

coupled electron-photon-positron simulations. At this point, ADEIS uses the same biasing for

electron and photon even though it is possible to bias them differently. Note that in ADEIS,

angular biasing is never performed on the positrons since they cannot be the particles of interest

as mentioned earlier. Moreover, because of the annihilation process, a positron traveling in any

direction can create secondary particles that might contribute to the ROI. To study, the impact of

angular biasing in coupled electron-photon-positron problems, the Chapter 5 reference case is

simulated. Table 6-13 gives the speedups obtained for the photon and electron tallies located in

the ROI using the same FOV described in the previous section.

Table 6-13. Electron and photon tally speedup using ADEIS with angular biasing
FOV Electron speedup Photon speedup
None
n/a n/a
(standard MCNP5)
None
131 12.7
(non-angular ADEIS)
[G e [0,1] 151 15.9
[t e [0.78,1] 27.6 8.57
et [0.95,1] 5.73 2.85
G e [0.98,1] 2.11 0.96
Space-dependent FOV 15.5 3.14


Above results indicate that for this case, the angular biasing for photon traveling in the

forward direction produces the highest increase in speedup. As shown previously, it appears that

space-dependent FOVs along the line-of-sight do not improve the efficiency of the ADEIS. The

same observation applies to the electrons in this problem.

Adjoint Source Selection

As discussed in Chapters 2 and 3, in adjoint calculations the source typically represents the

objective for which the importance is evaluated. Therefore, in the ADEIS methodology, the

adjoint source represents the objective toward which the simulation is biased. For coupled









electron-photon-positron, coefficients such as flux-to-dose conversion factors (flux-to-energy

deposition) are not readily available for all materials. It is expected that the use of any adjoint

source will not bias the simulation but might simply not produce significant speedup.

Therefore, ADEIS uses two automatically determined adjoint sources that are adequate for

energy deposition and flux calculations; i) a uniform spectrum to maximize the total flux in the

ROI, and ii) a local energy deposition response function to approximate energy deposition in the

ROI. Note that the ADEIS methodology uses a spatially uniform adjoint source over the whole

ROI.

In coupled electron-photon-positron simulation, it is possible to tally quantities associated

both with photons and electrons within the same model. At this point, ADEIS allows only the use

of a single cell as an objective since the use of different cells may reduce the gain in efficiency of

the simulation. ADEIS allows to bias electrons, photons or both within the same simulations

since the energy deposited in a given region can be influence by both types of particles. Even

though it is possible to bias only a single species in a coupled simulation, it is not typically done

because of the coupled nature of the physical processes. However, it is possible to define the

objective for only a single species of particles or for both species.

Consequently, it is useful to study the impact of having either a single tally/objective for a

given particle or two tallies for different particles with two objective particles. The reference

case defined in Chapter 5 is therefore used with three different combinations of tallies and

objective particles; i) electron tally and electron as the objective particle, ii) photon tally and

photons as the objective particle, and iii) electron and photon tallies with both particles as

objectives. Note that this is performed only for flux tallies, and therefore, the flat adjoint

spectrum is used.











Table 6-14. Electron and photon flux tally speedup using different objective particles
Objective particle Electron speedup Photon speedup
Electron 150.9 n/a
Photon n/a 14.9
Electron and photon 89.6 8.8


Table 6-14 indicates that it is obvious that larger speedups are obtained when a single

objective particle is used. The same analysis can be performed using the same reference case but

the energy deposited in the ROI is tallied instead of the flux. For this case, it can be seen in Table

6-15 that having both objective particles results in slightly larger speedup. However, it can also

be seen that almost the same speedup is achieved by using only the electrons as the objective

particles.

Table 6-15. Energy deposition tally speedup in the reference case for various objective particles
Objective particle Speedup
Electron 14.6
Photon 6.3
Electron and photon 15.1


Above finding can be explained by the fact that, physically, the photons deposit their

energy by creating electrons that are then more or less quickly absorbed. By using only electrons

as the objective particles, the photon adjoint solution will therefore represents the importance

toward producing electrons within the ROI, which corresponds closely to the physical process of

energy deposition. Moreover, in the Chapter 5 reference case, it is relatively easy for the photons

to reach the ROI since this reference case is based on a radiotherapy LINAC for which the

design goal was to have as much photons as possible reaching the ROI. Therefore, the additional

speedup provided by biasing the photon toward the ROI is smaller. Note that the use of the









uniform adjoint spectrum (equivalent to biasing toward the total flux in the ROI) results in

similar speedups for the energy deposition tally.

Conclusions

The analyses presented in this Chapter investigate strategies to improve the quality and

accuracy of the deterministic importance functions to maximize the speedups obtained from

ADEIS. These analyses considered a wide range of source energies and material average

z-numbers. To achieve this goal, these studies were performed on the selection of discretization

parameters for the different phase-space variables (space, energy and direction), as well as the

impact of the adjoint source and angular biasing.

First, it was shown that it is not necessary to accurately resolve the flux boundary layer at

each material and/or source discontinuity, and that the use of uniform mesh sizes within each

material region is sufficient. For each material region, it was shown that a mesh size based on the

source electron average depth of penetration before the first weight-window event occurs

resulted in near maximum speedups. This distance is evaluated using the path-length associated

with the first major energy step of the CH algorithm, and the detour factors derived from

empirical formulations.

Secondly, it was shown that the quality of the importance functions (and therefore the

speedup) is maximal at about 75 electron energy groups when using the first-order differencing

scheme for the CSD operator. For cases where the knock-on electrons contribute significantly to

the region of interest (ROI), the selection of more than 35 electron energy groups degrades

significantly the efficiency of ADEIS because of the larger amount of splitting and rouletting

occurring near the ROI. However, a significant fraction of the maximum speedup is already

obtained using 35 electron energy groups. For photon energy groups, it was shown that

maximum speedups are obtained in coupled electron-photon problems when a logarithmic









energy group structure resolving the bremsstrahlung is used. It is shown that for such group

structure, the maximum speedups are achieved with 15 energy groups.

Thirdly, it was shown that for problems where the gain in efficiency depends significantly

on the splitting game, the selection of discretization parameters is more critical. This can be

explained by the fact that accurate deterministic importance functions are required to properly

maintain the population of particles throughout the model. Alternatively, it was shown that

problems where the gain in efficiency is mainly a result of the rouletting of low-energy

secondary electrons, the speedups are relatively insensitive to the selection of the discretization

parameters.

It was shown that maximum speedups are obtained using an Ss quadrature set. Angular

biasing resulted in the largest increase in speedup when the FOV integrated all the directions in

the forward direction along the line-of-sight. It was also shown that for cases where flux is the

quantity of interest, higher speedups are obtained if the adjoint source is defined only for the

particle of interest. However, for problems where the energy deposition is the quantity of

interest, it was shown that maximum speedups are obtained when the adjoint source is defined

for both electron and photon. Moreover, it was shown that the major part of this speedup can be

obtained by defining an adjoint source for electron even when only photons reach the ROI. This

can be explained by the fact that, physically, the photons deposit their energy by creating

electrons. Therefore, by using only electrons as the objective particles, the photon adjoint

solution represents the importance toward producing electrons within the ROI, which

corresponds closely to the physical process of energy deposition.









CHAPTER 7
MULTIDIMENSIONAL IMPORTANCE FUNCTION

The analyses presented in Chapter 6 were performed using 1-D importance functions

generated with the CEPXS/ONELD package along the line-of-sight (LOS) in 3-D geometries.

From these analyses, a series of criteria to automatically select the discretization parameters were

developed. This Chapter investigates the generation and utilization of coupled electron-photon-

positron multi-dimensional importance functions for ADEIS. First, a series of analysis is

performed to study the computational cost and accuracy of 3-D importance functions generated

using the PENTRAN code. Secondly, the use of 2-D (RZ) importance functions generated by

PARTISN is studied. More specifically, the following four points are examined:

1. Generation of 1-D importance functions using the parallel SN PARTISN transport code;

2. Biasing along the LOS using the MCNP5 cylindrical weight-window;

3. Generation of 2-D (RZ) importance functions using PARTISN;

4. Speedup comparison between 1-D and 2-D (RZ) biasing.

Generation of 3-D Importance Function Using PENTRAN

This section presents studies on the level of accuracy and computational cost of an adjoint

solution obtained from a 3-D discrete ordinates calculation using CEPXS-GS cross-sections with

the 3-D discrete ordinates PENTRAN code. Even though ADEIS uses the MCNP5 cylindrical

weight-window, the study of 3-D Cartesian importance functions will provide information about

the computational cost and accuracy of generating 3-D importance functions in general. To

investigate the accuracy of the 3-D importance function generated with PENTRAN, a

comparison with the ONELD adjoint solution is performed. Comparing the 3-D importance

function generated from PENTRAN with the ONELD solution shows that PENTRAN can

achieve, at least, the level of accuracy required by ADEIS. Note that even though 1-D models









were considered by prescribing reflecting boundary conditions, PENTRAN effectively performs

3-D transport, i.e., various numerical formulations in 3-D are used. To examine the accuracy and

computation of obtaining a 3-Dimportance function using PENTRAN, three problem sets are

considered. More specifically, the impact on accuracy of the following numerical formulations in

PENTRAN is investigated:

* Differencing schemes: linear diamond (DZ), directional theta-weighted96 (DTW), and
exponential-directional weighted85 (EDW)

* Quadrature set order using level symmetric (LQN) up to S20 and Gauss-Chebyshev (PN-TN)
above S20.

Note that these studies required higher expansion orders of the scattering kernel that are

not typically needed for neutral particle transport. A new algorithm for the use of arbitrary PN

order and for pre-calculating all coefficients of the expansion was implemented into PENTRAN.

Problem #1

This first problem is designed to study the impact of various numerical formulations in a

3-D context for a low-Z material. Therefore, a problem with a uniform source (maximum energy

of 1 MeV) distributed throughout a beryllium slab is considered. A reference solution is obtained

with ONELD using the parameters given in Table 7-1. To emulate this 1-D problem using

PENTRAN, a cube with reflective boundary conditions is considered as illustrated in Figure 7-1.

Table 7-1. PENTRAN and ONELD simulation parameters for solving problem #1
PENTRAN ONELD
CEPXS-GS cross sections CEPXS cross sections
50 uniform meshes 50 uniform meshes
40 equal width electron groups 40 equal width electron groups
Level symmetric quadrature Gauss-Legendre quadrature
S16-P15 S16-P15
Linear diamond Linear discontinuous











Reflective BC
in 3-D model


Adjoint source
(flat spectrum)

Ber. Ilium




0.3 cm


Figure 7-1. Problem #1 geometry

By comparing the importance functions calculated by PENTRAN and ONELD, Figure 7-2

shows that the PENTRAN solution is similar (shape and magnitude) to the ONELD solution, and

therefore is adequate for use in ADEIS.




0.26


).24


1.22
ONELD
PENTRAN
0.2 -


o 1.18


J.16

0.05 0.1 0.15 0.2 0.25
X-axis [cm]

Figure 7-2. Importance function for fastest energy group (0.9874 MeV to 1.0125 MeV) in
problem #1

To examine the difference in the solutions of the two codes, Figure 7-3 shows the ratios of

the importance functions for different energy groups including; i) Group 10 (0.7618 MeV to










0.7869 MeV); ii) Group 20 (0.5112 MeV to 0.5363 MeV); iii) Group 30 (0.2606 MeV to

0.2856 MeV); and, iv) Group 40 (0.01 MeV to 0.0351 MeV).


1.1


1.05

Group 40
Group 10
0 Group 20 ----

0.9

0.9 3o Group 30
0.85 -

0.8
0.05 0.1 0.15 0.2 0.25
X-axis [cm]


Figure 7-3. Ratio of ONELD importance over PENTRAN importance for energy groups in
problem #1

From the ratios presented in Figure 7-3, it can be concluded that the shapes of the

importance functions for ONELD and PENTRAN are similar and that their magnitudes are

within 15%. This difference in magnitude can be attributed in part to the difference in quadrature

sets used in the codes. It is important to note that the observed unphysical oscillations occurs at

the boundary of the problem where the importance function drops significantly. This behavior

can be attributed to slightly less then adequate meshing (no refinement at the boundary), the

quadrature order and the use of the linear diamond differencing scheme in PENTRAN. Note that

the impact of the discretization scheme on the accuracy of the PENTRAN importance function is

studied using a second problem as presented in the following section. As mentioned earlier, the

simulation time for ONELD and PENTRAN are significantly different. For this problem,

ONELD required -1 second, while PENTRAN required 906 seconds. This is expected since, for

this problem, 3-D transport requires the solution of about 20 times more unknowns.










To study the impact of the mesh structure on the accuracy of the PENTRAN solution, the

meshing was modified to better resolve the boundary layers at the edges of the model. This new

mesh structure is illustrated in Figure 7-4.


zone 2:
40 meshes
0.26cm


zone 1 and 3:
20 meshes
0.02cm


Figure 7-4. Mesh refinement to resolve boundary layers at the edges of model for problem #1

Figure 7-5 shows that this mesh refinement reduces the observed oscillations. This is

especially evident for the slowest electron group (group 40).


1.1 -
Right half of the problem
1.05 (symmetric)

1 Group 40
Group 10
0.95 Group 20



09Group 30
0.85 -

0.8' I ------ I--II I-I--------
.15 0.2 0.25
X-axis [cm]


Figure 7-5. Ratio of ONELD importance over PENTRAN importance for four energy groups in
problem #1 with mesh refinement

For this problem, it is expected that the quadrature order (i.e., the level of accuracy of the

angular representation) will have a larger impact on the solution at the boundary of the model.










Figure 7-6. shows the ratios (PENTRAN to ONELD) of the importance functions for different

quadrature orders.

1.06-



1.04



1.02 -S16
S16



1 S32



0.98 I I
0.05 0.1 0.15 0.2 0.25
X-axis [cm]

Figure 7-6 Ratio of the ONELD and PENTRAN importance functions for group 1 obtained using
S16 and S32 quadrature order with mesh refinement in problem #1

Above figure indicates that the use of a higher quadrature order in conjunction with a refined

mesh at the boundary practically eliminates the observed oscillations.

Problem #2

Problem #2 is designed to study the impact of PENTRAN differencing schemes on the

accuracy of the importance function by introducing a source discontinuity in a high-Z material.

This source discontinuity will results in large variations in the importance function and therefore

is useful to study the impact of different differencing schemes. Figure 7-7 presents a schematic

of problem #2. Note that energy spectrum is uniform (with a maximum of 1 MeV) within the

source region shown in grey. Table 7-2 gives the various discretization parameters used for this

problem.











Tungsten /Reflective BC in
3D model







Adjoint source
4 ~(flat spectrum)

0.01 cm 0.02 cm


Figure 7-7. Problem #2 geometry

Table 7-2. PENTRAN and ONELD simulation parameters for solving problem #2
PENTRAN ONELD
CEPXS-GS cross sections CEPXS cross sections
50 uniform meshes 50 uniform meshes
25 equal width electron groups 25 equal width electron groups
Level symmetric quadrature Gauss-Legendre quadrature
S16-P15 S16-P15


To study the impact of the differencing scheme in PENTRAN, it is useful to define coarser

meshes in the zone of interest to estimate the improvement in the solution compared to the more

refined ONELD solution. Therefore, the mesh structure used for this problem is illustrated in

Figure 7-8.


.zone 1
ONELD: 15 meshes
PENTRAN: 5 meshes

zone 2
5 meshes




Figure 7-8. Mesh structure for problem #2










The adaptive differencing strategy in PENTRAN automatically shifts between three

differencing schemes; linear diamond with zero-flux fixup (DZ), directional theta-weighted

(DTW), and exponential directional-weighted (EDW). It is also possible to force the PENTRAN

code to use different differencing schemes within different regions. For this analysis, the code

was forced to use one of the aforementioned differencing throughout the model. Figure 7-9

shows the importance function for group 20 (0.2120 MeV to 0.2524 MeV) obtained using the

three differencing schemes.




ONELD
103
O DZ
S- DTW
EDW
10'4



1 l0


106 .

0.01 0.02
X-axis [cm]


Figure 7-9. Impact of differencing scheme on importance function for group 20 in problem #2

The above results clearly show that the use of an exponential differencing scheme (EDW)

can improve the accuracy of the importance function especially in regions where the importance

magnitude decreases significantly. For the adjoint problems of the type considered in the ADEIS

VR methodology, i.e., with highly localized adjoint source at a large distance from the actual

source, an exponential scheme seems especially appropriate. For this problem, even considering

the small number of meshes, the difference in simulation time between PENTRAN (3-D) and









ONELD (1-D) is still significant, i.e., about 200 seconds for PENTRAN vs. less than 1 second

for ONELD.

Problem #3

Problem #3 is design to verify the accuracy and computational cost of performing a 3-D

adjoint transport calculation to obtain a coupled electron-photon importance function. Therefore,

a problem with a uniform source (maximum energy of 1 MeV) distributed throughout a tungsten

slab is considered. A reference solution is obtained with ONELD using the parameters given in

Table 7-3. To simulate this 1-D problem using PENTRAN, a cube with reflective boundary

conditions is considered as illustrated in Figure 7-10.

Reflective BC
S in 3-D model


Adjoint source
(flat spectrum)
Tungsten



0.1 cm

Figure 7-10. Problem #3 geometry

Table 7-3. Other simulation parameters for problem #3
PENTRAN ONELD
CEPXS-GS cross sections CEPXS cross sections
50 uniform meshes 50 uniform meshes
50 equal width electron groups 50 equal width electron groups
30 equal width photon groups 30 equal width photon groups
Level symmetric quadrature Gauss-Legendre quadrature
S16-P15 S16-P15
Linear diamond Linear discontinuous


To verify that accurate coupled electron-photon importance functions can be obtained with

PENTRAN, it is useful to study the photon importance function resulting from a simulation of










the electron-photon cascade through up-scattering. To study in more details the difference

between the ONELD and PENTRAN importance functions, it is interesting to look at their ratios

for various energy groups. Figure 7-11 shows these ratios for the following photon groups: i)

Group 1 (0.9766 MeV to 1.01 MeV); ii) Group 10 (0.6766 MeV to 0.6933 MeV); iii) Group 20

(0.3433 MeV to 0.3766 MeV); and, iv) Group 30 (0.01 MeV to 0.4333 MeV).


1.05 -








Grupup 10
0.95 GiGrp1l



0.9 Group 20

0.02 0.04 0.06 0.08
X-axis [cm]

Figure 7-11. Ratio of ONELD importance over PENTRAN importance for four photon energy
group in problem #3

The above results indicate that accurate coupled electron-photon importance function can

be generated using PENTRAN, and that, for this problem, results are within 10%. Moreover, it is

interesting to note that similar unphysical oscillations shown in Figure 7-3 also affect the photon

importance functions. It is expected that proper meshing, adequate differencing scheme and

higher quadrature order would improve the accuracy. The PENTRAN computation time for this

problem is -108661 sec. To reduce this time, a detailed analysis was performed and it was

concluded that the upscattering algorithm was not efficient. Therefore, a more efficient up-scatter

algorithm was implemented. This new algorithm reduced the computation to -4764 seconds.

Note that the new upscattering algorithm is implemented for the parallel version of the code.









In conclusion, the above results indicate that the PENTRAN can solve for importance

functions using a 3-D geometry with adequate accuracy, however, significant computation time

is necessary. Therefore, the use of PENTRAN is limited to problems were three-dimensionality

is important.

Generation of 1-D Importance Functions Using PARTISN

Before performing biasing using 2-D (RZ) importance functions, it is useful to investigate

the use of PARTISN to generate 1-D importance functions. In addition to verifying the proper

implementation of new subroutines to automatically generate input files for PARTISN, these

studies show that another discrete ordinates solver other than ONELD can be used to generate

coupled electron-photon-positron importance functions within the context of ADEIS. In this

section, the reference case defined in Figure 5-1 and Table 5-1 is used to investigate the impact

on ADEIS efficiency of following four combinations of transport solver, cross sections, and

spatial differencing schemes:

* Case 1: ONELD, CEPXS cross sections and linear discontinuous (LD)

* Case 2: PARTISN, CEPXS cross sections and linear diamond (LZ)

* Case 3: PARTISN, CEPXS cross sections and linear discontinuous

* Case 4: PARTISN, CEPXS-GS cross sections and linear diamond

Note that the criteria defined in Chapter 6 are used to automatically select the discretization

parameters. Table 7-4 presents the dose tally speedups obtained for these four test cases.

Table 7-4. Energy deposition tally speedup for ONELD and PARTISN simulation of Chapter 5
reference case
Test case Speedup
Casel: (ONELD/CEPXS/LDa) 13.5
Case 2 (PARTISN/CEPXS/LZb) 15.1
Case 3 (PARTISN/CEPXS/LD) 13.4
Case 4 (PARTISN/CEPXS-GS/LZ) 15.6
a LD linear discontinuous
b LZ linear diamond









Table 7-2 indicates that it is possible to use PARTISN to generate 1-D importance

functions that are adequate within the context of the ADEIS VR methodology. It also appears

that the use of the linear discontinuous scheme is not as critical as in improving the quality of the

solution of adjoint problem in ADEIS. However, it must be noted that ADEIS uses rather

optimized discretization parameters to improve the quality of the importance function and in that

context, the use of a higher order spatial differencing scheme might not be necessary. Finally, it

is possible to observe that, as shown previously in Chapter 5, the CEPXS-GS cross sections are

adequate when used in conjunction with PARTISN and results in speedups comparable to

CEPXS. This is important for obtaining 2-D (RZ) importance functions since it is already known

that the CEPXS cross sections are inadequate for multi-dimension transport calculations.

Biasing Along the Line-of-Sight Using the MCNP5 Cylindrical Weight-Window

In order to use any importance functions evaluated along the line-of-sight in a more

general context than the 1-D-like problem (studied in Chapters 5 and 6), it is essential to use the

MCNP5 cylindrical weight-window transparent mesh illustrated in Figure 7-12.


\ .--\-"







\ \I





Figure 7-12. One-dimensional (R) and two-dimensional (RZ) weight-window mesh along the
line-of-sight in a 3-D geometry









In cases where the LOS is parallel to one of Cartesian frame of reference axis, the

Cartesian and cylindrical weight-windows are equivalent. However, if the LOS is not parallel to

one of the Cartesian axes, the cylindrical weight-window allows a more efficient and accurate

use of the 1-D importance function calculated along the line-of-sight by biasing though planes

perpendicular to the LOS as shown in Figure 7-12. The 2-D (RZ) importance functions generated

along the line-of-sight can be represented by concentric cylinders centered along the LOS and

require the use of the cylindrical weight-window. The analysis performed in this section was

intended to verify the implementation of the use of the MCNP5 cylindrical weight-window, and

to investigate the computational cost (and reduction in efficiency) of transforming the Cartesian

coordinates used during the MCNP5 particle tracking to the cylindrical coordinates system of the

weight-window. By comparing the speedups obtained for Case 4 (see Table 7-4) using the

Cartesian and cylindrical weight-window, it appears that a loss in speedup of about 10% occurs

when the cylindrical weight-window is used. Therefore, such a small decrease in efficiency does

not prevent the use of the cylindrical weight-window for all cases.

Generation of 2-D (RZ) Importance Functions Using PARTISN

This section presents the analysis performed to investigate the generation of 2-D (RZ)

importance functions using PARTISN. In PARTISN, it is possible to select various transport

solvers with different capabilities. It is assumed that these solvers are part of the PARTISN

system for historical reason as it evolved from DANTSYS. The solver used in these analyses was

chosen to maintain compatibility with ONELD and to take advantage of the various automated

processing tools already developed. However, this introduces some limitation to the scope of the

studies performed in this Chapter as discussed in the following paragraph.

The chosen solver uses a single level grid scheme where each axis is divided in coarse

meshes and each coarse mesh is assigned a fine mesh size. However, this implies that the same









fine mesh size is applied to all coarse meshes with the same coordinates along that axis, and

therefore, limits the possible automatic mesh refinements. The automatic criteria developed in

Chapter 6 will be used to define all parameters including the axial fine meshes (z-axis in

Figure 7-12). However, the minimum number of fine meshes per coarse mesh allowable by the

solver will be used for the radial coarse meshes. This choice, coupled with the fact that only a

linear diamond spatial differencing scheme is available for this solver, may not result in an

importance function of good quality. The use of the block-AMR (block adaptive mesh

refinement) solver available in PARTISN may resolve these issues.

With these discretization parameters, using the 2-D (RZ) importance functions to bias the

reference case resulted in a speedup of about 6, i.e., about 3 times less than what was achieved

with 1-D biasing along the LOS. The computation time required to obtain the 2-D (RZ)

importance function is about ten times larger than 1-D calculations (i.e., 41.8 seconds vs.

4.5 seconds) but still relatively short compared to the total computer time of the Monte Carlo

simulation (2148 seconds). Therefore, the decrease in efficiency can be attributed to the decrease

in the quality due to inadequate meshing.

Speedup Comparison between 1-D and 2-D (RZ) Biasing

It could be argued that the Chapter 5 reference case was highly one-dimensional and

therefore did not require the use of 2-D (RZ) importance functions. To address this issue, the

reference case was modified (as illustrated in Figure 7-13) by reducing the flattening filter to a

more realistic size (0.5 cm x 5 cm x 5 cm) and a simplified collimator was added. These

modifications slightly changed the nature of the Chapter 5 reference case to make it more

axisymmetric. Consequently, the use of 2-D (RZ) importance function should be more

appropriate for bias this new modified reference case.









7a
j------------


'/3


Beam 1 I EP


12 4 6 8
Figure 7-13. Modified reference case geometry

More detailed dimensions for each material zone are provided Table 7-5.

Table 7-5. Materials and dimensions of reference case
Zone Description Color Material Size (cm3)
1 Target Dark gray Tungsten 0.1 x 40 x 40
2 Heat dissipator Orange Copper 0.15 x 40 x 40
3 Vacuum White Low density air 8.75 x 40 x 40
4 Vacuum window Light gray Beryllium 0.05 x 40 x 40
5 Flattening filter Dark gray Tungsten 0.5 x 5 x 5
6 Collimator shield Dark gray Tungsten 2.0 x 40 x 40
Collimator hole White Air 2.0 x 5 x 5
7 Air White Air a) 40.95 x 40 x 40
b) 48 x 40 x 40
8 ROI (tally) Blue Water 0.1 x 40 x 40


Two ADEIS simulations are performed for this reference case: i) an ADEIS simulation

using a 2-D (RZ) importance function; and ii) an ADEIS simulation using a 1-D importance

function.

Table 7-6. Energy deposition tally speedup for 1-D and 2-D biasing
Test case Speedup
2-D (RZ) biasing 6.1
1-D biasing 15.1


Table 7-6 indicates that, in spite of the modifications, the biasing using 1-D importance

functions along the LOS still produce larger speedup. However, as mentioned earlier, this result

should be considered preliminary until better spatial discretization can be performed for the 2-D

(RZ) deterministic model, and more test cases are studied.


7b









Conclusions

From the analyses presented in this Chapter, it can be concluded that 3-D coupled electron-

photon importance functions can be generated using 3-D discrete ordinates methods. More

specifically, it was shown that PENTRAN/CEPXS-GS is adequate to evaluate coupled electron-

photon importance functions in low and high-Z materials given that the proper selection of

discretization parameter is made.

It can also be concluded that 3-D importance functions seem more sensitive to meshing

and exhibit oscillations not present in the 1-D solution. However, it was shown that these

unphysical behaviors can be mitigated with appropriate meshing. Moreover, it was showed that

3-D importance functions seem to require a higher quadrature order to have a proper angular

representation. It was also indicated that exponential differencing schemes seem useful to

decrease the computational cost associated with a given accuracy for the type of adjoint problem

associated with the ADEIS VR methodology.

Even though it was possible to obtain accurate enough importance functions using

PENTRAN, the computational cost limited the practical use of this approach in the context of the

ADEIS VR methodology. Therefore, the use of 2-D (RZ) importance functions was also studied.

The results presented in this Chapter show that 1-D and 2-D (RZ) importance functions could be

generated using PARTISN using CEPXS and CEPXS-GS cross sections. The 2-D (RZ)

importance functions were successfully used to perform biasing though the use of the cylindrical

weight-window mesh. For the reference case, these simulation resulted in speedups of about 6,

i.e., about 3 time smaller then the speedup obtained with 1-D importance functions. It was shown

that, in the context of the ADEIS VR methodology, the use of a linear discontinuous spatial

differencing scheme is not as critical for the 1-D importance function. However, as for the 1-D









importance functions, the selection of a more optimized mesh structure will result in an

importance of higher quality and may produce larger speedups.









CHAPTER 8
CONCLUSIONS AND FUTURE WORK

Conclusions

A new automated variance reduction methodology for 3-D coupled electron-photon-

positron Monte Carlo calculations was developed to significantly reduce the computation time

and the engineering time. This methodology takes advantage of the capability of deterministic

methods to rapidly provide approximate information about the complete phase-space in order to

automatically evaluate the variance reduction parameters. This work focused on the use of

discrete ordinates (SN) importance functions to evaluate angular transport and collision biasing

parameters, and accelerate Monte Carlo calculations through a modified implementation of the

weight-window technique. This methodology is referred to as Angular adjoint-Driven Electron-

photon-positron Importance Sampling (ADEIS).

For the problems considered in this work, the flux distributions can be highly angular-

dependent because: i) the source characteristics (e.g. high-energy electron beam); ii) the

geometry of the problem (e.g. duct-like geometry or large region without source); and iii) the

scattering properties of high-energy electrons and photons. For these reasons, ADEIS was based

on a slightly different derivation of the concept of importance sampling for Monte Carlo

radiation transport than its predecessor, CADIS32 33. In addition to more clearly illustrating the

separation between collision and transport biasing, this derivation uses:

* A different formulation of the approximated response in the region of interest to allow
angular surface sources;

* Angular-dependent lower-weight bounds based on the field-of-view (FOV) concept to
introduce this dependency without using a complete set of angular fluxes which requires an
unreasonable amount of memory;

* A different lower-weight bound definition that ensures that the highest energy source
particles are generated at the upper-weight bound of the weight-window, and maintains the









consistency between the weight-window and the source without having to perform source
biasing.

ADEIS was implemented into the MCNP5 code with a high degree of automation to

ensure that all aspects of the variance reduction methodology are transparent, and required only

the insertion of a tally-like card in the standard MCNP5 input. The accuracy and computational

cost of generating 3-D importance functions using PENTRAN was studied. However, the

computational cost limited the practical use of this approach in the context of the ADEIS VR

methodology. Therefore, to generate the angular importance functions, ADEIS used either the

ONELD (1-D) or PARTISN (2-D, RZ) code with cross sections generated from either the

CEPXS or CEPXS-GS code. Moreover, the implementation of ADEIS included the following

specific features to make it practical, robust, accurate, and efficient:

* The development and use of a driver (UDR) to manage the sequence of calculations
required by the methodology;

* A line-of-sight concept to automatically generate a deterministic model based on material
regions by tracking a virtual particle through the geometry;

* Capability to generate 1-D and 2-D (RZ) importance functions along the line-of-sight;

* The use of the MCNP5 cylindrical weight-window transparent mesh to bias along the line-
of-sight;

* On-the-fly generation of cross sections for each problem;

* Two automatically determined adjoint sources to circumvent the absence of appropriate
dose response coefficients; i) a local energy deposition response function to approximate
dose in the ROI, and ii) a uniform spectrum to maximize the total flux in the ROI;

* Development of criteria to automatically select discretization parameters that maximize
speedups for each problem;

* Selection of discretization parameters which reduce well-known unphysical characteristics
(oscillations and negativity) in electron/positron deterministic importance functions due to
numerical difficulties;

* Smoothing to ensure that no negative values remain in the importance functions;









* Explicit positron biasing using distinct importance functions in order to avoid an
undersampling of the annihilation photons and introducing a bias in the photon energy
spectra;

* Modification of the condensed-history algorithm of MCNP5 to ensure that the weight-
window is applied at the end of each major energy step and avoid introducing a bias in the
electron total flux and spectrum;

* Modification of the standard MCNP5 weight-window algorithm to allow for various
biasing configurations: i) standard weight-window; ii) angular-dependent weight-window
without explicit positron biasing; iii) explicit positron biasing without angular dependency;
and, iv) explicit positron biasing with angular dependency.

Future Work

To extend and continue this work, many avenues of research are possible. First, a more in-

depth study of the impact of the spatial mesh size (axial and radial) on the speedup for the 2-D

(RZ) model is required. Other issues affecting the quality of the importance function in RZ

simulations (e.g. source convergence acceleration technique, spatial differencing schemes and

quadrature set) should also be studied. Also, it might be also interesting to investigate the

possibility of using synthesis techniques to generate multi-dimension importance functions and

reduce the computational cost. Other possible improvements to ADEIS are listed below:

* Implement energy-dependent FOVs, especially since PARTISN allows for energy group-
dependent quadrature order.

* Further investigate weight checking frequency to verify if the current criterion is
appropriate for low-energy electrons, where the DRANGE is especially small.

* Implement a parallel algorithm in the weight-window algorithm to speedup the mesh index
search.

* Study the possibility of predicting the gain in efficiency using pre-calculated curves of
probability of transmission to the ROI versus speedup.

* Study the possibility of using angular flux moments rather the discrete angular flux to
calculate the FOV in order to circumvent issues arising when a FOV falls between two
direction cosines.









* Implement automatic source biasing for discrete and continuous sources by projecting it on
the discretized phase-space grid of the weight-window. This could be achieved by
sampling the actual source and tallying it over the weight-window.

* Further study the specific cause of the spectrum tail bias observed for cases where the
region of interest is located beyond the CSD range of the source particle.









APPENDIX A
VARIOUS DERIVATIONS

Selection of an Optimum Sampling Distribution in Importance Sampling

In the importance sampling technique, an optimum biased sampling distribution can result

in an estimator with a zero variance. In this section, it is shown that if a biased sampling

distribution is chosen to be proportional to the PDF of the random process, the resulting

estimator will have a zero variance. Let's consider a problem where the expected value can be

represented as Eq. A-1.

(g)= Jw(x)g(x)f(x)dx (A-l)

Where (g) is the estimated quantity, g(x) is a function of random variable x, f(x) represents

the biased sampling PDF, w(x) = f(x)/f(x) represents the weight of each contribution, and

f(x) represents the random process PDF. The variance of such an estimator is evaluated by the

Eq. A-2. where 02 is the variance.

2 = dx [w()g(x) (g)]2 (x) (A-2)


By assuming that the biased sampling PDF is proportional to the integrand of Eq. A-i,

i.e., a f(x) = f(x)g(x) -> w(x)g(x) = a, it is possible to write Eq. A-i as Eq. A-3 since the

integral of a PDF over the whole range of the random variable is equal to 1.

(g)= af (x)dx = a (A-3)

By replacing Eq. A-3 in Eq. A-2, it is possible to rewrite the expression of the variance as in

Eq. A-4.

-2 dx[a a]2 f(x) = 0 (A-4)

Eq. A-4 shows clearly that the estimators would have a zero-variance.









Biased Integral Transport Equation

As mentioned in the previous section of this appendix, it is possible to derive a formula for

the expected value using a more optimal sampling PDF. In the context of particle transport, this


is done by multiplying the Eq. A-5 by --P
R

W(P) = f( (P")C(P"->P')dP" + Q(P') ) T(P'->P)dP' (A-5)

Where (Y'P) represents an importance function associated to quantity being estimated; T(P)

represents the integral quantity being estimated; P, P' and P" are the respective phase-space

element (r,E,Q), (r',E', ') and (r",E", f"); C(P"-P') represents the collision kernel;

T(P'->P) represents the transport kernel; Q(P') represents the external source of primary

particles, and R is an approximated value of quantity being estimated (see Chapter 3).

Multiplying the resulting equation by 1 (dressed up in a tricky fashion), it is possible to obtain

Eq. A-6.





w(P) W'(P) y 'P)) ,P (P ') (p) (PP"
= ( (P ) C (P P ')dP T(P P) d) = C(P>P'



R R '


it is possible to rewrite Eq. A-6 as Eq. A-7.A-6)
(P) = (P ")C(P->P) dP"'(P'-P)P + (P) (A-7)
+JQ(P')T(P '9P) dP'


By combining the various terms as follow;




7 (P') ? '(P) (P ')
R '''(P ') cu'(p")

it is possible to rewrite Eq. A-6 as Eq. A-7.

Y(P) = J J Y(P")C(P"aP')dP"T(P'9P)dP' + JQ(P')T(P'9P)dP' (A-7)









Where '(P) represents the biased estimator, Q(P') represents the biased source, T(P'->P)

represent the biased transport kernel, and C(P"-P') represents the biased collision kernel.

Lower-weight Bounds Formulation and Source Consistency

To ensure that the source particles are generated at the upper bounds of the weight-window

when a mono-directional and mono-energetic point source is used, consider the formulations

given by Eqs A-8, A-9, and A-10.

R
w1 (, E)= (A-8)
q(tE) C,

R = dVdEdf Q(r, E, f) '(, E, 2) (A-9)

Q(F, E, Q) = 6(r ,) 6(E Eo)6i( 2+) (A-10)

In those equations, w,+ represents the lower-weight bound value; 6(r r) 6(E E) and

6(0f f ) are the Dirac delta functions representing a unit mono-energetic point source

emitting in a direction within the FOV. By replacing Eq. A-10 in Eq. A-9, the approximated

response R can be rewritten as Eq. A-12.

R = I dEdV 6(r-ro) 6(E-Eo,)(p = (pt(,,Eo). (A-11)

By replacing Eq. A-11 in Eq. A-8, the lower-weight bound formulation can be written as

Eq. A-12.

R ( +(,Eo) 1
w-,r ,EO) = (A-12)
w+ 0 (r0,E,) C. (p (r,E,) C.

Considering Eq. A-12 and the fact that that the upper bound of the weight-window is generally

defined as a multiple C, of the lower-weight bound, the formulation for the upper-bound can be

written as in Eq. A-13.

w .(,,Eo)= l. (A-13)









Since unbiased source particles generally have a weight of 1, Eqs A-12 and A-13 ensure the

consistency between the source and the weight-window in the absence of source biasing.

Determination of the Average Chord-Length for a Given Volume

For convenience, this section presents a standard derivation97 of the average chord-length

in a given arbitrary volume. Let us consider an arbitrary region of volume Vbounded by a

surface A with chords defined from an infinitesimal surface dA such that their number along a

given direction Q is proportional to Q The average length of these chords in the volume

can therefore be evaluated by Eq. A-14.

=Ir dQdA
r= h A (A-14)
JJfh .QddA


The integral over dQ is performed for h > > 0 since only chords going into the volume are

considered. The infinitesimal volume associated with each of these chords can be written as in

Eq. A-15.

dV = -dAdR (A-15)

In this equation, h Q > 0 and can be integrated to give the total volume of the region as shown

in Eq. A-16.

V= JfdV=fi h-dAdR= fr- ldA (A-16)

Replacing Eq. A-16 into Eq. A-14 and rewriting the denominator of Eq. A-14, it is possible to

obtain the formulation for the average chord-length of an arbitrary region given in Eq. A-17.

SVJdQ 4 V 4V V 4V
S1 -- (A-17)
fI dfidA dA JI h-dfi6 A 2dq5old A
o o









APPENDIX B
IMPLEMENTATION DETAILS

The following appendix contains sections providing additional details about the

implementation of the ADEIS methodology.

Universal Driver (UDR)

UDR was developed as a framework to manage any sequence of computational tasks. It

can be used as a library to manage a sequence of tasks independent of the parent code or as a

standalone application. It was essentially designed to replace script-based approaches and to

offer:

* a better task control by providing a single free-format input file for all tasks in the
sequence

* a better error and file management

* general and consistent data exchange between the tasks themselves or between the tasks
and the parent code

UDR was implemented as a FORTRAN90 module and contains the following major functions:

* udrhelp: utility to facilitate the creation and use of online help for tasks managed by UDR

* ffread: free-format reader that differentiate keyword and numerical inputs, store them in
separate buffers to be used by the task

* udropen/udrclose: automatically manage available file unit numbers and change filename
to prevent overwrite.
Ex. CALL adeisopen(udrlnk,'filename','OLD','READWRITE','FORMATTED')

* prgselect: manages calls to individual task following input processing

* Inkred/lnkrit: access the UDR data exchange file (link file) through the use of records.
Ex.: CALL lnkred(udrlnk,'dimension of deterministic calc',i)

By default, before the insertion of independent tasks, UDR can perform:

* stop: stop a sequence at any point











* $filename: if a $ is detected in the "option field" (see next section), the remainder of the
option field (i.e., up to the task termination character ";") is copied, line by line, to a file
filename.

An example of the UDR input file syntax is shown in Figure B-1.

! Performs link between ADEIS and MCNP5
mclnk: cutoff 0.025 nopositron nofirstorder
nge 43 lin
ngp 30 lin ;

!Generates CEPXS-GS cross-section
cepxs: notgs adjsrc flat :

Prepares input for transport solver
inprep: geomfile geofile ;

Solve adjoint problem for importance
trsprt: ;

Generates MCNP5 weight-window
wwgen: biasroi nosmooth
mufov 0.8 1.0
srcfile test
testl
1.0 1.0
1.0 0.0


stop: ;


Figure B-1. Example of UDR input file syntax

Performing an ADEIS Simulation

To perform an ADEIS simulation, it is necessary to use a script (adeisrun) which allows; i)

the use of a simplified the syntax to run MCNP5 in parallel, ii) to run ADEIS independently of

MNCP5 if necessary, iii) to generate soft links to the CEPXS/CEPXS-GS data files, iv) to clean

the various temporary files generated by ONELD, CEPXS or PARTISN, and v) to run MCNP5

without the ADEIS sequence. Figure B-2 shows two examples of calls to the adeisrun script for

a standard MCNP5 serial simulation and an ADEIS parallel simulation where the temporary

CEPXS files are kept.









A) adeisrun -mc i=test.inp o=test.out

B) adeisrun -uc cepxs -np 17 i=test.inp o=test.out


Figure B-2. Examples of calls to adeisrun A) for a standard MCNP5 run B) for an ADEIS run

To implement the ADEIS methodology, a new simulation sequence must used inside

MCNP5. This new sequence, illustrated in Figure B-3, requires the use of a new command line

option (ex.: mcnp5 a i=test.inp o=test. out). However, this is transparent to the user sue to

the use of the adeisrun script.

imcn xact adeismsh W adeismat


mcrun adeisww y adeis y adeispara


Figure B-3. New simulation sequence in MCNP5

ADEIS MCNP5 Input Card

This new sequence must used in conjunction with a new MCNP5 input card (ADEIS)

which constitutes the only task required from the user. This card is similar to a tally card with the

exception that, at this point, only one ADEIS card is allowed.

ADEIS:pl "variable specification"

pl = e or p or e,p: set the objective particle

Table B-1. The ADEIS keywords
Keyword Meaning Default
srcori Location of the source origin 0., 0., 0.
los Line-of-sight vector 1., 0., 0.
Dimension of the deterministic
dimen importance function ("ld", "2d" None
or "3d")
obj cel Cell number of the region of
interest (ROI)









APPENDIX C
ELECTRON SPECTRUM BIAS SIDE STUDIES

A series of studies were performed to investigate a small bias observed in the electron

energy spectrum tail. Even though they proved to be unrelated to the cause of the bias, they are

presented for completeness. For these studies, a reference case with the following characteristics

is considered:

* 2 MeV pencil impinging the left-side of a water cube with dimensions of about one range
on all side

* electron-only simulation is performed

* a weight-window of 50 uniform energy groups and 50 uniform meshes along the x-axis.

* the region of interest (ROI) is located slightly pass the range of the 2 MeV source electrons
and has a thickness of 2% of the range.

Note that these characteristics were chosen to clearly illustrate the bias. This appendix

presents analyses studying the impact, on spectrum tail bias, of the following aspects; tally

location, number of histories, source energy and energy cutoff, leakage, energy indexing scheme,

Russian roulette weight balance, knock-on electrons, knock-on electron collision biasing,

deterministic energy group structure

Impact of Tally Location

As a first study, it is interesting to analyze the impact of the tally location on the spectrum

tail bias. For a tally located at 70% of the range of the source electrons, Figure C-l shows that

the relative differences are larger in the spectrum tail but no systematic bias is present (relative

errors are within the 1-o statistical uncertainties). Note that the statistical uncertainty on the

relative differences is obtained through a typical error propagation formula.













1.E-01


4-Normalized spectrum
- Relative difference


2.0



1.5



1.0 L



0.5



0.0


,,-0.5
2.0E-01 4.0E-01 6.0E-01 8.OE-01 1.OE+00 1.2E+00
Energy [MeV]


Figure C-1. ADEIS normalized spectrum and relative difference with standard MCNP5 for tally
located at 70% of 2 MeV electron range


I.I11 -2.0
1.OE-01 2.0E-01 3.0E-01 4.0E-01 5.0E-01
Energy [MeV]


Figure C-2. ADEIS normalized spectrum and relative difference with standard MCNP5 for tally
located at 2 MeV electron range


1.E-02



u

g 1.E-03





1.E-04


1.E-05 -
0.OE+00


1.E-03







1.E-04






E05
Z 1.E-05


1.E-06 --
0.OE+00









However, Figure C-2 shows a small bias in the spectrum tail when the tally is located at a

larger depth within the target material (at about the range of the source electrons). Note that if a

99% confidence interval is used instead of the 68% confidence interval, the observed differences

are not statistically significant for the current precision. It can also be seen that this bias affect

only for spectrum values that are about two orders of magnitudes smaller then the mean of the

spectrum.

Impact of the Number of Histories on Convergence

The methodology samples more often particles that have large contributions to the integral

quantity, and therefore, for a limited number of histories, the particles contributing to the tail of

the distribution may not be properly sampled. If no bias is present, the tally spectra should

converge and the relative differences between the standard MCNP5 and ADEIS spectra should

decrease as the number of histories increases. On the other hand, if a bias is present, the relative

differences should stay relatively constant as the number of histories is increased. Therefore, it is

interesting to study the changes, as a function of the number of histories, in the relative

differences between the ADEIS and standard MCNP5. For these different numbers of histories, it

is interesting to study the relative differences as a function of energy. It is also interesting to look

at the i2 -norm (see Eq. C-l) of the relative differences since it provides a good indication of the

overall convergence of the tally,


=2_j_ 2 (C-l)


where E, is the relative difference associated with energy bin i, and Nis the total number of

energy bins in the tally. Figure C-3 shows that, for a tally located at 70% of the CSD range, the

relative differences decrease smoothly until they within each other 1-o statistical uncertainties.













16.0


14.0 --# of histories 2E6
-*-# of histories 8E6
12.0 -- # of histories 3.2E7

S10.0

S8.0

6.0

S4.0

2.0

0.0

-2.0

-4.0
0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.OE+00 1.2E+00
Energy [MeV]



Figure C-3. Relative differences between standard MCNP5 and ADEIS for tally located at 70%
of the 2 MeV electron range at various number of histories


As shown in Figure C-4, the convergence of the tally can be shown by looking at the


behavior of the 2 -norm of the relative differences as a function of the number of histories.


0.14


0.12


: 0.10








8 0.04


0.02


0.00
0.OE+00


5.0E+06 1.0E+07 1.5E+07 2.0E+07 2.5E+07 3.0E+07 3.5E+07
Number of histories


Figure C-4. Norm of relative differences between standard MCNP5 and ADEIS for tally located
at 70% of the 2 MeV electron range at various number of histories










By comparing Figure C-3 and Figure C-4, it can be concluded that the 2 -norm value is

dominated by the relative differences of spectrum tail. This is expected since the relative

differences at the others energies are extremely small. However, as shown in Figure C-5, when

the tally located beyond the CSD range of the 2 MeV electron, the 2 -norm of the relative

differences does not converge (or converge extremely).


0.16


0.14


0.12


15 0.10


1 0.08


S0.06


_' 0.04


0.02


fnn +-


1.0E+07 2.0E+07 3.0E+07 4.0E+07
Number of histories


5.0E+07 6.0E+07 7.0E+07


Figure C-5. Norm of relative differences between standard MCNP5 and ADEIS for tally located
at the 2 MeV electron range at various number of histories

Even though it cannot be concluded that this bias will not disappear after an extremely

large number of histories, it is very unlikely that it will considering the behavior of the 2 -norm

and the statistical uncertainties of the problematic energy bins.


0.OE+00










Impact of Electron Energy and Energy Cutoff

To understand the impact of the source electron energy and the energy cutoff, it is

interesting to compare the energy spectra obtained from a standard MCNP5 and ADEIS

calculations using the following parameters: i) 2 MeV electrons with 0.01 MeV cutoff; ii)

2 MeV electrons with 0.1 MeV cutoff; iii) 13 MeV electrons with 0.01 MeV cutoff; and, iv)

13 MeV electrons with 0.01 MeV cutoff

Figures C-6 and C-7 present the relatives differences in electron spectra obtained from a

standard MCNP5 and ADEIS simulations for these parameters. By comparing Figures C-6 and

C-7, it is possible to conclude that smaller biases are observed for higher source electron energies

and larger energy cutoff.


1.E+01


8.E+00


S6.E+00


S 4.E+00


a 2.E+00


SO.E+00


-2.E+00


-4.E+00 I 1I
O.OE+00 5.0E-02 1.OE-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy [MeV]


Figure C-6. Relative differences between the tally electron spectra from ADEIS and standard
MCNP5 for a 2 MeV electron beam at two energy cutoff











4.E+00

3.E+00 -Cutoff 0.01 MeV
-- Cutoff 0.1 MeV
2.E+00

1.E+00

O.E+OO _

-1.E+00

-2.E+00

-3.E+00

-4.E+00
0.OE+00 5.0E-01 1.OE+00 1.5E+00 2.0E+00 2.5E+00 3.0E+00
Energy [MeV]


Figure C-7. Relative differences between the tally electron spectra from ADEIS and standard
MCNP5 for a 13 MeV electron beam at two energy cutoff

By comparing in more details the different physical characteristics of each case, it can also

be concluded that the lateral leakage and amount of knock-on electron production are

significantly affected by the selection of the source electron energy and energy cutoff. It is

therefore interesting to study these two aspects.

Impact of Lateral Leakage

It could be argued that using 1-D importance functions to perform VR in a three-

dimensional model is introducing a small bias caused by inability of these function to properly

model the lateral leakage. Therefore, it is useful to evaluate the impact of the leakage on the

results of the ADEIS VR methodology. To study this aspect, the reference case is modified by

increasing the size of the cube along the y-axis and z-axis. These sides are increased to 1.96 cm

(twice the CSD range) and 2.94 cm (three times the CSD range). These modifications reduce the










leakage along these two directions and therefore make the problem more one-dimensional in

nature.


1.E+01
-4- 1R y- and z- sides
-- 2R y- and z- sides
8.E+00 --3R y- and z- sides


S6.E+00


H 4.E+00


2.E+00


O.E+ -


-2.E+00
O.OE+00 5.0E-02 1.OE-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy (MeV)


Figure C-8. Relative differences in spectrum between the standard MCNP5 and ADEIS for
various model sizes and an energy cutoff of 0.01 MeV

Figure C-8 shows that the 1-D importance functions inability to properly take into account

the lateral leakage is not responsible for introducing the bias.

Impact of Knock-On Electron Collision Biasing

To verify that the bias is not introduced by an implementation problem related to the

collision biasing of knock-on electrons, the standard MCNP5 and ADEIS were modified such

that no collision biasing is perform for those electrons. The relative differences between the

electron spectra of the standard MCNP5 and ADEIS are then compared with and without

collision biasing for knock-on electrons. By looking at Figure C-9, it is obvious that the

implementation of the collision biasing for knock-on electron is not responsible for the bias in

the spectrum tail.










1.2E+01


1.OE+01
--- No collision biasing for knock-on

8.0E+00


6.OE+00


g 4.0E+00


2.OE+00


O.0E+00


-2.0E+00
0.OE+00 5.0E-02 1.OE-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy [MeV]


Figure C-9. Relative differences in spectrum between the standard MCNP5 and ADEIS for the
reference case with and without collision biasing for knock-on electrons

Impact of Weight-Window Energy Group Structure

Considering that knock-on electrons are simulated according to physical properties

evaluated on a given energy grid (CH algorithm) and biased according to another one (weight-

window), it could be argued that the selection of the deterministic (weight-window) energy

group structure could affect the accuracy VR methodology. It is therefore interesting to study the

possible inconsistency between the predicted importance (from the deterministic calculation) and

the actual contribution (in the MC calculation) of a knock-on electron. Moreover, the fact that

this bias occurs near the CSD range of the source electron suggests that numerical straggling in

the deterministic solution (i.e. deviation from the one-to-one relationship between path-length

and energy loss due to the discretization approximations) might results in an importance function

of inadequate quality. Therefore, to study these two aspects, three test cases are considered: i)










same energy group structure as the CH algorithm; ii) 25 uniform energy groups; and, iii) 100

uniform energy groups. Figure C-10 shows the 2 -norm as a function of the number of histories

for the first where the CH algorithm and the weight-window energy group structure are the same.


0.25



0.2



0.15



o0.1



0.05


1.0E+07 2.0E+07 3.0E+07 4.0E+07
Number of histories


5.0E+07 6.0E+07 7.0E+07


Figure C-10. Norm of relative differences between standard MCNP5 and ADEIS for tally located
at the 2 MeV electron range at various number of histories with condensed-history
group structure

By comparing Figures C-5 and C-10, it is obvious that using the CH group structure does

not eliminate the bias. If the degradation of the importance quality caused by numerical

straggling was responsible for this possible bias, increasing the number of energy group should

reduce the bias. However, as it can be seen in Figure C- 1, the number of electron energy groups

as little impact of the observed bias. Therefore, it can be concluded that the weight-window

energy group structure is responsible for the bias.


O.OE+00











12.00


10.0025
-- 25 uniform energy groups
S50 uniform energy groups
8.00 --100 uniform energy groups


S6.00


4.00


2.00 T f


0.00


-2.00
0.OE+00 5.0E-02 1.OE-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy [MeV]


Figure C- 1. Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with various energy groups.

Impact of Knock-On Electrons

Considering that previous results suggested that knock-on electrons physical

characteristics, and not their biasing, might be responsible for the spectrum tail bias, it is

interesting to study the impact of the presence of these secondary electrons. To that effect, the

production of secondary electrons is disabled for both the standard MCNP5 and ADEIS

simulations. Figure C-12 shows that when the secondary electron production is disabled, the

spectrum tail bias disappears. This seems to suggest that, in ADEIS, the predicted importance of

low-energy electrons (created early on through knock-on production) toward a ROI located deep

within the target material is inconsistent with the actual contribution of these electrons.

Therefore, the remaining sections of this appendix will look at possible causes of this effect.














8Without kno n
-4-Without knock-on electrons
-U-With knock-on electrons

6



2





-2


-4
O.OE+00 5.0E-02 1.OE-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy [MeV]


Figure C-12. Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with and without knock-on electron production

Impact of the Energy Indexing Scheme

It is well known92 that the energy indexing in the CH algorithm can significantly affect the

dose (and spectrum) of electrons deep within a region of interest since methods that are not

consistent with the definition of the energy groups and their boundaries can lead to significant

errors. It is therefore interesting to verify the impact of different energy indexing algorithm

(MCNP and ITS) on the accuracy of the ADEIS methodology. Figure C-13 shows the relative

differences between the standard MCNP5 and ADEIS spectra when using both the MCNP and

ITS energy indexing scheme. It is obvious from these results that the energy indexing scheme is

not responsible for the possible bias.











14.00

12.00

10.00
-*-with ITS energy index
S8.00 with MCNP energy index

S6.00






0.00 -

-2.00
-2.00 .....--------------------------
0.E+00 1.E-01 2.E-01 3.E-01 4.E-01 5.E-01 6.E-01
Energy [MeV]


Figure C-13. Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range with the MCNP and ITS energy indexing scheme.

Impact of Russian Roulette Weight Balance

It is well known that the Russian roulette game does not preserves the total number of

particle for each VR event but rather preserves it over a large number of histories. Therefore,

even though the statistical uncertainty of an estimator can low, its value might not be accurate if

the weight creation and loss due to the Russian roulette do not balance out. To study the weight

creation and loss as a function of energy, the MCNP5 code was modified to add energy-

dependent ledgers that record weight creation and loss for each energy bins of the weight-

window. Note that since the simulation can be performed in parallel, these ledgers must be local

on each slave process before being accumulated by the master process. Figures C-14 to C-16

show the ratios of weight creation over weight loss for different number of histories (five

hundred thousands to hundred and twenty-eight millions).

















3.5


3


2.5


2
u

S1.5


. 5 1


0.5


0 1 1 1 1 1
0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 1.4E+00 1.6E+00 1.8E+00 2.0E+00

Energy (MeV)



Figure C-14. Ratios of weight creation over weight loss for 5x105 to 2x106 histories


4


3.5


3


2.5


2


1.5


1


0.5


0 I I ------------I
0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 1.4E+00 1.6E+00 1.8E+00 2.0E+00

Energy (MeV)



Figure C-15. Ratios of weight creation over weight loss for 4x106 to 1.6x107 histories











4
--# of histories 3.2E7
# of histories 6.4E7
3.5 --# of histories 1.28E8

3

2.5

2

1.5







0.OE+00 2.OE-01 4.0E-01 6.OE-01 8.0E-01 1.OE+00 1.2E+00 1.4E+00 1.6E+00 1.8E+00 2.OE+00
Energy (MeV)


Figure C-16. Ratios of weight creation over weight loss for 3.2x107 to 1.28x108 histories

Figures C-14 to C-16 shows that the ratio of weight creation over weight loss does

converge toward one as the number of histories increases. Three other major observations can

also be made from these figures; i) no Russian roulette game is played on particles above 1.8

MeV reflecting the importance of these particles to the tally, ii) the ratios converge much more

rapidly in the 0.8 to 1.6 MeV range, and iii) the range from the cutoff energy to 0.6 MeV

contains the largest fluctuations and is the hardest to converge. Even though that last energy

range contains the tally spectrum, the fact that most of the tally energy spectrum is not biased

suggests that this is not responsible from the observed bias. It can also be seen that the Russian

roulette did preserve the weight balance properly for most of the energy range of the problem.

Impact of Coupled Electron-Photon-Positron Simulation

It is possible to change the type of electrons contributing to the tally by performing a

coupled electron-photon simulation. It this mode, other secondary electrons, such as recoil

electrons from Compton scattering, will be created closer to the ROI and reduce the relative










contribution of the knock-on electrons created closer to the source. As expected, Figure C-17

shows that the bias essentially disappears. This reinforces the hypothesis that knock-on electrons

are related to the bias.



8 T

6


4 4




2 \
U T T T T T T --- T T ---- I-T T

-2 -.


-4
0.0E+00 5.0E-02 1.0E-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01 4.5E-01 5.0E-01
Energy [MeV]


Figure C-17. Relative differences between standard MCNP5 and ADEIS for tally located at the
2 MeV electron range in coupled electron-photon model

Conclusions

It was shown that a small possible bias in the electron spectrum tail (i.e., for energy bins

with flux values that are about two orders magnitude lower then the average flux) could be

observed for tallies located at depths near the CSD range, and for which the knock-on electrons

are the main contributors. Note that this bias is referred to as possible since, even though it is

statistically meaningful for the 68% confidence interval, it is not when the 99% confidence

interval is considered.

It was also shown that the inability of 1-D importance functions to provide an adequate

representation of the lateral leakage is not responsible for this bias. Further analyses also showed









that the collision biasing of knock-on electrons, the weight-window energy group structure, the

CH algorithm energy indexing scheme, and the Russian roulette weight balance were not

responsible for this bias. However, the results presented in this appendix suggest that, in ADEIS,

the transport of low-energy electrons over large distances might be slightly biased. Previous

studies91 suggested that differences in the straggling models could explain some discrepancies

between CEPXS and ITS for low-energy electrons. This suggests that the bias could be attributed

to an inconsistency between the predicted importance of these electrons and their actual

contributions due to differences in the straggling model. Finally, it must be mentioned that for

realistic cases requiring coupled electron-photon simulations, and where integral quantities are

estimated at location before the CSD range of the source electrons, this bias in the spectrum tail

does not affect the tallies.









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BIOGRAPHICAL SKETCH

I was born in 1974 in Rouyn-Noranda (QC), Canada. I went to the Universite de Montreal

and got my bachelor's degree in physics in 1997. I continued to graduate school at Ecole

Polytechnique de Montreal where I completed my master's degree in nuclear engineering under

Dr. Koclas and Dr. Teyssedou on coupled neutronic/thermal-hydraulic simulation in CANDU

reactor. In 2002, I moved to Florida to pursue my Ph.D. in nuclear engineering under Dr.

Haghighat.





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1 AUTOMATED VARIANCE REDUCTION TECHNIQUE FOR 3-D MONTE CARLO COUPLED ELECTRON-PHOTON -POSITRON SIMULATION USING DETERMINISTIC IMPORTANCE FUNCTIONS By BENOIT DIONNE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Benoit Dionne

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3 To my parents, Normand and Mireille, without whom this long road to a boyhood dream would not have been possible.

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4 ACKNOWLEDGMENTS I would like first to express m y thanks to Dr. Haghighat for his support and guidance over the years as well as to the committee members. I would also like to gratefully acknowledge the many members of the UFTTG next to whom I spe nd these many years. In particular, Mike and Colleen, whose discussions, friendships and willi ngness to listen to my rants gave me the support I needed.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.......................................................................................................................10 ABSTRACT...................................................................................................................................14 CHAP TER 1 INTRODUCTION..................................................................................................................16 Objective.................................................................................................................................17 Literature Review.............................................................................................................. .....18 2 THEORY................................................................................................................................28 Monte Carlo Transport Theory: General................................................................................ 29 Deterministic Transport Theory: Forward Transport ............................................................. 32 Deterministic Transport Theory: Backward Transport ...........................................................34 Electron, Photon and Positron Interactions............................................................................ 36 Numerical Considerations...................................................................................................... 40 3 ADEIS METHODOLOGY CONCEP TS AND FORMULATIONS ..................................... 49 Importance Sampling............................................................................................................ ..49 ADEIS Angular Transport Biasing.........................................................................................51 ADEIS Source Biasing...........................................................................................................55 ADEIS Collision Biasing........................................................................................................ 56 Criteria for Applyi ng Weight-W indow.................................................................................. 56 Selection of the Adjoint Source.............................................................................................. 57 Comparison with Methodologies in Literature Review.......................................................... 60 4 ADEIS METHODOLOGY IMPLEMENTATION................................................................ 65 Monte Carlo Code: MCNP5................................................................................................... 65 Deterministic Codes: ONELD, PARTISN and PENTRAN...................................................66 Automation: UDR...................................................................................................................67 Modifications to MCNP5.......................................................................................................67 Generation of the Deterministic Model.................................................................................. 68 Generation of the Weight-Window........................................................................................ 71 MCNP5 Parallel Calculations.................................................................................................72

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6 5 IMPACT OF IMPORTANCE QUALITY............................................................................. 74 Reference Case.......................................................................................................................76 Importance Function Positivity............................................................................................... 77 Positrons Treatment and Condensed-History in ADEIS........................................................ 85 Conclusions.............................................................................................................................98 6 IMPROVING THE QUALITY OF THE IMPORTANCE FUNCTION .............................100 Grid Sensitivity and Automatic Spatial Meshing Schem es.................................................. 102 Energy Group and Quadrature Order................................................................................... 111 Angular Biasing....................................................................................................................122 Coupled Electron-Photon -Position Sim ulation..................................................................... 126 Adjoint Source Selection......................................................................................................128 Conclusions...........................................................................................................................131 7 MULTIDIMENSIONAL IMPORTANCE FUNCTION......................................................133 Generation of 3-D Importance Function Using PENTRAN ................................................. 133 Generation of 1-D Importance Functions Using P ARTISN.................................................143 Biasing Along the Line-of-Sight Using the MCNP5 Cylindrical W eight-Window............. 144 Generation of 2-D (RZ) Importa nce Functions U sing PARTISN........................................ 145 Speedup Comparison between 1-D and 2-D (RZ) Biasing..................................................146 Conclusions...........................................................................................................................148 8 CONCLUSIONS AND FUTURE WORK ........................................................................... 150 Conclusions...........................................................................................................................150 Future Work..........................................................................................................................152 APPENDIX A VARIOUS DERIVATIONS................................................................................................. 154 Selection of an Optimum Sampling Distribution in Importance Sampling.......................... 154 Biased Integral Transport Equation...................................................................................... 155 Lower-weight Bounds Formulation and Source Consistency.............................................. 156 Determination of the Average C hord-Length for a Given Volum e...................................... 157 B IMPLEMENTATION DETAILS......................................................................................... 158 Universal Driver (UDR)....................................................................................................... 158 Performing an ADEIS Simulation........................................................................................ 159 ADEIS MCNP5 Input Card..................................................................................................160

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7 C ELECTRON SPECTRUM BIAS SIDE STUDIES.............................................................. 161 Impact of Tally Location...................................................................................................... 161 Impact of the Number of Histories on Convergence............................................................163 Impact of Electron Energy and Energy Cutoff.....................................................................166 Impact of Lateral Leakage.................................................................................................... 167 Impact of Knock-On Electron Collision Biasing.................................................................. 168 Impact of Weight-Window Energy Group Structure ............................................................169 Impact of Knock-On Electrons............................................................................................. 171 Impact of the Energy Indexing Scheme................................................................................ 172 Impact of Russian Roulette Weight Balance........................................................................ 173 Impact of Coupled Electron-Photon-Positron Simulation.................................................... 175 Conclusions...........................................................................................................................176 LIST OF REFERENCES.............................................................................................................178 BIOGRAPHICAL SKETCH.......................................................................................................185

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8 LIST OF TABLES Table page 3-1 Comparison of other varian ce reduction m ethodology with ADEIS.................................. 61 5-1 Materials and dimensions of reference case........................................................................ 75 5-2 Test case simulation parameters.......................................................................................... 75 5-3 Reference case spatial mesh structur e producing a positive importance function .............. 79 5-4 Average FOM and RFOM for all approaches........................................................................ 85 5-5 Impact of biasing on annihilation photons sampling........................................................... 87 5-6 Impact of explicit positron biasi ng on annihilation photons sampling ............................... 89 5-7 Impact explicit positron biasing on average FOM and RFOM..............................................91 6-1 Various test cases of the analysis plan .............................................................................. 101 6-2 Other reference case simulation pa ram eters of the analysis plan...................................... 102 6-3 CEPXS approximated detour factors................................................................................ 105 6-4 Calculated detour factors for each case of the analysis plan............................................. 106 6-5 Materials and dimensions of new si mplified test case......................................................107 6-6 Speedup for multi-layered geometries using automatic mesh criterion............................ 107 6-7 Speedup gain ratios from boundary la yers schem e #1 in Cases 1 and 9........................... 109 6-8 Speedup gain ratios from boundary la yers schem e #2 in Cases 1 and 9........................... 111 6-9 Total flux and relative e rror in the R OI for all case s of the analysis plan......................... 114 6-10 Average ratio of track created to tr ack lost for cases with sam e energy........................... 114 6-11 Speedup with angular biasing for Case 7 with a source using an angular cosine distribution and reduced size .............................................................................................125 6-12 Speedup as a function of the number of energy groups for Cases 6 and 9........................127 6-13 Electron and photon tally speedup us ing ADEIS with angular biasing ............................ 128 6-14 Electron and photon flux tally speedup using different objective particles ...................... 130

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9 6-15 Energy deposition tally speedup in the reference case for various objective particles ..... 130 7-1 PENTRAN and ONELD simulation pa ram eters for solving problem #1......................... 134 7-2 PENTRAN and ONELD simulation pa ram eters for solving problem #2......................... 139 7-3 Other simulation parameters for problem #3.................................................................... 141 7-4 Energy deposition tally speedup for ONELD and PARTISN s imulation of Chapter 5 reference case................................................................................................................. ...143 7-5 Materials and dimensions of reference case...................................................................... 147 7-6 Energy deposition tally spee dup for 1-D and 2-D biasing ................................................ 147 B-1 The ADEIS keywords....................................................................................................... 160

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10 LIST OF FIGURES Figure page 3-1 Field-of-View (FOV) concept............................................................................................54 4-1 Automated ADEIS flow chart............................................................................................ 65 4-2 Line-of-sight approach..................................................................................................... ..69 4-3 Two-dimensional model ge neration using line-of-sight ....................................................69 5-1 Reference case geometry................................................................................................... 75 5-3 Relative error and variance of varian ce in ADEIS with im portance function smoothing...........................................................................................................................78 5-4 Relative error and variance of variance in ADEIS with optimum mesh structure......... 80 5-5 Relative difference between importance with 1st and 2nd order CSD differencing............ 81 5-6 Relative error and variance of variance for ADEIS photon tally with 1st order CSD differencing scheme and 75 energy groups.......................................................................82 5-7 Relative error and variance of variance in ADEIS with 75 energy groups....................... 82 5-8 Relative error and variance of variance in ADEIS with CEPXS-GS................................83 5-9 Impact of large variation in im portance between positron and photon............................. 87 5-10 Electron and annihilation photon impo rtance function in tungsten target ......................... 88 5-11 Positron and annihilation photon impo rtance function in tungsten target ......................... 88 5-12 Surface Photon Flux Spectra at Tungsten-Air Interface.................................................... 89 5-13 Relative error and VOV in ADEIS with CEPXS and explicit positron biasing................ 90 5-14 Regions of interest consid ered in simplified test case ....................................................... 91 5-15 Relative differences between the standard MCNP5 and ADEIS total fluxes for three energy-loss approaches ...................................................................................................... 92 5-16 Relative differences between the st andard MCNP5 and ADEIS electron energy spectrum for Case 3 in the five regions of interest............................................................ 93 5-17 Relative differences in electron spectra for undivided and divided m odels...................... 94

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11 5-18 Impact of the modification of condens ed-history algorithm with weight-window on the relative differences in total fl ux between standard MCNP5 and ADEIS....................95 5-19 ADEIS normalized energy spectrum and relative differences with the standard MCNP5 at 70% of 2MeV electron range with the CH algorithm modification................ 96 5-20 Relative differences between the standard MCNP5 and ADEIS at various fraction of the range .............................................................................................................................97 6-1 Simplified reference model.............................................................................................. 101 6-2 Speedup as a function of FOD for Cases 1, 4 and 7........................................................ 103 6-3 Speedup as a function of FOD for Cases 2, 5 and 8........................................................ 103 6-4 Speedup as a function of FOD for Cases 3, 6 and 9........................................................ 104 6-5 Multi-layered geometries.................................................................................................107 6-6 Automatic boundary layer meshing scheme #1............................................................... 108 6-7 Automatic boundary layer meshing scheme #2............................................................... 110 6-8 Speedup as a function of the number of energy groups for Cases 1, 2 and 3.................. 112 6-9 Speedup as a function of the number of energy groups for Cases 4, 5 and 6.................. 112 6-10 Speedup as a function of the number of energy groups for Cases 7, 8 and 9.................. 113 6-11 Importance functions for source and knock-on electrons at a few en ergies for Case 7 ... 115 6-12 FOM as a function of the number of energy groups for m odified Case 9....................... 116 6-13 Number of knock-on electrons and their tota l statistical as function of the number of energy groups for Case 3.................................................................................................117 6-14 Splitted electron energy as a function pos ition for a 1000 source particles in Case 3. .... 118 6-15 Rouletted electron energy as a func tion positio n for a 1000 source particles in Case 3...............................................................................................................................118 6-16 Splitted electron weight as a function of position for a 1000 source particles in Case 3. ..............................................................................................................................119 6-17 Importance functions for 15 and 75 en ergy groups at 3.06 cm for Case 3. ..................... 119 6-18 Impact of discrete ordinates quadratur e set order on speedup for all cases of the analysis plan. ....................................................................................................................121

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12 6-19 Electron tracks for a 20 MeV electron pencil beam i mpinging on water (Case 7).......... 123 6-20 Electron tracks for a 20 MeV electron cosine beam i mpinging on tungsten (Case 7 with a source using an angular cosine distribution)......................................................... 124 6-21 Electron tracks for a 0.2 MeV electron pe ncil beam impinging on tungsten (Case 3).... 124 7-1 Problem #1 geometry....................................................................................................... 135 7-2 Importance function for fastest energy group (0.9874 MeV to 1.0125 MeV) in problem #1..................................................................................................................... ..135 7-3 Ratio of ONELD importance over PENTRAN i mportance for 4energy groups in problem #1..................................................................................................................... ..136 7-4 Mesh refinement to resolve boundary laye rs at the edges of model for problem #1.......137 7-5 Ratio of ONELD importance over PENTRAN importance for four energy groups in problem #1 with mesh refinement................................................................................... 137 7-6 Ratio of the ONELD and PENTRAN im portance functions for group 1 obtained using S16 and S32 quadrature order with mesh refinement in problem #1....................... 138 7-7 Problem #2 geometry....................................................................................................... 139 7-8 Mesh structure for problem #2......................................................................................... 139 7-9 Impact of differencing scheme on importance function for group 20 in problem #2...... 140 7-10 Problem #3 geometry....................................................................................................... 141 7-11 Ratio of ONELD importance over PENTRAN importance for four photon energy group in problem #3.........................................................................................................142 7-12 One-dimensional (R) and two-dimensi onal (RZ) weight-window mesh along the line-of-sight in a 3-D geometry........................................................................................ 144 7-13 Modified refere nce case geom etry................................................................................... 147 B-1 Example of UDR input file syntax................................................................................... 159 B-2 Examples of calls to adeisrun.......................................................................................... 160 B-3 New simulation sequence in MCNP5..............................................................................160 C-1 ADEIS normalized spectrum and relative difference with standard MCNP5 for tally located at 70% of 2 MeV electron range .........................................................................162

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13 C-2 ADEIS normalized spectrum and relative difference with standard MCNP5 for tally located at 2 MeV electron range ...................................................................................... 162 C-3 Relative differences between standard MCNP5 and ADEIS for tally located at 70% of the 2 MeV electron range at various num ber of histories............................................ 164 C-4 Norm of relative differences between st andard MCNP5 and ADEIS for tally located at 70% of the 2 MeV electron range at various num ber of histories............................... 164 C-5 Norm of relative differences between st andard MCNP5 and ADEIS for tally located at the 2 MeV electron range at various num ber of histories............................................ 165 C-6 Relative differences between the tally electron spectra from ADEIS and standard MCNP5 for a 2 MeV electron beam at two energy cutoff............................................... 166 C-7 Relative differences between the tally electron spectra from ADEIS and standard MCNP5 for a 13 MeV electron b eam at two energy cutoff............................................. 167 C-8 Relative differences in spectrum betw een the standard MCNP5 and ADEIS for various m odel sizes and an energy cutoff of 0.01 MeV.................................................. 168 C-9 Relative differences in spectrum between the standard MCNP5 and ADEIS for the reference case with and without co llision biasing for knock-on electrons ...................... 169 C-10 Norm of relative differences between st andard MCNP5 and ADEIS for tally located at the 2 MeV electron range at various num ber of histories with condensed-history group structure .................................................................................................................170 C-11 Relative differences between standard MCNP5 and ADEIS for tally located at the 2 MeV electron range with various energy groups. .........................................................171 C-12 Relative differences between standard MCNP5 and ADEIS for tally lo cated at the 2 MeV electron range with and w ithout knock-on electron production.......................... 172 C-13 Relative differences between standard MCNP5 and ADEIS for tally lo cated at the 2 MeV electron range with the MCNP and ITS energy indexing scheme.......................173 C-14 Ratios of weight creation over weight loss for 5x105 to 2x106 histories......................... 174 C-15 Ratios of weight creation over weight loss for 4x106 to 1.6x107 histories......................174 C-16 Ratios of weight creation over weight loss for 3.2x107 to 1.28x108 histories.................175 C-17 Relative differences between standard MCNP5 and ADEIS for tally lo cated at the 2 MeV electron range in coupl ed electron-photon model............................................... 176

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14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AUTOMATED VARIANCE REDUCTION TECHNIQUE FOR 3-D MONTE CARLO COUPLED ELECTRON-PHOTON -POSITRON SIMULATION USING DETERMINISTIC IMPORTANCE FUNCTIONS By Benoit Dionne December 2007 Chair: Alireza Haghighat Major: Nuclear Engineering Sciences Three-dimensional Monte Carlo coupled elect ron-photon-positron tran sport calculations are often performed to determine characteristics su ch as energy or charge deposition in a wide range of systems exposed to radiation field such as electronic circuitry in a space-environment, tissues exposed to radiotherapy linear accelerator beams, or radiation detectors. Modeling these systems constitute a challenging problem for the available computational methods and resources because they can involve; i) very large attenuation, ii) large number of secondary particles due to the electron-photon-positron cascade, and iii) la rge and highly forwar d-peaked scattering. This work presents a new automated varian ce reduction technique, referred to as ADEIS (Angular adjoint-Driven Electr on-photon-positron Importance Samp ling), that takes advantage of the capability of deterministic methods to ra pidly provide approximate information about the complete phase-space in order to automatically evaluate variance reduction parameters. More specifically, this work focuses on the use of disc rete ordinates importance functions to evaluate angular transport and collision biasing para meters, and use them through a modified implementation of the weight-window technique. Th e application of this new method to complex Monte Carlo simulations has resulted in spee dups as high as five orders of magnitude.

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15 Due to numerical difficulties in obtaini ng physical importance functions devoid of numerical artifacts, a limited form of smoothing was implemented to complement a scheme for automatic discretization parameters selection. Th is scheme improves the robustness, efficiency and statistical reliabi lity of the methodology by optimizi ng the accuracy of the importance functions with respect to the additional computational cost from generating and using these functions. It was shown that it is essential to bias diffe rent species of particles with their specific importance functions. In the case of electrons and positrons, ev en though the physical scattering and energy-loss models are similar, the im portance of positrons can be many orders of magnitudes larger than electron importance. More specifically, not e xplicitly biasing the positrons with their own set of importance functions results in an undersampling of the annihilation photons and, consequently, intr oduces a bias in the photon energy spectra. It was also shown that the implementation of the weight-window technique within the condensed-history algori thm of a Monte Carlo co de requires that the biasing be performed at the end of each major energy step. Applying the weight-w indow earlier into the step, i.e., before the last substep, will result in a biased electron en ergy spectrum. This bias is a consequence of systematic errors introduced in the energy-loss pr ediction due to an inappropriate application of the weight-window technique where the actual path-length differs from the pre-determined pathlength used for evaluating the en ergy-loss straggling distribution.

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16 CHAPTER 1 INTRODUCTION Three-dim ensional Monte Carlo coupled elect ron-photon-positron tran sport calculations are often performed to determine characteristics su ch as energy or charge deposition in a wide range of systems exposed to radiation fields, such as electronic circuitr y in a space-environment, tissues exposed to radiotherapy linear accelerator beams, or radiation detector. Modeling these systems constitutes a challenging problem for the available computational methods and resources because they can involve: i) very large attenuation (often referred to as deep-penetration problem), ii) large number of secondary partic les due to the electronphoton-positron cascade and iii) large and highly forward-p eaked scattering cross sections. Monte Carlo methods are generally consider ed more accurate since very complex 3-D geometries can be simulated without introducing systematic errors related to the phase-space discretization. However, even for problems where source particles have a reasonable probability of reaching the region of intere st, tracking large number of s econdary particles may result in unreasonable simulation times. Therefore, selecting the right particles through the use of variance reduction (VR) techniques can be hi ghly beneficial when su ch performing coupled electron-photon-positron simulations for complex 3-D geometries Note that hereafter, the expressions variance reduction and biasing will refer to fair-game techniques that achieve precise unbiased estimates with a reduced co mputation time unless implied otherwise by the context. Typical biasing methodol ogies often require a lot of experience and time from the user to achieve significant speedup while maintaining accurate and precise resu lts. It is therefore useful for a VR methodology to minimize the amount user involvement in order to achieve high efficiency.

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17 Deterministic approaches solving a discreti zed form of an equation representing the particle balance in phase-space have also been used successfully to model systems involving coupled electron-photon-positron tran sport. The current deterministic methods presented in the publicly available literature are limited in scope si nce they either accurate ly describe the physical processes only for 1-D or 2-D geometries, or by obtaining approximate solutions for 3-D geometries (e.g. fast semi-analyti c deterministic dose calculations1). The main advantage of these deterministic approaches is that they provide information about the whole phase-space relatively quickly compared to Monte Carlo (MC) methods. However, such methods require even more experience and time from the user to obtain accura te and precise results in reasonable amount of time. Therefore, despite the la rge increase in computer perfor mance, a methodology to perform more efficient and accurate coupled electron-photon-positron is needed. Objective Therefore, the objective of this work is to develop a new autom ated variance reduction methodology for 3-D coupled electron-photon-positron MC calculations that takes advantage of the capability of deterministic methods to rapidly provide approximate information about the complete phase-space in order to automatically evaluate the VR parameters. Such methodology significantly reduces the computation time (as show n previously in similar work performed for neutral particles2) and the engineering time if a sufficient amount of automation is implemented. Furthermore, it reduces the chance of unstable statis tical behavior associated with VR techniques requiring users to manually select VR parameters. More specifically, this work focuse s on the use of discrete ordinates (SN) importance functions to evaluate angular transport and co llision biasing parameters and accelerate MC calculations through a modified implementati on of the weight-window technique. This methodology is referred to as Angular adjo int-Driven Electron-photon-positron Importance

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18 Sampling (ADEIS) from hereon. To maximize the increase efficiency a nd reduce the amount of engineering time spent on evaluating ADEIS VR parameters, a high level of automation is implemented. As presented in the literature revi ew, the idea of using importance functions to accelerate MC calculations is not new, however as far as surveyed, no work has been done to perform angular transport and co llision biasing using determin istic importance functions in coupled electron-photon-positron problems. Literature Review This section presents a summary of pr evious work performed over the past few decades on coupled electron-photon-positron MC simulation and their associated variance reduction techniques, and on coupled electr on-photon deterministic methods. Note that some of the work presented in this section may address only el ectron transport simulation since the major difficulties in performing such calculations arise from modeling electron interaction with matter. Monte Carlo Monte Carlo (MC) m ethods were developed in the 1940s by scientists involved in nuclear weapon research. Based on their work, one of th e first accounts of the method was written by Metropolis and Ulam3 in 1949. Interestingly enough the auth ors suggested that the method is inherently parallel and should be applied to many computers working in parallel which seems to be becoming the standard approach. Nowadays, the term Monte Carlo refers to numerical methods based on the use of random numbers to solve physical and mathematical problems. Kalos and Whitlock4 provide a good general survey of vari ous MC techniques with applications to different fields. Radiation transport MC calculations simulate a finite number of particle histories by using pseudo-random numbers to sample from probability density functions (PDF) associated with the various kinds of physical processe s. Statistical averages and thei r associated variances are then

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19 estimated using the central limit theorem. Note th at an overview of the different aspects of the MC method in the context of radiation tr ansport is presented in Lewis and Miller5 as well as in Shultis and Faw6. Over the years many MC production codes have been developed to solve various problems involving coupled electron-photon-pos itron processes. Some of the major codes still in use in various fields are: MCNP57, developed at Los Alamos national Laboratory, ITS8, developed at Sandia National Laboratory, PENELOPE9, developed by university research groups in Spain and Argentina, MARS10, developed at Fermi National Accelerator Laboratory, EGS11, evolution of EGS3 developed by SLAC and other international agencies, GEANT12, developed as a worldwide effort and managed by CERN, DPM13, developed at the University of Michigan, FLUKA14, developed within INFN (National Institute of Nuclear Physics) TRIPOLI-415, developed at CEA (Commisariat de lEnergie Atomique) in France All these codes use a technique referred to as condensed history (CH) Monte Carlo in order to circumvent some of the difficulties created by the large interaction ra te of electrons (e.g., an electron slowing down from 0.5 MeV to 0.05 MeV will undergo between 105 to 106 collisions). This method developed by Berger16 condensed a large number of collisions in a single electron step To predict the change of energy and direction of motion at the end of each step cumulative effects of individual interactions are taken into account by analyt ical theories such as Bethe energy-loss theory17, Goudsmith-Saunderson18, 19 or Moliere scattering theory20, and Landau21 energy-loss straggli ng theory. Larsen22 has proven that in the limit of an infinitesimal step size,

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20 the CH approach is a solution of the Boltzmann transport equation. A good review and analysis of CH techniques is presented by Kawrakow and Bielajew23. Variance Reduction Many techniques have been developed to incr eas e the efficiency of MC calculations by reducing either the variance or th e computation time per history. As discussed more at length in Chapter 2, these techniques modify the physical la ws of radiation transport in an attempt to transport the particle toward a region of interest without introd ucing a bias in the statistical estimates. A complete summary of all the VR techniques is outside the scope of this review; therefore only techniques particle importance will be presented. Kalos24 described the importance sampling technique and its relation to the hypothetical zero-variance solution. Coveyou et al25 developed formulations invol ving the use of importance functions to reduce the va riance through source and tran sport biasing. Many studies26, 27, 28, 29, 30, 32, 33, 34, 36, 37, 38 were also carried out, with various degrees of success, to use a deterministic approximate importance functions to accelerate ne utral particles MC transport calculations. Tang and Hoffman26 used 1-D importance function along th e axial and radial direction of a shipping cask to bias the neutron reaction rates at two detectors (axi al and radial detectors) using a combination of source energy biasing, energy bi asing at collision site splitting and Russian roulette, and path-length stretc hing. This approach suggests that different importance functions should be used when the biasing ob jectives are significantly different. Mickael27 uses an adjoint diffusion solver embedded into MCNP to generate a deterministic neutron importance map. In this methodology, the group constants are generated by using an estimate of the flux from a short anal og MC calculation. However, this approach cannot be applied to electron and positron simulation sin ce a diffusion treatment is inadequate for highly angular phenomena such as hi gh energy electro n scattering.

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21 Turner and Larsen28, 29 developed the LIFT (Local Im portance Function Importance) VR technique for neutral particle transport. This method uses an analytic formulation that approximates the solution within each energy group and spatial cell of a MC model. This formulation is based on an approximate impor tance solution which can be estimated using various techniques su ch as diffusion, SN or SPN. This methodology biases the source distribution, the distance to collision, the selection of postcollision energy groups, and trajectories. Note that this approach approximates a theoretical zero-variance method. However, in this version of the methodology, only linear anisotropic scattering is considered, which makes it unsuitable for high-energy beam-like problems and mo st electron/positron simulations. Van Riper et al.30 developed the AVATAR (Automatic Variance and Time of Analysis Reduction) methodology for neutral particles. Th is methodology uses a 3-D importance function calculated from THREEDANT31 in order to evaluate lower-we ight bounds of the MCNP weightwindow using the basic inverse re lation between statistical weight and importance. An angular dependency was introduced into the MCNP weig ht-window by using an approximation to the angular importance function. Note that in Re f. 30, no mention was made of preserving the consistency between the source and weight-window definitions. Wagner and Haghighat32, 33 developed the CADIS (Consistent Adjoint-Driven Importance Sampling) for neutral particles. CADIS uses the concept of importanc e sampling to derive consistent relations for sour ce biasing parameters and wei ght-window bounds. This methodology implemented within a patch to MCNP is referred to as A3MCNP (Automated Adjoint Accelerated MCNP) and uses TORT34 to estimate the three-dimensional importance functions. A3MCNP automatically generates the input for th e 3-D transport code by using a transparent

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22 mesh and a back thinning technique. Cross-se ction mixing and various other tasks are handled through various scripts. Shuttleworth et al35 developed an inbuilt importance generator which uses the adjoint multigroup neutron diffusion equation to estimate the importance function. Note that even if special diffusion constants are used to provide a closer approximati on to transport theory, this is still inadequate for elect ron/positron transport. As reported by Both et al36 and Vergnaud et al37, a fairly automatic importance-generation technique was implemented in the MC code TRIP OLI. This technique uses exponential biasing, a form of splitting call quota sampling, and collision biasing. Their importance function is the product of three factors depending on space, ener gy and angle but essentially assumes a spatial exponential behavior. This method requires th at the user provides input values of a priori parameters. Giffard et al.38 developed a different variance re duction for TRIPOLI which does not require much a priori expertise. As many others, this me thodology uses a deterministic code to generate an approximate importance function. Howe ver, they use that information to create a continuous-energy importance function using an interpolation scheme. This methodology uses these importance functions for source biasing, tr ansport kernel biasing, Russian roulette and splitting. Wagner39 developed the ADVANTG (Automated Deterministic VAriaNce reducTion Generator) code based on the previously de veloped and proven CADIS methodology. The main difference between ADVANTAG and A3MCNP is that ADVANTG uses the standard MCNP interface file named wwinp

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23 The following methodologies do not use determin istic importance func tions but are still importance-based Goldstein and Greenspan40 developed a recursive MC (RMC) method where the importance is estimated by so lving the forward problem for ex tensively subdivided geometric regions. The methodology sprinkles the problem with test particles and tracks them until they score or die. Scoring particles make a contribut ion to the importance at their birth location. Booth41 developed an importance estimation te chnique known as the weight-window generator since it generates impor tance functions to be used in MCNP weight-window. In this methodology, it is considered that th e cell (or mesh) from which a pa rticle emerges after an event may be considered as the starting point for the remainder of the histor y. A contribution to the importance estimate is thus made in every cell (or mesh) that the par ticle passes through. Two others methods35 have been implemented in the MCBEND code. In the first method, the test particles are initially generated in a cell of the importance mesh that contains the detector and soon generates a reliable value for the importan ce. These particles are th en generated in cells adjacent to the target and then tracked. If they cr oss into the target, thei r expected score is known from the value of the importance that has alr eady been calculated. Processing then moves on to next layer of cells surrounding the ones alr eady completed and so on. The second method, named MERGE, uses an approach similar to weight -window generator. Ho wever, the partially completed importance function is then merged with an initial estimate of the importance. Murata et al.42 methodology is also similar to the weight-window generator with the exception that the contribution to the detector, and consequently th e importance at the location, is evaluated for each scattering point using the point detector tally. Note that the authors introduce the idea of using angular-dependent parameters in form of angular meshes, but do not present any results.

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24 Finally, a multigroup adjoint transport simulation can be performed in MCNP57 to obtain biasing functions which are then used through an energy-dependent but cell-based weightwindow. However, this capability requires the use of an undoc umented feature of the CEPXS package51 and the use of the unsupported CRSRD code to generate multigroup libraries that are suitable to be used in a MC code. Deterministic Methods A com plete review of all dete rministic methods is also out of the scope of this work, therefore only methods that are of interest fo r estimating deterministic coupled electron-photonpositron importance are discussed as well as so me early work that pioneered the field. Spencers approach43 was one of the first methodology performing numeri cal calculations of electron spatial distribution taking into accounts both energy-loss and change of direction. Spencers method is based on previous work from Lewis44 and considers monoenergetic and mono-directional sources distribut ed uniformly over an infinite plane in homogeneous media. This work recasts the Lewis equati on in terms of residua l ranges, and expands it into a series of spatial, angular and residual range moments. He then proceeded to numerically evaluate those moments and compare his results with experiments. Based on our survey, Bartine et al45 were the first to use the SN method to calculate the spatial distribution of low-energy electrons. In their approach, the scattering integral is rewritten using an asymptotic approach (Taylor series expansion) to obtain a continuous slowing down term for representing the small-angle ( soft ) inelastic collisions. Moreover, to reduce the number of moments required to represent the highly forward-peaked elastic scattering cross-sections still represented by an integral kernel the extended transport correction46 is applied. These modifications where implemented in the standard code ANISN47. Nowadays, this form of the

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25 transport equation is referred to as the Boltzmann-CS D and constitutes the basis, or is related to, most of the current practical work on deterministic electron transport. Melhorn and Duderstadt48 modified the TIMEX code49 to provide time-dependent FokkerPlanck (FP) solutions for one-dimensional slab and spherical geometries assuming that the scattering can be decomposed in a continuous energy-loss term and a continuous angular diffusion term. Note that the pure FP solutions are considered inad equate for electrons since they lack the ability to properly represent the hard collisions. Morel50 developed a method for performing Boltz mann-FP calculations using a standard SN production code. In this approach, Morel defines the scattering multigroup Legendre crosssection in terms of the FP functions. The SN quadrature set must be defined such that the Boltzmann solution converges to the Boltzmann-FP solution as the SN space-angle-energy mesh is refined. If the continuous a ngular diffusion term is neglected, this methodology threfore solves the Boltzmann-CSD transport equation as in the previous work from Bartine. Note that the Boltzmann-CSD equation is more amenable for el ectron transport simulation than either the Boltzmann and FP equations alone. Lorence and Morel used this approach to develop the CEPXS/ONELD package51. Przybylski and Ligou52 investigated two numerical appr oaches to solve the Boltzmann-FP equation using a discrete ordinates approach for the angular dependency of the angular flux and the angular diffusion term of the Fokker-Planck scattering kernel. They compared a multigroup approach to a method which uses a linear-dia mond scheme on space and energy. The goal was to mitigate numerical instabilities us ually resulting from the finite-difference approximations of the derivatives in the Fokker-Planck scattering kern el. Since the linear-diamond scheme is not

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26 guaranteed positive, they also developed a fix-up scheme that preserves the energy balance over the cell. Filippone53 developed a methodology, named SMART ma trix, to generate more accurate scattering matrices. This methodology allows the de finition of a scattering matrix that guarantees nearly exact calculations of el ectron angular distribution at eac h discrete direction. Using the Goudsmit-Saunderson theory18, 19, it is possible to find an integral expression for the scalar flux (as a function of the electron path-length) that can be integrat ed using a quadrature set. By comparing the scalar flux expression with the explicit expression obtaine d from the discretized SN transport equation (using a char acteristic differencing scheme); it is possible to derive an expression of the scattering kernel for which bot h formulations are identical. Fillipone also shows that smart matrices reducing energy and angular di scretization errors can be derived for any discretization scheme. Drumm54 improved a methodology developed by Morel50 by eliminating the need to exactly integrate a scattering kernel containing delta functions so that general quadrature sets can be used instead of the Gauss or the Galerk in sets. This methodology combined Morel and Filiponne53 approaches, resulting in better scattering cross-sections since they are more often positive, tend to exhibit smaller values than the true interaction cross-sections, and are not tied to any specific quadrature set. Note that this work is based on the Boltzmann-FP equation instead of the Spencer-Lewis equation43, 44 used by Filiponne. The author al so mentions that the use of these cross-sections eliminates some of th e well-known numerical os cillations present in CEPXS/ONELD results, and that the convergence ra te of the source iterat ion technique is also generally much faster.

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27 For completeness, it must be mentioned that besides these SN-like methods, there are other methodologies which have been used to perform deterministic electron tr ansport. For instance, Corwan et al55 derived a multigroup diffusion formula tion from the Fokker-Planck equation, Haldy and Ligou56 developed a code to calculate angular and spatial moments of the electron distribution based on the work by Spencer43, Honrubia and Aragones57 developed a methodology using a finite element discretization of the Bo ltzmann-FP equation where either the space-energy or space-energy-angle variables are treated in a coupled way, Prinja and Pomraning58 developed a generalized Fokker-Planck methodology which introduced higher order term s in the asymptotic expansion of the scattering kernel and finally, Franke and Larsen59 developed a methodology to calculate exact multidimensional information concerning the spreading of 3-D beams by solving a coupled set of 1-D transverse radial moment equations. This me thod is related to the method of moments developed by Lewis44. The methodology developed in this work, ADEIS, uses a modified version of the MCNP57 weight-window algorithm to implement an angu lar extension (similar to Ref. 42) of the CADIS32, 33 methodology to coupled electron-photon-pos itron simulations using beam sources. ADEIS uses the CEPXS/ONELD51, CEPXS-GS54 and PARTISN60 packages to calculate the deterministic importance functions required to ev aluate the VR parameters. Before presenting this methodology in more detail (see Chapter 3), it is useful to review the major theoretical concepts (see Chapter 2).

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28 CHAPTER 2 THEORY W hile the literature review presented in the previous chapter provided a general background to the current effort to develop a VR methodology, this chapter provides a more in-depth theoretical review of so me of the aforementioned concepts and formulations. A quick overview of the MC and deterministic approaches to radiation transport are presented followed by an intr oduction to the electron, photon and positron interaction physics. Finally, the remainder of this chapte r discusses various numerical considerations. To fully characterize all the particles, the positions and velocities of each particle before and after each collision must be know n. To accomplish this, it is necessary to describe these collisions in a six-dimensional space (three dimensions for position and three dimensions for the velocities) called th e phase-space. In a gene ral way, the transport of radiation through matter can be represented by Eq. 2-1. )()()( PPTPQPdPE (2-1) This equation describe s a source particle ()( PQ ) located at P in phase-space being transported ()(PPT ) to another location in the phase-space P, and contributing to an average quantity of interest ()(PE) at that location. In nuc lear engineering, it is customary to represent a phase-space element P as dEdrd where rd represents the position component of phase-space and dEd represents the velocity component of phase-space in terms of energy (dE) and direction ( d).

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29 Monte Carlo Transport Theory: General The MC m ethod could solve Eq. 2-1 by sampling the integrand using random numbers. However, the exact probability density function (PDF) of a complex process such as transporting particles through a 3-D geometry is never known, thus )( PPT is implicitly sampled by tracking all the microsc opic events in the histories of a large number of particles. In MC calculations, it is possible to estimate the expected value of the quantities of interest by cal culating average properties from a set of particle histories using laws of large number, e.g., the Strong Law of Large Number and the Central Limit Theorem. Using a simplified notation, a PDF, ) ( xf can be used to describe a particle being transported and contributing to the quantity of interest. Consequently, the expected value )( xE of that quantity would be calculated by Eq. 2-2. meantruedxxxfxE)()( (2-2) The true mean can then be estimated by the sample mean x calculated using Eq. 2-3. N i ix N x11 (2-3) In this equation, ix is the value of x selected from ) ( xf for the ith history and N is the total number of histories. This mean is equiva lent to the expected value since the Strong Law of Large Numbers states that if )(xE is finite, x will tend toward )(xE as N approaches infinity61. Note that the numerical operator used to estimates the mean is often referred as an estimator. Since in practical simulation N will be sm aller then infinity, it is necessary to evaluate the statistical uncerta inty associated with using x The variance of the

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30 population x values is a measure of their spread around the expected value and can be evaluated by Eq. 2-4. 2 2 2 2)]([)()()]([ xExEdxxfxEx (2-4) As with the true mean, the bias-correct ed variance of the po pulation can only be estimated based on the distribution of the sampled scores using Eq. 2-5. N i ixx N S1 2 2)( 1 1 (2-5) However, it is more useful to know the va riance associated with the average value ( x ) being calculated. If the Central Limit Th eorem is valid, the sample variance of x should be given by Eq. 2-6. N S Sx 2 2 (2-6) It is possible to define an estimated relative error to represent the statistical precision at 1 -level (i.e., x is within the interval xSx 68% of the time) as Eq. 2-7. Nx S x S Rx x. (2-7) It must be noted that there is an importan t difference between preci sion and accuracy of a MC simulation. The precision is a measure of the uncertainty associated with x due to statistical fluctuations, while the accuracy is related to th e fidelity of the model in representing the actual system and physics. In addition to the variance associated with each mean, it is important to verify that the tally is statistically we ll-behaved otherwise erroneous results could be obtained. Another useful quantity is the relative variance of variance (VOV) which is the estimated relative variance of the estimated R and theref ore is much more sensitive to large score

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31 fluctuations than R. Eq. 2-7 highlights a drawback of the MC method, i.e., the reduction of the statistical uncertainty requires a la rge number of histories. For example, to decrease the relative error associated with a converged result by a factor 2, the total number of histories must be increased by a factor of 4. When evaluating the efficiency of any MC simulation, three factors are important: i) the history scoring effici ency, ii) the dispersion of non-zero scores, and iii) the computer time per history. The first factor is essentially the fraction of source particles that contribute to a given tally, the second fact or is related to the sp read of the particle weights scores (and therefore the variance of a tally), and the third is related to the number of histories that can be simulated in a given unit of time. Therefore, the scoring efficiency, the ratio of the largest tally scor e to the average tally score and the number of simulated histories per minute can be used to take a detailed look at the performance of the simulation. However, these three factors are generally folded into a single metric to get a figure-of-merit (FOM) describing the perf ormance of the simulation. This FOM is defined by Eq. 2-8. T R FOM21 (2-8) This metric takes into acc ount the competing effects of the decreasing variance (as measured by the square of the relative error R2) and the increasing computation time (T) as a function of the number of histories. It is possible to get the speedup obtained from a variance reduction (VR) technique by comp aring the FOM from two simulations. By assuming that both simulations reach the same precision (relative error) in their respective time, an estimat e of the speedup can simply be obtained by Eq. 2-9.

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32 speedup T T FOM FOM 2 1 1 2 (2-9) It is important to note that if the relative errors are different, Eq. 2-9 still provide a good estimation of the speedup. Since the VOV is more sensitive to the stat istical fluctuations of an estimator, another useful metric used to measure the statistical reliability is the Figure-of-Reliability (FOR) defined by Eq. 2-10. T VOV FOR21 (2-10) Deterministic Transport Theory: Forward Transport By considering particles a nd the target atoms as two component of a gas, it is possible to write an equation, either in integral or integro-differential form, to characterize the behavior of particles. In that sense, the deterministic approach to radiation transport differs significantly from the MC approach since the average quantities of interest are calculated from the solution of that equation. Boltzmann resolved the task of writing an equation representing all the particles, with their respective positions and velocities, by assuming that it wa s only necessary to know precisely the state of mo tion within an infinitesimal volume element of the phasespace when you are interested in average m acroscopic properties. Consequently, the macroscopic states of the gas are not represented by point-wise functions; rather, they are represented by density functions. This equation constitutes a balance of the various mechanisms by which particles can be gained and lost from a phase-space element dEdrd 62. It is possible to write a linear form of the Boltzmann equation, someti mes called the forward transport equation, by assuming; i) that one component of the gas (particle) is considered having a very

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33 small density so that collisions of that component with itself can be neglected in comparison with the collisions with the other component (target atoms), and ii) the properties of the target material do not depe nd on the behavior of th e particle type of interest. The resulting time-i ndependent integro-differential form of the linear Boltzmann equation (LBE) for a non-multiplying media is expressed by Eq. 2-11 ),E,rQ(),E,r() E,E,r( Ed d ),E,r(,E) r( ),E,r( s t 04 (2-11) Where ),E,r( dEdrd is the angular flux with energy E within the energy range dE at position r within the volume element rd and in direction within the solid angle d. Similarly, ),E,rS( dEdrd is the angular external sour ce, i.e., the rate at which particles are introduced into the system in a given phase-space element. The double integral term, referred to as the scattering source, represents the sum of the particles scattered into dEd from all the dEd after a scattering collision represented by the double differential cross-section)E,E,r( s For simplicity, the transport equation (Equation 2-11) can be written in operator form. ),E,r(Q),E,r(H (2-12) The operator H in Eq. 2-12 is defined by Eq. 2-13. ) E,E,r( Ed d,E)r( Hs t 04 (2-13) Note that an analytical solution is possible only for very limited simple cases. It is therefore necessary to use numer ical methods for solving this equation. Such discussions are reserved for a later se ction of this chapter.

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34 Deterministic Transport Theory: Backward Transport At this point, it is interesting to discuss the concept of backward (adjoint) transport as it relates to the importance functions. Usi ng the operator notation in troduced at the end of the previous section, we can define the mathematical adjoint of the LBE by using the adjoint property62. H H (2-14) In the adjoint property shown in Eq. 2-14, denotes the adjoint and the Dirac brackets signify integration over all independent variables. For the adjoint property to be valid, vacuum boundary conditions for the angu lar flux and its adjoint (as shown in Eqs 2-15a and 2-15b) are required. 0 0 n ,r, ,E),r( (2-15 a) 0 0 n ,r, ,E),r( (2-15 b) In these equations is the boundary surface and n is an outward unit vector normal to the surface. Using this adjoint property, it can be proven that the adjoint to the LBE can be written in operator form. ),E,r(Q),E,r( H (2-16) Where, the operator H is defined by Eq. 2-17. s t ),E(E dEd H4 00 (2-17) By using a physical interpretation, it is possible to derive a balance equation for particle importance62, which is equivalent to the adjo int transport equa tion (Eq. 2-16 and Eq. 2-17) without the vacuum boundary restriction. Hence, the solution to the adjoint

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35 LBE relates to the physical importance of a particle toward a given objective represented by the adjoint sourceQ Adjoint Source and Objective Lets consider a transport problem repr esented by Eq. 2-12 and Eq. 2-16 and look at an example of the relationship between the source of the adjoint problem and the objective of the calculation. In this case, the objective of the simulation is to calculate the response of a detector in term of counts (related to the reaction rate in the detector). From the forward transport simulation, the response ca n be calculated according to Eq. 2-18 where R represents the detector response and d (cm-1) is the detector cross-section ,E)r(,E),r( Rd (2-18) It is possible to define a commutation relation by multiplying Eq. 2-12 and Eq. 2-16 by the adjoint function and angular flux respectively. Doing so, we obtain a commutation relation. QQ H H (2-19) Using the adjoint property shown in Eq 2-14 (i.e., assuming vacuum boundary conditions), we can rewrite Eq. 2-19: QQ (2-20) Using an adjoint source equal to the detector cross-section (Q =d ), Eq. 2-21 provides an alternative way to evaluate the reaction rate. Q R (2-21) This shows that by using the detector cross-section as the adjoint source, the resulting solution to the adjoint problem gives the importance toward producing a count

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36 in the detector. Note that for this case, an analysis of the units of Eq. 2-16 would show that the resulting adjoint function is unitless (or per count ). Electron, Photon and Positron Interactions This section will mainly be descriptive si nce a complete review of the electron, photon and positron interactions is out of the scope of this work. Note that for the details of the mechanisms the reader is referred to Evans63. An excellent review (even though it is relatively old) of electr on interactions and transport theory is given by Zerby and Keller64. Note that in the context of particle inte ractions, the incident particle can often be referred to as a source particle or a primary particle while particles resulting from the interactions are referred to as secondary particles. Electrons and Positrons The most important interactions of electrons and positrons below 10 MeV are elastic scattering, inelastic scattering from atomic electrons, bremsstrahlung, and annihilation for positrons. Typically, electron collisions ar e characterized either as soft or hard. This classification relates to the magnitude of the energy loss after the collision; interactions with a small en ergy loss are referred to as soft while interactions with large energy loss are referred as hard. However, the energy-loss thre shold that distinguishing a soft from a hard collision is arbitrary. Elastic scattering For a wide energy range (~100 eV to ~1 GeV), elastic collisions (sometimes referred to as Coulomb scattering) can be described as the scattering of an electron/positron by the el ectrostatic field of the atom wh ere the initial and final quantum states of the target atom stay the same. This type of interaction causes most of the angular deflections experienced by the electrons/positro ns as they penetrate matter. Note that

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37 there is a certain energy transfer from the proj ectile to the target, but because of the large target to projectile mass ratio, it is usually neglected. The electron/positron elastic scattering cro ss sections are large and concentrated in the forward directions resulting mostly in small deflections with an occasional largeangle scattering. Electron elastic scattering interactions are usua lly represented by the Mott65 cross-section with a screening correction from Moliere66. The positron elastic scattering cross-section is often approxima ted by the electron cross-section. This approximation is most accurate in low-Z mate rials and for small angles of deflection9. In high-Z material and for larger angles of deflection, the two cross sections can differ up to an order of magnitude. However the differe ntial cross-section for such large angle deflections is at least several orders of magnitude lower than for the smaller angles. Inelastic scattering Passing through matter, electrons and positrons lose small amount of energy due to their interactions with the electric fields of the atomic electrons. However, an electron colliding with another electron can exchange ne arly all its energy in a single collision and produce knock-on electrons (delta-rays). For en ergies below a few MeV, these processes are responsible for most of the energy losses. The fraction of these interactions resulting in hard events is often modeled through the Moller64 cross-section for the electrons and the Bhabba67 cross-section for the positrons. However, as it will be seen later, many of these collisions produce small energy lose s and are often repres ented by a continuous energy loss without angular deflection. This approach uses collisional stopping powers and related ranges. By comparing the va lues of stopping powers for electron and positron9 for various element and energies, it is possible to observe that the largest differences occurs (e.g., ~30% at 10 eV in gold) below 1 keV while above 100 keV the

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38 two quantities seem, from a practical standpo int, identical. Since most coupled electronphoton-positron simulations do not include pa rticles below 1 keV, using the same collisional stopping power valu es for electron and positron seem a valid approximation. This continuous slowing down (CSD) approach needs to be supplemented by an energyloss straggling model to correct for the fact that the CSD approximation forces a one-toone relationship between depth of penetration in the target material and the energy loss, while in reality the energy-loss is a stochast ic variable following a distribution, such as the Landau distribution21. Bremsstrahlung The sudden change in the speed of a char ged particle (in th is case electron or positron) as it passes th rough the field of the atomic nuclei, or the atomic electrons field, produces bremsstrahlung (braking ) photons. At very high energies most of the energy is lost through this proce ss. This process is often repres ented through the us e of radiative stopping powers. According to Ref. 9, even though the radiative stopping power of electrons differs significantly from positron at energies below 1 MeV (almost an order of magnitude at 10 keV), the differences in th e total range of the electron and positron is minimal. This can be explained by the fact that below 1 MeV, the collisional energy-loss dominates. It is therefore a valid approximation to use el ectron radiativ e stopping range for positron. Positron annihilation A typical way to model the positron annihi lation is to assume that it occurs only when the energy of the positron falls belo w the energy cutoff of the simulation. Upon annihilation, two photons are produced of equal energy are produced. This implicitly

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39 assumes that a positron annihilates only wh en absorbed and th erefore neglecting the small fraction of annihilation that occurs in flight. Photons For the photon energies considered in this work, the main interaction mechanisms are the photoelectric effect, Compton scat tering (incoherent s cattering) and pair production. Note that the fl uorescent photons and coherent scattering can also be important mechanisms in some cases. Photoelectric effect (and fluorescence) In the photoelectric effect, the photon inte racts with the atom as a whole, gets absorbed, and a photoelectron is emitted (usua lly from the K shell). As the vacancy left by the photoelectron is filled by an electron fr om an outer shell, either a fluorescence x-ray, or Auger electron may be emitted. Coherent scattering The coherent scattering (sometime referred to as Rayleigh scattering) results from an interaction of the incident photon with the electrons of an atom collectively. Since the recoil momentum is taken up by the atom as a whole, the energy loss (and consequently the change of direction) is real ly small and usually neglected. Incoherent scattering Incoherent scattering refers to a scattering event where the photon interacts with a single atomic electron rather then with all the electrons of an atom (coherent). The incident photon loses energy by tr ansferring it to this single el ectron (referred to as recoil electron) which gets ejected from the atom. This phenomenon is represented by the double differential Klein-Nishin a cross-section. However, th is cross-section was derived assuming scattering off a free electron, which is invalid when the kinetic energy of the

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40 recoil electron is comparable to its bindi ng energy. Therefore, a correction for the electron binding energy is usually applied usi ng a scattering form factor. Qualitatively, its effect is to decrease the Klein-Nishina cr oss-section (per elec tron) in the forward direction, for low E and for high Z, independently. Pair production In pair production, the incident photon is completely absorbed and an electronpositron pair is created. This intera ction has a threshold energy of 2mec2 (1.022 MeV) when it is a result of an interaction with the nucleus electric filed or 4mec2 when a result of an interaction with an el ectron electric field (also calle d triplet production). Note that the triplet production process is relatively not important, and therefore generally ignored. Numerical Considerations Some MC simulations cannot reach a certain statistical precision with a reasonable amount of time and therefore, it is necessa ry to use VR technique. A MC simulation using VR techniques is us ually referred to as non-analog since it uses unnatural probabilities or sampling distributions as opposed to the analog MC which uses the natural correct probabilities and distributions. Fo r completeness, the discussion on the importance-based VR presente d in the literature review is extended to include non importance-based VR techniques that are often used in coupled electron-photon-positron calculations to provide more theory about some of the other technique s used in this work. Ref. 68 provides a review of the variance reduction methods implemented in MCNP. Bielajew and Rogers69 as well as Kawrakow and Fippel70 present discussions of different techniques including electron-specific and photon-specific techniques. Note that the McGraths report71 provides a comprehensive list of variance reduction techniques.

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41 In the second part of this sect ion, a general overview of the SN method is presented since it is the numerical method used for solving the integro-differential form of the linear Boltzmann equation. Finally specific numerical techniques used to resolve issues arising when performing SN calculation for electron/positron will be discussed. Note that a comprehensive review of the deterministic methods for neutral particles is given by Sanchez and McCormick72. Monte Carlo Method: Variance Reduction In general, variance reduction technique s can be divided into four classes68: truncation methods, population control me thods, modified sampling methods, and partially-deterministic methods. The following list provides a short in troduction to most of the VR techniques used in coupled electron-photon-positron transport. 1. Truncation methods (note that truncati on methods usually introduce a degree of approximation and may reduce the accuracy of the calculation) a. Energy cutoff: an increase of the overall efficiency of the simulation can be achieved by increasing the energy cutoff because particles will then be tracked over a smaller energy range and less secondary particles will be produced. However, it is difficult to derive lim its on acceptable energy cutoff based on theoretical consideration, so it must be used carefully because improper cutoffs can result in the termination of important particles before they reach the region of interest. b. Discard within a zone (electron trapping): in this technique, improvement in the efficiency is obtained because, if the electron ranges are smaller than the closest boundary of the zone, they ar e not transported and their energy is deposited locally. Note that this approach neglects the creation and transport of bremsstahlung photons (or other secondary particles) which may have otherwise been created. c. Range rejection: this approach is similar to th e discard within a zone method with the exception that th e electron is discarded if it cannot reach some region of interest instead of the clos est boundary of the current zone. d. Sectioned problem: it is often possible to sepa rate a problem in various sections where different parts are modele d with different levels of accuracy. For example, one could model separately a complex geometry and store the phase-

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42 space parameters (energy, direction and pos ition) at the surfa ce of the geometry. It is then possible to use this informa tion repeatedly at no extra computational cost. Note that this is usually done at the price of some accuracy. 2. Population control methods a. Geometry splitting: if the particles are in an im portant region of the problem, it is often advantageous to increase thei r number (and decrease their weight accordingly) by splitting them. Commonly, this splitting is performed at boundary crossings. This increases the am ount of time per history but reduces the variance by having more par ticles scoring in the tally. b. Russian roulette: if the particles are in an unimportant region of the problem, or they are unimportant themselves (because of a small weight resulting from the use of another VR tec hnique), the particles can be terminated. Rather than simple termination, Russian roulette must be played to avoid introducing a bias in the estimators by not conserving the total number of particles. In this technique, a particle with a weight below a given thre shold has a set probability of being terminated and has its weig ht increased by the inverse of this probability if it survives. c. Weight-window: this technique provides a utility to administer splitting and Russian roulette within the same framew ork. This technique can be implement with various useful features; i) biasin g surface crossings and/or collisions, ii) controlling the severity of splitting or Russian roulette, and iii) turning off biasing in selected space or energy regions. The typical weight-window technique allows the use of spaceand energy-dependent weight bounds to control particle weights and population. 3. Modified sampling methods a. Implicit capture: This is probably the most unive rsally used VR technique. In this technique, the weight of the partic le is reduced by a f actor corresponding to its survival probability. Note that one must then provide a criterion for history termination based on weight. b. Source biasing: In this technique, the source di stribution is modified so more source particles are started in phase-space locations contributing more to the estimator. As always, to preserve unbiased estimators, the weight of the particles must be adjusted by the ratio of unbiased and biased source probabilities. c. Secondary particle enhancement: To enhance the number of certain secondary particles considered important for a given problem, it is possible to generate multiple secondary particles once a creation event has taken place. The secondary particles energy and direction are sampled to produce many

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43 secondary particles emanating from a si ngle interaction point. Note again that the weight must be adjusted to preserve unbiased estimators. d. Electron history repetition: This technique increases the efficiency of electron dose calculation by re-using a pre-calculat ed history in water. The starting positions and directions of the recycled electrons are different when they are applied to the patient geometry. 4. Partially-deterministic methods a. Condensed-history: This procedure uses analytical formulations to represent the global effect of multiple collisions as a single virtual collision therefore reducing the amount of time required to sample the excessively large number of single event collisions. It is useful to discuss in more details the CH method si nce it constitutes one of the major differences between neutral partic le and coupled elect ron-photon-positron MC simulations. As mentioned before, this appro ach uses an analytical theory to sum the effect of many small momentum transfers from elastic and inelastic collisions into a single pseudo-collision event often refer to as a CH step. The various flavors of implementations of this technique, developed by Berger16, can fall into the following two categories. From Refs. 73 and 74, these two classes can be described as: Class I: in this scheme, the particles move on a predetermined energy loss grid. This approach provides a more accurate tr eatment of the multiple elastic scattering but have disadvantages related to, i) the lack of correlation between energy loss and secondary particle production, and ii) in terpolation difficulties when CH step does not conform to the pre-determined energy grid because of interfaces and/or energyloss straggling. This scheme is implemented into ITS and MCNP5. Class II: in this scheme, the hard (or catastrophic) even ts, e.g., bremsstrahlung photon and Moller knock-on electrons, create d above a certain energy threshold are treated discretely, while sub-threshold (referred to as soft events) processes are accounted for by a continuous slowing down approximation. This class of scheme is implemented in EGS, DPM and PENELOPPE. Deterministic Discrete Ordinates (SN) Method One of the most widely used numerical methods to solve the integro-differential form of the transport equation is the SN method. In the nuclear community, the current

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44 method evolved from the early work of Carlson75. The method solves the transport equation along a set of discrete ordinates (directions) typically selected such that physical symmetries and moments of direction cosines ar e preserved. This se t of directions and associated weights are refe rred to as a quadrature set76. It is possible to use biased quadrature sets which are useful for specifi c applications requir ing highly directional information. However, as it will be discuss later, the selection of quadrature sets to perform electron can be limited. The energy variable is generally discretized into a finite number of energy groups and cross sections are averaged over these intervals. A large variety of approaches are used to discre tize the spatial variab le (including finite difference and finite element methods) resu lting in different representations of the streaming term. In order to numerically represent the behavior of particles in a spatial mesh, auxiliary equations, referred to as differencing schemes are needed. A good review of the main differencing schemes is provided by Sjoden77, 78. From a theoretical point of view, the lin ear Boltzmann equation is valid for charged particles transport79 but the usual numerical approaches fail for various reasons: i) the elastic scattering cro ss-section is so forwarded-peaked that a Legendre polynomial (or spherical harmonics) expansion would lead to an excessively large number of moments54 (~200), ii) the required quadrature order (>200) to accurately repres ent the large number of scattering kernel moments, and iii) the number of energy groups required to properly represent the small energy changes resulting from soft inelastic collision51 (>160). Therefore, various other nume rical and mathematical treatments have been studied.

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45 Fokker-Planck equation One approach consists of replacing the in tegral scattering operator of the LBE by a differential operator. This re sults in the Fokker-Planck (FP) equation, which can be written for a homogeneous and isotropic medium as Eq.2-22. ) ) )() )( ) 1 1 )1()( ) ) 2 2 2 2 2 2Q(E, E (E,ER E (E,ES (E, ET (E,(E) (E,a (2-22) where ),,()1( )(1 1 0 EE dEdETs (2-23) ),,()( 2)(1 1 0 EEEEdEd ESs (2-24) ),,()( 2)(2 1 1 0 EEEEdEd ERs (2-25) The left-hand terms of Eq. 2-22 represen t the streaming of particles and their absorption. The first term on the right-hand side represents the angular diffusion where T(E) can be considered as some sort of diffusion coefficient. This term causes the particles to redistribute in di rection without change in energy. The second and third terms (S(E) and R(E)) represent the energy-loss as a convective and diffusive process, respectively. Note that these last two terms cause the particles to redistribute in energy without directional change. Pomranning80 showed that this equa tion is an asymptotic limit of the Boltzmann equation that is valid when the deviation of the scattering angle from unity, the fractional energy change after a single scattering, and the scattering meanfree-path (mfp) are all vanishingly small. This asymptotic analysis also shows that one

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46 could set R(E) to zero and still get the leading order behavior in energy transfer. Therefore, the FP equation, by definition, doe s not include any large energy transfers and is inappropriate for a large number of problems involving lower energy electrons. Boltzmann-CSD equation To take into account the large deflection events, a scattering kernel combining the Boltzmann and FP formalisms was introduced and referred to as the Boltzmann-FP equation. This equation combines the advantag es of the usual tran sport equation (large energy transfer) with the FP formalism, which is very accurate for highly anisotropic collisions. However, a simplified form of the Boltzmann-FP equation, referred to as the Boltzmann-CSD (Continuous Slowing Down), is generally used. This form can be obtained by neglecting the diffusive terms in angle and energy and is given by Eq. 2-26. ) ) ,),( ) ( ~ ) ,) 0E,rQ( E E,r (ErS ,E,r) E,E,r( Edd E,r (E)r( E,r (s t (2-26) Note that integral limits of the scatteri ng kernel and the stopp ing power (S(E)) must reflect the energy boundary between the hard and soft collisions, and the tilde indicates that soft collisions have been exclud ed from the integral scattering kernel. Goudsmith-Saunderson equation The Goudsmit-Saunderson equation18, 19, as shown in Eq. 2-27, solves for the electron angular flux in an in finite homogeneous media. E (E,ES E,) (E,d (E,(E) el t ))( )( ) 0 (2-27) Eq. 2-27 takes into account the following physical phenomena: i) the elastic scattering for directional change without energy-loss, and ii) the soft inelastic scattering

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47 part for energy loss without significant dir ectional change. Note that this equation neglects the hard inelastic scattering. The major advant age of this formulation is that it can be solved exactly for a source free media as shown in Eq. 2-28. )()()( )( 2 12 ))(0 0 ]/)[( 10 0 ,E ESP ePd n (E,ESn S Ed n nE E n elel (2-28) CEPXS methodology To perform deterministic electron tran sport calculations, CEPXS generates effective macroscopic multigroup Legendre scattering cr oss sections which, when used in a standard SN code, effectively solves the Boltz mann-CSD equation. To achieve this, CEPXS uses the following treatments: A continuous slowing-down (CSD) approximati on is used for soft electron inelastic scattering interactions and radiative events resulting in small-energy changes, i.e., restricted stopping powers are used. The extended transport correction46 is applied to the forward-peaked elastic scattering cross-section. Interactions resulting in hard events are treated through the use of differential cross sections. First or second-order energy differencing scheme is applied to the restricted CSD operator. These pseudo cross-sections are unphysical si nce they do not posses associated microscopic cross sections and require the use of specific quadrat ure sets, e.g. GaussLegendre, not available in multi-dimension. The efficiency and accuracy of this technique highly depends on the proper sel ection of the discretization parameters. CEPXS-GS methodology The CEPXSGS (Goudsmit-Saunderson) appr oach combines the elastic scattering and CSD in a single downscatter operator that is less anisotropic than the scattering cross section. From the Goudsmit-Saunderson equati on (Eq. 2-28), it is possible to define

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48 multigroup Legendre scattering cross sections by doing the following; i) specify E0 as the upper bound of energy group g, ii ) relate the angular flux at the energy group boundary to the group average through an energy-differe ncing scheme, iii) divide the resulting discretized equation by the energy group width, iv) truncate the Lege ndre expansion, and v) compare it with the standard multigroupLegendre expansion to obtain the various terms. It is then possible to use these cross sections in a standard SN code to perform coupled electron-photon-positron in multi-dime nsional geometries since they do not depend, as with the CEPXS cross-sections, on the quadrature set. These cross sections are an improvement on the CEPXS cross sections since they result in a faster convergence, eliminates some well known numerical oscillat ions, require smaller expansion orders, and can be guaranteed positive under certain conditions.

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49 CHAPTER 3 ADEIS METHODOLOGY CONCEPTS AND FORMULATIONS The ADEIS (Angular adjoint-Driven Electron-photon-positron Im portance Sampling) methodology is based on the same principles as the CADIS (Consistent Adjoint Driven Importance Sampling) methodology32. In both these methodologies, importance sampling is used to performed transport and collision biasing through the weight-window technique using deterministic importance functions to determine variance reduction parameters. However, in order to address issues relate d to coupled electronphoton-positron transport, many specific features had to be developed and implemented in ADEIS. Before discussing these features, it is useful to present in more detail the concept of importance sampling, and the diffe rent mathematical formulations used in ADEIS. Importance Sampling The general idea of the importance sampling technique is to take into account that certain values of a random va riable contribute more to a given quantity being estimated and consequently, sampling them more fre quently will yield an estimator with less variation. Therefore, the basic approach is to select a biased sampling distribution (PDF) which encourages the sampling of these important values, while weighting these contributions in order to preserve the correct estimator. Using a simplified notation, this concept can be represented mathematically by introducing a biased PDF in the formulation of the unbiased expected value shown in Eq. 3-1.

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50 b adxxfxgg )()( (3-1) It is also necessary to introduce a biased contribution function to preserve the expected value as shown in Eq. 3-2. b adxxfxgxwg )( ~ )()(, (3-2) In this equation, g is the estimated quantity, )(xg is a function of the random variable x defined over the range [a,b], )(xf represents the sampling PDF, )( ~ xf represents the biased sampling PDF, and )( ~ )()( xfxfxw represents the weight of each contribution. As shown in Appendix A, choosing an optimum biased sampling PDF with the same shape as ) ()( xgxf will yield a zero-variance solution. However, this implies an a priori knowledge of the solution defying the purpose of performing the simulation. However, this suggests that an approximation to that optimum biased sampling PDF can be used to reduce the variance with a minimal increase in computation time per history. It also suggests that the closer th at approximated integrand is to the real integrand; the more the variance should be reduced. To apply this methodology to a particle transport problem it is useful to use a more detailed fo rm of the equation representing the transport process (Eq. 2-1) as shown in Eq. 3-3. PP)dP()P(P)dPP()P((P) TQ C (3-3) In this equation, (P) represents the integral quantity being estimated, )PP( C represents the collision kernel, P)P( T represents the transport kernel, and )P( Q represents the external source of primary particles. The collision kernel describes the particles emerging from a phase-space element after either a scatteri ng or the creation of

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51 a secondary particle. The transport kernel simply reflects the change in phase-space location due to streaming and collision. As s hown in Appendix A, it is possible to use the concept of importance sampling to write the equation representing the biased transport process as shown in Eq. 3-4. Note that Eq. 3-4 and its derivation (as presented in Appendix A) are slightly different the previous work23, 33, and show more clearly the separation between collision and transport biasing. PP)dP()P(PP)dP(P)dPP()P((P) T Q T C (3-4) In this equation, (P) represents the biased estimator, )P( Q represents the biased source, P)P( T represent the biased transport kernel, and )PP( C represents the biased collision kernel. Eq. 3-4 shows that performing importance sampling on the integral transport equation is equivalent to performing transport, source and collision biasing in a consistent manner. The following three sections present ADEIS approach to these three type of biasing. ADEIS Angular Transport Biasing From Appendix A, it can be seen that the biased transport kernel is described by Eq. 3-5. )P( (P) P)P(P)P( T T (3-5) From a physical point of view, this biased transport kernel can be seen as an adjustment of the number of particles emer ging from a phase-space element according to the ratio of importance of the original and final phase-space elements. Since no explicit PDF of P)P( T is available to be modified, it is possible to achieve a modification of the transfers by creating extra particles when the original particle transfers from a less to

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52 a more important region of phase-space or by destroying a particle when it transfers from a more to a less important region of phase-space. This creation / destruction can be performed using the standard splitting/roul etting VR techniques and following the rules given in Eqs 3-5 and 3-6. createdare particles )P( (P) 1 (splitting) (3-5) destroyed are particles )P( (P) 1 (Russian roulette) (3-6) As discussed in the previous section a nd in Appendix A, there is an inverse relationship between the biased sampling distribution (in ADEIS case the importance functions) and the resulting weight of the contribution to a given estimator. Consequently, by associating the importance of a given phase-space element to a corresponding weight, it is possible to force the statistical weight of a particle to corresponds to the importance of the phasespace region by following the new set of rules presented in Eqs 3-7 and 3-8. createdare particles )Pw( w(P) 1 (splitting) (3-7) destroyed are particles )Pw( w(P) 1 (Russian roulette). (3-8) However, the cost of performing splitti ng and rouletting every time a particle changes phase-space location could offset the be nefit gained by this technique. Therefore, it is generally useful to define a range of weights that are accept able in a given phasespace element. Consequently, the weight-win dow technique allows for particles with weights within a given window (range) to be left untouche d, while others are splitted

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53 and/or rouletted to be forced back into the window The standard approach is to define the lower-weight bounds of this window and set the upper bound as multiple of the lowerweights. Note that more statistically reliable results are possible because a better control over the weights scored by indivi dual particles is achieved. It is possible to write the formulation for the lower-weight bounds by using the expression for the biased angular flux and the conservation law shown in Eq. 3-9. (P)w(P)w (3-9) The resulting expression for the lower-w eight bounds is given in Eq. 3-10. u l(P) C R (P)w (3-10) Where (P)wl is the lower-weight bound, R is the approximated estimator, (P) is the importance function value, and Cu is the constant multiplier linking the lower and upper bound of the weight-window. Even though Eq. 3-10 states that the lower-weight depends on all phase-space variables ) ,,( Er, it typically depends only on space and energy. Note that this formulation is slightly diffe rent then the formulation used in previous work32, 33. However, for most problems considered in this work, the flux distributions can be highly angular-dependent because: i) the sour ce characteristics (e.g. high-energy electron beam); ii) the geometry of the problem (e.g. duct-like geometry or large region without source); and iii) the scattering properties of high-energy electrons and photons. Therefore, to achieve a higher efficiency, it is expected that the weight-window bounds should be also angular-dependent. However, in the context of a deterministic importancebased VR technique, it is important to be able to introduce this angular dependency

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54 without using a complete set of angular flux es which requires an unreasonable amount of memory. To address this issue, ADEIS uses the concept of field-of-view (FOV) where the angular importance is integrated within a field-of-view subtending the region of interest. Figure 3-1 illustrates simplified space-dependent FOVs in 1-D and 2-D geometries. Figure 3-1. Field-of-View (FOV) concept. A) in 1-D geometry B) in 2-D geometry It is therefore necessary to calculate tw o sets of lower-weight bounds for directions inside and outside the FOV as shown in Eqs 3-11 and 3-12. Note that the nFOV represents the field-of-view associ ated with a given particle type n since it may be useful to bias differently various particle species. Note that in principle, the FOV could be dependent on energy; however, it is not considered for this version of ADEIS. Corresponding lower-weights for positive and negative directions on the FOV are defined by: ,E) Cr( R ,E)r(wu l, (3-11) n n FOV: andFOV: )E,,r(d,E)r( (3-12) A ) B )

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55 ADEIS Source Biasing A formulation for a better sampling of th e source probability distribution can be developed by using the relative contributi on of the sampled source phase-space element to the estimated quantity. From Appe ndix A, it can be seen that this biased source is described by Eq. 3-12., R )P( )P()P( QQ (3-13) More specifically, in ADEIS, the biased source would be calculated using the formulation presented in Eq. 3-14. R ,E)rQ(,E)r( (r,E)Q (3-14) Where ,E)r( the FOV-integrated angular impor tance function and R is is the approximated estimator. Again, to preserve th e expected number of particles, the weight of the biased source particles would be ad justed according to Eq.3-15. ),( ),( ErQ ErQw w (3-15) However, all examples studied in this work are mono-energetic pencil beam and therefore, no source biasing was implemented at this time and is only shown for completeness. In ADEIS, the approximated value R is calculated according to either one of the two formulations shown by Eqs 3-16 and 3-17. ,E)r(,E)r(QdVdE,E)r(,E)r(QdVdER (3-16) ), ~ ), ), ~ ), 0 0 Er(Er(nddEd Er(Er(nddEdR FOV n FOV n (3-17)

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56 Where ,E)r(Q and ,E)r( ~ are the projections of th e volumetric and surface source over the discretized phase-space of the deterministic calculation, and n is the outward normal to the surface Eq. 3-16 is used to calculate the approximated estimator value for cases with a volumetric source and vac uum boundaries while Eq. 3-17 is used for cases with an incoming source at a boundary. As shown in Appendix A, using Eq. 3-11 to calculate the lo wer-weight bound also ensures that the source particles are generated at the upper-weight bounds of the weightwindow if a mono-directional, mono-energetic point source is used. This is a useful characteristic since, for such problems, it is possible to maintain the consistency between the weight-window and the s ource without having to perf orm useless source biasing operations that would not increase th e efficiency of the simulation. ADEIS Collision Biasing Collision biasing can be achieved by pl aying the weight-window game at every collision, on the primary part icles and on most of the second ary particles before they are stored into the bank. This is the standard a pproach used in MCNP5. Note that in electron transport, the term collision can be interprete d as the end of each major energy step in the CH algorithm. Criteria for Applying Weight-Window To minimize the amount of computational overhead associated with comparing the particle weight to the transparent mesh associated with the weight-window, it is important to optimize the frequency of these checks. Each check against the transparent weight-window mesh has a computational cost associated with the binary search algorithm. This increased cost has to be as small as possible in order to maximize the

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57 increase in FOM. Earlier studies2, 33 found that checking the wei ght every mean-free-path (mfp) was the near-optimum criterion for neutra l particles. For electr ons, it is necessary to check the weight against the weight-window at most after each major energy step of the electron condensed-history (i.e., at every pseudo-collision) in order not to introduce a bias in the electron spectrum. Selection of the Adjoint Source As discussed in Chapter 2, the importance function is related to the objective for which the user wishes to bias the simulation. In a similar approach developed for neutral particle transport81, the objectives were typically reacti on rates in a small detector (e.g. multigroup detector response crosssection from the BUGLE-96 library82). Consequently, the objectives of these calculations were de fined by the library containing the detector response cross sections. For coupled electronphoton-positron, MC calculations are often performed to determine energy deposition (dose) profile and accurate (i.e. adjusted to fit experimental data) coefficients such as flux-to-dose conversion factors may not be readily available for all materials. To circumve nt this difficulty, ADEIS allows the use of two automatically determined adjoint sour ces; i) a local energy deposition response function to approximate dose in the ROI, and ii) a uniform spectrum to maximize the total flux in the ROI. Note that the ADEI S methodology uses a spatially uniform adjoint source over the whole ROI. Local Energy Deposition Response Function A local energy deposition response function can be use as an adjoint source for problems where the objective is related to th e energy deposited (MeV) in the ROI. From the forward transport simulation, the energy depo sition can be defined by Eq. 3-22.

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58 ,E)r(E,E),r( Rx (3-22) Where ,E)r(x is some sort of energy deposition coefficient. By comparing Eq. 2-14 and Eq. 3-22, it is possible to deduce that if ,E)r(Ex (MeV cm-1) is used as an adjoint source, the im portance function units would be MeV per count Consequently, the solution of the adjoint problem represents the importance of a particle toward energy deposition. To evalua te these coefficients for photons, different assumptions can me made which then results in different coefficients6 as listed below. Linear absorption: assumes that when a photoel ectric or pair production event occurs, all the energy is deposited locally, i. e., no energy is re-emitted in the form of fluorescence x-rays, bremsstrahlung annihilation photon or other secondary particles, Linear pseudo-energy-transfer: similar to the linea r absorption with the exception that the energy re-emitted in the form of annihilation photon, Linear energy-transfer: similar to the linear pseudo-energy-transfer with the exception that energy is also re-emitted in the form of fluorescence x-rays, Linear energy absorption: similar to the linear energy-transfer with the exception that energy is re-emitted from bremsstrahlung through radiation. It may be argued that by a phenomenon of error compensation, these various approximations result in almost the same dose6 when multiplied with the appropriate fluence. However, in the context of a VR technique, an approximate objective can be used since only an approximate importance f unction is needed. Therefore, at this point, ADEIS uses the absorption cross-section multiplied by the energy of the group as an adjoint source as s hown in Eq. 3-23. (E)E(E)E(E)EEQs t a )( (3-23) In this equation, t the total collision cross-section, and s is the total scattering crosssection. It must be noted th at using the absorption crosssection as an energy deposition

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59 coefficient contains the additional assumption that no energy is deposited when a Compton scattering event occurs. This adjoin t source spectrum was chosen because these cross sections are readily available from CEPXS. For electrons, the situation is slightly mo re problematic since such coefficients are not readily available because of the use of the continuous slowing-down approximation. However, it is possible to conclude from the previous discussion on photons that any quantity that represents the deposited energy per unit pa th-length (MeV cm-1) would constitute a sufficient approximation in the co ntext of a VR technique. Therefore, it is possible to define such a quantity as the energy imparted in a volume divided by the mean chord length of the volume. By energy im parted, it is usually meant the sum of the energies of all charged and neutral ionizing part icles entering the volume minus the sum of the energies of all charged and neutral ionizing particles leaving the volume. The adjoint source in ADEIS approximates this qu antity based on the energy deposited by an electron in the ROI divided by the average chord length. For each electron energy group, a surface source, with an angular distribution proportional to the cosine of the angle and with an energy corresponding to the midpoint of the group, is assumed, so an average chor d length can be calculated (see Appendix A). Using a CSD approximation, the ener gy deposited can be approximated by: 1. Subtract the average chord-length ( r ) from the range of the electron in the energy group (gR ) being considered, i.e., rRRgg 2. If 0 gR an amount of energy corresponding to the middle point of the energy group g is assumed deposited (gdEE ). 3. If0 gR the energy group g corresponding to that resi dual range if found and an interpolation if performed to find the energy (gE ) corresponding to that range. The

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60 difference between the midpoint of the or iginal energy group and that remaining energy is assumed deposited (ggdEEE ). 4. Finally, the adjoint source is defined as r E EQd )( (MeV cm-1). Uniform Spectrum In theory, an adjoint source uniform spect rum could be used for a problem where the objective is related to the total flux in the ROI. This objective may not be optimal for problems concerned with energy deposition, but could be sufficient to produce significant speedups. From the forward transport simulation, the average total flux over the ROI can be defined as )( 1 1 1 rdV V ,E),r( ddEdV V ,E),r( V (3-23) By comparing Eq. 2-14 and Eq. 3-23, it is possi ble to deduce that if a uniform spectrum, equal to the inverse of th e volume of the ROI, is used as an adjoint source (1 V,E),r(Qcm-3), the importance function units would be per count per cm2. Consequently, the solution of the adjoint prob lem represents the importance of a particle toward the average total flux in the ROI. Comparison with Methodologies in Literature Review Most techniques described in Chapter 1 differs significantly from ADEIS either because they; i) focus exclusively on neutral pa rticle transport, ii) use diffusion or linear anisotropic scattering approximations, iii) use importance f unctions generated from the convolution of various functions for each phase -space variable. It is therefore difficult to compare the ADEIS methodology with such appr oaches. However, it is possible to do a more detailed point-by-point comparison wi th four approaches that share common features with ADEIS: i) multigroup adjoint tr ansport in MCNP5 (referred to as MGOPT);

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61 ii) weight-window generator (ref erred to as WWG); iii) ADVANTAG/A3MCNP/CADIS (referred to as CADIS); and, iv) AVATAR. Note that ADVANTAG, A3MCNP and CADIS are grouped together since A3MCNP and ADVANTAG are both rather similar implementations of the CADIS methodology. These four methodologies were chosen b ecause they all take advantage of the weight-window technique implemented in va rious versions of MCNP (either in an original or modified form) to perform tran sport and collision bias ing using a transportbased importance functions. The first two approa ches were also chosen because they can be used to perform coupled electronphoton biasing even though the importance functions are determined using MC simulations rather then deterministic. Alternatively, the last two approaches were chosen because the importance functi ons are deterministicbased even though they were developed for neut ral particle biasing. Table 3-1 presents a summarized point-by-point comparison of th e four methodologies and is followed by a slightly more in-depth di scussion of certain points. Table 3-1. Comparison of other vari ance reduction methodology with ADEIS ADEIS MGOPT WWG CADIS AVATAR Coupled electron/photon biasing Deterministic importance Explicit positron biasing Angular biasing Source biasing 3-D importance function Automation Mesh-based weight-window

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62 Deterministic Importance Function Despite some difficulties related to disc retization, cross-s ections, input files generation and transport code management, th e use of deterministic importance functions constitute a significant advantage since in formation for the whole phase-space of the problem can be obtained relatively rapidly. The MC approach to generate importance functions is limited by the fact that obtaining good statistics for certain region of phasespace can be extremely difficult, hence the need for VR methodology for the forward problem. This difficulty is often circumve nted to some extent by generating the importance function recursively, i.e., usi ng the incomplete phase-space information generated in the prior iteration to help obtaini ng better statistics fo r the current iteration and so on until the user is satisfied with the quality of his importance functions. This requires a lot of engineering time and expertise to be used efficiently. Explicit Positron Biasing Currently, ADEIS in the only methodology th at generates a distinctive set of importance functions for the positron and bias es them independently of the electrons. Further discussion on this topic is provided in Chapters 4 and 5. Angular Biasing The major issue with angular biasing is th e amount of information that is possibly required. ADEIS and AVATAR use completely different schemes to circumvent this issue. While AVATAR uses an approximation to the angular importance function, ADEIS uses the concept of field-of-view to introduce an angular dependency for each of weight-window spatial mesh and for each partic le species. Further discussion of this topic is provided in Chapters 4 and 6.

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63 3-D Importance Functions Since ADEIS is based on a modified ve rsion of the mesh-based weight-window implemented in MCNP5, it is theoretically possible to use 3-D importance functions. However, because of the computational cost and difficulties (large number of groups, possible upscattering, high orders for quadratur e and scattering expans ion, optically thick spatial meshes) of generating the 3-D c oupled electron-photon-positron importance functions, it was chosen that only 1-D and 2-D (RZ) importa nce functions would be used. This can be justified by the following argum ents, i) large computational cost of a generating a 3-D importance functi on may offset the gain in variance, ii) a large class of problem of interest in coupl ed electron-photon-positron can be adequately approximated by 1-D and 2-D models, and iii) the line-of-sight approach introduces an additional degree of freedom to better approximate a 3-D geometry. Obviously, highly threedimensional problem by nature might not be properly approximated by such treatments and may require 3-D importance functions. Automation Even though a small degree of automation is incorporated into WWG and MGOPT; the user-defined spatial mesh structure and energy group structure, the necessary renaming of files and manual iterative process to generate statistically reliable importance functions still requires too much engineering time and expertise to really qualify as automated. Note that the AVATAR package wa s not marked as automated either since from the available papers, it is difficult to judge the extent of automation implemented in the code. The A3MCNP implementation of the CADIS methodology was automated to a large extent since the deterministic model wa s automatically generated, the energy group

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64 structure determined from the cross-secti on library and the data manipulation handle through scripts. The degree of automation was ex tended in ADEIS in order to ensure that all aspects of the VR methodol ogy are transparent to the us er; only a simple input card and command-line option are requi red to use the VR methodology. Mesh-Based Weight-Window The use of the MGOPT option in MCNP5 seems to generate importance function for the cell-based weight-window. This constitutes a significant disadvantage, since to generate and use 3-D importance functions, it is necessary to subdivide the geometry in many sub-cells. In addition to the additional engineering ti me required to perform this, the presence of additional su rfaces can considerably slow the particle tracking process and introduce a systematic error due to the us e of a class-I CH algorithm in MCNP (see Chapter 5 for further discussion on this topic).

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65 CHAPTER 4 ADEIS METHODOLOGY IMPLEMENTATION Im plementing a deterministic importance-base d VR technique such as ADEIS requires various processing tasks such as generating th e deterministic model and the lower-weight bounds for the weight-window. All implementations choices described in this chapter are made with automation in mind since a large degree of automa tion is required for th is technique to be efficient and practical. Figure 4-1 shows the flow chart of the automated ADEIS. Figure 4-1. Automated ADEIS flow chart The following sections are addressing topics rela ted the different parts of the automated ADEIS flow chart. More details about cert ain aspects are given in Appendix B Monte Carlo Code: MCNP5 Since a large number of MC codes are availa ble to the nuclear engineering community, it was necessary to select a single Monte Carlo c ode to implement ADEIS. For this work, MCNP5 was selected for the following reasons: It is well known and benchmarked. The availability of a weight-window algor ithm with a transparent mesh capability. Start MCNP5 New command line option New card: adeis Link file U D R Generate input files Generate xs file Generate importance Generate wwinp call mcrun Calls ADEIS as a shared library End MCNP5 wwinp adeisinp Calls CEPXS/CEPXS-GS as a shared library Calls ONELD/PARTISN as a shared library

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66 Uses a standard interface file for the weight-window. The experience with previous version of the code using similar techniques30, 32. The availability and clarity of the source code and documentation. Deterministic Codes: ONELD, PARTISN and PENTRAN A few deterministic transport code systems are available to the community such as DANTSYS31, DOORS34, PARTISN60, and PENTRAN83, 84. ONELD is a special version of the 1-D SN code ONEDANT code (part of the DANTSYS package) that includes a spatial linear-discont inuous differencing sche me that lessens the constraints on the numerical meshes. This selection was motivated by the fact that CEPXS/ONELD51, a package developed at Sandia National Laboratory, already made use of this transport code to perform coupled electron-pho ton-positron transport simulation. Moreover, CEPXS/ONELD has a few advantages including: It is well known and benchmarked. It has some degrees of automation which faci litated testing and verification of the earlier non-automated versions of ADEIS. However, it is not possible to generate multi-dimensional importance functions using ONELD; therefore the PARTISN and PENTRAN codes were considered. PARTISN was selected for the following reasons: It is an evolution of the DANTSYS system so input file and cross-section formats remain the same as in ONELD. Linear discontinuous differencing scheme is also available. It contains 1-D, 2-D and 3-D solvers so it may be possible to perform all the required transport simulations within one framework. PENTRAN was selected for the following reasons: Familiarity and experience with the code to perform 3-D transport calculations.

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67 The availability of different quadrature sets and an adaptive differencing strategy including a family of exponential differencing schemes85, 86 which might be useful for electron transport. The availability of preand post-processing tools. The capability of performing full domain deco mposition (space, a ngle and energy) and memory partitioning in parallel environments. Note that the expansion to mu ltidimensional calculations requi res the use of the CEPXS-GS version of cross-section generator. Automation: UDR To reach a high degree of automation, a Universal DRiver (UDR) was developed to manage the different processing tasks required by the implementa tion of ADEIS within a single framework. UDR is a library that can be linke d (or shared) with any pre-existing computer program to manage an independent sequence of calculations. In addition to the automation, UDR allows for more input flexibility through a free-fo rmat input file, better error management and a more consistent structure than a simpler script-based approach. Moreover, UDR has utilities that allows for general data exchange between the va rious components of the sequence and the parent code. In the context of this wo rk, this implies that ADEIS is a sequence of operations managed by UDR and called by MCNP5. Additional deta ils about UDR are given in Appendix B. Modifications to MCNP5 A standard MCNP5 simulation involves processing the input, the cross sections, and performing the transport simulation. However, an ADEIS simulation requires a few other tasks before performing the actual transport simulation. To address this issue, a new command line option was implemented into MCNP5. By usi ng this option, MCNP5 performs the following tasks: i) process input and cross sections; ii) generate the deterministic model; iii) extract material information and other necessary parameters; iv) run the independent ADEIS sequence;

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68 v) process the modified weight-windo w information; and, vi) perform the biased transport simulation. In addition to this new command line option, an adeis input card has also been implemented in MCNP5. Additional details a bout the command line option, the new MCNP5 simulation sequence and the adeis card are given in Appendix B. An important change to the MCNP5 code c oncerns the weight-window algorithm, which was modified to take into account various combination of biasing configuration: i) standard weight-window; ii) angular-dependent weight-window wit hout explicit positron biasing ; iii) explicit positron biasing without angular dependency; and, iv) explicit positron biasing with angular dependency. The application of the wei ght-window within the CH algorithm was also modified to ensure that the electron spectrum would not be biased (see Chapter 5). Generation of the Deterministic Model Because of the high computational cost associated with performing exclusively 3-D deterministic transport simulation for coupled electron-photon-positron problems, ADEIS allows in principle the use of 1-D, 2-D and 3-D de terministic transport simulation to obtain the importance functions. Other consid erations such as the material compositions and the energy group structure are also automatically managed by ADEIS before performing the deterministic transport simulation. 1-D Model (X or R) Generation In order to automatically generate appli cable 1-D (X or R) importance functions, a line-ofsight approach is used. In this approach, the user defines a line-of-sight between the source origin and the region of interest (ROI). A model is then generated along that line by tracking through the geometry and detecting material discont inuities as illustrated in Figure 4-2.

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69 Figure 4-2. Line-of-sight approach This approach is better suited for problem types in which the beam is relatively well collimated and the overall behavior of the solution is 1-D-like. 2-D Model (XY or RZ) Generation In order to automatically generate an applic able 2-D model (XY or RZ), a perpendicular direction to the line-of-sight is defined either by default or by the user. While the model is being generated by tracking along the line-of-sight the tracking algorithm recursively branch along the perpendicular direction each time a material disc ontinuity is encountered. That new direction is tracked and material discontinuities are reco rded until a region of zero importance is encountered. At this point, the algorithm return s to the branching point, and continues along the line-of-sight as illustrated in Figure 4-3. Figure 4-3. Two-dimensional mode l generation using line-of-sight LOSx0x1x2 y1y2y3y4 LOSx0x1x2 y1y2y3y4 LOS MC Deterministic0.00.10.25 9.09.05 10.1510.25 100.0110.0 BEAM LOS MC Deterministic0.00.10.25 9.09.05 10.1510.25 100.0110.0 BEAM

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70 These various material regions can then be regrouped into coarse meshes and automatically meshed as described in the next section. No te that this methodology also allows for the generation of 3-D (XYZ) models by branching along a third perpendicular direction at each material discontinuity. Obviously, models genera ted by such an approach are approximate, but they are sufficient for the purposes of generating relative importance functions to be used in the context of a VR methodology. Automatic Meshing of Material Regions Earlier studies87, 88, 89 required a significant amount of engineering time to determine an appropriate spatial mesh stru cture. Moreover, another study90 showed that an improper meshing can introduce unphysical oscillations in the importa nce function (especially with the use of the CEPXS package) and be partly responsible for statistical fluctuations in the photon tallies obtained from the coupled electron-photon-pos itron simulation. Therefore, automating the selection of a proper mesh density in each materi al region constitute an important consideration, both from practical (less e ngineering time) and technica l (reducing possible unphysical oscillations in the importance functions) perspectives. Different automated meshing schemes have been implemented and studied (see Chapter 6): i) uniform mesh size; ii) selective refinement of a boundary layer at material and source discontinuities; and, iii) material region mesh size based on partia l range associated with electron energy. Moreover, for all these automatic mesh ing schemes, the approximate rule-of-thumb91 shown in Eq. 4-1 is respected. )(1.0 )(1 1ERx G ER (4-1) In this equation, R(E1) is the range associated with el ectron in the fastest energy group, G is the total number of groups and is the ratio of the mean vector range to the CSD range. This rule

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71 ensures that possible fluctuations in the energy domain are not transmitted to the spatial domain by maintaining a mesh size larger than the partia l range associated with the slowing down of electrons from one group to the next. This is especially useful when using the CEPXS cross sections. Material Composition The composition of each material region is ex tracted from MCNP5 after the input file has been processed. A special attention is given to the fact that different MCNP5 cells can have the same material but different densit ies and that certain material can be gaseous (important for the density correction of the stopping power). This information is then used to automatically generate an input file for either of the CEPXS or CEPXS-GS codes. Energy Group Structure ADEIS is highly flexible and allows any group st ructure to be used since the cross sections are generated on-the-fly for each problem using CEPXS or CEPXS-GS. Because of the absence of resonance regions in the cross sections, the acc uracy of the results is not as sensitive to the multigroup structure as in determin istic neutron transport. Theref ore, a uniform or logarithmic distribution of the energy group width is generally sufficient. Generation of the Weight-Window The generation of the weight-window require s additional tasks addressing practical concerns related to the implementation of the methodology described in Chapter 3. Importance Function Treatment Because of the numerical difficulties inherent in the deterministic coupled electron-photonpositron transport calculations, th e importance functions may exhib it undesirable characteristics which make them inappropriate to calculate phys ical quantities such as the lower-weight bounds. A few possible problems have been identified: i) the use of CEPXS cross sections may lead to

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72 unphysical and negative values for the importance f unction; ii) extremely sma ll or large values of the importance; and, iii) numeri cal round-off resulting in importanc e values of zero. To address these various possible problems, the following steps are taken to eliminate undesirable numerical artifacts. First, ADEIS eliminates the negative values by a simple smoothing pr ocedure based on the knowledge that in most cases, the averag e of those oscillations is correct91. For each negative value detected within an energy group, the im portance value can either be interpolated or extrapolated from the closest neighbor points, depending on the locations of the negatives value within the model. Many smoothing passes may be pe rformed to ensure that no negative values remain. In order to avoid nume rical problems with extremely small and large numbers during the Monte Carlo simulation, the impor tance function values are limited to the same values used in MCNP fur huge (1036) and tiny (10-36) numbers. Finally, importance va lues that are equal to zero are set to the minimum value of the importance of that energy group. This is necessary since weight-windows bounds equal to zero are usually us ed to indicate a region in phase-space where no biasing is required. MCNP5 Parallel Calculations The MCNP5 code can perform parallel calcula tions, i.e., distributing the simulation over many processors. In the case of Monte Carlo simula tions, the parallelization of the tasks is quite natural considering that each part icle history can be simulated i ndependently of the others. More specifically, MCNP5 parallelized the simulation by breaking the total of number of particle histories over the total number of processors. In this work, th e simulations are performed on a parallel machine (cluster) usi ng essentially a distributed memory architecture where each processor has access to its own independent memory In the MCNP5 version used in this work, the communications between the various proces sors are handled through the use of message

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73 passing via the MPI library. A complete discussion of the various aspect s of parallel computing and its implementation within MCNP5 are beyond the scope of this work, and the reader is referred to the MCNP5 users manual7. It is however important to mention that it was necessary to implement the following ADEIS feature within the parallel framework of MCNP5 in order to be able to perform parallel ca lculations; the possibi lity of using field-of-view (FOV) depending on particle type and space requir ed additional message passing at the onset of the simulation to communicate the FOV to all processors. Note that the addition of new ledgers to tally the amount of weight created and lost th rough splitting and Russian roul ette over the weight-window transparent mesh also required a parallel implementation.

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74 CHAPTER 5 IMPACT OF IMPORTANCE QUALITY In Appendix A, it was shown that a bi ased sampling distribution wi th the exact shape of the integrand would result in a zero-variance so lution, therefore it is expected that a biased PDF that only approximates that shape would still yield a reduction in variance. Consequently, it can also be expected that the more accurate the im portance function, the larger the reduction in variance. However, obtaining and using more acc urate importance functions has a computational cost that can offset the gain in variance and results in the reduction of the FOM. This implies that for a given problem, there is a combination of the importance function accuracy and cost that should result in a maximum increase in FOM. This combination might be difficult to find and, most of the time, a given accuracy of the importa nce function is arbitrarily chosen. The accuracy of the importance function may also affect the statistical reliability of the estimators and introduce statistical fluctu ations that delay or even prevent th e convergence of the estimator. It is therefore possible to refer to the importance function quality i.e., the desirable characteristics to produce accurate and statistically we ll behaved tallies when used for biasing in ADEIS. In previous work on neutral particle2, 33, it was shown that methodologies similar to ADEIS produce significant speedup with relatively approximate importance functions. From these studies, it appears that the quality of the importance function was not a cr itical issue for neutral particle. It is, however, important to verify how the quality of coupled electron-photon-positron importance functions impacts the efficiency a nd accuracy of the ADEIS methodology. To study this impact, a reference case with a poor quality importance function was deliberately chosen. This reference case c onsiders a mono-energetic 6 MeV electron beam impinging a tungsten target 100 cm away from a re gion of interest (ROI) composed of water.

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75 The heterogeneous geometry illustrated in Figure 5-1 represents a simplified accelerator head and patient. Figure 5-1. Reference case geometry The details associated with each zone (1 to 8) are presented in Table 5-1. Table 5-1. Materials and di mensions of reference case Zone Description Color Material Size (cm) # of meshes 1 Target Dark gray Tungsten 0.1 2 2 Heat dissipator Orange Copper 0.15 3 3 Vacuum White Low density air8.75 175 4 Vacuum window Light gray Beryllium 0.05 1 5 Air White Air 1.1 22 6 Flattening filter Dark gray Tungsten 1.0 20 7 Air White Air 88.85 1775 8 ROI (tally) Blue Water 0.1 2 The other simulation parameters for this re ference case are presented in Table 5-2. Table 5-2. Test case simulation parameters Monte Carlo Electron-Photon Adjoint Transport Energy-loss straggling is not sampled CEPXS cross sections Mode: Electrons and photons 43 uniform electron groups 30 uniform photon groups Energy cutoff at 0.025 MeV Energy cutoff at 0.025 MeV Default value for ESTEP in CH algorithm S16-P15 Flat adjoint source spectrum No smoothing Different factors suspected of influencing the quality of the importance function are then varied and the statistical behavi or of the tallies as a function of the number of histories is 12 35 46 7 8 12 35 46 7 8Beam ROI

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76 investigated using two parameters; i) the relative error of the total flux, and ii) the variance of variance of the total flux. The energy spectra are al so studied to verify that no bias is introduced by the use of an importance function of poor quality Reference Case The behavior of the relative error and the vari ance of variance of a surface flux tally at the air-tungsten interface are studied as a function of the number of histories for a standard MCNP5 simulation without variance reduction. It is possible to see in Figure 5-2 that the tally is statistically well behaved since it is rapidly converging (FOM of 1798) to a low relative error and VOV. These values then smoothly decrease as the number of histories increases. 0.0001 0.001 0.01 0.1 1 10 0.E+005.E+051.E+062.E+062.E+063.E+063.E+064.E+064.E+06# of histories Relative Error Variance of Variance Figure 5-2. Relative error and va riance of variance for a statisti cally stable photon tally in a standard MCNP5 simulation For the ADEIS simulation, a uniform spatial mesh of 0.05 cm (size of the smallest material region, zone 4, vacuum window) is used throughout the model. The selection of this mesh size obviously assumes that the user would have no knowledge or experien ce with deterministic methods. This exercise is however usef ul to illustrate the impact of the quality of the importance function and the need to automate the process and encapsulate within the code the knowledge

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77 about generating importance functions of good quality It is important to mention that simulations performed with the parameters gi ven in Table 5-2 result in importance function values that are negative for larg e portions of the model and therefor e, cannot be used to calculate any physical quantities such as the weight-window bounds. It is therefore essential to ensure that the importance function values are po sitive everywhere in the model. Importance Function Positivity In deterministic electron transport, significan t numerical constraints can be imposed on the differencing scheme since most practical mesh si ze can be considered optically thick because of the large electron total cross s ections. These constraints can produce oscillating and negative solutions when a lower-order differencing scheme, such as linear-diamond, is used. The spatial linear-discontinuous scheme used in ONELD reduces these constraints by introducing some additional degrees of freedom. Moreover, the introduction of a differential operator to represent part of the scattering allows similar c onstraints to produce oscillations in the energy domain which, in certain cases, can propagate into the spatial domain For all these reasons, the use of a deterministic method to obtain the electron-photon-positron importance function can result in solutions of poor quality (negative and oscillating) if prope r care is not given to, among other things, the selection of the discretization para meters. In the context of an automated VR procedure, the robustness of the methodology is especially important to minimize users intervention. As mentioned earlier, it is essential to ensure, as a minimum, the positivity of the importance function. However, the chosen appro ach to ensure positivit y should not excessively degrade the importance functions quality or increase significan tly the computation time. Importance Function Smoothing The first solution, considered to address this issue, was to smooth the importance function to remove negative and zero values. The importan ce function values are also limited to prevent

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78 numerical problems with extremely small or large numbers. After applying the smoothing procedure, it is possible to bias the reference case using the parameters presented in Table 5-2. However, large statistical fluctuations are observe d in the photon tallies as shown in Figure 5-3. 0.0001 0.001 0.01 0.1 1 10 0.0E+005.0E+051.0E+061.5E+062.0E+062.5E+063.0E+063.5E+064.0E+06# of histories Relative Error Variance of Variance Figure 5-3. Relative error and variance of va riance in ADEIS with importance function smoothing By looking at the relative e rror and VOV, it is obvious that small values are rapidly obtained (FOM of 6648 after 3.5x105 histories) but as the simu lation progresses, statistical fluctuations degrade the performance of the tally (FOM of 329 after 3.5x106 histories). The presence of these fluctuations is, as expect ed, especially visible in the VOV. Obviously, by simulating an extremely large number of historie s it would be possible to obtain a converged tally, but this would defy the purpose of using a VR technique. Before addressing this issue of statistical fl uctuation, it is useful to investigate other methods to obtain positive importance functions wi thout numerical artifacts since it is possible that an importance function of better quality would resolve this issue. However, the importance smoothing approach will be kept since it provides more robustness to the methodology and is not incompatible with other methods.

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79 Selection of Spatial Meshing Early studies of the CEPXS methodology showed86 that the numerical oscillations in the energy domain have a wavelength that is equal to twi ce the energy group width and since the CSD operator forces a correlation between the pa th-length and energy-loss, these oscillations could propagate in the spatial domain The approximate rule-of-thumb described in Eq. 4-1 was developed to ensure that the mesh size exceeds the path-length associated with the oscillations in the energy domain and therefore mitigate these oscillations. It is possible to manually select a mesh structure meeting that criterion and ther efore generate importan ce functions of higher quality Table 5-3 presents a mesh structure for th e reference case that produces a positive importance function throughout the m odel and for all energy groups. Table 5-3. Reference case spatial mesh st ructure producing a positive importance function Zone # of meshes 1 5 2 25 3 10 4 5 5 5 6a (10.15cm to 11.0cm) 20 6b (11.0cm to 11.15cm) 20 7 10 8 5 In addition to the engineering time, the design of this mesh structure requires a more indepth knowledge of deterministic methods and a certain familiarity with the CEPXS/ONELD package. Moreover, if thin regions are considered it may not be possible to respect the criterion for all cases. This definitely highlights the need for an automatic mesh generator and for complementary techniques to further ensure the robustness of the methodology. Even though positivity is obtained for this spatial mesh structure, statistical fluctuations are still present in the photon tallies as shown in Figure 5-4

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80 0.0001 0.001 0.01 0.1 1 10 0.0E+005.0E+051.0E+061.5E+062.0E+062.5E+063.0E+063.5E+064.0E+06# of histories Relative Error Variance of Variance Figure 5-4. Relative error and variance of vari ance in ADEIS with optimum mesh structure In this case, it is also po ssible to observe that the re lative error and VOV reach small values rapidly (FOM of 5699 after 3.5x105 histories) but as the simula tion progresses, statistical fluctuations degrade the performance of the simulation (FOM of 1306 after 3.5x106 histories). However, these statistical fluctuations are slight ly smaller then those observed in the previous section. From this, it can be c oncluded that in addition to help achieving positivity, the selection of the mesh structure also aff ect the statistical behavior of the tallies by influencing the quality of the importance functions used by ADEIS. First-Order Differencing of the CSD Scattering Term By default, CEPXS uses a second-order differencing scheme for the restricted CSD operator since it provides a more accurate solution by reducing the amount of numerical straggling. Note that numerical st raggling refers to the variation in the electron energy loss due to the discretization approximation rather than the physical process. However, this differencing scheme is responsible for the spurious oscillations in the energy domain These oscillations can be suppressed by selecting a first-order differe ncing scheme. However, this criterion is not

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81 sufficient to ensure positivity of the importance function, since negative importance function values are still obtained when the first-order scheme is selected for the reference case mesh structure and it was necessary to perform smoothing on the importance functions. Alternatively, the use of the mesh structure described in Table 5-3 in conjunction with the first-order differencing scheme for the CSD operator produc es an importance functions which is too inaccurate. As shown in Figure 5-5, the relative difference between the importance functions of certain energy groups obtained with the first an d second-order differencing scheme of the CSD operator are significant. Position[cm] RelativeDifference[%] 11 11.05 11.1 11.15 -80 -60 -40 -20 0Group42 Group39 Group36 Group34 Group31 Group1 Figure 5-5. Relative difference between importance with 1st and 2nd order CSD differencing The lower-order differencing scheme is well-known91 to produce large numerical straggling degrading the accuracy of the transport solution. This translates in poor performance when the importance function is used in ADEIS. However, increasing the number of energy groups should improve that solution, since smaller energy group widths are more appropriate for first-order differencing scheme.

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82 0.0001 0.001 0.01 0.1 1 10 0.0E+005.0E+051.0E+061.5E+062.0E+062.5E+063.0E+063.5E+064.0E+06# of histories Relative Error Variance of Variance Figure 5-6. Relative error and variance of variance for ADEIS photon tally with 1st order CSD differencing scheme and 75 energy groups. Even though statistical fluctuations are still presents, Figure 5-6 shows that they are significantly smaller. Therefore, using the first-or der differencing scheme and a larger number of group seems to improve the quality of the importance. This is reinforced by the fact that performing the simulation with 75 energy gr oups and using the second order differencing scheme results in a significantly worse statisti cal behavior of the photon tallies as shown in Figure 5-7. 0.0001 0.001 0.01 0.1 1 10 0.0E+005.0E+051.0E+061.5E+062.0E+062.5E+063.0E+063.5E+064.0E+06# of histories Relative Error Variance of Variance Figure 5-7. Relative error and variance of variance in ADEIS with 75 energy groups

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83 In the context of an automated VR tec hnique where only approximated solutions are required, the robustness provided by the first or der differencing scheme could be a useful advantage. However, using a la rger number of energy groups ma y prove to be computationally too expensive in some cases. Therefore, further studies are required to investigate ways to improve the quality of the importance at a minimum computational cost CEPXS-GS Methodology In addition to the capability to perfor m multidimensional co upled electron-photonpositron, the CEPXS-GS methodology elim inates the oscillations in the energy domain54 even for very small mesh size. However, CEPXS-GS w ith the reference or improved mesh structures still results in negative importance functions requiring the use of the smoothing technique. This is understandable since these nega tive values can also be the resu lts of the spatial differencing scheme and/or optically thick regions. To pe rform a fair comparison between CEPXS-GS and CEPXS, simulations using a first-order diffe rencing scheme and 75 energy groups were performed. Figure 5-8 shows that the relative error and the VOV using CEPXS-GS are not significantly different from the ones obtained with CEPXS (see Figure 5-6). 0.0001 0.001 0.01 0.1 1 10 0.0E+005.0E+051.0E+061.5E+062.0E+062.5E+063.0E+063.5E+064.0E+06# of histories Relative Error Variance of Variance Figure 5-8. Relative error and variance of variance in ADEIS with CEPXS-GS

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84 Impact of Importance Quality on Statisti cal Fluctuations: Preliminary Analysis In previous sections, the approaches consider ed to obtain usable importance functions (i.e., functions of sufficient quality ) ensured the positivity either by themselves or in combination with smoothing and resulted in various degrees of statistical fluctuations. Since none of these approaches resolved completely the problem of statistical fluctuations, the comparison presented in this section must be considered preliminar y. Even though it is po ssible to qualitatively compare the various approaches by comparing, as previously done, the curves of the relative error and VOV as a function of hist ories, it would be interesti ng to have a more quantitative criterion. The figure-of-merit (FOM) is t ypically used to indicate the ef ficiency of a simulation tally and consequently, it can be assu med that the higher the FOM, the better the quality of the importance functions. However, since all results pr esented earlier are not fully converged and the final value of the FOM cannot be used, it beco mes interesting to look at how the FOM changes as a function of the number of hi stories. To simplify the analysis, it seems pertinent to look at the average FOM (characterize overall performance) for different number of histories and the relative variation of these FOMs (characteri ze statistical fluctuation). Eq. 5-1 shows the formulation used to calculate the relative variation of the FOM. FOM FOM FOMx S R (5-1) FOMS is the standard deviation of the FOM, and it is estimated from the FOMs obtained at various number of histories during the simulation. FOMx is estimated by calculating the average of these FOMs.

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85 Table 5-4. Average FOM and RFOM for all approaches Case Average FOM RFOM Standard MCNP5 1516 0.139 Optimal Mesh Structure 3907 0.422 1st order differenci ng scheme for CSD + optimal mesh + 75 energy groups + smoothing 3776 0.247 CEPXS-GS + 1st order differencing scheme for CSD + optimal mesh + 75 energy groups + smoothing 4711 0.405 Table 5-4 indicates that that the standard MCNP5 results have the smallest efficiency (smallest average FOM) and the smallest amount of statistical fluctuations (smallest RFOM). The use of first-order differencing scheme and CEPXS seems to produce the least amount of statistical fluctuation while the use of CEPXS-GS with first-or der differencing scheme produces the larger increase in FOM. Besides the gain in FOM from ADEIS, it is also possible to observe that none of those approaches test ed completely eliminate the excessi ve statistical fluctuations. It is therefore important to further study the root ca use of these statistical fluctuations before any other conclusions can be drawn from this comparison. Positrons Treatment and Condensed-History in ADEIS The quality of the importance function, as defined earli er, is related to the characteristics of the function that results in accurate and statistically reli able tallies. However, the quality of an importance can appear poor if improperly used within the MC code because of various implementation considerations. Therefore, this se ction presents studies evaluating the impact of positron treatment during the simulation, and im plementation of the ADEIS weight-window based methodology within the context of the CH algorithm. Positron Biasing in ADEIS MCNP5 follows the traditional approach of using the same scattering physics for electrons and positrons, but only flags the particle as a positron for spec ial purposes such as annihilation

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86 photon creation and charge deposition. Note that CEPXS follows a similar approach by using the same scattering laws and stopping powers for elec trons and positrons. Therefore, following the traditional approach, the ADEIS used the electron impor tance function to bias both the electrons and positrons. However, this treatment revealed to be inappropriate within the context of the ADEIS VR methodology as shown in the following studies. Generally, the types of statisti cal fluctuations presented in the previous sections are an indication that undersampli ng of an important physical process is occurring. In ADEIS, the large differences (at certain location in phase-spa ce) between the electron importance function and photon importance functions are in part respons ible for this undersam pling and statistical fluctuations. More precisely, such variations betw een the importance functi ons (ratio larger then 5 orders of magnitude) produces statistical fluctuations in th e photon tallies because positrons surviving Russian roulette game see their wei ght increased significantly because of the low importance predicted by the electron wei ght-window bounds. Conse quently, annihilation photons generated by the survivi ng positrons may result in infr equent high weight scores, therefore leading to statistical fluctuations in the photon tallies. More specifically, for the reference case, statistical fl uctuations occur because, 1. due to the low importance of the positrons in the flattening filter as predicted by the electron importance function, most positrons are killed by Russian roulette, 2. but, infrequently, a positron will survive Russian roulette and therefore its weight increased significantly to balance the total number of positrons in the simulation, 3. however this positron will a nnihilate quickly and produce high weight annihilation photons, 4. which, because of the geometry of the problem are likely to contri bute directly to the tallies at the surface of the flat tening filter or in the ROI, 5. and increase the spread of the scores distribut ion which affect the variance of variance and possibly, the variance itself.

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87 In the physical process illu strated in Figure 5-9, the thickness of the arrows represents the weight of the particles. Figure 5-9. Impact of large variation in importance between positron and photon By comparing the average weight per source pa rticle created as a nnihilation photons in a standard MCNP5 and ADEIS simulati on, it is possible to observe th at this excessive rouletting of the positrons results in the annihila tion photons being undersampled in ADEIS. Table 5-5. Impact of biasi ng on annihilation photons sampling Case Annihilation Photon Weight / Source Particle Standard MCNP5 1.646E-02 ADEIS 3.51E-04 However, this artificial effect stems from the use of the electron importance function to bias the positrons. This phenomenon can be easily understood by comparing the electron importance function with the importance function of the annihilation photons. By definition, the importance of a particle should include its own importance toward the objective and the sum of the importance of all its progeni es including secondary particles. However, near the cutoff energy (i.e., the energy at which positrons annihilate), the el ectron importance function is significantly smaller than the annihilation photo n importance function as shown in Figure 5-10. Consequently, the electron importance function cannot be used to represent the positron importance for which the annihilation photons are progenies. Tungsten Water Air Photon Annihilation photon Positron

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88 Figure 5-10. Electron and annihilation photon importance function in tungsten target Moreover, the electron importance function gr eatly underestimate the importance of the positrons and, as observed, result s in excessive rouletting of th e positrons and undersampling of the annihilation photons. A more realistic and physical positron importance function calculated by CEPXS/ONELD is compared to the annihila tion photon importance function in Figure 5-11. Figure 5-11. Positron and annihilation photon importance function in tungsten target Electron Importance 0.E+00 1.E-02 2.E-02 3.E-02 4.E-02 5.E-02 6.E-02 10.1510.1610.1710.1810.1910.210.2110.2210.2310.2410.25Position [cm]Photon Importance1.E-36 1.E-34 1.E-32 1.E-30 1.E-28 1.E-26 1.E-24 1.E-22 1.E-20 1.E-18 1.E-16 1.E-14 1.E-12 1.E-10 1.E-08 1.E-06 1.E-04 1.E-02 1.E+00Electron ImportanceAnnihilation Photon Importance 0.0E+00 2.0E-02 4.0E-02 6.0E-02 8.0E-02 1.0E-01 1.2E-01 10.1510.1610.1710.1810.1910.210.2110.2210.2310.2410.25Position [cm]ImportancePositron Importance Annihilation Photon Importance

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89 As expected, the positron importance functi on values are slightly larger than the annihilation photon. Using a modified version of the MCNP5 wei ght-window algorithm, importance sampling is therefore performed us ing a distinct importance function for the positrons (explic it positron biasing ). Table 5-6 shows that performing such biasing eliminates the annihilation photons undersampling. Table 5-6. Impact of exp licit positron biasing on annihilation photons sampling Case Annihilation Photon Weight / Source Particle Standard MCNP5 1.646E-02 ADEIS 1.652E-02 It is also interesting to examine the surf ace photon flux spectrum at the interface between regions 6 and 7, i.e., at the surface of the flat tening filter. By examining the spectrum coming out of this region, it is possible to better observe the impact of the positron biasing through the annihilation photons before this effect is smeared by scattering in the rest of the model. 0.0E+00 2.0E-06 4.0E-06 6.0E-06 8.0E-06 1.0E-05 1.2E-05 1.4E-05 1.6E-05 1.8E-05 2.0E-05 0.00.20.40.60.81.01.21.41.61.82.0Energy [MeV]Normalized Surface Flux [#/cm2] ADEIS ADEIS with positrons Standard MCNP5 Figure 5-12. Surface Photon Flux Spec tra at Tungsten-Air Interface Figure 5-12 shows that wh en positrons are not biased explicitly, a single energy bin presents a bias (~35% smaller and not within the st atistical uncertainty). Agai n, this effect is due 0.5 to 0.525 MeV

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90 to the undersampling of the annihilation photons since that energy bin (0.5 MeV to 0.525 MeV) tallies mainly the 0.511 MeV annihilation photons. Note that the 1statistical uncertainty on these results is smaller than the size of the points. Finally, it is interesti ng to see in Figure 5-13 that all statistical fluctuations in the relative error and the variance of variance disappear when the positrons are explicitly biased. 0.0001 0.001 0.01 0.1 1 10 0.E+005.E+051.E+062.E+062.E+063.E+063.E+064.E+064.E+06# of histories Relative Error Variance of Variance Figure 5-13. Relative error and VOV in ADEIS with CEPXS and explicit positron biasing Impact of Importance Quality on Stat istical Fluctuations: Final Analysis Since the major statistical fluctuations have been eliminated through explicit positron biasing, it is to compare again the approaches listed in Table 5-4 by examining the average FOM and its relative variation as a function of histories. By compar ing Tables 5-4 and 5-7, it is possible to observe that the RFOM are decreased to about the same value as the standard MCNP5 simulation and that the average FOM is increased significantly. It can also be observed that no significant gains in FOM or statistical stability are obtained from using either the CEPXS or CEPXS-GS. However, a significant impr ovement in the average FOM and RFOM is observed when the 2nd-order CSD operator and a smaller number of energy groups are used.

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91 Table 5-7. Impact explicit posit ron biasing on average FOM and RFOM Case Average FOM RFOM Standard MCNP5 1516 0.139 Optimal Mesh Structure + Smoothing 8754 0.111 1st order differenci ng scheme for CSD + optimal mesh + 75 energy groups + smoothing 6842 0.166 CEPXS-GS + 1st order differencing scheme for CSD + optimal mesh + 75 energy groups + smoothing 6930 0.150 This behavior could be attributed to various factors affecting the quality of the importance function. It is possible that; i) the number of energy groups is too small for the first-order differencing scheme to have the same accuracy as the second-order scheme, and ii) the larger number of energy groups in creases the computational cost Further studies on this topic and the optimization of other discretization para meters are presented in Chapter 6. Condensed-history algorithm and weight-window in ADEIS This section presents studies related to the accuracy of elect ron tallies and the implementation of the ADEIS weight-window ba sed VR methodology within the context of the CH algorithm. However, to better understand th e impact of the implementation of the weightwindow within the CH algorithm, it is useful to simplify the test case. Therefore, an electrononly simulation is performed in a simple cube of water with a 13 MeV pencil beam impinging on the left surface. As illustrated in Figure 5-14, five regions of interest are considered; 0.15 cm starting at about 70%, 80%, 84%, 92% and 100% of the CSD range of the source electrons. Figure 5-14. Regions of interest c onsidered in simplified test case

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92 In the MCNP5 implementation of the CH algorithm, all distributions are evaluated on a predetermined energy-loss grid at the beginning of the simulation. Path-lengths associated with the major steps are used to model the energy loss using the CSD expected value and the Landau/Blunck-Leisegang distribution for energy-l oss straggling. In CEPXS, the energy-loss is modeled through the use of a di fferential cross-section for hard collisions and restricted stopping powers for the soft collisions (no energy-loss straggling is considered for soft collisions). Therefore, for each ROI, the electron total flux and spectrum are estimated with three different energy-loss approximations: Case 1: CSD expected value of the energy loss in MC and unrestricted stopping power in deterministic Case 2: CSD expected value of the energy loss in MC and implicitly modeled energy-loss straggling using differential cross-section for hard collisions in deterministic Case 3: CSD expected value of the energy lo ss and sampling of the Landau/BlunckLeisegang energy-loss distribution in MC a nd implicitly modeled energy loss straggling using differential cross-section for hard collisions in deterministic. Figure 5-15 shows the percentage of relative difference between the standard MCNP5 and ADEIS electron total fluxes in the ROI for the three energy-loss approximations. -3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 0.60.70.80.91.01.1Depth [fraction of CSDA range]Relative Difference [%] Case 1: With CSDA Only Case 2: With Energy-Loss Straggling in Deterministic Only Case 3: With Energy-Loss Straggling Figure 5-15. Relative differences between the st andard MCNP5 and ADEIS total fluxes for three energy-loss approaches

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93 By comparing the relative differences in total fluxes for the Cases 1 a nd 2, it is possible to conclude that discrepancies in th e energy-loss models are not res ponsible for the bias observed in Figure 5-15. This conclusion can be reached since the relative difference behaviors of these two cases are not significantly different in spite of having significantly differe nt energy-loss models. Therefore, it can be implied that this bias is somewhat related to the energy-loss straggling sampling within the CH algorithm and the us e of the weight-window in the ADEIS VR methodology. Consequently, it is in teresting to look further at th e relative differences between the electron energy spectra in the ROI from th e standard MCNP5 and ADEIS calculations for Case 3. -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00 0.01.02.03.04.05.06.07.08.0Energy [Mev]Relative Difference [%] 0.69 CSDA Range 0.79 CSDA Range 0.84 CSDA Range 0.92 CSDA Range 1.00 CSDA Range Figure 5-16. Relative differences between th e standard MCNP5 and ADEIS electron energy spectrum for Case 3 in the five regions of interest Figure 5-16 indicates that syst ematic errors are introduced in the spectra from the ADEIS VR methodology and that those erro rs seem to increase with incr easing penetration depths. This behavior is analogous to the systematic errors introduced in the energy spectrum when an electron track is interrupted by cell boundaries92 in the class-I CH algorithm as implemented

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94 within MCNP5. This can be simply shown by dividing the simplified test base into small subregions and calculating the relati ve difference between the resulting spectrum and the spectrum obtained from the undivided model. Note that thes e simulations are performed in an unmodified standard version of the MCNP5 code with the ROI located at about the range of the source particles. -90.0 -80.0 -70.0 -60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 0.01.02.03.04.05.06.0Energy [MeV]Relative Difference [%] With 5 subregions With 9 subregions With 18 subregions Figure 5-17. Relative differen ces in electron spectra for undivided and divided models Comparison of Figures 5-16 and 5-17 shows that the relative difference behaviors are similar. However, it can be observed that thes e systematic errors introduced by the ADEIS VR methodology are much smaller. These systematic errors are introduced when a major step is interrupted by a cell boundary and the real path-length is shorter th an the predetermined pathlength used in the pre-determination of the en ergy-loss straggling distribution in that step. Obviously, this systematic error increases with the increasing number of surface crossings. In a similar manner, it is important that the weight-w indow be applied at the end of major step of a class-I algorithm otherwise a similar systematic e rror will be introduced since particles that are

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95 splitted or have survived Russian roulette will have experienced an energy-loss based on the full length of the step rather then partial le ngth where the weight-window is applied. A review of the MCNP5 CH algorithm reveal ed an indexation error resulting in the weight-window being applied before the last substep rather than afte r. Because of this error, all particles created through the weight-window tec hnique have inaccurate energy losses due to the small truncation of the full path-length of the ma jor step. This error is small but accumulates as particles penetrate deeper into the target ma terial. Therefore, the MCNP5 CH algorithm was modified to ensure that the weight-window is ap plied at the end of each major step. To further illustrate the impact of applying the weight-w indow before the end of a major CH step, Figure 5-18 shows the impact on the total flux at different depth of performing the bias at three locations within the major step: i) just before the second-to-last substep, ii) just before the last substep, and iii) just after the last substep. -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 0.6 0.7 0.8 0.9 1.0 1.1Depth [fraction of CSDA range]Relative Difference [%] Weight-window applied just before the second-to-last substep Weight-window applied just before the last substep Weight-window applied just after the last substep Figure 5-18. Impact of the modification of c ondensed-history algorithm with weight-window on the relative differences in total fl ux between standard MCNP5 and ADEIS

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96 As expected, Figure 5-18 shows that the system atic error is larger for the case where the weight-window is applied before the second-to-l ast substep since the error in the energy-loss prediction is larger. It can also be seen that no bias is introduced when the weight-window is applied after the last substep. Figure 5-19 shows that the large systematic bias in the electron tally spectra shown in Figure 5-16 disappears when the weight-wi ndow is applied after the last substep of each major step. 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 0.E+001.E+002.E+003.E+004.E+005.E+006.E+007.E+008.E+00Energy [MeV]Normalized spectrum-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Relative difference [%] Normalized spectrum Relative difference Figure 5-19. ADEIS normalized energy spectrum a nd relative differences with the standard MCNP5 at 70% of 2MeV electron range with the CH algorithm modification Above results also indicate that as expected, the la rgest relative differe nces are located at the tail of the spectrum. For most problems a nd tally locations, no bias is observed in the spectrum, as shown in Figure 5-20. Note that th e uncertainties in the re lative difference were obtained using a standard error propagation formula.

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97 -10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00 0.01.02.03.04.05.06.07.08.0Energy [Mev]Relative Difference [%] 0.69 CSDA Range 0.79 CSDA Range 0.84 CSDA Range 0.92 CSDA Range 1.00 CSDA Range Figure 5-20. Relative differences between the st andard MCNP5 and ADEIS at various fraction of the range Even though the total flux and its spectrum ar e unbiased for most cases, it is important to note that a small bias (within 1statistical uncertainty) can remain in the tail of the spectrum for tallies located pass the range of the source particle. At this point, the source of this small bias is not fully understood but it is possible that difference in straggling models between the deterministic importance and the MC simulati on could be responsible. In previous studies91, it was also supposed that discrepancies between electron MC and deterministic results might be caused by the differences in straggling models. Mo reover, no bias can be seen with the 99% confidence interval and this small bias has a ne gligible impact on the integral quantities of interest such as energy deposition. No further st udies of this aspect will be presented here; however, for completeness, a series of analyses in search of the specific cause of this behavior are presented in Appendix C.

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98 Conclusions First, the analyses presented in chapter showed that the CEPXS methodology can be used to generate importance functions for coupled electron-photon-positron transport and collision biasing. However, numerical difficulties in ob taining physical importan ce functions devoid of numerical artifacts were encount ered. Our studies indicate th at a combination of limited smoothing, a proper selection of the mesh structure and the use of a first-order differencing scheme for the CSD operator (EO ) circumvents some of the num erical difficulties but large statistical fluctuations remain in the photon ta llies. Note that the need for smoothing and selecting a proper spatial mesh highl ights the fact that automation must be an essential aspect of this methodology in order to be practical. Secondly, it was shown that it is essential to bias different species of particles with their specific importance function. In the specific ca se of electrons and positrons, even though the physical scattering and energy-loss models are similar, the importance of positrons can be many orders of magnitudes larger then the electr on importance functions due to the creation of annihilation photon from positrons. More specifically, it was show n that not explicitly biasing the positrons with their own impor tance functions results in an undersampling of the annihilation photons, and consequently introduces a bias in the photon energy spectra. Therefore, in ADEIS, the standard MCNP5 weight-window algorit hm was modified to perform explicit biasing of the positrons with a distinctive set of importance functions. It is importa nt to note that the computational cost of generating coupled electron-photon-positron importance functions may become noticeable in multidimensi onal problems due to upscattering. Thirdly, it was shown that the implementati on of the weight-window technique within the CH algorithm, as implemented wi th MCNP5, requires that the biasing be performed at the end of

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99 each major step. Applying the weight-window earlier into the step, i.e., before the last substep, results in a biased electron energy spectrum. This bias is a consequence of systematic errors introduced in the energy-loss prediction due to an inappropriate implementation of the weightwindow. More specifically, these e rrors occur if the path-lengths between weight-window events differ from the pre-determined path-lengths us ed for evaluating the energy-loss straggling distribution. Therefore, in ADEIS, the standard MCNP5 CH algorithm was modified to ensure that the weight-window is applied after the last substep of each major step. Finally, in general, it can be concluded that improving the quality of the importance function could improve the statistical reliabi lity of the ADEIS methodology. However, the analyses in this chapter did not address in de tail an important reason of performing non-analog simulations; i.e., achieving speedups. Therefore, various strategies to further improve the quality of the importance function are studied in Chapter 6. These strategies are aim at improving and/or maintaining the statistica l reliability (robustness of the met hodology) of the ta llies as well as maximizing the speedup.

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100 CHAPTER 6 IMPROVING THE QUALITY OF THE IMPORTANCE FUNCTION An increas e in accuracy of the importance f unction may result in larger decrease in variance, but depending on the choice of phase -space discretization, it may also have a computational cost that may offsets the gain in variance. In theory this implies that for a given problem, there is a combination of accuracy and cost of the importance function that should result in a maximum increase in FOM and statistica l reliability. Such importance functions could be referred to as importance functions of good quality The previous chapter highlighted the need for an automatic discretization scheme to encap sulate within the code the knowledge necessary to obtain an importance function of good quality Moreover, it was concluded that such automatization schemes increase the robustness and st atistical reliability of the methodology while reducing the amount of engine ering time necessary to use ADEIS. Therefore, the present chapter studies strate gies to automatically select discretization parameters that improve the quality of the importance function. This problem is two-fold: i) the selection of discretizatio n parameters that generates a positive importance function of sufficient accuracy, and ii) the maximization of the va riance reduction in the MC simulation while minimizing the computational overhead cost Note that for most cases using 1-D deterministic importance functions, the overhead cost associated with performing the deterministic calculation (a few seconds) is negligible compared to MC simulation time (tens of minutes at the least). Therefore, most of the conclusions presented in this chapter refl ect primarily the impact of the accuracy of the importance functi on used in the VR technique. To simplify the analyses, a reference case re presenting a cube of a single material (one layer), with an impinging monoene rgetic electron pencil beam and a region of interest (ROI) located slightly pass the range of the source particle is considered. The ROI has a thickness of

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101 approximately 10% of the range of the source part icle. The geometry of this reference case is illustrated in Figure 6-1. Figure 6-1. Simplified reference model It is well known that the accuracy and effi ciency of a coupled electron-photon-positron deterministic discretization scheme depends on th e energy of the source particles as well on the atomic number (Z) of the material. These two pa rameters influence: i) the anisotropy of the scattering, ii) the total interaction rate, and ii i) the yield of seconda ry particles creation. Therefore, different combinations of source particle energies and materials are considered as part of the analysis plan presented in Table 6-1. The other reference case simulation parameters such as the number of energy groups, the energy cutoff values, the quadrature order or the Legendre expansion order are presented in Table 6-2. Table 6-1. Various test ca ses of the analysis plan Case Energy (MeV) Material Average Z Thickness (cm) 1 0.2 Water 8 0.0450 2 0.2 Copper 29 0.0075 3 0.2 Tungsten 74 0.0045 4 2.0 Water 8 0.9800 5 2.0 Copper 29 0.1550 6 2.0 Tungsten 74 0.0840 7 20.0 Water 8 9.3000 8 20.0 Copper 29 1.1700 9 20.0 Tungsten 74 0.5000 BEAM

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102 Table 6-2. Other reference case simulati on parameters of th e analysis plan Monte Carlo Electron Adjoint Transport Energy-loss straggling is sa mpled CEPXS cross sections Mode: electrons only 50 equal width electron groups Energy cutoff at 0.01 MeV Energy cutoff at 0.01 MeV Default value for ESTEP in CH algorithm S8-P7 No angular biasing Flat adjoint source spectrum 1st order CSD operator discretization Smoothing Grid Sensitivity and Automatic Spatial Meshing Schemes As discussed in Chapter 5, the automatic selection of a spatial mesh structure is essential for the robustness and ease of use of the me thodology. As described in Chapter 4, the deterministic models are automatically created by tracking the material discontinuities along a line-of-sight between the source and the region of interest (ROI). This section presents a series of studies aimed at identifying the impact of the me sh size within each of those material regions on ADEIS efficiency. This aspect is essential to the development of an automatic discretization scheme. Uniform Mesh Size The simplest approach is to select a default mesh density to be applied throughout the model. Even though this approach is not believed to be the most e fficient, it provides a better understanding of the impact of different mesh si zes on the efficiency and accuracy of the ADEIS methodology. The optimum mesh size should be rela ted to the energy of the source particle and the average Z of the material, and consequently it should be problem-dep endent (i.e., different for each case of the analysis plan). Since the wei ght-window is applied at the end of each major energy step of the CH algorithm, it might be more insightful to study the speedup as a function of the ratio of the mesh size to the partial range associated with a major energy step (referred to as DRANGE in MCNP5) as given by Eq. 6-1.

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103 1)(DRANGE x FOD DRANGE ofFraction (6-1) Moreover, by calculating this ratio using th e DRANGE of the first major energy step, this parameter becomes a function of the source partic le energy and the average Z, and therefore is problem-dependent. Figures 6-2 to 6-4 show th e ADEIS speedup obtained as a function of this ratio for all the test cases in the analysis plan. 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.0E-021.0E-011.0E+001.0E+01Fraction of DRANGESpeedup Case 1: 0.2 MeV Case 4: 2.0 MeV Case 7: 20.0 MeV Average Z = 8 Figure 6-2. Speedup as a function of FOD for Cases 1, 4 and 7 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.0E-021.0E-011.0E+001.0E+01Fraction of DRANGESpeedup Case 2: 0.2 MeV Case 5: 2.0 MeV Case 8: 20.0 MeV Average Z = 29 Figure 6-3. Speedup as a function of FOD for Cases 2, 5 and 8

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104 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.0E-021.0E-011.0E+001.0E+01Fraction of DRANGESpeedup Case 3: 0.2 MeV Case 6: 2.0 MeV Case 9: 20.0 MeV Average Z = 74 Figure 6-4. Speedup as a function of FOD for Cases 3, 6 and 9 In all these cases, it can be observed that the speedup increases as the mesh size decreases until a plateau is reached at, or below, mesh sizes close to the DRANGE. It is interesting to note that, in Figures 6-2 to 6-4, th e inconsistent ups and downs in speedup are produced by the fact that the numerical artifacts are not completely removed by the smoothing. For 1-D deterministic transport, the overhead computational cost associated with generating and using the importance function is always significantly smaller than the computational cost associated with the actual MC simu lation. This explains why the speedup plateau covers such a wide range of mesh sizes. However, for multi-dimensional importance, it is expected that, as the mesh size is further reduced, the speedup would decrease due to the computational cost of generating and using such detailed importance functions. Therefore, it is important to select a mesh size as large as possible to reduce the computational cost in future multi-dimensional deterministic transport simulati on. Therefore, the onset of the speedup plateau is indicative of the criter ion that should be used to automati cally select the appropriate mesh size in each material region of the problem. It can be observed in Figures 6-2 to 6-4 that the onset of

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105 the speedup plateau occurs at FOD that are similar to the approximate detour factor91, 93, 94 values presented in Table 6-3, which are define d as the ratio of the projected rangea to the CSD range. Table 6-3. CEPXS approximated detour factors Z number Detour factor Low (Z<6) 0.5 Medium (630) 0.1 These results imply that, for a given problem, a near optimal mesh size correspond to about the average depth-of-penetration along the line-o f-sight before the first weight-window event evaluated by multiplying the detour factor by the CSD range. An alternative scheme using the average energy of an electron in a coarse mesh (calculated using a CSD approximation and the distance traveled from the source ) to evaluate DRANGE was also implemented. This scheme is not discussed in details since it resulted in extremely high mesh density near ROIs located deep within the model and therefore led to significant reduction the e fficiency of the methodology. Note that the onsets of the speedup plateau do not exactly occur as predicted by the detour factors presented in Table 6-3. In part, this can be explained by the fact that these approximated detour factor values are mainly valid for ener gies below 1 MeV where they are independent of the energy of the particle. A more recent and accurate set of semi-empirical formulas for electron detour factors95 can be used to determine the detour factors for each case of the analysis plan presented in Table 6-4. These vari ous detour factors are plotted as vertical lines in the previous Figures 6-2 to 6-4 and it can be seen that they agree slightly be tter with the onset of the speedup plateau, and it is probable that a more accurate estimate of the average depth-of-penetration along the line-of-sight would provi de a better indication of the onset of the speedup plateau. a The projected range is the projection of the vector distance from the starting point to the end point of a trajectory along the initial direction of motion

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106 Table 6-4. Calculated detour factors for each case of the analysis plan Case Detour factor 1 0.49 2 0.28 3 0.18 4 0.54 5 0.32 6 0.20 7 0.78 8 0.56 9 0.40 It appears that, for Cases 7 and 8, the speedups are somewhat insensitiv e to the selection of the mesh size, and that for Case 9, the onset of the speedup plateau occurs at FODs much larger then one. This can be explained by the following facts: i) the gain in efficiency for these cases results mainly from rouletting low-energy elect rons, and ii) Russian roulette is much less sensitive to the selection of the discretization pa rameters as it will be shown later. Therefore, based on all these analyses, ADEIS will use the empirical formulas presented in Ref. 90 and the DRANGE of the first major energy step to au tomatically determine the mesh size for each material region. Multi-layered geometry Since most realistic cases are composed of mo re than one material, it is important to study the automatic meshing scheme for such problems. Therefore, two new test case with a 2 MeV electron beam impinging on three mate rial layers are considered as illustrated in Figure 6-5. Table 6-5 provides more detailed information a bout these new test case geometries while the other simulation parameters are the same as given in Table 6-2. Using the spatial mesh criterion described in the previous section, simulations are performed for these multi-layered geometries and the results are presented in Table 6-6.

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107 Figure 6-5. Multi-layered geometries. A) Tungsten-Copper-Water B) Copper-Tungsten-Water Table 6-5. Materials and dimensi ons of new simplified test case Zone Color Material Size (cm) 1 Dark Gray Tungsten 0.035 2 Orange Copper 0.055 3 Blue (ROI) Water 0.295 Table 6-6. Speedup for multi-layered geomet ries using automatic mesh criterion Case Code FOM Speedup Standard MCNP5 7.1x10-3 W-Cu-H2O ADEIS 62 8732 Standard MCNP5 3.2x10-3 Cu-W-H2O ADEIS 33 10313 These results clearly indicate that the automatic meshing criterion is applicable for geometries with multiple materials and produce significant speedup. Boundary Layer Meshing In certain deterministic transport problem involving charge deposition near material discontinuities or photoemission curr ents, it is important to sel ect a mesh structure that can resolve the boundary layer near the material and source discontinuities, i.e., the region near a discontinuity where rapid changes in the flux occu r. In CEPXS, this is achieved by generating a logarithmic mesh structure where the coarse mesh size decreases as depth increases and material/source discontinuities are approached. This approach is well suited for problems involving a source on the left-hand side of the model but may not be adequate for ADEIS needs. Beam 12 3 12 3 A ) 12 3 12 3 BeamB )

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108 It is therefore useful to study various boundary layer meshing sc hemes and measure their impact on the robustness and efficiency of the ADEIS methodology. However, in the ADEIS methodology, the resolution of the b oundary layers may affect both the forward MC simulation (i.e., using accurate values of the importance for biasing when approaching the material and source discontinuities from the source side) and the backward (adjoint) deterministic simulation (i.e., using appropriate meshing when approaching the material and source discontinuities from the ROI side to generate accura te importance functions). Theref ore, the automatic boundary layer meshing scheme allows for appropriate meshing on either or both side of each discontinuity. Automatic scheme #1 This automatic scheme has two steps. The first step is similar to the CEPXS approach, where the coarse mesh size is decreased as the di stance to a material discontinuity is decreasing. In the second step, the fine mesh density in each coarse mesh is automatically selected based on the criterion described earlier. This scheme is illustrated in Figure 6-6. Figure 6-6. Automatic boundary layer meshing scheme #1 MC geometry Deterministic coarse meshes Source sideROI side MC geometry Deterministic coarse meshes Source sideROI side

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109 This scheme is first used while performing simulations for Cases 1 and 9 in order to evaluate the impact of properly modeling the boundary layers at the edge of the adjoint source region (ROI) and source interface. Using this sc heme with five and ten coarse meshes, the changes in speedup with respect to the plateau sp eedup (see Figures 6-2 to 6-4) are presented in Table 6-7. It is obvious that th e change in speedup obtained by resolving the boundary layers at the source region interface is rather small and can even results in a slight decrease in efficiency. Table 6-7. Speedup gain ratios from bounda ry layers scheme #1 in Cases 1 and 9 Case # of coarse meshes Forward Backward Speedup gain ratio Yes No 1.09 5 No Yes 1.09 Yes No 1.15 1 10 No Yes 1.14 Yes No 0.94 5 No Yes 0.93 Yes No 0.97 9 10 No Yes 0.99 Then this scheme was applied to the multi-la yer geometry described in Figure 6-5 A) and Table 6-5. The resulting speedup was reduced by a factor 3 compared to the results shown in Table 6-6. It is therefore concl uded that any gain obtained by refi ning the meshes at the material or source discontinuities is lost because of the extra computational cost of searching through the many additional coarse meshes. Automatic scheme #2 This second scheme is based on the knowledge that if the selected mesh size can resolve the lowest energy group flux near th e material boundary then all th e fluxes for all energies will be resolved in that region. However, su ch a refined meshing would be extremely computationally costly, and therefore should be used only with in a short distance of material or source discontinuities. Even though this distance is somewhat arbitrary, it is possible to make an

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110 informed selection. This distance is chosen as the distance along the line-of-sight between the boundary and a fraction of the partial range re presenting the slowing down of the fastest electrons to the next adjacent en ergy group. Finer meshes in that region should properly describe the exponential drop of the highe r energy fluxes and the buildup of the lower energy fluxes as illustrated in Figure 6-7. Figure 6-7. Automatic boundary layer meshing scheme #2 As with the automatic scheme #1, this scheme is first applied to the test cases 1 and 9. Table 6-8 clearly shows that the gain in speedup from resolving th e boundary layer at the source region interface using the automated meshing scheme #2 is minimal, and can even results in a slight decrease of performance. Application of this scheme also resulted in a decrease of efficiency for the multi-layered geometry. It is interesting to note that the change in speedups using the automated scheme #2 is quite similar to the previous automated scheme both in the case of source region and ma terial discontinuities. Partial range Faster flux Source sideROI side Partial range Slower flux Partial range Faster flux Source sideROI side Partial range Slower flux

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111 Table 6-8. Speedup gain ratios from bounda ry layers scheme #2 in Cases 1 and 9 Case Size of refined region Forward Backward Speedup gain ratio Yes No 1.09 0.5 R1 No Yes 1.10 Yes No 1.09 1 R1 No Yes 1.09 Yes No 0.95 0.5 R1 No Yes 0.94 Yes No 0.94 9 R1 No Yes 0.94 Conclusions Even though it was shown in previous studies89 that manually adjusting the mesh structure to resolve the boundary layer at certain material discontinuity had a positive impact of the efficiency and statistical reliability of the tallies, this section showed that a systematic and automatic approach to perform such a task does not appear to yiel d any improvement, and therefore, will not be used by default in ADEIS. Energy Group and Quadrature Order Performing discrete ordinates simulations also require the selection of the number and structure of the energy groups as well as the quadr ature set order. It is therefore important to study the impact of these parameters on the ADEIS methodology speedup in order to properly select a criterion for the automatic scheme. Number of Energy Groups To study the impact of the number of energy gr oups, the test cases of the analysis plan presented in Table 6-1 are used. Using a mesh si ze equal to the crow-flight distance associated with the slowing down from the first to second en ergy group, each test case is simulated with different number of energy groups ranging fr om 15 to 85 energy groups of equal width. Figures 6-8 to 6-10 present the speedup obtaine d from all cases of the analysis plan.

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112 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 0102030405060708090Number of GroupsSpeedup Case 1: Z=8 Case 2: Z=29 Case 3: Z=74 Energy = 0.2 MeV Figure 6-8. Speedup as a function of the num ber of energy groups for Cases 1, 2 and 3 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 0102030405060708090Number of GroupsSpeedup Case 4: Z=8 Case 5: Z=29 Case 6: Z=74 Energy = 2.0 MeV Figure 6-9. Speedup as a function of the num ber of energy groups for Cases 4, 5 and 6

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113 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 0102030405060708090Number of GroupsSpeedup Case 7: Z=8 Case 8: Z=29 Case 9: Z=74 Energy = 20.0 MeV Figure 6-10. Speedup as a function of the num ber of energy groups for Cases 7, 8 and 9 A few observations can be made about Figures 6-8 to 6-10; i) no clear optimal values seems to apply to all cases, ii) the dependency on the number of energy gr oup is rather weak for cases with an average low Z number (Cases 1, 4 and 7), iii) the dependency on the number of energy group is rather weak for cases with high energy source electrons (C ases 7, 8 and 9), iv) speedup can vary by a few order of magnitude s depending on the number of energy groups, which clearly illustrates the need for an automatic selection of the discretization parameters, v) higher maximum speedups are obtained for cases with larger average Z, and vi) the speedup plateau seems to be reached at about 65 energy groups for Cases 5 and 6. A clear optimal value for the number of energy groups is difficult to pinpoint in Figures 6-8 to 6-10, because these test cases are inherently different as demonstrated by the ROI total fluxes shown in Table 6-9.

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114 Table 6-9. Total flux and relative error in the ROI for all cases of the analysis plan Case Total flux (cm-2) Relative error 1 1.5405E+00 0.0034 2 9.9798E-02 0.0111 3 1.6629E-04 0.0349 4 2.9529E-03 0.0033 5 5.0038E-05 0.0190 6 1.6312E-08 0.0900 7 2.3149E-03 0.0009 8 4.9834E-02 0.0011 9 4.5018E-02 0.0015 To explain this behavior, it is useful to first remember th at the Russian roulette and splitting games improve the simulation efficiency in completely different ways; Russian roulette reduces the time per history by killing time consuming unimportant particles but increases the variance while splitting decreases variance by multiplying important particles but increases the time per history. Therefore, for a given problem these two mechanisms compete to produce an increase in efficiency. This can be seen by l ooking at Table 6-10 showi ng the average ratio of tracks created from splitting to tracks lost from Russian roulette of the cases with same source electron energy given in Table 6-10. Table 6-10. Average ratio of track created to track lost for cases with same energy Electron energy (MeV) Ratio of track created to track losta 0.2 0.64 2.0 0.17 20.0 0.04 a Tracks are created through splitting and lost through Russian roulette For example, Figure 6-10, as well as Tables 6-9 and 6-10, indicate that cases with a 20 MeV electron beam have a low ratio of track created to track lost, lower speedups, higher total fluxes and a rather weak dependency on the number of energy groups. This suggests that, for these cases, Russian roulette dominates because; i) a significant amount of secondary electrons will have to be rouletted, and ii) it is re latively easy for a source particle to reach the

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115 ROI as shown by the total flux. Figure 6-11 show s the importance of the source particles and knock-on electrons at a few energi es. Note that in certain cases the ADEIS smoothing algorithm limits the importance function values. Positionalongx-axis[cm] Importance 2.0 4.0 6.0 8.0 10.0 10-23 10-18 10-13 10-8 10-319.3MeVto20.7MeV 8.28MeVto9.66MeV 2.76MeVto4.14MeV 0.01MeVto1.38MeV Figure 6-11. Importance functions for source and knoc k-on electrons at a few energies for Case 7 Figure 6-11 clearly shows that the importance of the source electrons (in red) will differ significantly from the importance of the knock-on electrons created in the other energy ranges. Considering that 99% of the knock-on generated from the source electrons will be created below 1.38 MeV, it is obvious by looking at the importance that the very large majority of them will be rouletted. However, for cases (e.g. 5 and 6) where splitting is more important (lower total flux in ROI and larger ratio of track creation to track loss) the use of a larger number of energy groups increases the quality of the importance function used in ADIES. As seen previously for the spatial meshing, an increase in accuracy is accompanied by an increase in speedup until a plateau is reached. Once again this plateau may not be as wide for multi-dimensional calculations.

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116 To further demonstrate this effect, Case 9 was modified to increase the importance of splitting by tallying at larger depths within the target material. As expected, Figure 6-12 shows a smaller total flux in the ROI, a larger ratio of track created to track lo st and stronger dependency on the number of energy groups. 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 0102030405060708090100Position along the x-axis [cm]Figure-of-merit (FOM)Tracks created / tracks lost = 0.064 Total flux in ROI = 6.1265E-07 Figure 6-12. FOM as a function of the numbe r of energy groups for modified Case 9 It can also be observed in Figure 6-12 that the onset of the speedup plateau for this modified case is reached at about 65 energy groups as seen previously for Cases 5 and 6. Finally, it is interesting to study in more details Case 3 for which the efficiency is reduced rather then increased as the number of energy groups increase s. The reduction in efficiency is caused by additional splitting produced by the increase in the number of energy gr oups, resulting in a significant increase in the number of secondary electrons. Th is can be demonstrated by examining the changes in the number of knock-on elec trons and their total statistical weight as a function of the number of energy groups as shown in Figure 6-13.

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117 1.E+05 1.E+06 1.E+07 1.E+08 01020304050607080Number of energy groupsNumber of knock-on electrons0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Statistical weight Number of knock-on electrons Total weight of knock-on electrons Figure 6-13. Number of knock-on elect rons and their total statistical as function of the number of energy groups for Case 3. Figure 6-13 clearly shows the large increase in the number of knockon electrons as the number of energy group increases, while the total sum of their weights remains almost constant to conserve the total number of pa rticles. Therefore, part of the decrease in efficiency observed for Case 3 in Figure 6-8 can be expl ained by the additional computational cost associated with simulating these additiona l secondary electrons. Another part of the decrease in efficiency comes from the overhead computational cost associated with performing additional splitting and rouletting. To illustrate this fact, it is interesting to look at scatter plots of the energy of electrons that ha ve been splitted or rouletted as a function of their position in the model. Note that in Figures 6-14 to 6-16, the grid represents the spatial and energy discretiza tion of the weight-window.

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118 Figure 6-14. Splitted electron energy as a functio n position for a 1000 source particles in Case 3. A) for a weight-window using 15 energy groups B) for a weight-window using 75 energy groups Figure 6-15. Rouletted electron energy as a function position for a 1000 source particles in Case 3. A) for a weight-window using 15 energy groups B) for a weight-window using 75 energy groups By comparing Figures 6-14 and 6-15, it can be se en that slightly more biasing is performed on high-energy electrons near the source for the ca se with 15 energy groups while significantly more splitting and rouletting of low-energy electrons is performed near the ROI (deep within the target material) for the case with 75 energy groups. Figures 6-16 A) and B) examine the statistical weight of the splitted electrons as a function of the position in the model for a weightwindow with 15 and 75 energy groups. A) B) A) B)

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119 Figure 6-16. Splitted electron weight as a function of position for a 1000 source particles in Case 3. A) for a weight-window using 15 energy groups B) for a weight-window using 75 energy groups. There is obviously no physical ju stification for this change in behavior as a function of the number of energy groups. It is therefore possible that the quality of the importance function might be responsible. Consequently, it is in teresting to compare the importance functions spectrum obtained with 15 and 75 energy groups. In Figure 6-17, the impor tance function for the case with 75 groups shows a significant amount of unphysical oscillations that are obviously degrading the quality of the function and the efficiency. Energy[MeV] Importance 0.05 0.1 0.15 10-5 10-3 10-1 101 10315energygroups 75energygroups Figure 6-17. Importance functions for 15 and 75 energy groups at 3.06 cm for Case 3. A) B)

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120 Conclusions It can be concluded that for problems where Ru ssian roulette is the dominant factor in the improvement in efficiency, the speedup is not significantly affected by the number of energy groups. On the other hand, for most cases wher e the splitting is the dominant factor, the quality of the importance function improves as the number of energy group increased. A speedup plateau is reached around 65 groups, where both the accuracy and efficiency are optimum. The increased computational cost associated with a larger number of energy groups is not strongly influencing the efficiency when using 1-D importance functions. It was also shown that a high number of energy groups can be significantly detrimental to the efficiency of an ADEIS simulation in certain cases because of the additional computational cost from the unnecessary splitting and rouletting near the ROI. ADEIS simu lation should therefore be performed with a weight-window using at most 35 energy groups for which the speedup plateau is almost reached and no degradation in efficiency was noticed. Moreover, it is expected a smaller number of energy groups will results in significant computation time savings in multi-dimensional simulations. Quadrature Order The quadrature order represents the number of discrete directions used to solve for the deterministic importance functions. In the nonangular biasing, the angular importance along these directions is integrated into a scalar importance functions. However, to properly model the angular behavior of the solution be fore integration, it is important to have a number of directions that adequately represents th e physics of the problem. Highe r anisotropy requires a larger number of directions. Ty pically, unbiased quadrature sets are symmetric along (over the unit sphere in the case of 3-D simulations) and have an even number of directions equal to the order (e.g. S4 correcponds to 4 angles). Acco rding to previous studies86 performed using the

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121 CEPXS/ONELD package, an S16 quadrature order is required to properly model the highly angular behavior of an elect ron beam. However, these studies also show that an S8 quadrature order is sufficient to model problems w ith a distributed volumetric source. These recommendations can easily be extended to th e adjoint calculations performed in the ADEIS simulations. It is therefore expected that an S8 quadrature set should be sufficient since the adjoint source is usually distribut ed over the ROI. For completeness, the various test cases of the analysis plan are simulated with three different SN order (quadrature order); S4, S8, and S16 corresponding to 4 angles, 8 angles, and 16 angles Note that the Gauss-Legendre quadrature set is used to meet the limita tion of the CEPXS methodology. 1 10 100 1000 10000 100000 024681012141618SN orderSpeedup case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 case 9 Figure 6-18. Impact of discrete ordinates quadr ature set order on speedup for all cases of the analysis plan. Figure 6-18 shows that speedups do not strongly depend on the number of directions. However, the cases exhibiting the larger variations in speedup are the cases where splitting is the dominating VR game. From these results, it can be concluded that for cases where Russian

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122 roulette dominates, the speedup dependency on quadrat ure order is rather w eak. It can also be concluded that the quality of the importance functions obtained with 4 directions is not optimal and would be probably even less adequate for a smaller ROI. As expected, it does not seem necessary to increase the order to S16. This is a desirable characteristic for the calculations of multi-dimensional importance functions where the number of directions for a given order is much larger (e.g., an S16 level symmetric quadrature set have 288 directions in 3-D). Therefore, by default, ADEIS will use an S8 quadrature set until the impact of the parameter on multidimensional importance function calculations is observed. Angular Biasing It is also important to study th e angular aspect of the biasing to verify if the field-of-view (FOV) approach is appropriate for all cases and for all particles. For th ese studies, Case 1 and Case 7 were simulated by using various cons tant and changing FOVs as listed below: [0, 1]: the FOV for truly 1-D geometries is equivalent to biasing in the forward direction. [0.78, 1]: Calculating subtending the ROI from the location where the beam impinges on the face of the model gives a FOV of [0.89, 1]. However this direction falls between two directions of the S8 quadrature set. This FOV integrates all the directions of the quadrature set that have smaller s and the next immediate direction. [0.95, 1]: Calculating subtending the ROI from the location where the beam impinges on the face of the model gives a FOV of [0.89, 1]. However this direction falls between two directions of the S8 quadrature set. This FOV integrates all the directions of the quadrature set that have smaller s. [0.98, 1]: For completeness a more forward-peaked biasing is analyzed. Note that the quadrature order had to be in creased to S16 to have a F OV subtending a smaller solid angle. This highlights one of the limitations of the FOV methodology since the size of ROI and the quadrature set order are linked. This lim itation will be further discussed in Chapter 8. Space-dependent -FOV: As shown in Figure 3-1 A), it is possible to define different subtending the ROI at different depth and us e them to calculate space-dependent FOVs.

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123 In Figure 6-19, tracks from a standard MC NP5 and non-angular ADEIS simulations for Case 7 are shown. Note that the four space-i ndependent FOVs studied in this analysis are overlaid on Figure 6-19 B). By looking at the tracks, it is obvious that some angles of travel are already favored in the non-angular ADEIS since the lower energy electrons (less forward-peaked particles) are already ro uletted. Therefore, the out-of-FOV bi asing will not be responsible for a significant of amount of rouletting since few electrons are naturally out of the selected FOV. The major impact of the angular bias ing should therefore be to increas e the splitting of the particles traveling within the FOV. Figure 6-19. Electron tracks for a 20 MeV electron pencil beam impinging on water (Case 7) A) standard MCNP5 B) ADEIS. However, compared to non-angul ar biasing, a loss in effici ency (between 25% and 50% of the non-angular speedup) was obtained from additional angular biasing in this case since most electrons are naturally traveling within the FOV. Therefore, the additional computational cost of the extra splitting with the FOV provides little reduction in variance. To further study the usefulness of the angular biasing, the source of Case 7 was slig htly modified to introduce an angular dependency in the form of a cosine distribution along B) A) FOV: [0,1] FOV: [0.78,1] FOV: [0.95,1] FOV: [0.98,1]

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124 Figure 6-20. Electron tracks for a 20 MeV electr on cosine beam impinging on tungsten (Case 7 with a source using an angular cosine distribution). A) standard MCNP5 B) ADEIS. Figure 6-20 shows that because of the source angular profile, the non-angular ADEIS biasing does not favor a directi onal behavior. Therefore the loss in efficiency compared to nonangular biasing is not as larg e, between 70% and 85% of the non-angular speedup. However, in Case 7, the spreading of the beam is similar to the FOV subtending the ROI. It can be therefore supposed that the increase of efficiency from angular biasing should occur if a significant amount of particles are traveling outside the FOV. First, angular biasing is performed for Case 3 since the spreading of the beam is significantly smaller then the FOV as shown in Figure 6-21. Figure 6-21. Electron tracks for a 0.2 MeV elec tron pencil beam impinging on tungsten (Case 3) A) standard MCNP5 B) ADEIS. B) A) FOV: [0,1] FOV: [0.78,1] FOV: [0.95,1] FOV: [0.98,1] B) A) FOV: [0,1] FOV: [0.78,1] FOV: [0.95,1] FOV: [0.98,1]

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125 Again, as expected, angular biasing reduces the gain in speedup compared to non-angular biasing by 50% since most electrons biased by the non-angular version of ADEIS reaches the ROI. It is interesting to note, in Figures 6-21 A) and B), the difference in behavior between the unbiased and biased electrons. It is clear that in the ADEIS simulation the higher energy electrons (red) are significantly sp litted close to the source so mo re of them can reach the ROI. Secondly, Case 7 is further modified to reduc e by 75% the size of the ROI along the z-axis and locate it at larger depth in the target material This obviously decreases the number of electrons naturally traveling within the FOV to the ROI. Table 6-11 gives the various speedups obtained from the different FOVs for this case at two di fferent depths; i) ROI at 10.5cm, ii) ROI at 11.5 cm. Table 6-11. Speedup with angular biasing for Case 7 with a s ource using an angular cosine distribution and reduced size FOV Speedup: case i) Speedup: case ii) None (standard MCNP5) n/a n/a None (non-angular ADEIS) 122 378 (386/362) [0,1]137 334 (464/490) [0.78,1] 138 335 (464/495) [0.95,1] 137 334 (464/496) [0.98,1] 152 463 Space-dependent FOV 17 44 Based on these results, it seems that the a ngular biasing improve the efficiency of nonangular ADEIS simulations for cases with a sign ificant number of electrons remaining outside the FOV if only space-energy biasing is performed. It can also be seen that for these two cases, the largest improvement is obtained from highly forward-peak biasing. It seems that having space-dependent FOV along the line-of-sight sign ificantly reduces the speedup, and might be useful only when multi-dimensional problems are studied.

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126 It is important to discuss the fact that there are some issues related to the quality of the angular information. It is a well known fact that angular fluxes are gene rally less accurate than the scalar fluxes because of the errors compensation. A similar error compensation phenomenon occurs when calculating the partially integrated values of the FOVs importance. However, the integration is performed over a small fraction of th e unit sphere resulting in values less accurate than the scalar fluxes. Consequently, the ADEIS angular importance functions contain a much larger fraction of negative values requiri ng smoothing, which may further decrease the quality. In Table 6-11, this is obvious by looking at the speedup values in parentheses. These values were obtained by calculating the importance func tion with higher quadrature orders, S16 and S32 respectively. As mentioned before, the computational cost associated with calculating the importance functions is minimal and therefore the increase in speedup es sentially reflect the increase in accuracy. Coupled Electron-Photon-Position Simulation Most realistic simulations require the mode ling of the complete cascade and therefore necessitate coupled electron-phot on-positron simulations. In su ch coupled problems, ADEIS uses weight-window spatial mesh determined for electrons because, i) the same spatial meshing must be used in ONELD and MCNP5 to bias all particles, and ii) the accuracy of the electron importance is much more sensitive to the mesh size as discussed previously. However, there is no need to use the same energy group structure for photon and electrons, th erefore the number of energy groups considered for the weight-window should be optimized. Note that, because of the CEPXS methodology, the positrons energy group structure must be the same as the electrons. Moreover, in ADEIS, the positrons cannot be used as the particles of inte rest since they cannot be tallied in MCNP5. Therefore, the adjoint source is set equal to zero for the positrons energy groups if they are present in the simulation.

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127 To study the impact of the photon energy group structure in co upled electron-photon simulations, Cases 6 and 9 were modified to ta lly the photon flux in the ROI rather then the electron flux and accordingly, a flat adjoint spectrum is defined only for the photon energy groups. These two cases were selected because the medium and high-energy electrons interacting with tungsten will create photons through bremss trahlung and therefore create a model where electrons and photons are tightly coupled. The number of energy groups for the electrons and the other discretization parameters are kept identical to cases of the analysis plan while the number of photon energy groups is varied. Two group stru cture are also studied for these different number of photon energy groups; linear and logarithmic. Previous studies86 using the CEPXS package showed that photon groups with a logari thmic structure describe more accurately the bremsstrahlung by reducing the group width at lower energies. This should be useful in the context of the ADEIS VR methodology since for the same computational cost the accuracy of the importance function, and conseque ntly, the speedup c ould be increased. Table 6-12. Speedup as a function of the number of energy groups for Cases 6 and 9 # of energy groups Speedup: Case 6 Speedup: Case 9 15 linear 15 logarithmic 0.97 5.48 0.51 8.37 25 linear 25 logarithmic 0.46 5.85 0.38 8.47 35 linear 35 logarithmic 1.30 5.55 0.63 8.01 45 linear 45 logarithmic 6.13 6.00 0.92 8.20 From Table 6-12, two interes ting observation can be made; i) the logarithm energy group structure has a significant imp act on the efficiency of the AD EIS methodology, and ii) 15 energy groups seems to be sufficient if the logarithmic group structure is used. Consequently, by default, ADEIS will use these parameters.

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128 It is also important to study the impact of the ADEIS angular biasing methodology for coupled electron-photon-positron si mulations. At this point, ADEI S uses the same biasing for electron and photon even though it is possible to bi as them differently. Note that in ADEIS, angular biasing is never performed on the positrons since they cannot be th e particles of interest as mentioned earlier. Moreover, because of the a nnihilation process, a po sitron traveling in any direction can create secondary part icles that might contribute to th e ROI. To study, the impact of angular biasing in coupled el ectron-photon-positron problems, th e Chapter 5 reference case is simulated. Table 6-13 gives the speedups obtaine d for the photon and electron tallies located in the ROI using the same FOV descri bed in the previous section. Table 6-13. Electron and photon tally spee dup using ADEIS with angular biasing FOV Electron speedupPhoton speedup None (standard MCNP5) n/a n/a None (non-angular ADEIS) 131 12.7 [0,1]151 15.9 [0.78,1] 27.6 8.57 [0.95,1] 5.73 2.85 [0.98,1] 2.11 0.96 Space-dependent FOV 15.5 3.14 Above results indicate that for this case, th e angular biasing for photon traveling in the forward direction produces the highest increase in speedup. As shown previously, it appears that space-dependent FOVs along the line-of-sight do not improve the efficiency of the ADEIS. The same observation applies to the electrons in this problem. Adjoint Source Selection As discussed in Chapters 2 and 3, in adjoint calculations the source typically represents the objective for which the importance is evaluated. Therefore, in the ADEIS methodology, the adjoint source represents the objective toward which the simulation is biased. For coupled

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129 electron-photon-positron, coefficients such as flux-to-dose conversion fa ctors (flux-to-energy deposition) are not readily availabl e for all materials. It is expect ed that the use of any adjoint source will not bias the simu lation but might simply not produce significant speedup. Therefore, ADEIS uses two automatically determ ined adjoint sources that are adequate for energy deposition and flux calculati ons; i) a uniform spectrum to maximize the total flux in the ROI, and ii) a local energy deposition response function to approximate energy deposition in the ROI. Note that the ADEIS methodology uses a spatially uniform adjoin t source over the whole ROI. In coupled electron-photon-positr on simulation, it is possible to tally quantities associated both with photons and electrons wi thin the same model. At this point, ADEIS allows only the use of a single cell as an objective sinc e the use of different cells may re duce the gain in efficiency of the simulation. ADEIS allows to bias electrons, photons or both w ithin the same simulations since the energy deposited in a given region can be influence by both types of particles. Even though it is possible to bias only a single species in a coupled simulation, it is not typically done because of the coupled nature of the physical pro cesses. However, it is possible to define the objective for only a single species of particles or for both species. Consequently, it is useful to study the impact of having either a sing le tally/objective for a given particle or two tallies for different par ticles with two objective particles. The reference case defined in Chapter 5 is therefore used with three different combinations of tallies and objective particles; i) electron ta lly and electron as the objective particle, ii) photon tally and photons as the objective particle and iii) electron and photon tal lies with both particles as objectives. Note that this is performed only fo r flux tallies, and therefore, the flat adjoint spectrum is used.

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130 Table 6-14. Electron and photon flux tally speedup using different objective particles Objective particle Electron speedup Photon speedup Electron 150.9 n/a Photon n/a 14.9 Electron and photon 89.6 8.8 Table 6-14 indicates that it is obvious that larger speedups are obtained when a single objective particle is used. The same analysis can be performed using the same reference case but the energy deposited in the ROI is tallied instead of the flux. For this case, it can be seen in Table 6-15 that having both objective particles results in slightly larger speedup. However, it can also be seen that almost the same speedup is achieve d by using only the electrons as the objective particles. Table 6-15. Energy deposition tally speedup in the reference case for various objective particles Objective particle Speedup Electron 14.6 Photon 6.3 Electron and photon 15.1 Above finding can be explained by the fact that, physically, the photons deposit their energy by creating electrons that are then more or less quickly absorbed. By using only electrons as the objective particles, th e photon adjoint solution will theref ore represents the importance toward producing electrons within the ROI, which corresponds clos ely to the physical process of energy deposition. Moreover, in th e Chapter 5 reference case, it is relatively easy for the photons to reach the ROI since this reference case is based on a radiotherapy LINAC for which the design goal was to have as much photons as possible reaching the ROI. Therefore, the additional speedup provided by biasing the photon toward the ROI is smaller. Note that the use of the

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131 uniform adjoint spectrum (equivalent to biasin g toward the total flux in the ROI) results in similar speedups for the energy deposition tally. Conclusions The analyses presented in this Chapter investigate strategies to improve the quality and accuracy of the deterministic importance functions to maximize the speedups obtained from ADEIS. These analyses considered a wide range of source energies and material average z-numbers. To achieve this goal, these studies we re performed on the sel ection of discretization parameters for the different phase-space variable s (space, energy and direction), as well as the impact of the adjoint source and angular biasing. First, it was shown that it is not necessary to accurately re solve the flux boundary layer at each material and/or source discontinuity, and th at the use of uniform mesh sizes within each material region is sufficient. For each material region, it was shown that a mesh size based on the source electron average depth of penetration be fore the first weight-window event occurs resulted in near maximum speedups. This distance is evaluated using the path-length associated with the first major energy step of the CH algorithm, and the detour factors derived from empirical formulations. Secondly, it was shown that the quality of the importance func tions (and therefore the speedup) is maximal at about 75 electron energy groups when using the first-order differencing scheme for the CSD operator. For cases where th e knock-on electrons contribute significantly to the region of interest (ROI), the selection of more than 35 electron energy groups degrades significantly the efficiency of ADEIS because of the larger amount of splitting and rouletting occurring near the ROI. However, a significan t fraction of the maximum speedup is already obtained using 35 electron energy groups. Fo r photon energy groups, it was shown that maximum speedups are obtained in coupled el ectron-photon problems when a logarithmic

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132 energy group structure resolving the bremsstrahl ung is used. It is shown that for such group structure, the maximum speedups are achieved with 15 energy groups. Thirdly, it was shown that for problems where the gain in efficiency depends significantly on the splitting game, the selection of discretiza tion parameters is more critical. This can be explained by the fact that accurate determinis tic importance functions are required to properly maintain the population of particles throughout th e model. Alternatively, it was shown that problems where the gain in efficiency is main ly a result of the rouletting of low-energy secondary electrons, the speedups ar e relatively insensitiv e to the sel ection of the discretization parameters. It was shown that maximum speedups are obtained using an S8 quadrature set. Angular biasing resulted in the largest in crease in speedup when the FOV integrated all the directions in the forward direction along the li ne-of-sight. It was also shown that for cases where flux is the quantity of interest, higher sp eedups are obtained if the adjoin t source is defined only for the particle of interest. However, for problems where the energy depositi on is the quantity of interest, it was shown that maximum speedups ar e obtained when the adjoint source is defined for both electron and photon. Moreover, it was show n that the major part of this speedup can be obtained by defining an adjoint source for electr on even when only photons reach the ROI. This can be explained by the fact that, physically the photons deposit their energy by creating electrons. Therefore, by using only electrons as the objective particles, the photon adjoint solution represents the importance toward producing electrons with in the ROI, which corresponds closely to the physical process of energy deposition.

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133 CHAPTER 7 MULTIDIMENSIONAL IMPORTANCE FUNCTION The analyses presen ted in Chapter 6 were performed using 1-D importance functions generated with the CEPXS/ONELD package along the line-of-sight (LOS) in 3-D geometries. From these analyses, a series of criteria to auto matically select the discretization parameters were developed. This Chapter investigates the ge neration and utilization of coupled electron-photonpositron multi-dimensional importance functions fo r ADEIS. First, a series of analysis is performed to study the computational cost and accuracy of 3-D impor tance functions generated using the PENTRAN code. Secondly, the use of 2-D (RZ) importance functions generated by PARTISN is studied. More specifically, the following four points are examined: 1. Generation of 1-D importance functions using the parallel SN PARTISN transport code; 2. Biasing along the LOS using the MCNP5 cylindrical weight-window; 3. Generation of 2-D (RZ) importa nce functions using PARTISN; 4. Speedup comparison between 1-D and 2-D (RZ) biasing. Generation of 3-D Importan ce Function Using PENTRAN This section presents studies on the level of accuracy and computational cost of an adjoint solution obtained from a 3-D discrete ordinates calculation using CEPXS-GS cross-sections with the 3-D discrete ordinates PENTRAN code. Ev en though ADEIS uses the MCNP5 cylindrical weight-window, the study of 3-D Cartesian importance functions will provide information about the computational cost and accuracy of generating 3-D impor tance functions in general. To investigate the accuracy of the 3-D importance function generated with PENTRAN, a comparison with the ONELD adjoint solution is performed. Compari ng the 3-D importance function generated from PENTRAN with th e ONELD solution shows that PENTRAN can achieve, at least, the level of accuracy require d by ADEIS. Note that even though 1-D models

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134 were considered by prescribing reflecting bound ary conditions, PENTRAN effectively performs 3-D transport, i.e., various numerical formulati ons in 3-D are used. To examine the accuracy and computation of obtaining a 3-Dimportance func tion using PENTRAN, three problem sets are considered. More specifically, the impact on accuracy of the following numerical formulations in PENTRAN is investigated: Differencing schemes: linear diamond (DZ), directional theta-weighted96 (DTW), and exponential-directional weighted85 (EDW) Quadrature set order using level symmetric (LQN) up to S20 and Gauss-Chebyshev (PN-TN) above S20. Note that these studies require d higher expansion orders of the scattering kernel that are not typically needed for neutral particle transport. A new algorithm for the use of arbitrary PN order and for pre-calculating all coefficients of the expansion was implemented into PENTRAN. Problem #1 This first problem is designed to study the im pact of various numerical formulations in a 3-D context for a low-Z material. Therefore, a problem with a uniform source (maximum energy of 1 MeV) distributed throughout a beryllium slab is considered. A refere nce solution is obtained with ONELD using the parameters given in Ta ble 7-1. To emulate th is 1-D problem using PENTRAN, a cube with reflective boundary conditions is considered as illustrated in Figure 7-1. Table 7-1. PENTRAN and ONELD simulati on parameters for solving problem #1 PENTRAN ONELD CEPXS-GS cross sections CEPXS cross sections 50 uniform meshes 50 uniform meshes 40 equal width electron groups 40 equal width electron groups Level symmetric quadrature Gauss-Legendre quadrature S16-P15 S16-P15 Linear diamond Linear discontinuous

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135 Figure 7-1. Problem #1 geometry By comparing the importance functions cal culated by PENTRAN and ONELD, Figure 7-2 shows that the PENTRAN solution is similar (sha pe and magnitude) to th e ONELD solution, and therefore is adequate for use in ADEIS. Figure 7-2. Importance function for fastest energy group (0.9874 MeV to 1.0125 MeV) in problem #1 To examine the difference in the solutions of the two codes, Figure 7-3 shows the ratios of the importance functions for different energy gr oups including; i) Group 10 (0.7618 MeV to Reflective BC in 3-D model Beryllium 0.3 cm Adjoint source (flat spectrum) X-axis[cm]AdjointFunction0.050.10.150.20.25 0.16 0.18 0.2 0.22 0.24 0.26ONELD PENTRAN Importance Function

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136 0.7869 MeV); ii) Group 20 (0.5112 MeV to 0. 5363 MeV); iii) Group 30 (0.2606 MeV to 0.2856 MeV); and, iv) Group 40 (0.01 MeV to 0.0351 MeV). X-axis[cm]Ratio0.050.10.150.20.25 0.8 0.85 0.9 0.95 1 1.05 1.1Group40 Group20 Group30 Group10 ~3% X-axis[cm]Ratio0.050.10.150.20.25 0.8 0.85 0.9 0.95 1 1.05 1.1Group40 Group20 Group30 Group10 ~3% Figure 7-3. Ratio of ONELD importance over PENTRAN importance for 4energy groups in problem #1 From the ratios presented in Figure 7-3, it can be concluded that the shapes of the importance functions for ONELD and PENTRAN are similar and that their magnitudes are within 15%. This difference in magnitude can be attr ibuted in part to the difference in quadrature sets used in the codes. It is important to note that the observed unphysical oscillations occurs at the boundary of the problem where the importance function drops significantly. This behavior can be attributed to s lightly less then adequate meshing (no refinement at the boundary), the quadrature order and the use of the linear diamon d differencing scheme in PENTRAN. Note that the impact of the discretization scheme on the accuracy of the PENTRAN importance function is studied using a second problem as presented in the following section. As mentioned earlier, the simulation time for ONELD and PENTRAN are significantly different. For this problem, ONELD required ~1 second, while PENTRAN require d 906 seconds. This is expected since, for this problem, 3-D transport requires th e solution of about 20 times more unknowns.

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137 To study the impact of the mesh structure on the accuracy of the PENTRAN solution, the meshing was modified to better resolve the boundary layers at the edges of the model. This new mesh structure is illustrated in Figure 7-4. Figure 7-4. Mesh refinement to resolve boundary layers at the edges of model for problem #1 Figure 7-5 shows that this mesh refinement reduces the observed oscillations. This is especially evident for the sl owest electron group (group 40). X-axis[cm]Ratio0.15 0.2 0.25 0.8 0.85 0.9 0.95 1 1.05 1.1Group20 Group10 Group30 Group40 ~3%Right half of the problem (symmetric) X-axis[cm]Ratio0.15 0.2 0.25 0.8 0.85 0.9 0.95 1 1.05 1.1Group20 Group10 Group30 Group40 ~3%Right half of the problem (symmetric) Figure 7-5. Ratio of ONELD importance over PE NTRAN importance for four energy groups in problem #1 with mesh refinement For this problem, it is expected that the quadr ature order (i.e., the level of accuracy of the angular representation) will have a larger impact on the solution at the boundary of the model. zone 2: 40 meshes 0.26cm zone 1 and 3: 20 meshes 0.02cm

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138 Figure 7-6. shows the ratios (PENTRAN to ONELD ) of the importance functions for different quadrature orders. X-axis[cm]Ratio0.050.10.150.20.25 0.98 1 1.02 1.04 1.06S32 S16Group 1 X-axis[cm]Ratio0.050.10.150.20.25 0.98 1 1.02 1.04 1.06S32 S16Group 1 Figure 7-6 Ratio of the ONELD and PENTRAN im portance functions for group 1 obtained using S16 and S32 quadrature order with mesh refinement in problem #1 Above figure indicates that the use of a higher qu adrature order in conjunction with a refined mesh at the boundary practically elim inates the observed oscillations. Problem #2 Problem #2 is designed to study the impact of PENTRAN differencing schemes on the accuracy of the importance functi on by introducing a source discontinuity in a high-Z material. This source discontinuity will results in large va riations in the importance function and therefore is useful to study the impact of different differencing schemes. Figure 7-7 presents a schematic of problem #2. Note that energy spectrum is uniform (with a maximum of 1 MeV) within the source region shown in grey. Tabl e 7-2 gives the various discretiza tion parameters used for this problem.

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139 Figure 7-7. Problem #2 geometry Table 7-2. PENTRAN and ONELD simulati on parameters for solving problem #2 PENTRAN ONELD CEPXS-GS cross sections CEPXS cross sections 50 uniform meshes 50 uniform meshes 25 equal width electron groups 25 equal width electron groups Level symmetric quadrature Gauss-Legendre quadrature S16-P15 S16-P15 To study the impact of the differencing scheme in PENTRAN, it is useful to define coarser meshes in the zone of interest to estimate the improvement in the solution compared to the more refined ONELD solution. Therefore, the mesh structure used for this problem is illustrated in Figure 7-8. Figure 7-8. Mesh structure for problem #2 0.02 cm Reflective BC in 3D model Tungsten 0.01 cm Adjoint source (flat spectrum) zone 2 5 meshes zone 1 ONELD: 15 meshes PENTRAN: 5 meshes

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140 The adaptive differencing strategy in PENTRAN automatically shifts between three differencing schemes; linear diamond with zer o-flux fixup (DZ), direct ional theta-weighted (DTW), and exponential directional-weighted (EDW ). It is also possible to force the PENTRAN code to use different differencing schemes within different regions. For this analysis, the code was forced to use one of the aforementione d differencing throughout the model. Figure 7-9 shows the importance function for group 20 (0 .2120 MeV to 0.2524 MeV) obtained using the three differencing schemes. Figure 7-9. Impact of differe ncing scheme on importance function for group 20 in problem #2 The above results clearly show that the use of an exponential differencing scheme (EDW) can improve the accuracy of the importance func tion especially in regions where the importance magnitude decreases significantly. For the adjoin t problems of the type considered in the ADEIS VR methodology, i.e., with highl y localized adjoint source at a large distance from the actual source, an exponential scheme seem s especially appropriate. For th is problem, even considering the small number of meshes, the difference in simulation time between PENTRAN (3-D) and X-axis[cm]AdjointFunction0.010.02 10-610-510-410-3ONELD DZ DTW EDW Group 20 X-axis[cm]AdjointFunction0.010.02 10-610-510-410-3ONELD DZ DTW EDW Group 20 Importance Function

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141 ONELD (1-D) is still significant, i.e., about 200 seconds for PE NTRAN vs. less than 1 second for ONELD. Problem #3 Problem #3 is design to verify the accuracy and computational cost of performing a 3-D adjoint transport calcul ation to obtain a coupled electron-phot on importance function. Therefore, a problem with a uniform source (maximum ener gy of 1 MeV) distributed throughout a tungsten slab is considered. A reference solution is obtai ned with ONELD using the parameters given in Table 7-3. To simulate this 1-D problem us ing PENTRAN, a cube with reflective boundary conditions is considered as illustrated in Figure 7-10. Figure 7-10. Problem #3 geometry Table 7-3. Other simulation parameters for problem #3 PENTRAN ONELD CEPXS-GS cross sections CEPXS cross sections 50 uniform meshes 50 uniform meshes 50 equal width electron groups 30 equal width photon groups 50 equal width electron groups 30 equal width photon groups Level symmetric quadrature Gauss-Legendre quadrature S16-P15 S16-P15 Linear diamond Linear discontinuous To verify that accurate coupled electron-phot on importance functions can be obtained with PENTRAN, it is useful to study the photon import ance function resulting from a simulation of Reflective BC in 3-D model Tungsten 0.1 cm Adjoint source (flat spectrum)

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142 the electron-photon cascade through up-scattering To study in more details the difference between the ONELD and PENTRAN importance functions it is interesting to look at their ratios for various energy groups. Figure 7-11 shows th ese ratios for the following photon groups: i) Group 1 (0.9766 MeV to 1.01 MeV); ii) Group 10 (0.6766 MeV to 0.6933 MeV); iii) Group 20 (0.3433 MeV to 0.3766 MeV); and, iv) Group 30 (0.01 MeV to 0.4333 MeV). X-axis[cm]Ratio0.020.040.060.08 0.9 0.95 1 1.05Group20 Group30 Group10 Group1 Figure 7-11. Ratio of ONELD importance ove r PENTRAN importance for four photon energy group in problem #3 The above results indicate that accurate c oupled electron-photon importance function can be generated using PENTRAN, and that, for this problem, results are within 10%. Moreover, it is interesting to note that similar unphysical oscilla tions shown in Figure 7-3 also affect the photon importance functions. It is expected that prope r meshing, adequate differencing scheme and higher quadrature order would im prove the accuracy. The PENTRAN computation time for this problem is ~108661 sec. To reduce this time, a detailed analysis was performed and it was concluded that the upscattering al gorithm was not efficient. Theref ore, a more efficient up-scatter algorithm was implemented. This new algorithm reduced the computation to ~4764 seconds. Note that the new upscattering algorithm is implem ented for the parallel version of the code.

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143 In conclusion, the above results indicate that the PENTRAN can solve for importance functions using a 3-D geometry with adequate accuracy, however, significant computation time is necessary. Therefore, the use of PENTRAN is limited to problems were three-dimensionality is important. Generation of 1-D Importan ce Functions Using PARTISN Before performing biasing using 2-D (RZ) impor tance functions, it is useful to investigate the use of PARTISN to generate 1-D importance functions. In a ddition to verifying the proper implementation of new subroutines to automatica lly generate input files for PARTISN, these studies show that another discre te ordinates solver ot her than ONELD can be used to generate coupled electron-photon-positron impo rtance functions within the context of ADEIS. In this section, the reference case define d in Figure 5-1 and Table 5-1 is used to investigate the impact on ADEIS efficiency of following four combinati ons of transport solver cross sections, and spatial differencing schemes: Case 1: ONELD, CEPXS cross sections and linear discontinuous (LD) Case 2: PARTISN, CEPXS cross s ections and linear diamond (LZ) Case 3: PARTISN, CEPXS cross sections and linear discontinuous Case 4: PARTISN, CEPXS-GS cross sections and linear diamond Note that the criteria defined in Chapter 6 are used to automatically select the discretization parameters. Table 7-4 presents the dose tally speedups obtained for these four test cases. Table 7-4. Energy deposition tally speedup for ONELD and PARTISN simulation of Chapter 5 reference case Test case Speedup Case1: (ONELD/CEPXS/LDa) 13.5 Case 2 (PARTISN/CEPXS/LZb) 15.1 Case 3 (PARTISN/CEPXS/LD) 13.4 Case 4 (PARTISN/CEPXS-GS/LZ) 15.6 a LD linear discontinuous b LZ linear diamond

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144 Table 7-2 indicates that it is possible to use PARTISN to generate 1-D importance functions that are adequate with in the context of the ADEIS VR methodology. It also appears that the use of the linear disc ontinuous scheme is not as critical as in improving the quality of the solution of adjoint problem in ADEIS. However, it must be noted that ADEIS uses rather optimized discretization parameters to improve the quality of the importance function and in that context, the use of a higher orde r spatial differencing scheme might not be necessary. Finally, it is possible to observe that, as shown previously in Chapter 5, the CEPXS-GS cross sections are adequate when used in conjunction with PART ISN and results in speedups comparable to CEPXS. This is important for obtaining 2-D (RZ) importance functions since it is already known that the CEPXS cross sections are inadequate for multi-dimension transport calculations. Biasing Along the Line-of-Sight Using the MCNP5 Cylindrical Weight-Window In order to use any importance functions ev aluated along the line-o f-sight in a more general context than the 1-D-like problem (studied in Chapters 5 and 6), it is essential to use the MCNP5 cylindrical weight-window transp arent mesh illustrated in Figure 7-12. Figure 7-12. One-dimensional (R) and two-dime nsional (RZ) weight-w indow mesh along the line-of-sight in a 3-D geometry L O SZ R W e i g h t w i n d o wModel L O SZ R W e i g h t w i n d o wModel

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145 In cases where the LOS is parallel to one of Cartesian frame of reference axis, the Cartesian and cylindrical weight-w indows are equivalent. However, if the LOS is not parallel to one of the Cartesian axes, the cylindrical wei ght-window allows a more efficient and accurate use of the 1-D importance function calculated along the line-of-sight by biasing though planes perpendicular to the LOS as shown in Figure 712. The 2-D (RZ) importance functions generated along the line-of-sight can be represented by concentric cyli nders centered along the LOS and require the use of the cylindrical weight-window The analysis performe d in this section was intended to verify the implementation of the us e of the MCNP5 cylindrical weight-window, and to investigate the computational cost (and reduction in efficiency) of transforming the Cartesian coordinates used during the MCNP5 particle tracking to the cylindrical coordinates system of the weight-window. By comparing the speedups obt ained for Case 4 (see Table 7-4) using the Cartesian and cylindrical weight-w indow, it appears that a loss in speedup of about 10% occurs when the cylindrical weight-window is used. Ther efore, such a small decrease in efficiency does not prevent the use of the cylindrical weight-window for all cases. Generation of 2-D (RZ) Import ance Functions Using PARTISN This section presents the analysis performe d to investigate the ge neration of 2-D (RZ) importance functions using PARTISN. In PARTISN, it is possible to sele ct various transport solvers with different capabilities. It is assume d that these solvers are part of the PARTISN system for historical reason as it evolved from DANTSYS. The solver used in these analyses was chosen to maintain compatibility with ONELD a nd to take advantage of the various automated processing tools already developed. However, this introduces some limitation to the scope of the studies performed in this Chapter as discussed in the following paragraph. The chosen solver uses a single level grid sc heme where each axis is divided in coarse meshes and each coarse mesh is assigned a fine me sh size. However, this implies that the same

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146 fine mesh size is applied to al l coarse meshes with the same coordinates along that axis, and therefore, limits the possible automatic mesh re finements. The automatic criteria developed in Chapter 6 will be used to define all parameters including the axial fine meshes (z-axis in Figure 7-12). However, the minimum number of fine meshes per coarse mesh allowable by the solver will be used for the radial coarse meshes This choice, coupled with the fact that only a linear diamond spatial differencing scheme is ava ilable for this solver, may not result in an importance function of good quality. The use of the block-AMR (block adaptive mesh refinement) solver available in PARTISN may resolve these issues. With these discretization parameters, using the 2-D (RZ) importance functions to bias the reference case resulted in a spee dup of about 6, i.e., about 3 times less than what was achieved with 1-D biasing along the LOS. The comput ation time required to obtain the 2-D (RZ) importance function is about ten times larger than 1-D calculations (i .e., 41.8 seconds vs. 4.5 seconds) but still relatively short compared to the total computer time of the Monte Carlo simulation (2148 seconds). Therefore, the decrease in efficiency can be attributed to the decrease in the quality due to inadequate meshing. Speedup Comparison between 1-D and 2-D (RZ) Biasing It could be argued that the Chapter 5 re ference case was highly one-dimensional and therefore did not require the use of 2-D (RZ) im portance functions. To address this issue, the reference case was modified (as illustrated in Fi gure 7-13) by reducing th e flattening filter to a more realistic size (0.5 cm x 5 cm x 5 cm) and a simplified collimator was added. These modifications slightly changed the nature of the Chapter 5 reference case to make it more axisymmetric. Consequently, the use of 2-D (RZ) importance function should be more appropriate for bias this ne w modified reference case.

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147 Figure 7-13. Modified reference case geometry More detailed dimensions for each mate rial zone are provided Table 7-5. Table 7-5. Materials and di mensions of reference case Zone Description Color Material Size (cm3) 1 Target Dark gray Tungsten 0.1 x 40 x 40 2 Heat dissipator Orange Copper 0.15 x 40 x 40 3 Vacuum White Low density air 8.75 x 40 x 40 4 Vacuum window Light gray Beryllium 0.05 x 40 x 40 5 Flattening filter Dark gray Tungsten 0.5 x 5 x 5 6 Collimator shield Collimator hole Dark gray White Tungsten Air 2.0 x 40 x 40 2.0 x 5 x 5 7 Air White Air a) 40.95 x 40 x 40 b) 48 x 40 x 40 8 ROI (tally) Blue Water 0.1 x 40 x 40 Two ADEIS simulations are performed for this reference case: i) an ADEIS simulation using a 2-D (RZ) importance function; and ii) an ADEIS simulation using a 1-D importance function. Table 7-6. Energy deposition tally speedup for 1-D and 2-D biasing Test case Speedup 2-D (RZ) biasing 6.1 1-D biasing 15.1 Table 7-6 indicates that, in spite of the m odifications, the biasing using 1-D importance functions along the LOS still produce larger speedup. However, as mentioned earlier, this result should be considered preliminar y until better spatial di scretization can be performed for the 2-D (RZ) deterministic model, and more test cases are studied. Beam ROI 1 2 3 5 4 6 7b 8 7a 1 2 3 5 4 6 7b 8 7a

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148 Conclusions From the analyses presented in this Chapter, it can be concluded th at 3-D coupled electronphoton importance functions can be generated using 3-D discrete or dinates methods. More specifically, it was shown that PE NTRAN/CEPXS-GS is adequate to evaluate coupled electronphoton importance functions in low and high-Z ma terials given that th e proper selection of discretization parameter is made. It can also be concluded that 3-D importance functions seem more sensitive to meshing and exhibit oscillations not pr esent in the 1-D solution. However, it was shown that these unphysical behaviors can be mitigated with approp riate meshing. Moreover, it was showed that 3-D importance functions seem to require a high er quadrature order to have a proper angular representation. It was also i ndicated that exponential differenc ing schemes seem useful to decrease the computational cost associated with a given accuracy for the type of adjoint problem associated with the ADEIS VR methodology. Even though it was possible to obtain accurate enough importance functions using PENTRAN, the computational cost limited the practical use of this approach in the context of the ADEIS VR methodology. Therefore, the use of 2-D (RZ) importance functions was also studied. The results presented in this Chapter show that 1-D and 2-D (RZ) importance functions could be generated using PARTISN using CEPXS and CEPXS-GS cross sections. The 2-D (RZ) importance functions were successf ully used to perform biasing though the use of the cylindrical weight-window mesh. For the reference case, thes e simulation resulted in speedups of about 6, i.e., about 3 time smaller then the speedup obtaine d with 1-D importance functions. It was shown that, in the context of the ADEIS VR methodology, the use of a linea r discontinuous spatial differencing scheme is not as critical for the 1-D importance function. However, as for the 1-D

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149 importance functions, the selection of a more optimized mesh structure will result in an importance of higher quality and may produce larger speedups.

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150 CHAPTER 8 CONCLUSIONS AND FUTURE WORK Conclusions A new automated variance reduction met hodology for 3-D coupled electron-photonpositron Monte Carlo calculations was developed to significantly reduce the computation time and the engineering time. This methodology takes advantage of the capability of deterministic methods to rapidly provide approximate informa tion about the complete phase-space in order to automatically evaluate the variance reduction pa rameters. This work focused on the use of discrete ordinates (SN) importance functions to evaluate angu lar transport and collision biasing parameters, and accelerate Monte Carlo calculations through a m odified implementation of the weight-window technique. This methodology is re ferred to as Angular adjoint-Driven Electronphoton-positron Importance Sampling (ADEIS). For the problems considered in this work, the flux distributions can be highly angulardependent because: i) the source characteristi cs (e.g. high-energy electron beam); ii) the geometry of the problem (e.g. duct-like geometry or large region without source); and iii) the scattering properties of high-energy electrons a nd photons. For these reasons, ADEIS was based on a slightly different deriva tion of the concept of importance sampling for Monte Carlo radiation transport than its predecessor, CADIS32, 33. In addition to more clearly illustrating the separation between collision and transport biasing, this derivation uses: A different formulation of the approximated response in the region of interest to allow angular surface sources; Angular-dependent lower-weight bounds base d on the field-of-view (FOV) concept to introduce this dependency without using a complete set of a ngular fluxes which requires an unreasonable amount of memory; A different lower-weight bound de finition that ensures that the highest energy source particles are generated at the upper-weight bound of the weight-window and maintains the

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151 consistency between the weight-window and the source without having to perform source biasing. ADEIS was implemented into the MCNP5 c ode with a high degree of automation to ensure that all aspect s of the variance reduction methodology ar e transparent, and required only the insertion of a tally-like card in the standa rd MCNP5 input. The accuracy and computational cost of generating 3-D importance functions using PENTRAN was studied. However, the computational cost limited the practical use of this approach in the c ontext of the ADEIS VR methodology. Therefore, to generate the angular importance functions, ADEIS used either the ONELD (1-D) or PARTISN (2-D, RZ) code with cross sections generated from either the CEPXS or CEPXS-GS code. Moreover, the impl ementation of ADEIS included the following specific features to make it practical robust, accurate, and efficient: The development and use of a driver (UDR) to manage the sequen ce of calculations required by the methodology; A line-of-sight concept to automatically generate a deterministic model based on material regions by tracking a virtual pa rticle through the geometry; Capability to generate 1-D and 2-D (RZ) importance functions al ong the line-of-sight; The use of the MCNP5 cylindrical weight-window transparent mesh to bias along the lineof-sight; On-the-fly generation of cross sections for each problem; Two automatically determined adjoint sources to circumvent the absence of appropriate dose response coefficients; i) a local energy deposition response function to approximate dose in the ROI, and ii) a uniform spectrum to maximize the total flux in the ROI; Development of criteria to automatically se lect discretization parameters that maximize speedups for each problem; Selection of discretization parameters which reduce well-known unphysical characteristics (oscillations and negati vity) in electron/positr on deterministic importance functions due to numerical difficulties; Smoothing to ensure that no negative values remain in the importance functions;

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152 Explicit positron biasing using distinct im portance functions in order to avoid an undersampling of the annihilation photons a nd introducing a bias in the photon energy spectra; Modification of the condensed-h istory algorithm of MCNP5 to ensure that the weightwindow is applied at the end of each major energy step and avoid introducing a bias in the electron total flux and spectrum; Modification of the standard MCNP5 wei ght-window algorithm to allow for various biasing configurations: i) standard weight-window; ii) angular-dependent weight-window without explicit positron biasing; iii) explicit positron biasing without angular dependency; and, iv) explicit positron biasing with angular dependency. Future Work To extend and continue this work, many avenues of research are possible. First, a more indepth study of the impact of the spatial mesh si ze (axial and radial) on the speedup for the 2-D (RZ) model is required. Other issues affecting the quality of the importance function in RZ simulations (e.g. source convergence acceleration technique, spatial differencing schemes and quadrature set) should also be studied. Also, it might be also interesting to investigate the possibility of using synthesis techniques to ge nerate multi-dimension importance functions and reduce the computational cost. Other possible improvements to ADEIS are listed below: Implement energy-dependent FOVs, especially since PARTISN allows for energy groupdependent quadrature order. Further investigate weight ch ecking frequency to verify if the current criterion is appropriate for low-energy electrons, wher e the DRANGE is especially small. Implement a parallel algorithm in the wei ght-window algorithm to speedup the mesh index search. Study the possibility of predicti ng the gain in efficiency usi ng pre-calculated curves of probability of transmission to the ROI versus speedup. Study the possibility of using angular flux moments rather the discrete angular flux to calculate the FOV in order to circumvent i ssues arising when a FOV falls between two direction cosines.

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153 Implement automatic source biasing for discre te and continuous s ources by projecting it on the discretized phase-space grid of the weight-window. This could be achieved by sampling the actual source and tallying it over the weight-window. Further study the specific cause of the spectrum tail bias observed for cases where the region of interest is located beyond th e CSD range of the source particle.

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154 APPENDIX A VARIOUS DERIVATIONS Selection of an Optimum Sampling Distribution in Importance Sampling In the importance sampling technique, an optimum biased sampling distribution can result in an estimator with a zero variance. In this section, it is shown that if a biased sampling distribution is chosen to be proportional to the PDF of th e random process, the resulting estimator will have a zero variance. Lets consid er a problem where the expected value can be represented as Eq. A-1. dxxfxgxwg)( ~ )() ( (A-1) Where g is the estimated quantity, ) (xg is a function of random variable x, ) ( ~ xf represents the biased sampling PDF, )( ~ )()(xfxfxw represents the weight of each contribution, and )(xf represents the random process PDF. The variance of such an estimator is evaluated by the Eq. A-2. where 2 is the variance. )( ~ )()(2 2xfgxgxwdx (A-2) By assuming that the biased sampling PDF is proportional to the integrand of Eq. A-1, i.e., )()()()()( ~ xgxwxgxfxf, it is possible to write Eq. A-1 as Eq. A-3 since the integral of a PDF over the whole range of the random variable is equal to 1. dxxfg)( ~ (A-3) By replacing Eq. A-3 in Eq. A-2, it is possible to rewrite the expression of the variance as in Eq. A-4. 0)( ~ 2 2 xf dx (A-4) Eq. A-4 shows clearly that the estima tors would have a zero-variance.

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155 Biased Integral Transport Equation As mentioned in the previous se ction of this appendix, it is po ssible to derive a formula for the expected value using a more optimal sampling PD F. In the context of particle transport, this is done by multiplying the Eq. A-5 by R (P) PP)dP()P(P)dPP()P((P)TQ C (A-5) Where (P) represents an importance function associated to quantity being estimated; (P) represents the integral quantity being estimated; P P and P are the respective phase-space element ) ,,(Er, ) ,,( Erand ) ,,( Er; )PP( C represents the collision kernel; P)P( T represents the transport kernel; )P( Q represents the external source of primary particles, and R is an approximated value of quantity being estimated (see Chapter 3). Multiplying the resulting equation by 1 (dressed up in a tricky fashion), it is possible to obtain Eq. A-6. Pd )P( )P( R (P) P)P(PQ Pd )P( )P( )P( )P( R (P) P)P(P)dPP()P( R (P)(P) T)( T C (A-6) By combining the various terms as follow; R (P) (P)(P) R )P( )P()P( R )P( )P()P( QQ )P( (P) P)P(P)P( T T )P( )P( )PP(P)P( C C it is possible to rewrite Eq. A-6 as Eq. A-7. PP)dP()P(PP)dP(P)dPP()P((P) T Q T C (A-7)

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156 Where (P) represents the biased estimator, )P(Q represents the biased source, P)P( T represent the biased transport kernel, and )PP( C represents the biased collision kernel. Lower-weight Bounds Formulation and Source Consistency To ensure that the source pa rticles are generated at the uppe r bounds of the weight-window when a mono-directional and mono-energetic point source is used, consider the formulations given by Eqs A-8, A-9, and A-10. ,E) Cr( R wu l E),r(, (A-8) )Er(Er dVdEdR,,),,(Q (A-9) ) ()E (E)rr (Er 0 0 ),,(Q (A-10) In those equations, ,lw represents the lower-weight bound value; 0)rr ( )E (E0, and ) ( are the Dirac delta functions repres enting a unit mono-ener getic point source emitting in a direction within the FOV. By replacing Eq. A-10 in Eq. A-9, the approximated response R can be rewritten as Eq. A-12. ),Er( ) (E-E)r-r (dEdVR00 0 0 (A-11) By replacing Eq. A-11 in Eq. A-8, the lowe r-weight bound formulation can be written as Eq. A-12. u u u lC ) C,Er( ),Er( ) C,Er( R ),Er(w100 00 00 00, (A-12) Considering Eq. A-12 and the fact that that th e upper bound of the weight -window is generally defined as a multiple Cu of the lower-weight bound, the fo rmulation for the upper-bound can be written as in Eq. A-13. 100, ),Er(wu (A-13)

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157 Since unbiased source particles generally have a weight of 1, Eqs A-12 and A-13 ensure the consistency between the source and the weight -window in the absence of source biasing. Determination of the Average Chord-Length for a Given Volume For convenience, this section pr esents a standard derivation97 of the average chord-length in a given arbitrary volume. Let us c onsider an arbitrary region of volume V bounded by a surface A with chords defined from an infinitesimal surface dA such that their number along a given direction is proportional to n. The average length of these chords in the volume can therefore be evaluated by Eq. A-14. dAdn dAdnr r (A-14) The integral over d is performed for 0 n since only chords going into the volume are considered. The infinitesimal volume associated w ith each of these chords can be written as in Eq. A-15. dAdRndV (A-15) In this equation, 0 n and can be integrated to give the total volume of the region as shown in Eq. A-16. dAnrdAdRndVV (A-16) Replacing Eq. A-16 into Eq. A-14 a nd rewriting the denominator of Eq. A-14, it is possible to obtain the formulation for the av erage chord-length of an arbi trary region given in Eq. A-17. A V ddA V dndA V dAdn dV rn4 4 4 2 0 1 0 0 (A-17)

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158 APPENDIX B IMPLEMENTATION DETAILS The following appendix contains sections providing additional deta ils about the implementation of the ADEIS methodology. Universal Driver (UDR) UDR was developed as a framew ork to manage any sequence of computational tasks. It can be used as a library to manage a sequence of tasks independent of the parent code or as a standalone application. It was essentially designed to replace sc ript-based approaches and to offer: a better task control by provi ding a single free-format input file for all tasks in the sequence a better error and file management general and consistent data exchange between the tasks themselves or between the tasks and the parent code UDR was implemented as a FORT RAN90 module and contains th e following major functions: udrhelp: utility to facilitate the creation a nd use of online help for tasks managed by UDR ffread: free-format reader that differentiate keyword and numerical inputs, store them in separate buffers to be used by the task udropen/udrclose: automatically manage availa ble file unit numbers and change filename to prevent overwrite. Ex. CALL adeisopen(udrlnk,filename ,'OLD','READWRITE','FORMATTED') prgselect: manages calls to individual task fo llowing input processing lnkred/lnkrit: access the UDR data exchange file (link file) through the use of records. Ex.: CALL lnkred(udrlnk,'dimensi on of deterministic calc',i) By default, before the insertion of independent tasks, UDR can perform: stop: stop a sequen ce at any point

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159 $filename: if a $ is detected in the option field (see next section), the remainder of the option field (i.e., up to the task termination character ;) is copied, line by line, to a file filename. An example of the UDR input file syntax is shown in Figure B-1. Figure B-1. Example of UDR input file syntax Performing an ADEIS Simulation To perform an ADEIS simulation, it is necessary to use a script (adeisrun) which allows; i) the use of a simplified the syntax to run MCNP5 in parallel, ii) to run ADEIS independently of MNCP5 if necessary, iii) to generate soft links to the CEPXS/CEPXS-GS data files, iv) to clean the various temporary files generated by ONELD CEPXS or PARTISN, and v) to run MCNP5 without the ADEIS sequence. Figure B-2 shows two examples of calls to the adeisrun script for a standard MCNP5 serial simulation and an ADE IS parallel simulation where the temporary CEPXS files are kept.

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160 Figure B-2. Examples of calls to adeisrun A) for a standard MCNP5 run B) for an ADEIS run To implement the ADEIS methodology, a new simulation sequence must used inside MCNP5. This new sequence, illustrated in Figur e B-3, requires the use of a new command line option (ex.: mcnp5 a i=test.inp o=test.out). However, this is transparent to the user sue to the use of the adeisrun script. Figure B-3. New simulation sequence in MCNP5 ADEIS MCNP5 Input Card This new sequence must used in conjunction with a new MCNP5 input card (ADEIS) which constitutes the only task required from the us er. This card is similar to a tally card with the exception that, at this point, only one ADEIS card is allowed. ADEIS:pl variable specification pl = e or p or e,p: set the objective particle Table B-1. The ADEIS keywords Keyword Meaning Default srcori Location of the source origin 0., 0., 0. los Line-of-sight vector 1., 0., 0. dimen Dimension of the deterministic importance function (d, d or d) None objcel Cell number of the region of interest (ROI) None A) adeisrun mc i=test.inp o=test.out B) adeisrun uc cepxs np 17 i=test.inp o=test.out adeismat mcrun imcn xact adeismsh adeispara m adeis adeisww

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161 APPENDIX C ELECTRON SPECTRUM BIAS SIDE STUDIES A series of studies were perf orm ed to investigate a small bias observed in the electron energy spectrum tail. Even though they proved to be unrelated to the cause of the bias, they are presented for completeness. For these studies, a reference case with the following characteristics is considered: 2 MeV pencil impinging the left-side of a wate r cube with dimensions of about one range on all side electron-only simulation is performed a weight-window of 50 uniform energy groups and 50 uniform meshes along the x-axis. the region of interest (ROI) is located slight ly pass the range of the 2 MeV source electrons and has a thickness of 2% of the range. Note that these characteristics were chosen to clearly illustrate the bias. This appendix presents analyses studying the impact, on spectrum tail bias, of the following aspects; tally location, number of histories, source energy a nd energy cutoff, leakage, energy indexing scheme, Russian roulette weight balance, knock-on electrons, knock-on electr on collision biasing, deterministic energy group structure Impact of Tally Location As a first study, it is interesting to analyze the impact of the tally location on the spectrum tail bias. For a tally located at 70% of the rang e of the source electrons, Figure C-1 shows that the relative differences are larger in the spectrum tail but no systematic bias is present (relative errors are within the 1statistical uncertainties). Note th at the statistical uncertainty on the relative differences is obtained through a typical error propagation formula.

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162 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+00Energy [MeV]Normalized spectrum-0.5 0.0 0.5 1.0 1.5 2.0 2.5Relative difference [%] Normalized spectrum Relative difference Figure C-1. ADEIS normalized spectrum and relati ve difference with standard MCNP5 for tally located at 70% of 2 MeV electron range 1.E-06 1.E-05 1.E-04 1.E-03 0.0E+001.0E-012.0E-013.0E-014.0E-015.0E-01Energy [MeV]Normalized spectrum-2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0Relative difference [%] Normalized spectrum Relative difference Figure C-2. ADEIS normalized spectrum and relati ve difference with standard MCNP5 for tally located at 2 MeV electron range

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163 However, Figure C-2 shows a small bias in the spectrum tail when the tally is located at a larger depth within the target material (at about the range of the source electrons). Note that if a 99% confidence interval is used instead of the 68% confidence interval, the observed differences are not statistically significant for the current precisi on. It can also be seen that this bias affect only for spectrum values that are about two orde rs of magnitudes smaller then the mean of the spectrum. Impact of the Number of Histories on Convergence The methodology samples more often particles that have large contributi ons to the integral quantity, and therefore, for a limited number of histories, the particles contributing to the tail of the distribution may not be properly sampled. If no bias is pr esent, the tally spectra should converge and the relative differences between the standard MCNP5 and ADEIS spectra should decrease as the number of histories increases. On th e other hand, if a bias is present, the relative differences should stay rela tively constant as the nu mber of histories is incr eased. Therefore, it is interesting to study the changes, as a function of the number of histories, in the relative differences between the ADEIS and standard MCNP5. For these different numbers of histories, it is interesting to study the relative differences as a function of en ergy. It is also interesting to look at the 2-norm (see Eq. C-1) of the relative differen ces since it provides a good indication of the overall convergence of the tally, N i i 1 2 2 (C-1) where iis the relative difference associated with energy bin i, and N is the total number of energy bins in the tally. Figure C-3 shows that, for a tally locate d at 70% of the CSD range, the relative differences decrease smoothly until they within each other 1statistical uncertainties.

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164 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+00Energy [MeV]Relative difference [%] # of histories 5E5 # of histories 2E6 # of histories 8E6 # of histories 3.2E7 Figure C-3. Relative differences between standard MCNP5 and ADE IS for tally located at 70% of the 2 MeV electron range at various number of histories As shown in Figure C-4, the convergence of the tally can be shown by looking at the behavior of the 2-norm of the relative diffe rences as a function of the number of histories. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0E+005.0E+061.0E+071.5E+072.0E+072.5E+073.0E+073.5E+07Number of historiesl2-norm of relative differences Figure C-4. Norm of relative differences betwee n standard MCNP5 and ADEIS for tally located at 70% of the 2 MeV electron range at various number of histories

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165 By comparing Figure C-3 and Figure C4, it can be concluded that the 2-norm value is dominated by the relative differences of spectrum tail. This is expected since the relative differences at the others energies are extremely small. However, as shown in Figure C-5, when the tally located beyond the CSD ra nge of the 2 MeV electron, the 2-norm of the relative differences does not converge (or converge extremely). 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.0E+001.0E+072.0E+073.0E+074.0E+075.0E+076.0E+077.0E+07Number of historiesl2-norm of relative differences Figure C-5. Norm of relative differences betwee n standard MCNP5 and ADEIS for tally located at the 2 MeV electron range at various number of histories Even though it cannot be concluded that this bias will not disappear after an extremely large number of histories, it is very unlikely that it will considering the behavior of the 2-norm and the statistical uncertainties of the problematic energy bins.

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166 Impact of Electron Energy and Energy Cutoff To understand the impact of the source el ectron energy and the energy cutoff, it is interesting to compare the energy spectra obtained from a standard MCNP5 and ADEIS calculations using the following parameters: i) 2 MeV electrons with 0.01 MeV cutoff ; ii) 2 MeV electrons with 0.1 MeV cutoff; iii) 13 Me V electrons with 0.01 MeV cutoff; and, iv) 13 MeV electrons with 0.01 MeV cutoff. Figures C-6 and C-7 present the relatives differences in electron spectra obtained from a standard MCNP5 and ADEIS simulations for thes e parameters. By comparing Figures C-6 and C-7, it is possible to conclude that smaller biases are observed for higher source electron energies and larger energy cutoff. -4.E+00 -2.E+00 0.E+00 2.E+00 4.E+00 6.E+00 8.E+00 1.E+01 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy [MeV]Relative difference [%] Cutoff 0.01 MeV Cutoff 0.1 MeV Figure C-6. Relative differences between the ta lly electron spectra from ADEIS and standard MCNP5 for a 2 MeV electron beam at two energy cutoff

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167 -4.E+00 -3.E+00 -2.E+00 -1.E+00 0.E+00 1.E+00 2.E+00 3.E+00 4.E+00 0.0E+005.0E-011.0E+001.5E+002.0E+002.5E+003.0E+00Energy [MeV]Relative difference [%] Cutoff 0.01 MeV Cutoff 0.1 MeV Figure C-7. Relative differences between the ta lly electron spectra from ADEIS and standard MCNP5 for a 13 MeV electron beam at two energy cutoff By comparing in more details the different physi cal characteristics of each case, it can also be concluded that the late ral leakage and amount of knoc k-on electron production are significantly affected by the selection of the s ource electron energy and energy cutoff. It is therefore interesting to study these two aspects. Impact of Lateral Leakage It could be argued that using 1-D importa nce functions to perform VR in a threedimensional model is introducing a small bias ca used by inability of these function to properly model the lateral leakage. Therefore, it is useful to evaluate the impact of the leakage on the results of the ADEIS VR methodology. To study th is aspect, the referen ce case is modified by increasing the size of the cube along the y-axis and z-axis. These sides ar e increased to 1.96 cm (twice the CSD range) and 2.94 cm (three times th e CSD range). These modifications reduce the

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168 leakage along these two directi ons and therefore make the pr oblem more one-dimensional in nature. -2.E+00 0.E+00 2.E+00 4.E+00 6.E+00 8.E+00 1.E+01 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy (MeV)Relative difference (%) 1R yand zsides 2R yand zsides 3R yand zsides Figure C-8. Relative differences in spectrum between the standard MCNP5 and ADEIS for various model sizes and an energy cutoff of 0.01 MeV Figure C-8 shows that the 1-D importance functi ons inability to properly take into account the lateral leakage is not responsible for introducing the bias. Impact of Knock-On Electron Collision Biasing To verify that the bias is not introduced by an implementation problem related to the collision biasing of knock-on electrons, the st andard MCNP5 and ADEIS were modified such that no collision biasing is perform for those electrons. The relative differences between the electron spectra of the standard MCNP5 and ADEIS are then compar ed with and without collision biasing for knock-on electrons. By looki ng at Figure C-9, it is obvious that the implementation of the collision biasing for knock-on electron is not responsible for the bias in the spectrum tail.

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169 -2.0E+00 0.0E+00 2.0E+00 4.0E+00 6.0E+00 8.0E+00 1.0E+01 1.2E+01 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy [MeV]Relative difference [%] Collision biasing for knock-on No collision biasing for knock-on Figure C-9. Relative differences in spectrum be tween the standard MCNP5 and ADEIS for the reference case with and without co llision biasing for knock-on electrons Impact of Weight-Window Energy Group Structure Considering that knock-on electrons are simulated according to physical properties evaluated on a given energy grid (CH algorithm ) and biased according to another one (weightwindow), it could be argued that the selecti on of the deterministi c (weight-window) energy group structure could affect the accuracy VR met hodology. It is therefore interesting to study the possible inconsistency between the predicted importance (from the deterministic calculation) and the actual contribu tion (in the MC calculatio n) of a knock-on electron. Moreover, the fact that this bias occurs near the CSD range of the sour ce electron suggests that numerical straggling in the deterministic solution (i.e. deviation from the one-to-one relationship between path-length and energy loss due to the discretization approxima tions) might results in an importance function of inadequate quality. Therefore, to study these two aspects, three test cases are considered: i)

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170 same energy group structure as the CH algorit hm; ii) 25 uniform energy groups; and, iii) 100 uniform energy groups. Figure C-10 shows the 2-norm as a function of the number of histories for the first where the CH algorithm and the we ight-window energy group st ructure are the same. 0 0.05 0.1 0.15 0.2 0.25 0.0E+001.0E+072.0E+073.0E+074.0E+075.0E+076.0E+077.0E+07Number of historiesl2-norm of relative differences Figure C-10. Norm of relative differences betwee n standard MCNP5 and ADEIS for tally located at the 2 MeV electron range at various num ber of histories with condensed-history group structure By comparing Figures C-5 and C-10, it is obvious that usin g the CH group structure does not eliminate the bias. If the degradation of the importance quality caused by numerical straggling was responsible for this possible bi as, increasing the number of energy group should reduce the bias. However, as it can be seen in Figure C-11, the number of electron energy groups as little impact of the observ ed bias. Therefore, it can be c oncluded that the weight-window energy group structure is responsible for the bias.

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171 -2.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy [MeV]Relative difference [%] 25 uniform energy groups 50 uniform energy groups 100 uniform energy groups Figure C-11. Relative differences between standa rd MCNP5 and ADEIS for tally located at the 2 MeV electron range with various energy groups. Impact of Knock-On Electrons Considering that previous results sugg ested that knock-on electrons physical characteristics, and not th eir biasing, might be res ponsible for the spectrum tail bias, it is interesting to study the impact of the presence of these secondary electrons To that effect, the production of secondary electr ons is disabled for both th e standard MCNP5 and ADEIS simulations. Figure C-12 shows that when the secondary electron production is disabled, the spectrum tail bias disappears. This seems to suggest th at, in ADEIS, the predicted importance of low-energy electrons (created ea rly on through knock-on production) toward a ROI located deep within the target material is inconsistent w ith the actual contributi on of these electrons. Therefore, the remaining sections of this appe ndix will look at possible ca uses of this effect.

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172 -4 -2 0 2 4 6 8 10 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy [MeV]Relative difference [%] Without knock-on electrons With knock-on electrons Figure C-12. Relative differences between standa rd MCNP5 and ADEIS for tally located at the 2 MeV electron range with and w ithout knock-on electron production Impact of the Energy Indexing Scheme It is well known92 that the energy indexing in the CH algorithm can significantly affect the dose (and spectrum) of electrons deep within a region of interest since methods that are not consistent with the definition of the energy group s and their boundaries ca n lead to significant errors. It is therefore interesting to verify the impact of different energy indexing algorithm (MCNP and ITS) on the accuracy of the ADEIS methodology. Figure C-13 shows the relative differences between the standard MCNP5 and ADEIS spectra when using both the MCNP and ITS energy indexing scheme. It is obvious from th ese results that the energy indexing scheme is not responsible for the possible bias.

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173 -2.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 0.E+001.E-012.E-013.E-014.E-015.E-016.E-01Energy [MeV]Relative difference [%] with ITS energy index with MCNP energy index Figure C-13. Relative differences between standa rd MCNP5 and ADEIS for tally located at the 2 MeV electron range with the MCNP and ITS energy indexing scheme. Impact of Russian Roulette Weight Balance It is well known that the Russian roulette game does not preserves the total number of particle for each VR event but rather preserves it over a large number of histories. Therefore, even though the statistical uncertainty of an esti mator can low, its value might not be accurate if the weight creation and loss due to the Russian ro ulette do not balance ou t. To study the weight creation and loss as a function of energy, th e MCNP5 code was modified to add energydependent ledgers that record weight creation and loss for eac h energy bins of the weightwindow. Note that since the simula tion can be performed in parallel, these ledgers must be local on each slave process before being accumulated by the master process. Figures C-14 to C-16 show the ratios of weight creation over wei ght loss for different number of histories (five hundred thousands to hundred and twenty-eight millions).

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174 0 0.5 1 1.5 2 2.5 3 3.5 4 0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+001.8E+002.0E+00Energy (MeV)Ratio Weight Creation / Weight Loss # of histories 5.0E5 # of histories 1.0E6 # of histories 2.0E6 Figure C-14. Ratios of weight cr eation over weight loss for 5x105 to 2x106 histories 0 0.5 1 1.5 2 2.5 3 3.5 4 0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+001.8E+002.0E+00Energy (MeV)Ratio Weight Creation / Weight Loss # of histories 4.0E6 # of histories 8.0E6 # of histories 1.6E7 Figure C-15. Ratios of weight cr eation over weight loss for 4x106 to 1.6x107 histories

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175 0 0.5 1 1.5 2 2.5 3 3.5 4 0.0E+002.0E-014.0E-016.0E-018.0E-011.0E+001.2E+001.4E+001.6E+001.8E+002.0E+00Energy (MeV)Ratio Weight Creation / Weight Loss # of histories 3.2E7 # of histories 6.4E7 # of histories 1.28E8 Figure C-16. Ratios of weight cr eation over weight loss for 3.2x107 to 1.28x108 histories Figures C-14 to C-16 shows that the ratio of weight creation ov er weight loss does converge toward one as the number of historie s increases. Three other major observations can also be made from these figures; i) no Russian roulette game is played on particles above 1.8 MeV reflecting the importance of these particles to the tally, ii) the ratios converge much more rapidly in the 0.8 to 1.6 MeV range, and iii) the range from the cutoff energy to 0.6 MeV contains the largest fluctuations and is the hardest to converge. Even though that last energy range contains the tally spectrum, the fact that most of the tally energy spectrum is not biased suggests that this is not responsible from the obser ved bias. It can also be seen that the Russian roulette did preserve the weight balance properly for most of the energy range of the problem. Impact of Coupled Electron-P hoton-Positron Simulation It is possible to change the type of elect rons contributing to th e tally by performing a coupled electron-photon simulation. It this m ode, other secondary electrons, such as recoil electrons from Compton scattering, will be created closer to the ROI and reduce the relative

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176 contribution of the knock-on electrons created clos er to the source. As expected, Figure C-17 shows that the bias essentially disappears. This reinforces th e hypothesis that knock-on electrons are related to the bias. -4 -2 0 2 4 6 8 0.0E+005.0E-021.0E-011.5E-012.0E-012.5E-013.0E-013.5E-014.0E-014.5E-015.0E-01Energy [MeV]Relative difference [%] Figure C-17. Relative differences between standa rd MCNP5 and ADEIS for tally located at the 2 MeV electron range in c oupled electron-photon model Conclusions It was shown that a small possible bias in the electron spectrum tail (i.e., for energy bins with flux values that are about two orders magnitude lower then the average flux) could be observed for tallies located at depths near the CSD range, and for which the knock-on electrons are the main contributors. Note that this bias is referred to as possible since, even though it is statistically meaningful for th e 68% confidence interval, it is not when the 99% confidence interval is considered. It was also shown that the inability of 1-D importance functions to provide an adequate representation of the lateral leakag e is not responsible for this bias Further analyses also showed

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177 that the collision biasing of knock-on electrons, the weight-wi ndow energy group structure, the CH algorithm energy indexing scheme, and the Ru ssian roulette weight balance were not responsible for this bias. However, the results pr esented in this appendix suggest that, in ADEIS, the transport of low-energy elec trons over large distances might be slightly biased. Previous studies91 suggested that differences in the straggli ng models could explain some discrepancies between CEPXS and ITS for low-energy electrons. This suggests that the bias could be attributed to an inconsistency between the predicted im portance of these elec trons and their actual contributions due to differences in the straggling model. Finally, it must be mentioned that for realistic cases requiring coupled electron-photon simulations, and where integral quantities are estimated at location before the CSD range of th e source electrons, this bias in the spectrum tail does not affect the tallies.

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184 85. Sjoden, G. E., and Haghighat, A., The Exponential Directional Weighted (EDW) SN Differencing Scheme in 3D Cartesian Geometry, Proceeding of Joint. Int. Conf. on Mathematical Methods and Supercom puting in Nuclear Applications, Saratoga Springs (1997). 86. Sjoden, G. E., An Efficient Exponential Dire ctional Iterative Differencing Scheme for Three-Dimensional Sn Computations in XYZ Geometry, Nucl. Sci. Eng., 155 (2), 179189 (2007). 87. Dionne, B., and Haghighat, A., Application of a CADIS-like Variance Reduction Technique to Electron Transport, Proc. of 6th Inter. Conf. in Sim. Meth. in Nucl. Eng., Canadian Nuclear Society, Montreal (2004). 88. Dionne, B., and Haghighat, A., Variance Reduction of Electron Transport Calculations Using 1-D Importance Functions, Proc. ANS Annual Winter Meeting, American Nuclear Society, Washington (2004). 89. Dionne, B., and Haghighat, A., Developm ent of The ADEIS Variance Reduction Methodology for Coupled Electron-Photon Transport, Proc. of The Monte Carlo Method: Versatility Unbounded In A Dynamic Computing World, American Nuclear Society, Chattanooga (2005). 90. B. Dionne, A. Haghighat, Impact Of Importa nce Quality In Coupled Electron/Photon Simulation Using Splitting/Rouletting VR Tech niques, Proceedings of Mathematics and Computation ANS Topical Meeting, Monterey, CA, USA (2007). 91. Lorence, L.J., Morel, J.E., and Valdez, G.D., Results Guide to CEPXS/ONELD: A OneDimensional Coupled Electron-Photon Discrete Ordinates Coed Package Version 1.0, SAND89-2211, Sandia National Laboratories (1990). 92. Schaart, Dennis R., Jansen, Jan Th. M., Zoetelie f, Johannes, and Leege, Piet F. A. de, A comparison of MCNP4C electron transport w ith ITS 3.0 and experiment at incident energies between 100 keV and 20 MeV: infl uence of voxel size, substeps and energy indexing algorithm, Phys. Ned. Biol., 47, 1459-1484 (2002). 93. Journal of the ICRU, Report 73, Oxford University Press, 5 (1) (2005) 94. MacCallum, C. J. and Dellin, T. A., Photo-Compton in unbounded media, J. Appl. Phys., 44 (4), 1878 (1973). 95. Tabata, T. and Andreo, P., Semiempirical form ulas for the detour factor of 1to 50-MeV electrons in condensed material, Rad. Phys. Chem., 53, 353-360 (1998). 96. Petrovic, B., and A. Haghighat, Analysis of I nherent Oscillations in Multidimensional SN Solutions of the Neutron Transport Equation, Nucl. Sci. Eng., 124, 31-62 (1996). 97. Dirac, P. A. M., Approximate Rate of Neutron Multiplication for a Solid of Arbitrary Shape and Uniform Density. Briti sh Report MS-D-5, Part I (1943).

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185 BIOGRAPHICAL SKETCH I was born in 1974 in Rouyn-Noranda (QC), Canada I went to the Universite de Montreal and got m y bachelors degree in physics in 1997. I continued to gra duate school at Ecole Polytechnique de Montreal where I completed my masters degree in nuclear engineering under Dr. Koclas and Dr. Teyssedou on coupled neut ronic/thermal-hydraulic simulation in CANDU reactor. In 2002, I moved to Florida to pursu e my Ph.D. in nuclear engineering under Dr. Haghighat.


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