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HYBRID DISCRETE ORDINATES AND CHARACTERISTICS METHOD FOR SOLVING
THE LINEAR BOLTZMANN EQUATION
By
CE YI
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
O 2007 Ce Yi
To my Mom, Dad and Brother
ACKNOWLEDGMENTS
I thank my advisor Dr. Alireza Haghighat for his instructions and guidance. Without his
support, the accomplishment of this research work would have not been possible. And I am
grateful to Dr. Glenn Sjoden for his insightful suggestions. His PENTRAN code manual has
served as a constant source of knowledge throughout my studies. I also wish to express my
gratitude to other committee members, Dr. David Gilland, Dr. John Wagner, Dr. Jayadeep
Gopalakrishnan, and Dr. Shari Moskow, for their help and support.
I would like to gratefully acknowledge Mr. Benoit Dionne, Mr. Mike Wenner, and other
members of the transport theory group at University of Florida for their support, especially
Benoit for his understanding of this work. The discussions with him on various topics inspired
me finding ways to improve the performance of my transport code.
TABLE OF CONTENTS
page
ACKNOWLEDGMENT S .............. ...............4.....
LI ST OF T ABLE S ............ ..... ._ ...............8....
LIST OF FIGURES .............. ...............10....
AB S TRAC T ............._. .......... ..............._ 13...
CHAPTER
1 INTRODUCTION ................. ...............15.......... ......
O verview .................. ... ............. ...............15.......
Linear Boltzmann Equation (LBE) ................. ...............15................
Numerical Methods to Solve the LBE ................. ...............18...............
Discrete Ordinates Method ................. ...............18........... ....
Method of Characteristics (MOC) ................. ...............19................
Ray-Effects in Low Scattering Region............... ...............20.
Hybrid Approach .............. ...............2 1....
2 THEORY AND ALGORITHMS .............. ...............23....
Multi-Block Framework Overview .............. ...............23....
Discrete Ordinates Formulations .............. ...............24....
Source Iteration Process .............. ...............26....
Differencing Scheme .............. ...............27....
Characteristics Formulations .............. ...............30....
Block-Oriented Characteristics Solver .............. ...............33....
Backward Ray-Tracing Procedure .............. ...............34....
Advantage of Backward Ray-Tracing ................. ...............36........... ...
Ray Tracer ............... ............. ...............37......
Interpolation on the Incoming Surface .....__ ................ ............... 38.....
Quadrature Set ................ ......... ...............4
Level-symmetric Quadrature............... ...............4
Legendre-Chebyshev Quadrature ................. ........_ ...._.._ .......... 4
Rectangular and PN-TN Ordinate Splitting .............. ...............46....
3 PROJECTIONS ON THE INTERFACE OF COARSE MESHES ................. ................ ..49
Angular Proj section ................. ...............49......_.._ ....
Spatial Proj section ....__. ................. ........_.._.........5
Proj section Matrix ....__. ................. ........_.._.........5
4 CODE STRUTURE............... ...............56
BI ock Structure .............. ...............56....
Processing Block ............... ....... ... .... .........5
First Level Routines: Source Iteration Scheme ................. .......... ............... 59. ...
Second Level Routines: Sweeping on Coarse Mesh Level ............... ...................6
Third Level Routines: Sweeping on Fine Mesh Level ................ .........................63
Data Structure and Initialization Subroutines ................. ...............65........... ...
Coarse and Fine Mesh Interface Flux Handling ................. ...............66........... ..
5 BENCHMARKING ................ ...............70.................
Benchmark 1 A Uniform Medium and Source Problem ................. .......... ...............70
Benchmark 2 A Simplified CT Model .............. ...............73....
Monte Carlo Model Description............... ..............7
Deterministic Model Description .............. ...............75....
Comparison and Analysis of Results.................... .......... ...............76.....
Benchmark 3 Kobayashi 3-D Problems with Void Ducts............... ...............79.
Problem 1: Shield with Square Void ............ ......__ ...............80
Problem 2: Shield with Void Duct .............. ...............84....
Problem 3: Shield with Dogleg Void Duct. .....__.....___ .........._ ..........8
Analysis of Results ............... .... .. ... ... ..._ .... ..........8
Benchmark 4 3-D C5G7 MOX Fuel Assembly Benchmark ................. .......................89
M odel Description .................. ...............89.................
Pin Power Calculation Results .............. ...............91....
Eigenvalue Comparison............... ...............9
Analysis of Results ................. ...............95................
6 FICTITIOUS QUADRATURE ............ ..... ._ ...............97...
Extra Sweep with Fictitious Quadrature ................. ...............97........... ...
Implementation of Fictitious Quadrature ................. ...............99................
Extra Sweep Procedure............... ...............9
Implementation Concerns............... ...............10
Iteration structure .............. ...............101....
Direction singularity ................. ...............101................
Solver compatibility ................. ...............102......... ......
Heart Phantom Benchmark ................. ...............102...............
M odel Description ................ ............ .. ........... ..... ..........10
Photon Cross Section for the Phantom Model .............. ...............104....
Performance of Fictitious Quadrature Technique ................. ............................106
7 PENTRAN INTEGRATION AND LIMITATION STUDIES OF THE
CHARATERISTICS SOLVER ......__................. ..........._..........11
Implementation of the Characteristics Solver in PENTRAN ................. ............ .........1 10
Benchmarking of PENTRAN-CM .............. ... ........... ...............112 ....
Meshing, Cross Section and Quadrature Set ................ ...............112..............
Benchmark Results and Analysis ................ ...............114...............
Investigation on the Limitations of Characteristics Solver............... .................1 16
M emory Usage .............. ..... .. ......... ...............116......
Limitation on the Spatial Discretization ....._ .....___ .........___ ............1
2-D meshing on the coarse mesh boundaries ....._____ .......___ ...............118
Coarse mesh size limitation for the characteristics solver .................. ...............122
Possible Improvements and Extendibility of the Characteristics Solver. ................... ...127
8 CONCLUSIONS AND FUTURE WORK ................. ...............128........... ...
Conclusions............... ..............12
Future Work................ ...............129
Acceleration Techniques ................. ...............129......... ......
Parallelization ................. ... .. _.._. ......................13
Improvements on Characteristics Solver ................. ...........__........131.__ ....
Other Enhancements ..........._.._. ....._... ...............13. 1....
APPENDIX
A SCATTERING KERNEL IN LINEAR BOLTZMANN EQUATION ............... .... .........._.132
B NUMERICAL QUADRATURE ON UNIT SPHERE SURFACE ............... ... ........._.._. 142
C IS FORTRAN 90/95 BETTER THAN C++ FOR SCIENTIFIC COMPUTING? ...............152
D TITAN I/O FILE FORMAT ................. ...............166.......... ....
LIST OF REFERENCES ................. ...............171................
BIOGRAPHICAL SKETCH ................. ...............175......... ......
LIST OF TABLES
Table page
5-1 CT model run time and error norm comparison with the MCNP reference case .............78
5-2 Kobayashi problem 1 point A set flux results for case 1 .............. .....................8
5-3 Kobayashi problem 1 point B set flux results for case 1 .............. .....................8
5-4 Kobayashi problem 1 point C set flux results for case 1 .............. .....................8
5-5 Kobayashi problem 1 point A set flux results for case 2 .............. .....................8
5-6 Kobayashi problem 1 point B set flux results for case 2 .............. .....................8
5-7 Kobayashi problem 1 point C set flux results for case 2 .............. .....................8
5-8 Kobayashi problem 1 point A set flux results for case 3 .............. .....................8
5-9 Kobayashi problem 1 point B set flux results for case 3 .............. .....................8
5-10 Kobayashi problem 1 point C set flux results for case 3 .............. .....................8
5-11 CPU time and memory requirement for SN and hybrid methods............... ................8
5-12 Kobayashi problem 2 point A set flux results for case 3 .............. .....................8
5-13 Kobayashi problem 2 point B set flux results for case 3 .............. .....................8
5-14 Kobayashi problem 3 point A set flux results for case 3 .............. .....................8
5-15 Kobayashi problem 3 point B set flux results for case 3 .............. .....................8
5-16 Kobayashi problem 3 point C set flux results for case 3 .............. .....................8
5-17 Pin power calculation results for the unrodded case............... ...............92..
5-18 Pin power calculation results for the rodded A case ................. ................ ......... .93
5-19 Pin power calculation results for the rodded B case ................. ................ ......... .94
5-20 Eigenvalues for three cases of C5G7 MOX benchmark problems ................ ................95
6-1 Materials list in the heart phantom model ................. ...............104.............
6-2 Group structure of cross section data for the heart phantom benchmark ........................ 105
6-3 Material densities and compositions used in CEPXS ................ .......... ...............105
6-4 Directions in the fictitious quadrature set for the heart phantom benchmark ..................1 06
6-5 TITAN calculation errors relative to the SIMIND simulation............... ................0
7-1 Memory structure differences between PENTAN and TITAN ................. ................. 111
7-2 Comparison of the characteristics solver in PENTAN-CM and TITAN ................... ......11 1
7-3 One group cross section used in the CT benchmark with TITAN ................. ...............112
7-4 Two group cross section used in the CT benchmark with PENTRAN-CM ................... .113
7-5 Characteristics solver calculated detector response by PENTRAN-CM and TITAN.....114
7-6 Characteristics solver performance in PENTRAN parallel environment ................... .....115
7-7 Error comparison with different z meshing .............. .....................121
7-8 Characteristics solution relative difference to SN solution with different scattering
ratios and coarse mesh size ................. ...............126........... ...
C-1 Run time comparison of the sample FORTRAN and C++ codes............... ................154
D-1 TITAN input file list ................. ...............166........... ...
D-2 TITAN output file list ................. ...............169........... ...
LIST OF FIGURES
FiMr page
1-1 Angular flux formulation of the integral transport equation ................. ........._.._.......20
2-1 Coarse mesh/fine mesh meshing scheme ................. ...............23......_.._...
2-2 Differencing scheme on one fine mesh............... ...............27..
2-3 Schematic of characteristic rays in a coarse mesh using the characteristics method.......31
2-4 A coarse mesh with characteristics solver assigned .............. ...............34....
2-5 Characteristic rays for one fine mesh on one outgoing surface............_ ..........._..__...35
2-6 Bilinear interpolation for the incoming flux ................ ...............38........... ..
2-7 Schematic of the Slo level-symmetric quadrature set in one octant ................. ...............43
2-8 PN-TN quadrature of order 10............... ...............45...
2-9 Ordinate splitting technique ................. ...............46.._._._ .....
3-1 Angular proj section ........._.._.. ...._... ...............49...
3-2 Theta weighting scheme in angular domain. ............. ...............50.....
3-3 Mismatched fine-meshing schemes on the interface of two adj acent coarse meshes........53
4-1 Code structure flowchart............... ...............58
4-2 Pseudo-code of the source iteration scheme ................. ...............59........... ..
4-3 Pseudo-code of the coarse mesh sweep process ................. ...............62..............
4-4 Pseudo-code of the fine mesh sweep process ................. ...............63..............
4-5 Frontline interface flux handling .............. ...............67....
5-1 Uniform medium and source test model ................ ...............71........... ..
5-2 Group 1 calculation result............... ...............71.
5-3 Group 2 calculation result ................. ...............72........... ...
5-4 Group 3 calculation result............... ...............72.
5-5 Computational tomography (CT) scan device ................ ...............73...............
10
5-6 A simplified CT model .............. ...............74....
5-7 MCNP model of the simplified CT device ................. .................. ................75
5-8 SN solver meshing scheme for the CT model .............. ...............75....
5-9 Hybrid model meshing for the CT model .............. ...............76....
5-10 SN simulation results without ordinate splitting............... ...............7
5-11 Quadrature sets used in the CT benchmark .............. ...............77....
5-12 Hybrid and SN simulation results with ordinate splitting ................. ................ ...._.78
5-13 Kobayashi Problem 1 box-in-box layout ................. .....__. ....__. ..........8
5-14 Kobayashi Problem 2 first z level model layout ......____ ..... ... .__ ........_......8
5-15 Kobayashi Problem 3 void duct layout ................. ...............86........... ..
5-16 Relative fluxes for Kobayashi problem 1 .............. ...............87....
5-17 Relative fluxes for Kobayashi problem 2 .............. ...............88....
5-18 Relative fluxes for Kobayashi problem 3 .............. ...............88....
5-19 C5G7 MOX reactor layout............... ...............89.
5-20 3-D C5G7 MOX model .............. ...............90....
5-21 Eigenvalue convergence pattern for the rodded A configuration ................ ................ .95
6-1 Extra sweep procedure with fictitious quadrature .............. ...............100....
6-2 Heart phantom model ........._._.._......_.. ...............103...
6-3 Activity distribution in the phantom model ........._._.. ....__.. ...._.._._..........0
6-4 Globally normalized proj section images calculated by TITAN and SIMINTD.................. 107
6-5 Individually normalized proj section images calculated by TITAN and SIMINTD............107
7-1 Characteristics coarse mesh boundary meshing based on flux resolution requirement...119
7-2 Detector response relative errors with different number of z fine meshes for the
characteristics solver............... ...............120
7-3 Detector response relative errors with different number of z fine meshes for the SN
solver ................. ...............120................
7-4 Detector response sensitivity to the fine mesh number along y axis .............. ..... .........._121
7-5 Detector response comparison between SN and characteristics solver in pure absorber
media............... ...............123.
7-6 Detector response comparison between SN and characteristics solver in media with
different scattering ratio............... ...............124.
7-7 Characteristics solutions with different coarse mesh size along x axis ...........................125
B-1 Chebyshey roots (N =4) on a unit circle ................ ...............146........... .
D-1 A 3 by 3 coarse mesh model on one z level ................. ...............167...........
D-2 A sample bonphora. inp input file ................ ...............167........... ..
D-3 C5G7 MOX benchmark problem bonphora. inp input file ................. ......................168
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
HYBRID DISCRETE ORDINATES AND CHARACTERISTICS METHOD FOR SOLVING
THE LINEAR BOLTZMANN EQUATION
By
Ce Yi
August 2007
Chair: Alireza Haghighat
Major: Nuclear Engineering Sciences
With the ability of computer hardware and software increasing rapidly, deterministic
methods to solve the linear Boltzmann equation (LBE) have attracted some attention for
computational applications in both the nuclear engineering and medical physics Hields. Among
various deterministic methods, the discrete ordinates method (SN) and the method of
characteristics (MOC) are two of the most widely used methods. The SN method is the traditional
approach to solve the LBE for its stability and efficiency. While the MOC has some advantages
in treating complicated geometries. However, in 3-D problems requiring a dense discretization
grid in phase space (i.e., a large number of spatial meshes, directions, or energy groups), both
methods could suffer from the need for large amounts of memory and computation time.
In our study, we developed a new hybrid algorithm by combing the two methods into one
code, TITAN. The hybrid approach is specifically designed for application to problems
containing low scattering regions. A new serial 3-D time-independent transport code has been
developed. Under the hybrid approach, the preferred method can be applied in different regions
(blocks) within the same problem model. Since the characteristics method is numerically more
efficient in low scattering media, the hybrid approach uses a block-oriented characteristics solver
in low scattering regions, and a block-oriented SN solver in the remainder of the physical model.
In the TITAN code, a physical problem model is divided into a number of coarse meshes
(blocks) in Cartesian geometry. Either the characteristics solver or the SN solver can be chosen to
solve the LBE within a coarse mesh. A coarse mesh can be filled with fine meshes or
characteristic rays depending on the solver assigned to the coarse mesh. Furthermore, with its
obj ect-oriented programming paradigm and layered code structure, TITAN allows different
individual spatial meshing schemes and angular quadrature sets for each coarse mesh. Two
quadrature types (level-symmetric and Legendre-Chebyshev quadrature) along with the ordinate
splitting techniques (rectangular splitting and PN-TN splitting) are implemented. In the SN solver,
we apply a memory-efficient 'front-line' style paradigm to handle the Eine mesh interface fluxes.
In the characteristics solver, we have developed a novel 'backward' ray-tracing approach, in
which a bi-linear interpolation procedure is used on the incoming boundaries of a coarse mesh. A
CPU-efficient scattering kernel is shared in both solvers within the source iteration scheme.
Angular and spatial proj section techniques are developed to transfer the angular fluxes on the
interfaces of coarse meshes with different discretization grids.
The performance of the hybrid algorithm is tested in a number of benchmark problems in
both nuclear engineering and medical physics fields. Among them are the Kobayashi benchmark
problems and a computational tomography (CT) device model. We also developed an extra
sweep procedure with the fictitious quadrature technique to calculate angular fluxes along
directions of interest. The technique is applied in a single photon emission computed tomography
(SPECT) phantom model to simulate the SPECT proj section images. The accuracy and efficiency
of the TITAN code are demonstrated in these benchmarks along with its scalability. A modified
version of the characteristics solver is integrated in the PENTRAN code and tested within the
parallel engine of PENTRAN. The limitations on the hybrid algorithm are also studied.
CHAPTER 1
INTTRODUCTION
Overview
The linear Boltzmann equation (LBE) (also called neutron transport equation) describes
the behavior of neutral particles in a system (e.g. a nuclear reactor, a radiological medical
device). LBE is derived based on the physics of particle balance in a phase space composed of
energy, spatial and angular domains. By solving the LBE, we can acquire some insights into the
characteristics of the system. In this work, we developed a hybrid transport algorithm to solve
the LBE, specifically for application to problems containing regions of low scattering. A new
deterministic transport code (TITAN) has been developed based on the new hybrid approach.
The code, over 16,000 lines at present, is written in FORTRAN 95 with some language
extensions of obj ect-oriented features (part of the FORTRAN 2003 standard). TITAN is
benchmarked for several problems.
Linear Boltzmann Equation
The original Boltzmann equation is derived for molecular dynamics of sufficiently dilute
gas, in which only binary interaction is considered.l
8 Fd 8 f
(-+9 V +-) f (F, t) =( )catasso, (1-1)
dt m dv dt
Where,
9 = the velocity of gas molecules.
F'= the position of gas molecules.
f(', v, t)cfidf= the expected number of gas molecules in phase space drdv .
F = external force on molecules.
m = mass of molecules.
Since only binary collision is considered, the collision term on the right side of Eq. 1-1 can
be written as:
dt (1-2)
Where ;, and v'' are the velocities prior to collision, &(6, II, 9) represents the
probability of two molecular collision, and the f (F, 9, ', t) f (F, v'', t) and f (F, 9 t) f (F, v9, t) terms
represent the gain and loss of molecules in the phase space, respectively. Note that the gain and
loss terms are quadratic. Thereby, the original Boltzmann equation (Eq. 1-1) is nonlinear. In
order to solve the equation numerically, one has to linearize the equation first, which could
introduce some system errors to the physics of the real problem to be solved. Usually Monte
Carlo approach is used to solve gas dynamics problems without solving the equation directly.
Fortunately, the neutron, or in general, neutral particle transport phenomenon, can be
simulated with the linear form of the Boltzmann equation. The linear form is valid, because
neutron-neutron interactions are negligible compared to neutron-nucleus interactions. For
example, in a typical reactor, the neutron number density is usually ~15 orders of magnitude less
than the number density of surrounding medium. In such systems, only the neutron-nucleus
interaction is considered, and the medium remains unchanged within the time scope of neutron
transport. This assumption reduces the collision term on the right side of Eq. 1-1 from quadratic
to a linear term. Further simplification can be achieved by assuming F = 0, which is true in most
situations, because neutrons or neutral particles are not affected by electric or magnetic field, and
gravitational force is negligible because of the negligible weight neutrons and zero weight of
gamma rays. With these simplifications to Eq. 1-1, the neutron transport equation, or the linear
Boltzmann equation (LBE), can be written as:
1 8 1C(F~, E, 02, t)
+a,(rFE)(y(F,E, O, t) + DV c(~, E, O,t) =
v St
dE' n's(FEsE~s6)(FE,6't)+(1-3)
4i ~6dE' Sdn' va, (F, E')y(F', E', 0', t) + S(F, E, i2, t)
Where v is the speed of neutron, ry(F', E, 02, t) = v n(F, E, 02, t) is the angular flux,
n(Fl, E, 02, t) is expected number of neutrons in the phase space of d~d~da2, a,, as and at are
total, scattering and fission cross sections of the nuclei in the medium, respectively, X(E) is the
fission spectrum, and S(F',E~, 0,t) is the independent source. The time-independent linear
Boltzmann equation can be written as:2, 3
o, (F,~ E)7(F', E, 0) + 0Z V cy(F, E, 0) =
4x-4
Equation 1-4 is the fundamental equation we are to solve with our code. It represents two
basic types of problems. In operator form, they are:
* Fixed source problem: Hry = S,
* Eigenvalue problem: Hry = Fry
Where the transport operator H and the fission operator F are defined as:
H = 6 V + ,(F,) 6E) d' dn' as (F, E' 4 E, 6' 6 (1-5)
F = E' ive,(FE')(1-6)
And k is the eigenvalue of the system. The fixed source problem is also referred to as
shielding problem, and the eigenvalue problem often is called criticality problem in the area of
reactor physics.
Numerical Methods to Solve the LBE
In the past fifty years, numerous numerical methods have been developed to solve the
transport equation. Two of the most widely used methods are:
* Discrete ordinates method (SN).
* Method of characteristics (MOC).
These methods are often referred to as deterministic methods, as opposed to the M~onte
Calrlo method, in the sense that they are to solve the LBE or its derived formulations directly by
numerical methods. To numerically solve a differential equation, it is required to discretize the
equation in its phase space. In the LBE, the angular flux y/(Fl, E, 02) is defined in a phase space
composed of three domains: spatial, energy and angular domain. In deterministic methods,
generally, the energy domain variable is discretized using the multigroup approximation.4 The
angular domain variables are discretized using the numerical quadrature technique.2 And in the
spatial domain, different methods may take their individual approaches in various geometry
systems. For example, one can divide space into structured or unstructured meshes (SN) with
finite differencing or finite element approach, or arbitrary-shaped material regions (MOC).
Discrete Ordinates Method
The SN method was first introduced by Carlson into the nuclear engineering field in 1958.5
It has been one of the dominant deterministic methods for its efficiency and numerical stability.
In the SN method, Eq. 1-4 is the fundamental equation to solve. And the angular flux is only
calculated in a number of discrete directions. In other words, if we consider the angular flux as a
function defined on the surface of a unit sphere, the SN method evaluates function values at
discrete points on the surface, which are carefully chosen by a quadrature set in order to conserve
the flux moments. In the spatial domain, numerical differencing schemes are required in the SN
method to evaluate the streaming term.
Method of Characteristics (MOC)
Recently with the advancements in computing hardware, MOC has drawn more and more
attentions in both the nuclear engineering and medical physics communities.6, 7 A number of
2D/3D MOC codeS 8, 9 have been developed for reactor physics and medical applications.
Among advantages of the MOC, its ability to treat arbitrary geometrical bodies is an attractive
feature, especially for medical applications, in which the Monte Carlo approach is still dominant.
MOC usually uses the same quadrature technique as the SN method to accomplish angular
discretization. It solves the LBE along parallel straight lines (referred to as the characteristic
rays) instead of discretized meshes as in SN method. The angular flux along a characteristic ray
can be described by the formulation of the integral transport equation:
yr(F,E,A)= g(E) IRdl dE'v (F-l~E)y(F-li~,E',h6)e "Ei
+ dlSx(F -i, ,E,6)e E'""'' B +p~(F R,E,6!)e E''"E
Where TE, (~,-K) ddl'a,( -l'id,E) is the optical path length along the characteristic
ray for particles with energy of E. Figure 1 -1 illustrates the terms in Eq. 1-6, which is the
fundamental formulation for the MOC.
Figure 1-1. Angular flux formulation of the integral transport equation.
The streaming term in the LBE disappears in Eq. 1-6 because of the integration over the
characteristic ray. Therefore, as one benefit, differencing schemes are not required in the MOC.
However, MOC requires a sufficient number of rays in order to adequately cover the spatial
domain. The main disadvantage of the MOC is the need of a large amount of memory to store
the geometry information for the characteristic rays. Since the 3-D MOC could be very
expensive,'o some synthesis methods, coupled 2-D MOC with 1-D nodal/transport method,l
have been developed based on the fact that in most reactor system, flux profile changes relatively
slowly along z axis, comparing to rapidly changing profile over the x-y plane.
Ray-Effects in Low Scattering Region
One numerical difficulty for the deterministic methods is the so-called ray-effects,12
especially in the SN method with structured meshing in a low scattering medium, where the
uncollided flux is dominant. As the distance between a localized source and a region of interest
increases, the number of discrete ordinates that intersect each distant spatial mesh is reduced,
resulting in unphysical oscillations of the scalar flux. Generally, the meshing and the quadrature
set in the SN method should remain consistent. Otherwise, in a system where spatial and angular
domains are tightly coupled, the mismatch between discretization grids in the two domains may
cause the ray-effects.
The ray-effects can be alleviated naturally by increasing isotropic scattering or fission,
since fission is always considered isotropic and an isotropic influence tends to "flatten" the flux
in the angular domain. However, the ray-effects become worse in low scattering medium or a
highly absorbing medium, where the flux is usually highly angular dependent. Therefore, particle
transport problems in low-scattering media often present a difficulty for deterministic methods.
Hybrid Approach
Both the SN and characteristics methods have been studied intensively, and utilized into
many codes. The goal of this work is to solve the LBE efficiently by taking a hybrid SN and
characteristics approach for problems containing low scattering regions. Such problems are very
common in medical physics applications and in many shielding problems.
Both methods numerically solve the LBE by discretizing the angular flux in the spatial,
angular and energy domains. However, they solve different formulations of the LBE, which in
return leads to different spatial discretization approaches. In the general SN method, the
resolution and accuracy of flux distribution depends on the mesh size and the differencing
scheme. In the characteristics method, the resolution of flux distribution depends on the sizes of
flat source regions. And the accuracy of the flux for each region relies on the densities of
characteristic rays. Although the two methods use different discretization methods in the spatial
domain, the same discretization approaches (energy group and discrete quadrature set) can be
used in both methods in the energy and angular domains. Therefore, it is possible to combine
both methods into one code.
The SN method and the MOC are two of most efficient techniques to solve the LBE.
However, in 3-D problems requiring a dense grid in phase space discretization (i.e. a large
number of spatial meshes, directions, or energy groups), both techniques could suffer from the
need for large amounts of memory and computation time. In this work, we developed a new
transport code (TITAN) with a hybrid discrete ordinates and characteristic method, specifically
for application to problems containing regions of low scattering. In this hybrid approach,
different methods can be applied to solve the LBE for a given spatial block (coarse mesh) in a
physical model. The hybrid approach can take advantages of both methods by applying the
preferred method in different regions (blocks) based on the problem physics. Since the
characteristics method is numerically more efficient in low scattering media, the hybrid approach
uses a block-oriented characteristics solver in low scattering regions, and uses a block-oriented
SN solver in the remainder of the physical model.
CHAPTER 2
THEORY AND ALGORITHMS
Multi-Block Framework Overview
To numerically solve the LBE with a deterministic method, discretization schemes are
required in the energy, angular and spatial domains. Once the discretization grid is built in the
phase space, one can evaluate the angular flux on each node by sweeping the grid in a specific
order repeatedly via an iteration scheme (e.g., the source iteration scheme) until solution
convergence is achieved.
The hybrid method is built on a multi-block spatial meshing scheme, which is also used in
the PENTRAN code.13 The meshing scheme divides the whole problem model into coarse
meshes (blocks) in the Cartesian geometry. And each coarse mesh is further filled with uniform
fine meshes or characteristic rays depending on which solver is assigned to the coarse mesh.
Figure 2-1 shows the multi-block framework of the hybrid approach.
Figure 2-1. Coarse mesh/fine mesh meshing scheme.
The multi-block framework leads to an important feature of the hybrid code: both the SN
and characteristics solvers are coarse-mesh-oriented. They are designed to solve the transport
equation on the scope of a coarse mesh. A coarse mesh can be considered as a relatively
independent coding unit with its own spatial discretization grid (fine meshes or characteristic
rays) and angular discretization grid (quadrature set). Users can assign either solver to each
coarse mesh.
We provide the formulations for the block-oriented SN and characteristics solvers, and
demonstrate the two solvers on the multi-block framework. We also discuss the angular
quadrature sets used in the TITAN code along with the ordinate splitting technique.
Discrete Ordinates Formulations
Here, we apply the multigroup theory to discretize the LBE in the energy domain. And we
rewrite Eq. 1-5 in the Cartesian geometry as:3
ax Sy 8:
g =1 l=0l k=1 (l +k). (2-1)
[ ,:(x, y, z) cos(kp) + 7R, (x, y, z) sin(kp7)]) +
va,,,(x, y, z),~, c x ,z o x(,yz ,
ko g=1
Where, pu, r ,and 5 are the x, y and z direction cosines for the discrete ordinates, 8,9 are
the polar and azimuthal angles, respectively. (pu,9) or (u, r,5) specifies a discrete ordinate,
where pu= cos(0), r= sin(0) cos(cp, (=- sin(0) sin~cp). P,,o,,, p(p) is the th Legendre polynomial (for 1 =1,
L where L is Legendre expansion order). And li~k(pu) is the th, kth associated Legendre
polynomial, ryg(x, y, z, pu, ) is the group g angular flux (for g=1, G where G is the number of
groups) at the position of (x, y, z-) and in the direction of (pu, y) #g'ot is the .&th Legendre scalar
flux moment for group g'. g R,I(x, y, z) is (ith kth COsine associated L~egendre scalar flux
moment for group g', and 4P :(x,y y, ) is eth kth Sine associated Legendre scalar flux moment
for group g'at the position of (x, y, z) These flux moments are defined as:
Pi dp'~ 21 dp'1~i
(x )= 12 2x ,, (x, y, z-, p', 9') (2-2)
; ~(x,~"= y, z) u = ~ (-1~ 21 do 2xcos(kp' )lyg (x, y, Z, p'1, 9') (2-3)
Sd'2 2= dp
; x y ) 1 o xsin(k7' )CI/R (x, y, z, pu', 9') (2-4)
And other variables are:
a,: total group macroscopic cross section
as,, ,: .eth moment of the macroscopic differential scattering cross section from g's g .
X,: group Hission spectrmm
ko: criticality eigenvalue
vf, : group Hission production
S,*(x, y, z, u, G) : external source on the position of(x,yi,z) and in the direction of (pl, p)
We can make several observations on Eq. 2-1. First, obviously it accomplishes the
discretization in the energy domain by utilizing the multigroup theory. As a result, ry(~, E, 0)
becomes ry,(x, y, z, pu, ) Secondly in the angular domain, no further discretization is required,
since we solve for the angular flux in a number of discrete directions of (pu,,, ,) n = 1, N where
Nis the total number of directions. The discrete directions are carefully chosen by the quadrature
set so that we can conserve the integral quantities such as scalar fluxes. Thirdly, if we compare
Eqs. 1-5 and 2-1, the most challenging term is the scattering term, in which we convert the
integration over energy and angular domain into numerical summations for energy groups and
Legendre expansion terms. Derivations of the scattering kernel are given in Appendix A. It is
important to note that in Eq. 2-1, the scattering kernel, as well as the fission term, does not
explicitly depend on the angular flux, but on the flux moments. The relationships between the
angular flux and the flux moments are defined by Eqs. 2-2 to 2-4. Finally the streaming term in
Eq. 1-5 becomes a differential term in Cartesian geometry. In order to numerically evaluate the
differentials, differencing scheme is required in the SN method.
Source Iteration Process
Since the terms on the right hand side of Eq. 2-1, including scattering term, fission term
and fix-source term, are not explicitly dependent on the angular flux, we can further simplify Eq.
2-1 by combining all the source terms into one source term.
(PU + 9 +5 z) (x, y, z, pU, 3)+ c, (x, y, z)IYg (x, y, z, pU, 9) = Q, (x, y, z, pU, 9) (2-5)
where Q, = Sscatterin + Sysson or S~x Sscattring, Sysson and Sax represent the three terms on
the right hand side of Eq. 2-1 respectively. Eq. 2-5 can be viewed as a numerical iteration
equation, which usually is called 'source iteration' scheme (SI).2 In this iteration process, Qs is
calculated from previous iteration results. Therefore, we can solve Eq. 2-5 for the angular flux by
taking Q, as a constant. Flux moments can be evaluated by Eqs. 2-2 to 2-4 with the latest
angular flux, then we can use the flux moments to update Q, for the next iteration. This process
is repeated until the 0 'th flux moment is converged under some convergence criterion. The
iteration process for each group (g) can be illustrated as follows:
Step 1: Solve Eq. 2-5 for angular flux y, (x, y, z, pu, 9) .
Step 2: Evaluate flux moments based on Eqs. 2-2 to 2-4.
Step 3: Update the scattering source.
Step 4: Repeat the process from Step 1, until max(| 2- ) <; tolerence .
In Step 1, r7, is calculated for every fine mesh along a given direction, which is referred to
as 'one direction sweep'. After sweeps for every direction are completed, flux moments can be
updated in Step 2. The group iteration (g=1, G) needs to repeat only once for fixed source
problems with only down-scattering, because the scattering source for the current group only
depends on the converged upper group flux moments. The summation over groups in the
g-1 G
scattering term can be reduced to C instead of C However, for problems with up-
g'=1 g'=1
scattering, an outer iteration is required since the scattering source is coupled with lower energy
groups. For eigenvalue problems, another outer loop is necessary so that the fission source and k-
effective can be updated in between two successive outer iterations.
Differencing Scheme
From Eq. 1-5 to Eq. 2-1 to Eq. 2-5, we are finally one step away to numerically solving the
LBE, which is the evaluation of the differencing (streaming) term in Eq. 2-5 by various
differencing schemes.14 As shown in Figures 2-2, Eq. 2-5 applies on a spatial domain of a fine
mesh with the sizes of Ax, Ay and Az on three axes.
Figure 2-2. Differencing scheme on one fine mesh."
Here, we solve for the average flux on the fine mesh.
dx dy dzy,(x1, yzr u ,
Uk ax ay A
Where i, j, k are the fine mesh indices, g is the group index, and n is the direction index.
I'k = Ax~yAZ is the volume of the fine mesh. Now, we can finally complete the discretizations
on all three domains in the phase space. To calculate (7 I,, we integrate Eq. 2-5 over the fine
mesh volume l'k
rasy J raz do X (Ax yz) -1 :(0, y, z)]
+7xdx z d xA, ) x,0,z
(2-7)
+4( dx o d~y (x, yA)- (x ,0
rax ~ay az rax ~ay raz
+cryk d~i dyn dzy"() (x, yz) = dxI dy~ dzg")(x, y,z)
We assume cross sections are constant inside the fine mesh. In a similar way as Eq. 2-6,
we define the fluxes on the three incoming boundaries and the three outgoing boundaries as:
x n = ) dy dzy l") (0, y,/ z)
x u dy d/(/-! (Ax, y, z-)
Ayx Azy
(2-8)
y ou dx J.-~ (x, Ay, z)
axn AzJ -:~
~xy AxA
And the angular source for the fine mesh can be defined as:
Q -r dx dy d~zQ (x, y,z,u ,c) (2-9)
zyk ax ay A
We can divide both sides of Eq. 2-7 by F',, then substitute Eqs. 2-6, 2-8 and 2-9 into Eq.
2-7, and obtain Eq. 2-10.
(Wx out Vxi) (W u yi ut zi a (2-10)
In Eq. 2-10, the three incoming fluxes ( yx n, ryinm and ryzn) can be obtained from the
finnemesh bourlndry cor+ndiins, atn then three, inomngsrfces. Themrefo+nre,toclculate I.: and
the three outgoing fluxes, we need three additional equations, which are provided by the
differencing scheme. One of the simplest schemes is the linear diamond (LD) differencing
expressed by:
x out = 27C., x in -
Iy out = 2 I//,, -/ B (2-11)
When moving in positive directions (as shown in Figure 2-2), we may eliminate the
outgoing fluxes in Eq. 2-10 by using Eq. 2-11 to obtain Eq. 2-12.
~18~"(2-12)
The original LBE (Eq. 1-5) finally reduces to a set of linear equations of Eqs. 2-11 and 2-
12. Note that the incoming surfaces change for different directions. The fine mesh sweeping
order is decided by the octant number of the direction. The same principle is also applied to
coarse meshes: we always try to calculate the outgoing fluxes by solving the LBE based on the
incoming fluxes. In this sweeping process, the outgoing fluxes will be the incoming flux for the
next adj acent fine/coarse mesh along the direction. If the incoming or outgoing boundaries of the
fine/coarse mesh are aligned with the model boundaries, model boundary conditions are applied.
However, for the coarse mesh sweep, flux proj sections are required on the interface of two
adj acent coarse meshes if the two coarse meshes use different spatial and angular discrtetization
grids. The proj section techniques are discussed in Chapter 3.
In Eq. 2-12, the terms of--,-- and C"are always positive, since we always sweep fine
Ax A y Az
meshes along the direction defined by the direction cosines(p,, q,, r,,~), i.e., p,, and Ax either
both are positive, or both are negative. The incoming fluxtes, Qh and o-zyk are positive with their
phys~Cica mening. As a reslt,l I is always positive. However, the outgoing fluxes calculated
by Eq. 2-11 of the linear diamond differencing scheme could be negative, which conflicts with
its physical meaning. In order to avoid negative fluxes, flux zero fix-upl4 is usually applied in the
diamond differencing scheme. Furthermore, the diamond differencing scheme introduces
artificial oscillations in certain contiditions.l For this reason, and to facilitate increasing
accuracy with adaptive differencing, more advanced differencing schemesl7 I, such as DTW19,
EDW20, and EDI21 are implemented in the PENTRAN code. Currently, the diamond and DTW
differencing schemes are applied in the TITAN code.
Characteristics Formulations
Now we further discuss the formulations for the MOC used in the TITAN code. MOC
solves the transport equation for the angular flux along characteristic rays with region-wise
discretization grid (i.e. coarse mesh) in the spatial domain. Since a region can be any shape,
MOC has the ability to treat the geometry of a model exactly. Similar to the coarse/fine mesh
sweep process in the SN method, in the MOC, we still calculate the outgoing flux based on the
incoming flux for each region, and the outgoing flux will be the incoming flux for the next
adj acent region. In the angular domain, we perform this sweeping process for a number of
directions chosen by a quadrature set. Within one region, we assume constant cross sections and
calculate the average flux for the region by filling the region with characteristic rays along the
directions in a quadrature set. Figure 2-3 shows the parallel characteristic rays along direction n
in a square region i.
Outgoing Boundary
Incoming
Boundary
Sink
ly/ ", Region i
Fine mesh centers on the outgoing boundary
*Non fine mesh centers on the incoming boundary
Figure 2-3. Schematic of characteristic rays in a coarse mesh using the characteristics method.
For a given ray of k with a path length of snk, We Solve the transport equation for
'y glnk (1) 0 < 1 < snk Which is the angular flux for group g, along direction n, at position I
alog ay i reio i.Wedentey'nk lg~nk (0) and lyo rk =pg~nk (Slnk) The transport equation
along ray k can be written as:
An v, gInk ,~w~) 9 gk (2-13)
Where Q,,, = Sscatterin + Sys,,, or S~x is the total angular source in region i along direction n
for group g. We assume a constant angular source for each ray in region i along direction n. The
streaming term in Eq. 2-13 can be viewed as flux gradient' s proj section along direction n, which
is the directional derivative of the angular flux. Therefore, Eq. 2-13 can be rewritten as:
+ C,~ lglnk (I gzn (2-14)
Where, I is the path length. Eq. 2-14 can be solved analytically if we know the incoming
flux ry",'k Igink (0) as a boundary condition.
The outgoing flux can be calculated as follows.
SgI
In order to calculate the average angular flux in region i, first we use Eqs. 2-15 and 2-16 to
evaluate the average angular flux for each parallel ray along direction n, which is given by:
(2-17)
egmn glnk g inuk gn Bg nk
~gl Slnk gl ~gl Slnk gz
Where Agink g~7:nk rytk. Then, we evaluate the average angular flux for region i by
summation of average angular fluxes for all the parallel rays along direction n, with a weighting
factor of 6\nk 3AnkSink, Where 3Ank is the width (in 2-D) or the cross sectional area (in 3-D)
which ray (i,n,k) represents. The average angular flux along direction n is expressed by:
gi inkko g Sin nkM Sin (S ink inkR
k i 7 i gi (2-18)
gi in ikikSnkg i ikSn
k k k
Note that the volume (in 3-D) or the area (2-D) for region i can be represented as
V ~ 1Sa enk~Sznk, f 6Ank is Small enough. Since MArk represents the distance between
k k
two adj acent parallel rays, denser rays are required to cover region i as 6Ank decreases.
Therefore, in order to get an accurate region-averaged angular flux with Eq. 2-18, two conditions
are necessary:
* Region i is small, or flux changes slowly over the region.
* Rays are dense enough to cover the region.
Note that similar conditions are required in the SN method in the sense of spatial domain
discretization approach. Generally, in the SN method finer meshes are required to get a more
accurate flux distribution.
The source iteration scheme can be applied to the MOC similarly as in the SN method. Eqs.
2-16 and 2-18, as Eqs. 2-11 and 2-12 in the SN method, are the fundamental equations for Step 1
(the 'sweep' process) in the source iteration scheme, except that the fine-mesh-averaged angular
flux in the SN method becomes region-averaged angular flux in the MOC.
Block-Oriented Characteristics Solver
The block-oriented characteristics solver is different from the general MOC approach, in
the sense that we only apply the solver on an individual block within the multi-block framework.
For a characteristics coarse mesh, we build uniform fine meshing on the boundaries, and draw
the characteristic rays from the fine mesh centers along quadrature directions. We consider the
characteristics coarse mesh as one region. And the coarse mesh space is covered with
characteristic rays. The boundary fluxes with uniform fine meshing grid are used to
communicate with adjacent blocks, since coarse meshes are coupled on their interfaces in the
sweep process.
Backward Ray-Tracing Procedure
Figure 2-4 shows a typical coarse mesh with 5 x 5 fine meshes on the 6 surfaces. Note that
fine meshing is only applied on the surfaces of a coarse mesh to which the characteristics solver
is assigned. The same coarse-mesh volume could be divided into 5 x 5x5 fine meshes if the SN
solver is assigned.
fine esan fu
Figure 2-4. A coarse mesh with characteristics solver assigned.
Now we can demonstrate how we set up rays in a coarse mesh shown in Figure 2-4. In the
'sweep' process, our goal is to calculate the outgoing flux based on the incoming flux. In Figure
2-4, the front surface becomes one of the three outgoing surfaces for the directions in four of
eight octants in a quadrature set. For the other four octants, it becomes one of the three incoming
surfaces. For demonstration purposes, we assume the front surface in Figure 2-4 is one of the
outgoing surfaces. Now we need to calculate the outgoing angular flux for each fine mesh on the
surface for each direction in the four octants. Figure 2-5 shows the characteristic rays associated
with the center fine mesh on the front surface.
Figure 2-5. Characteristic rays for one fine mesh on one outgoing surface.
As shown in Figure 2-5, we draw 12 rays backward from the center of one fine mesh
(located on the front surface) to the incoming surfaces across the coarse mesh. The four different
color rays in Figure 2-5 represent the directions in four octants. Since the intersection positions
are not necessarily at the centers of fine meshes on the incoming boundary, an interpolation
scheme is required to calculate the incoming fluxes at the intersection positions based on the
known incoming fluxes at the fine-mesh centers. Here, we consider an S4 quadrature set which
provides three directions per octant. For directions in 4 of the 8 octants, the front surface is one
of the three outgoing surfaces. Therefore, 12 rays for each fine mesh on the front surfaces are
required. The overall characteristic ray density to cover the coarse mesh depends on both the fine
mesh grid densities on the outgoing boundaries and the number of directions in the quadrature
set. Figure 2-3 also illustrates the characteristic ray drawing procedure in 2-D. The green dots on
the outgoing boundary in Figure 2-3 are located on the centers of the fine meshes. While the red
dots, which represent the intersection points on the incoming boundary, are off-centered.
Advantage of Backward Ray-Tracing
In the characteristic ray drawing procedure, we could choose a 'forward' approach:
drawing the characteristic rays from the fine mesh centers on the incoming boundary to the
outgoing boundary. The outgoing boundary will experience rays intersecting its fine meshes in a
scattered manner. After the outgoing angular fluxes are calculated, an interpolation procedure is
required to proj ect the scattered outgoing flux onto the fine mesh centers.
In a ray drawing procedure, we can always choose a fine mesh center, either on the
incoming boundary or on the outgoing boundary, as one node of each characteristic ray to avoid
interpolations on that boundary. The other node of the ray will be scattered onto the other
boundary, on which interpolations are required regardless since we are interested in the fluxes
only on the centers of the fine mesh grid. An interpolation procedure on the incoming boundary
needs to evaluate the angular flux at the incoming node of each characteristic ray based on the
known incoming fluxes at the structured fine mesh centers. On the other hand, an interpolation
procedure on the outgoing boundary needs to evaluate the outgoing flux at the center of each fine
mesh based on the calculated fluxes at the scattered outgoing nodes of the rays. The difference
between the two choices is: on the incoming boundary, the interpolation procedure is carried on
from structured data points (incoming fluxes on the fine mesh centers) to scattered data points
(incoming fluxes for the rays), while on the outgoing boundary, the procedure is carried on from
scattered data points (outgoing fluxes from the rays) to structured data points (outgoing fluxes on
the fine mesh centers).
In the block-oriented characteristics approach, we choose to fix the interpolations on the
incoming boundary, because it is numerically more accurate and efficient to interpolate scattered
points from structured points than the other way around. For interpolations on the outgoing
boundary, the scattered outgoing nodes of the rays are the known base points. These scattered
points could be too few, or too badly non-uniformly scattered on the boundary, to complete a
relatively accurate interpolation to evaluate the flux on the center of every Eine mesh. For
interpolations on the incoming boundary, the structured, uniformly distributed Eine mesh center
fluxes are the known data points. Four closest Eine mesh centers to any scattered point can
always be found to complete a bi-liner interpolation. Clearly an interpolation procedure on the
incoming boundary is a better choice. The backward ray-tracing facilitates the integration of the
block-oriented solvers.
Ray Tracer
In order to calculate the outgoing flux by using Eq. 2-16, we need to evaluate the incoming
flux, which is located on the other end of the rays on the incoming surfaces. The incoming flux is
known from the boundary conditions if the incoming surface is part of the model boundaries, or
from the outgoing flux for the adjacent coarse mesh in the coarse mesh sweep process. We
assume these fine-mesh-averaged incoming angular fluxes are located on the center of each Eine
mesh on the incoming surface. However, the intersection point on the incoming surface is not
necessarily on the center of a Eine mesh. Therefore, we need to determine the intersection
position of the ray with the incoming surface, and to evaluate the flux at the intersection point by
some interpolation method from the fine-mesh-centered incoming flux array.
In a MOC code, a ray tracer subroutine is required to calculate the intersection point of a
ray with a surface. The coordinates of the points along a ray can be defined as:
x =xo+t-pu
y = yo +t (2-19)
z = zo+t-f
Where (xo,, 0, zo ) is the starting point of the ray, t is path length along the ray, and
(pu, r, ) are the direction cosines. We can substitute Eq. 2-19 into a region boundary surface
oP(2,t)
A --- --- -- -
function to evaluate the coordinates of the intersection points of the ray with that surface and the
path length t (i.e., sl, in Eqs. 2-16 and 2-18). In the MOC, it can be very expensive, in terms of
computer memory, to store the geometry information if the number of rays and the number of
regions are very large. For this reason, 3-D MOC could be prohibitive for a large model. The
block-oriented characteristics solver considers the whole coarse mesh as one region. Therefore,
for Eq. 2-19, the region boundaries become the coarse mesh surfaces. Because the characteristics
solver is designed for solving the transport equation in a low scattering medium, across which we
can expect that the angular flux along the ray does not change significantly, it is possible to use a
relatively large region (i.e. a coarse mesh) for a flat-source MOC formulation.
Interpolation on the Incoming Surface
Based on the positions of the intersection points of rays on the incoming surface of a
coarse mesh, we can further evaluate the averaged flux for each fine mesh by interpolation. As
shown in Figure 2-6, points A, B, C, and D denote the closest 4 neighbors to point P, which is the
intersection point of a characteristic ray across one incoming boundary. We need to evaluate the
angular flux at point P based on the fluxes at the 4 neighboring points.
Figure 2-6. Bilinear interpolation for the incoming flux.
For simplification, we assume the coordinates for the 4 neighbors and point P are A(-1,-1),
B(1, -1), C(1, 1), D(-1, 1) and P(s, t), where s, t are evaluated by the ray tracer. Note that the
actual positions of the Eine mesh centers and point P are proj ected into the coordinates shown in
Figure 2-6, in which A, B, C, D and P are located at (-1,-1), (1,-1), (1,1), (-1,1) and (s, t) for the
interpolation. Two interpolation techniques are applied in the TITAN code. Either of them can be
used to estimate the incoming flux at point P.
* closest neighbor.
ryP is equal to the angular flux at the closest neighbor. For example, in Figure 2-6 ryP Will
be equal to the ry, under the closest neighbor approach.
* bilinear interpolation.
A bilinear interpolation formulation is applied:22
(1- )(1 t)(1- s)(1 + t)
y~st) =y(-1 -1)+ ry(-1, +1)
4 4
(2-20)
(1+ s)(1 t) (1+ s)(1 + t)
+ r(+1, 1) + r(+1, +1)
4 4
Where ry(-1, -1)= r,, ry(1, -1)= ry,, ry(+1, +1)= rye, ry(-1, +1)= ry, and ry(s, t)= ryP'
The truncation error indicates the bilinear approach is a second order interpolation. And it should
be more accurate than the first approach, which is a first order interpolation. However, we should
note that these point-wise angular fluxes are actually averaged values: Eine-mesh-centered fluxes
(ry,, ry,, ry, and ryD) are the averaged fluxes on the Eine meshes, and the ray intersection-
point flux ( rP,) is the averaged flux on the cross sectional area (3Ank in Eq. 2-18) of the volume
the ray represents. An assumption is made that the averaged flux happens at the center of the Eine
mesh, or at point P of the ray cross section area. This assumption is reasonable if the fine mesh is
small. Therefore, our ray solver may require a relatively finer meshing on the coarse mesh
surfaces, which leads to denser rays in the coarse mesh and longer computer time and memory
requirements. On the other hand, if the fine mesh is relatively large, the closest neighbor
interpolation scheme is not necessarily less accurate than the advanced bilinear interpolation.
The most suitable interpolation scheme could depend on the problem and its modeling. By
default, the bilinear interpolation scheme is used in the TITAN code.
In the characteristics solver, the cross sectional area represented by each ray (defined in
Eq. 2-18) can by calculated by the following formulation:
6A,., = S,., x cos(0) (2-21)
Where S,, is the fine mesh area on the outgoing boundary, and B is the angle between the
ray direction and the direction normal to the boundary. Even with a uniform fine meshing
applied on the surfaces of a coarse mesh for the characteristics solver, rays are not necessarily
distributed uniformly within the coarse mesh volume, because rays along a certain direction can
form different angles with the normal directions of the three incoming surfaces of the coarse
mesh. Non-uniform ray distribution could lead to the requirement of denser rays and/or smaller
coarse meshes to maintain accuracy of the bi-linear interpolation.
Quadrature Set
We discussed the formulations for the SN and characteristics solver, respectively. Our
focus has been on the Step 1 of the source iteration scheme, which is to solve the transport
equation for the angular flux. For Steps 2 and 3, the formulations are fundamentally the same for
both solvers because of the following similarities between two methods:
* Calculate the angular flux, although with different formulations.
* Apply the same energy and angular domain discretization approaches.
* Use the source iteration scheme.
The maj or difference between the two methods is the discretization method in spatial
domain. Both block-oriented solvers share the same goal to calculate the outgoing angular fluxes
for a block. However, they complete the task with different formulations of the original LBE.
Now we can further demonstrate Step 3 of the source iteration scheme. In both methods, we
denote the source term in Eq. 2-5 or Eq. 2-15 by:
Q = Sscatteng +Sso,,,, or Sysi (2-22)
For simplification, we omit the index for energy group, direction, and fine mesh (SN) or
region (MOC). In Eq. 2-22, Sax is known as external source. Sscattenn and Susso,, can be
evaluated from flux moments calculated from the results of the previous iteration.
g= =1 =0 k1(+).(2-23)
[ X,(') rnob/ cos ky,() + #'sin(ky,,)])
Where i is the iteration index, g is the energy group index, I and k are the Legendre
expansion indices, (pu,,, 9,) specifies direction n in the quadrature set, ~X"' k' ( and Xk ')
are the flux moments calculated from the last iteration, which is indexed by i-1 here, and x is the
fine mesh index in the SN formulation, or the region index in the MOC formulation.
The scattering kernel defined by Eq. 2-23 can be expanded to an arbitrary Legendre order
if the same order of cross section data is provided. The isotropic fission source and the k-e~ffctive
can be evaluated by Eqs. 2-24 and 2-25 from an outer iteration.
fission k(II E r ', #g O,x e (2-24)
g'=1
`fissl~ o
Where <> denotes the integration over the entire phase space. Note that j is the outer
iteration index, while in Eq. 2-23 i is the inner iteration index. Scattering source is updated after
one sweep is completed for each group, while the fission source is updated only after all groups
are converged based on the previous fission source.
Equations 2-23 and 2-24 are the formulations for Step 3 in the source iteration scheme. For
Step 2, we use a quadrature set to evaluate the integral over angular domain defined in Eqs. 2-2
to 2-4 for flux moments.
1 ",M ~~~
8 ,, 2
k 1 7 p, ~~y, (2-26)
8 =
8 =
Here, for simplification, we drop the indices for energy group and fine mesh or region.
Direction n can be specified by (pu,,,p,) where -1< p,, <1,0 O <9, < 27<, or
(p,,, r,,, () where -1 < p,,, r,,, <1 p,2 + 9,, + 4,, = 1 In order to preserve symmetries, a
quadrature set only specifies directions in the first octant (0 < p,, q,, r,,~ <; ), directions in the
other octants can by acquired by changing the signs of p,,, 9,,, and/or (,. For
example, (-pu,,,-r,,,-(,) specifies the opposite direction corresponding to directions, {)
in another octant. Direction (pu,,r, ,,,,) and all its seven corresponding directions in other
octants have the same weight (w,,). Usually, we keep the total weight for all directions in one
octant equal to one. These directions and the associated weights (w,,) are carefully chosen by a
quadrature set, so that we can accurately evaluate the moments of direction cosines and the flux
moments defined by Eq. 2-26. Other concerns related to the physics of the problems can affect
the choice of the directions too. Further discussions are given in Appendix B. Currently, in the
TITAN code, we have two types of quadrature sets available: the level-symmetric quadrature5
and the Legendre-Chebyshev quadrature.23
Level-symmetric Quadrature
Figure 2-7 shows a level-symmetric quadrature with an order of 10 (Sl0). We use a point
on the unit sphere to represent a direction. The xyz coordinates of the point are the three direction
cosines of the direction. These directions are ordered with a 'triangle shape' formation. To
generate a quadrature set, we need to find the direction cosines and the weights for all the
directions.
Figure 2-7. Schematic of the Slo level-symmetric quadrature set in one octant.
Slo specifies 15 directions in the first octant on 5 levels. Directions in the other seven
octants are chosen to be symmetric to the directions in the first octant. Therefore, the total
number of directions on the unit sphere is 15 x 8 = 120 for all 8 octants. Generally, for a level-
symmetric quadrature with an order of N, we can calculate the number of levels L, and total
number of directions 2~in the first octant by:
N N x (N + 2)
L = ,M = (2-27)
2' 8
To keep a symmetric layout of the directions, N is always chosen from even numbers. The
level-symmetric quadrature set is widely used in the SN codes for its rotation invariance property
and preservation of moments. Rotation invariance keeps the quadrature directions unchanged
after 90 degree rotation along any axis. In other words, if (pu,, ,q,,,) is one direction in the first
octant of the quadrature set, any combinations of p,,, 9,,, and (,, such as (u,,~, (, q,) or
({,, q,,, p,,) are also defined in the first octant of the quadrature set. Note that rotation invariance
is different from octant symmetry of the directions, where (fip,,f,fmi,) defines the eight
symmetric directions in the eight octants. Rotation invariance is very desirable in many real
problems to keep the symmetry, especially when reflective boundary conditions are applied.
However, it also places a strict constraint on the choice of the quadrature directions. The
symmetry condition requires pu,, r,,~ (for 1 <; i, j, k <; following the same sequence.
f t, = 77, = rk for / j, k = 1, 2, -, N/ 2
ft, =ft, C~i- 1)(2-28)
C = 2(1 3 p,")
N-2
In Eq. 2-28, only pu, is free of choice. The remaining degrees of freedom on direction
weights are used to conserve the odd and even moments of pu, r, and 5.1o
ij .~=~ .: ij (2-29)
w ,, = w ,,9 w,,, ,' = 0 for nodd
w,,,i~ p,"t =C W,,,q =C W,i,,'= f or n even~n < L
m~l ml m~l n+1
The directions and their associated weights can be calculated by Eqs. 2-28 and 2-29. Level-
symmetric quadrature only can conserve moments to an order of maximum L=N/2 because of the
symmetry condition. Another disadvantage of level-symmetric quadrature is that Eqs. 2-28 and
2-29 lead to negative weights if Nis greater than 20. Negative weights are not physical.
Therefore, they cannot be used. This means that the order of Level-Symmetric quadrature is
limited to 20.
Legendre-Chebyshev Quadrature
The Legendre-Chebyshev quadrature,23 also called PN-TN quadrature, aims to conserve
moments to a maximum order without the constraints of the symmetry condition. Figure 2-8
shows a P -TN S1o quadrature layout.
Figure 2-8. P -TN quadrature of order 10.
The Legendre-Chebyshev quadrature conserves moments to the order of 2L-1, instead ofL
in the level-symmetric quadrature set (L=N/2), at the cost of lack of rotation invariance.
Moments in Eq. 2-28 cannot be conserved strictly in the PN-TN quadrature.24 Note that Figures 2-
7 and 2-8 share a similar triangle-shaped direction layout on the unit sphere, because Eq 2-27
still holds in the PN-TN quadrature. The direction weights are positive definite in the PN-TN
quadrature. Therefore, unlike the level-symmetric quadrature set, the PN-TN quadrature order is
unlimited mathematically, except for the limitation of computer memory limitation.
We have derived the procedure on how to build the PN-TN quadrature on the unit sphere.
Based on the procedure, it can be shown that the PN-TN quadrature is the best choice in
mathematically conserving higher moments. We also have proved the positivity of weights in
PN-TN quadrature. Details of the above derivations are given in Appendix B. To build a PN-TN
quadrature set, it is required to find the roots of an even order Legendre polynomial. These roots
are used as level positions of the quadrature. A modified Newton' s method is applied. Details of
the algorithm also are given in Appendix B.
Rectangular and PN-TN Ordinate Splitting
Ordinate splitting is a technique associated with a quadrature set.25 A selected direction in
a quadrature set can be further split into a number of directions. The total weight of the split
directions is equal to the weight of the original direction in the quadrature. We apply the ordinate
splitting techniques to solve problems with highly peaked angular-dependent flux and/or source.
Two splitting methods, rectangular splitting and PN-TN splitting are available in the TITAN code.
Figure 2-9 depicts the two splitting directions for one direction of an Slo quadrature set. Note that
ordinate splitting technique is independent of choice of quadrature set type or order, and can be
applied to as many directions as necessary.
A B *
Figure 2-9. Ordinate splitting technique. A) Rectangular splitting. B) P -TN splitting.
In the rectangular splitting technique, the split directions are uniformly distributed within a
box-shape region centered at the original quadrature direction. In the TITAN code, the size of the
box can be defined by users. The total number of splitting directions can be calculated from the
user-specified splitting order with Eq. 2-30.
s = (21 -1)2 (2-3 0)
Where s is the total number of splitting directions, I is the splitting order. Figure 2-9A
shows the 25 split directions for a rectangular splitting with an order of 3. All the splitting
directions are equal-weighted, defined as ws = -w,,, where w,, is the weight of the original
direction, which remains in the quadrature set after splitting with a reduced weight.
The rectangular-shaped layout of the split direction may not be efficient in conserving the
moments. We developed the Legendre-Chebyshev (PN-TN) splitting technique based on the
regional angular refinement (RAR) technique.26 In the PN-TN splitting, the original direction can
be associated with a local area on the unit sphere surface centered on the original direction. And
the range of the area can be decided by users as in the rectangular splitting. The technique
proj ects the directions in the first octant of a regular PN-TN quadrature set with an order of 21 (1is
the splitting order), into the local area. For a regular PN-TN quadrature, usually there is only one
direction on the top level as shown in Figure 2-8. For the local PN-TN quadrature fitted in the
splitting technique, users can specify the number of directions on the top level. The number of
directions on the following levels increases by one from the previous level, as for a general PN-
TN quadrature. Therefore, the total number of split directions can be calculated by:
(2t +1- 1)-1I
s = (2-3 1)
Where t is user-specified number of directions on the top level, and I is the splitting order.
The weights of the split directions are calculated in the same way as a general PN-TN quadrature,
except that we normalize the total weight to the original direction weight, instead of unity as in a
general PN-TN quadrature. The split direction weights is calculated by Eq. 2-32.
w, = w,, wS P ST, (2-32)
Where w,, is the original weight of the splitting direction, wS P and ws r are the level
weight and the Chebyshev weight, respectively for one split direction in the local PN-TN
quadrature. Note that unlike the rectangular splitting, the original splitting direction is dropped
off after splitting in the PN-TN splitting technique. However, the split directions could be more
'uniformly' distributed within the splitting region than the rectangular splitting, since it is formed
'uniformly' on a sphere surface instead of a rectangular region, and also the PN-TN quadrature
conserves integration more accurately than an equal-weighting formulation. In Chapter 5, we
will use the splitting techniques on one benchmark problem.
At the end of this chapter, we quote a comment on different deterministic methodologies
by Weinberg and Wigner.27 The comment was made about half a century ago, yet even today, it
provides us some insights on this matter.
At present, 11 ithr so much of the practical work of reactor design being done 11 ithr large digital
computers, the arguments in favor of one method of approximation rather than another tend to
center around the question of how well suited the method is for digital computers. Actually, as
the computers become larger, the choice between methods becomes less and less clear: any
method which converges will do if the computer is large enough. This viewpoint certainly has
practical merit; however, convenience for a digital computer is hardly a substitute for intrinsic
nzathentatical beauty or physical relevance. hz this respect the spherical harmonics method is
perhaps most satis~iing; its first order is identical as ithr diffusion theory, and its higher orders
show the deviations fions diffusion theory very clearly.
Alvin M~ Weinberg & Eugene P. Wigner, 1958
CHAPTER 3
PROJECTIONS ON THE INTERFACE OF COARSE MESHES
The TITAN code is built on the multi-block framework with the source iteration scheme.
Both the block-oriented SN and characteristics solvers can apply an individual quadrature set and
fine-meshing scheme on each coarse mesh. Transport calculations can benefit from the multi-
block framework, which provides users more options on the choices of discretization grids in
different regions of a problem model. However, the benefits are not free in term of
computational cost. In Step 1 of the source iteration scheme, while sweeping across the interface
of two coarse meshes, we need to proj ect the angular flux on the interface from one frame to the
other, if the two coarse meshes use different quadrature sets and/or fine-meshing schemes.
Therefore, angular and spatial proj section techniques are developed to transfer the interface
angular fluxes in the coarse-mesh-level sweep process.
Angular Projection
Angular proj section is triggered by the two adj acent coarse meshes with different
quadrature sets. Figure 3-1 shows the layout of directions in two quadrature sets.
A B
Figure 3-1. Angular projection. A) Level-symmetric Slo (red) to PN-TN S1o (green). B) Slo to Ss.
Figure 3-1A compares the directions for the level-symmetric and PN-TN quadrature sets of
order 10. Figure 3 -1B presents a more general situation of angular proj section: from a higher
order quadrature to a lower order quadrature, or vice versa. In general, an angular proj section
from quadrature P to quadrature Q is used to evaluate the angular fluxes for the directions in
quadrature Q for each fine mesh on the interface, based on the angular fluxes from quadrature P.
For each direction 0,, in quadrature Q, we search for the closest three neighboring directions in
quadrature P to 02, The angular flux for 0,, can be calculated by a --weighting scheme,
where m is a positive integer, and 6 is the angle between 0,, and one neighbor direction in
quadrature P. Note that 6 also represents the shortest distance between 0,, and its neighbor on
the surface of a unit sphere. As shown in Figure 3-2, P1, Pt, and Pg are the three closest
neighbors in quadrature P to 0,, in quadrature Q.
P2 xL P3
Figure 3-2. Theta weighting scheme in angular domain.
If we consider that the distances between 0,, and the three closest neighbors are 8 ,,
and 6,, respectively, then the angular flux at 0,, can be written as:
ci, if 6, = min(Oz, 8 83,O) <10
ii 1` (3-1)
Where f(")is the m 'th normalization factor and defined as f(") = + -+ Note
that we set the angular flux at 0,, equal to the closest neighbors, if the minimum distance is less
or equal than 104 radians.
The 0 'th moment (scalar flux) and the first moment (flux current) of the angular flux have
to be conserved after an angular proj section. Therefore, we need to maintain:
=1- ]=1
Where, N and M~ are the total number of directions in one octant in quadratures P and Q,
respectively. p,~ is the cosine of the angle between the interface normal direction and direction i
in quadrature P. p,~ is the cosine of the angle between the interface normal direction and
direction j in quadrature Q. And w's are the direction weights. Note that the total weights are set
to one for both quadrature sets (f~~ w [w i = 1). In order to evaluate (, ca, while conserving
=1- ]=1
the scalar flux and the current, we assume (y,, is a linear combination of cy," and cy,7
Where, ,," and 7' are calculated with Eq. 3-1 with m=1, 2, respectively. And a and P
are the linear coefficients, which can be evaluated by substituting Eq. 3-4 into Eqs. 3-2 and 3-3.
a=1 ,=1 ,=1 (3=1
M M)
]=1 ]=1 ]=1 ]=1
Once r ),,re 2, a, and p are evaluated by Eqs. 3-1, 3-5, and 3-6, ry, can be calculated by
Eq. 3-4. Under this angular proj section scheme, the scalar flux and the first flux moment remains
the same for each fine mesh on the interface before and after the proj section. It is also possible to
conserve higher moments at additional computational cost. We can always introduce higher
1 1
order weighting schemes with Eq. 3-1 (e.g. 3 4), then more terms and coefficients can be
added in Eq. 3-4. In order to calculate the linear combination coefficients (a, P, y etc.), higher
moment conservation equations can be introduced besides Eqs. 3-2 and 3-3. Although the
scattering source term defined by Eq. 2-23 is calculated with all flux moments up to the order of
L, generally it is not necessary to conserve flux moments with an order higher than one on the
interface, since only the 0 'th and first moments carry physical meanings (scalar flux and flux
current), other than just a mathematical term.
In the TITAN code, we also apply a negative fix-up rule to keep the positivity of angular
fluxes by relaxing the 0 'th and/or the first moment conservation rule if necessary. The angular
proj section can be used with any type of the quadrature set. It is also compatible with the ordinate
splitting technique. In order to perform a relatively efficient angular proj section, it is
recommended that both proj ecting and proj ected quadrature sets have at least three directions per
octant (i.e. at least S4). If there is only one direction in one octant (i.e. S2), the direction can be
91'1,1 Z st3.!] iFJe,~)
~III~ I I r~2,1)
~4 ~21) 1 I rt2~2)
8~!,3) 1 gI~3) 1 ~~.~ 3)i
considered as three directions with the same position and only one-third of the original weight,
so the above angular proj section procedure still can be performed without any modifications.
Spatial Projection
Spatial proj section is triggered if the fine-meshing schemes mismatch on the interface of
two adjacent coarse meshes. Figure 3-3 shows a projection situation between a 3x3 meshing
scheme and a 2x2 meshing scheme.
.e
,
,r
,-
,r
A B
Figure 3-3. Mismatched fine-meshing schemes on the interface of two adjacent coarse meshes.
A) 3-D layout. B) 2-D layout.
In Figure 3-3B, we denote the 3x3 fine meshes on the green surface as g(1,1), g(2,1) ..
g(3, 3), the 2x2 fine meshes on the red surface as r(1,1), r(2,1) ... r(2,2). The average angular
fluxes on these fine meshes can be referred to as r (1,1(~) E U(,(3,3) and ry, (1,1) a ry, (2,2).
Assuming a green-to-red proj section, we need to calculate ry(, (1,1) a ry(, (2,2) based on
'ya,(1,1) a y,(3,3) by an area weighting scheme. Here, we only demonstrate how to calculate
the angular flux on fine mesh r(1,1). The rest of the red meshes can be evaluated based on the
same approach.
w (1,1) A(1,1)+ lyg(1,2) Al + lyg(2,2) Az + ryg(2,1) A,
A, (1,1) + A, + A, + A3 (3-7)
= f,(1,1)- gg(,(1,1)+ fg (1,2)- gyg,(1,2)+ fg (2,2)- gyg,(2,2)+ fg (2,1)- gyg,(2,1)
Where A y, Az, and As are the shade areas in Figure 3-3B. Ag(1,1) is the area of Eine mesh
g(1,1). Since Eine meshes are uniformly distributed on either surface, we can denote
Ag (1,1) = Ag Note that Ay = Ag (1,1) + A1 + Az + A3 is the area of fine mesh r(1,1). Therefore, the
factor f p, can be denoted as:
A, A, A, A,
A)~g(1,1)= A1 f',(1,2)= Al ',(2,2)=A -, f ,,(2,1)= (3-8)
If we assume a red-to-green proj section, lyg(1, 1) lyg(3, 3) will be evaluated based on
y,g (1, 1) a ry,g (2, 2) The same area weighting scheme can be applied:
A, Ag A3 (3-9)
The area weighting scheme can conserve the angular flux for each fine mesh, assuming a
flat flux distribution within fine meshes. Therefore, the total angular flux over the entire interface
is conserved automatically. The post re-normalization process described in the angular projection
is not necessary in spatial proj section. In the TITAN code, we separate the 2-D proj section to two
single 1-D proj sections in order to reduce computation cost. For example, a 2-D 3 x 8 4 6 x 4
proj section can be separated as a 3 4 6 proj section along x axis, and an 8 4 4 proj section along y
axis, because x and y proj sections are actually independent of each other. Generally, a projection
pair, n am and na 4 n, require 2 x n x na memory units to store the geometry meshing factors
(f/,, fev)). However, since most of the factors are zeros, we store only the non-zero factors with
a sparse matrix for each proj section pair. Note that the factors in an n am proj section remain the
same whether they are applied in an x or y axis proj section.
Projection Matrix
Both angular and spatial proj sections could be expensive in the source iteration scheme,
because for every iteration, they are performed whenever the 'sweep' processes cross the
interface of two coarse meshes with different angular or spatial frame. If both proj sections are
required on an interface, we perform the angular proj section first, then the spatial proj section. A
proj section from coarse mesh A to coarse mesh B on the interface can be described as
We = P I *A(3-10)
Where P,4B is a proj section matrix, which stores all the necessary geometry information on
the interface. Since proj section matrices are independent of angular fluxes, they can be calculated
and stored before the sweep process starts.
CHAPTER 4
CODE STRUCTURE
The fundamental structure of the TITAN code is built on the four steps of the Source
Iteration (SI) scheme with the multi-block framework. And the SN and characteristics solver
kernels are integrated in Step 1, in which we apply the 'sweep' process to solve the LBE for
angular fluxes. 'Sweep' is a process to calculate the outgoing flux from the incoming flux for
a coarse mesh, a Eine mesh (SN), or ai region (characteristics) by simulating the particle
transport along certain directions. The Eine mesh/region averaged angular fluxes are updated
during the process. In Step 2, we evaluate the flux moments based on the angular flux
calculated in Step 1 by a numerical quadrature set, then use the flux moments to update the
source in Step 3 for next iteration. The iteration process continues until fluxes are converged
based on a convergence criterion.
In this chapter, first we introduce the overall block structure of the code. Then, we
further discuss the transport calculation block, with some details of several key subroutines.
Finally, the front-line style sweep process is presented.
Block Structure
The TITAN code is composed of three maj or blocks: input, processing, and output. The
input block loads the input decks to initialize the model material and the Eixed source
distribution, meshing scheme, and some control variables. The processing block performs the
transport calculation. And the output block handles the calculation results. In this section, we
introduce the input and output blocks. The processing block is discussed in the next section.
The input decks include the cross-section data file, PENMSH-style input Hiles to build
up the model geometry,28 29 and a block-structured input fie (bonphora.inp), to setup some
control variables such as quadrature sets and solvers for each coarse mesh. By default, the
output block writes up the material number, the source intensity and the calculated scalar flux
for each fine mesh into a TECPLOT-format binary data file. The data in this file is organized
by coarse meshes. Each data point/fine mesh is composed of an array of values: xyz
coordinates of the center of the fine mesh, material number and fixed source intensity in the
fine mesh, and the average scalar flux for each energy group. Comparing to the ASCII format
of the TECPLOT data file, the binary file is smaller in size and faster to load by TECPLOT
for various plotting. As an option, the output block can also prepare the input deck for the
PENTRAN code. More details about TITAN I/O file format are given in Appendix D.
Processing Block
The subroutines in the processing block can be roughly arranged in four levels. The
lower level routines are called only by the immediate upper level routines. The top level (0th
level) routines choose the corresponding module for different types of problems (shielding or
criticality). The first level routines setup the source iteration schemes for all energy groups.
The second level routines complete one system sweep for all the directions in the quadrature
sets for one group. The third level routines only handle one sweep for all the directions in one
octant for one coarse mesh and one group. Finally on the forth level, we apply the SN or
MOC formulations discussed in Chapter 2 to calculate the angular flux in one fine mesh (SN)
or one region (characteristics). Figure 4-1 shows the maj or subroutines within the four-level
code structure. In the following sections, we further discuss some of the routines on each
level .
LO. 2 Processing Block
LO.21 TransCal LO.22 UpScaCal
LO.23 Ksearch LO.24 Ksearch up
L1.2 GetInMnt G L1.3 SolverSn_L1_S1 L1.4 Undates~S cal 15Fsinr
LO.21 loop : group=1,num grp
LO.22 outer loop for upscattering
LO.23 or LO.24 k outer loop for criticality problems
II L2.3-2 L2.5-1 L2.5-2
SInitCM Ray I I I FreeCM Snl I FreeCM Ray
Figure 4-1. Code structure flowchart.
On the top level, TITAN has a simple three-block structure: input block, processing block,
and output block. In the processing block, four kernel subroutines are available for different
types of problems:
LO.21 TransCal: fixed source problem with only down scattering.
LO.22 UpScaCal: fixed source problem with upscattering.
LO.23 Ksearch: criticality problem with only down scattering.
LO.24 Ksearch~up: criticality problem with upscattering.
Based on some parameters from the input block, we choose one of the four subroutines to
perform the transport calculation. TransCal provides the fundamental loop structure of the
source iteration scheme. Here, we assume that the source iteration scheme starts from the energy
group loop. The other three subroutines require one (LO.22 and LO.23) or two (LO.24) additional
outer loops besides the fundamental source iteration scheme loop structure (LO.21). They are
designed for problems with upscattering and/or criticality problems.
First Level Routines: Source Iteration Scheme
The flowchart on the first level demonstrates the structure of the processing block. The
subroutines on this level can be illustrated in the following pseudo-code.
!! Pseudocode: processing block (TransCal, UpScaCal, Ksearch, Ksearch_up)
Call InitSn
Loop outer~k k loop(power iteration) if eigenvalue problem
Loop outer_g outer_ g loop ifupscattering presents
For g=1, num~group group loop
call GethInntG(g)
while (flux not converged) within group loop
call SolverSNL1_S1(g)
call w..I.. ck..T l.u-
end wti group loop
end outer~g loop ifupscatten'ng presents
call FissionSrc ifk loop presents
End outer~k loop
Figure 4-2. Pseudo-code of the source iteration scheme.
Subroutine L1.11 nlithG is designed to complete the initialization works before the transport
calculation starts. This initialization includes loading cross section data, allocating memory for
interface fluxes, angular fluxes, and flux moments, and initialization of the quadrature sets and
proj section matrices.
Subroutine L1.2 Getln2~ntG is called at the beginning of each group loop. And it has only
one input argument: group index g. Getln2~ntG(g) calculates the flux moment summation for
all other groups other than group g, which we call scattering-in-moments, or in-moments. In-
moments are used to efficiently calculate the scattering source, which is performed in Step 3 of
the source iteration scheme. By applying the in-moments, we can rewrite Eq. 2-23 by switching
the group and Legendre order expansion.
S uttering = (+1s,g g,tx{Pll\~n/)V ') +2 (1k)Pl
g= =1 =0 k=1 (1 + k).
= 2+1 P P) aG iX") +usg x- 1(41
l=0 g =1
2 (1- k) lkn) COs(ky n) -[C as~R,rg g,lt. +asgg~~
k=1 (1+k). '1sggl~~Cglx
2 (1 k) Pk n) -Sin(ky7 )-[C a;,G ,g,l,xV gXk(l +uses ,(il x
k=1 (l+ k). '1sgglxV ~
In Eq. 4-1, the terms of a O x ,2,adgG~ ~ :1 r
g =1 g =1 g =1
g'R g'R g'R
defined as zero in-moments, cosine in-moments and sine in-moments. Mathematically, this
formulation seems more complicated than Eq. 2-23. However, it is more efficient to evaluate
scattering source. The in-moments can be pre-calculated before the within-group starts, since
they are independent of group g moments, which are the only changing moment terms between
the within-group loops. Therefore, once the in-moments are pre-calculated by the subroutine
Getln2~ntG, the summation process over all groups inside the within-group loop reduces to a
two-term summation: in-moments plus the group g moments.
Inside the subroutine Getln2~nt G, we calculate the in-moments for all the coarse meshes.
If the characteristics solver is assigned to a coarse mesh, Subroutine L1.2-2 Getln2~nt ray is
called to calculate the in-moments for each region in the coarse mesh. Otherwise, L1.2-1
Getln2~nt Sn is called to calculate the in-moments for each Eine mesh within the coarse mesh.
Subroutine L1.3 SolverSnL is the kernel subroutine on this level, which completes one
system sweep for a given group g. Its structure is illustrated on the next level. Subroutine L1.4
UpdateScaFlx is used to calculate the scalar fluxes for the current iteration, and evaluate the
maximum difference from the previous iteration. SolverSnL and UpdateFlx are the two maj or
subroutines of the within-group loop. They are repeatedly called until the maximum scalar flux
difference between two interations satisfies the user-defined convergence criterion.
L1.5 FissionSrc is called at the end of each k-effective loop (power iteration) to update the
fission source and the k-effective for the next power iteration. The fission source is considered as
an isotropic Eixed source for all the other inner loops (within-group loop and upscattering loop).
Fission source is evaluated for each Eine mesh. Then, the k-effective is calculated by using Eq. 2-
25. More advanced formulas derived from power iteration acceleration techniques can be
investigated and applied within the scope of this subroutine.
Second Level Routines: Sweeping on Coarse Mesh Level
The subroutines on this level are called by the kernel subroutine SolverSN LS1 of the
first level. Two inner loops, octant loop and coarse mesh loop are constructed in
SolverSN LS1. Its structure can be illustrated in the following pseudo code.
!! Pseudocode: SolverSn_L1_S1 (group) !group: energy group index
For octant=1, 8 octant loop
call MapBnd2inter(octant,group)
call SweepOrder~cn(octant)
for cmjijk in the sweeping order !coarse mesh loop
if (MOC solver is assigned to cm~ijk)
call InitCnlRay(cn1ijk)
call SolverRayL2_S1(cn1_ijk, octant, group)
call FreeCnlRay(cn1ijk)
else
call InitCmSn(cn1ijk)
call SolverSnL2_S1(cn1ijk, octant, group)
call FreeCmSn (cn1ijk)
endif
end cni loop
call MapInter2Bnd(octant,group)
end octant loop
call CalMnt(group)
Figure 4-3. Pseudo-code of the coarse mesh sweep process.
Subroutines L2. 4-1 SolverRayL2_S1 and L2. 4-2 SolverSn_2S1 are the kernel
subroutines, which complete the sweep process within the scope of one coarse mesh for
directions in one octant and for a given group by using either the characteristics solver or the SN
solver. The detail structures of the two subroutines are illustrated in the next section.
Subroutines L2.1Ma2pBnd2inter and L2. 6Maphiter2Bnd are used in the sweep process on
the system level. The sweep process starts from the three incoming boundaries of the model for
the directions in a given octant, and ends at the three outgoing boundaries. At the incoming
surfaces, model boundary conditions need to be applied. And if the outgoing surfaces are
reflective or albedo boundaries, the outgoing angular fluxes need to be reflected back as
incoming fluxes for directions in another octant. Therefore, at the beginning of the system sweep
process, MapBnd2inter is called to map the incoming system boundary conditions to a system
interface flux array, while at the end of the sweep process, Maplnter2Bnd is called to map the
system interface flux back to the model boundary.
Subroutine L2. 2 SweepOrderC2~initializes the coarse mesh sweep order for directions in
a given octant before the coarse mesh loop starts. Subroutines L2. 31nitC2~and L2. 5 FreeC2~are
designed to allocate and free memory for the interface flux array within one coarse mesh. More
details about the interface flux array will be discussed later. Both InitCM~ and FreeC2~have two
versions corresponding to the characteristics and SN solver kernel.
Subroutine L2. 7 Cal2~nt is called after the system sweep completes. The subroutine is used
to evaluate the flux moments (source iteration scheme: Step 2) based on the angular fluxes
calculated by the system sweep (source iteration scheme: Step 1).
Third Level Routines: Sweeping on Fine Mesh Level
Two sets of routines are built on this lowest level for the characteristics and SN solvers,
respectively. Both calculate angular fluxes within the scope of one coarse mesh, one octant, and
one group. Their structures can be illustrated by the following pseudo code.
!! Pseudocode: SolverSn_L2_S1 (cm_ijk, octant, group)
call ProjectionH() (cm~ijk, octant) angular projection
call ProjectionD() (cmiijk, octant) spatial projection
call SweepOrderfin(cm~ijk, octant)
For direc=1, num~direc direction loop within one octant
call MapSys2CM(cmiijk, direc)
call GetFmSrc_CMin(cm~ijk, octant, direc, group)
for fm~ijk in the sweeping order !fine mesh loop
call DiffScheme
end fine mesh loop
call MapCM2Sys(cm~ijk direct)
end direction loop
!! Pseudocode: SolverRay_L2_S1 (cm_ijk, octant, group)
call ProjectionH() (cm~ijk, octant) angular projection
call ProjectionD) (cmiijk, octant) spatial projection
For direc=1, num~direc direction loop within one octant
call GetZnSrcCMin(cm_ijk, octant, direc, group)
for each parallel ray ray loop
call GetBakFlx
call GetRayAvg
end ray loop
call GetZnAvg
call MapCM2Sys(cm~ijk direct)
end direction loop
Figure 4-4. Pseudo-code of the fine mesh sweep process.
Subroutines L3.1~ProjectionHO and L3.2 ProjectionDO complete angular and spatial
proj section procedures. The two subroutines, called within SolverSnL2_S1 and
Solver Ray L2_S1, remap the incoming flux array onto the same frame (in the angular domain
and spatial domain) as the current coarse mesh by the proj section techniques. Note that here
angular proj section is performed first if both proj sections are required.
For the SN solver, Subroutine L3. 3 SweepOrder jmn initializes the fine mesh sweep order
for the following Eine mesh loop. L3. 4MapSys2C2~and L3.7MapC M~2SSy are similar to their
counterparts, L2.1~ and L2. 7, on the second level. However, here we need to map between the
system interface flux array and the coarse mesh interface array, instead of between the model
boundaries and the system interface flux array.
Subroutine L3.5 GetFmSrc C2~in calculates the total source term for each Eine mesh
before the Eine mesh loop starts. Within the fine mesh loop, L3. 6 Diff~cheme is called to
calculate the outgoing flux and fine-mesh-averaged flux based on the incoming flux by a
differencing scheme. The diamond-differencing and direction-theta-weighted differencingl9
schemes are implemented. Other differencing schemes can be added into this subroutine.
The characteristics subroutine set is similar to the SN set with a two-level loop structure:
direction loop and parallel ray loop, instead of fine mesh loop in the SN solver. L3. 8
Gl'ril.CrcC2~in as its counterpart L3. 5 for the SN solver, calculates the total source term for
each zone, instead of each fine mesh. For each parallel ray, L3. 9 GetBakFlx evaluates the
incoming flux by the bilinear interpolation scheme. L3.10O GetRayAvg calculates the average
angular flux for the current ray. After all the parallel ray average fluxes are updated, L3.11~
GetZnAvg is used to calculate the average flux for the zone/coarse mesh. And the coarse mesh
outgoing flux is mapped back onto the system interface flux array.
Data Structure and Initialization Subroutines
The 4-level code flowchart, as outlined in the previous section, is built on the data
structure, which organizes of the data arrays, such as angular fluxes and flux moments. In the
TITAN code, a number of derived data types are defined by applying the paradigm of obj ect-
oriented programming (OOP). These user-defined data obj ects, such as coarse mesh obj ect,
quadrature obj ect, and proj section objects, are initialized in subroutine I Lr.11it at the
beginning of transport calculation. In recent years, OOP has already evolved into one standard
paradigm for modern coding language for computer applications. While FORTRAN 90/95,
designed mainly for scientific computing, generally is not considered as an object-based
language. However, FORTRAN 90/95 does provide some tools and language extensions to allow
users to utilize some concepts of OOP. And the OOP support is further enhanced in the new
FORTRAN 2003 standard.
In the TITAN code, coarse mesh is treated as a relatively independent obj ect, within which
a number of parameters, arrays, and sub-obj ect are defined. Among these parameters are
Solver ID, QuadID, Mat~matrix, Src~matrix, and angular flux and flux moment sub-objects.
Solver ID and QuadID specify the solver and quadrature set for the coarse mesh, respectively.
Mat matrix and Src matrix are the material and source distributions within the coarse mesh,
respectively. And the angular flux and moments for the coarse mesh are defined as sub-obj ects
for each group and octant. They are initialized in subroutine L1. 1-4 InitC2~flux.
Quadrature set is another essential obj ect, which contains the direction cosine values and
the weights associated with the directions for each direction in one octant. L1. 1-3 Createuad'
generates all the quadrature sets with ordinate splitting used in the model. For the level-
symmetric quadrature, direction cosines and weights are preset for quadrature order from 2 to 20.
For the PN-TN quadrature set, since the quadrature order is not limited to 20 as level-symmetric
quadrature, directions cosines and weights are pre-calculated by a polynomial root-finding
subroutine. After one SN or PN-TN quadrature is created, another subroutine is called to build up
the splitting ordinates on top of the regular quadrature set.
As described by Eq. 2-43, the projection matrix should be pre-calculated in both spatial
and angular domain. In the spatial domain, L1.1~-51InitProjection scans all the coarse mesh
interfaces and analyzes all the proj sections on the interfaces of coarse meshes. Since a 2-D
proj section is defined by two separated 1-D proj sections, only a 3 4 5 proj section matrix is
necessary for a proj section of3 x 3 4 5 x 5 The 2-D proj section matrix is built implicitly by the 1-
D component projection matrix. Furthermore, 1-D projection matrix is always stored in pair, e.g.
3 4 5 and 5 4 3, because they always happen together on the same coarse mesh interface
depending the sweeping direction. Note that since the same proj section could happen in a number
of interfaces, it is not necessary to build one proj section matrix for every coarse mesh interface. In
such case, only one proj section matrix is stored to reduce the memory cost. And a proj section ID is
assigned to each coarse mesh interface to specify the associated proj section matrix. The angular
proj section matrix is built in a similar way, but with a subroutine to find the three closest neighbor
directions in one quadrature set to every direction in the other quadrature set. Afterwards, the
three neighboring direction indices and the distance weights are stored in an angular proj section
matrix.
Coarse and Fine Mesh Interface Flux Handling
In the sweeping process, the fine-mesh interface flux propagates along the sweep direction.
Instead of storing all the interface fluxes for each fine mesh, we only store the fluxes on the
propagation frontline. As shown in Figure 4-2, for a 2-D coarse mesh with 4 by 4 fine meshes,
two one dimensional interface arrays,1Inter~x( and Intery(), can be allocated to store the
frontline interface flux, both with a size of 4.
Inter y(:) ;
1 2 3 4 Inter x(:)
Figure 4-5. Frontline interface flux handling.
At the beginning of the direction n sweep process, Inter~x and Inter] are assigned to the
incoming fluxes at the bottom and left boundary, respectively. This task is completed by
subroutine L3. 3 MapSys2CM. The sweep process starts from FM (1,1) by using Inter}(1) and
Inter~x(1 as incoming fluxes. After the average flux for FM(1,1) is updated, we assign the
outgoing flux for FM(1,1) back into lnter}(1) and Inter~x(1. And the rest of elements of
Inter~x and Inter]y remain the same. Therefore, for FM(1,2), 1nter~x(1 and Inter}(2) become
the incoming fluxes. Generally speaking, for FM~m,n),1Inter~x(m) and Inter j(n) always store
the incoming fluxes before the sweep begins, and the outgoing fluxes afterwards. For example,
after the sweep process updates the fluxes for the first 6 fine meshes, the blue line becomes the
propagation frontline. At this point, Inter~x stores the interface fluxes on the horizontal lines
along the blue front line, while InterJ stores all the interface flux on the vertical lines. After all
the fine meshes are processed, Inter~x and InterJ store the outgoing fluxes for the coarse mesh
at the top and right boundaries, respectively.
The front-line approach to handle the Eine-mesh interface fluxes can be extended to the
sweep process in a 3-D coarse mesh. We use three 2-dimentional arrays to store the interface
fluxes: Inter xy(-,.), Inter~xz(-,.), and Inter yz(-,.), instead oflInter~x( and Inter j() in a 2-D
coarse mesh. The front-line shown in Figure 4-2 becomes 'front-surface' in 3-D along x, y and z
axes.
The front-line approach is memory-efficient compared to the straightforward process to
store the interface fluxes for all the fine meshes. Under this approach, only the interface fluxes
on the marching front-line are stored. For the case shown in Figure 4-2, the frontline approach
only requires 8 memory units, while 40 memory units are necessary otherwise. For a 3-D coarse
mesh with i x j xk fine meshes, a total of ix j x(k +1)+ i x(j +1)x k +(i +1)x j xk memory
units are required if all the interface fluxes are stored. While the front-line approach only
requires ix j+i xk+ j xk memory units. Another benefit of the frontline approach is to avoid
'memory jumps' for the fine mesh incoming fluxes during the sweep process. As shown in
Figure 4-2, the interface flux arrays, Inter~x( and Inter j(, are always accessed sequentially
as the frontline marches forward, which is much more efficient than 'memory jumps', especially
when handling large size arrays.
The same approach can be applied on the coarse mesh sweep process, in which a coarse
mesh is considered as the finest unit. However, each element of the interface flux array becomes
another array, or an obj ect, instead of a scalar value as in the fine mesh sweep process. Here we
use another set of obj ect arrays, called system interface arrays Inter xy_cm(-,.), Inter~xz~cm(',.,
and Inter yz~cm(-,.), which are similar to Inter xy(-,.), Inter~xz-,.), and Inter yz(-,.). They can
be considered as an array of arrays, or an array of obj ects on the system level, which means each
element in Inter xy_cm(-,.) is another array, instead of a scalar value as in a regular array.
Inter xy_cm(-,:) represents the front-line coarse mesh fluxes on the xy plane in the global sweep
process, as Inter xy(-,.) represents the front-line fine mesh fluxes in a coarse mesh sweep
process. The system interface arrays are initialized by Subroutine L1.1~-21Initlnter, and connected
to coarse mesh interface flux arrays by subroutines L3. 3MapSys2C2~and L3.7MapC M~2SSys
which performs two mapping actions:
* Mapping one system array element to the corresponding coarse mesh interface array as the
coarse mesh incoming flux before the fine mesh sweep process starts.
* Mapping the coarse mesh interface array back onto the system array element afterwards as
the outgoing flux.
CHAPTER 5
BENCHMARKING
We carefully chose a number of benchmark problems to test the performance of the
TITAN code:
* A uniform medium and Eixed source problem, to test the SN solver.
* A simplified CT model, to test the hybrid approach with the ordinates splitting technique.
* The Kobayashi benchmark, to test both the SN and hybrid formulations.
* The C5G7 MOX benchmark, to test eigenvalue problems.
These benchmark problems are used to examine different aspects of the code. In this
chapter, we present the results of the TITAN code on these benchmark problems, and provide
some analysis on the results.
Benchmark 1 A Uniform Medium and Source Problem
This benchmark is a test problem designed to examine the accuracy of the SN solver of the
hybrid algorithm. A 15x15x15 cm3 water cube is divided into 3x3x3 coarse meshes of size of
5x5x5 cm. Each coarse mesh is divided by 5x5x5 Eine meshes. The entire model, as shown in
Figure 5-1, is composed of 15x15x15 fine meshes in 27 coarse meshes. The Eine mesh size is
lxlxl cm3. The vacuum boundary condition is applied on all the six surfaces of the water box.
The cross section data is extracted from the SAILOR-96 library by the GIP code.30 We only use
the first 3 neutron group cross section data from the SAILOR-96 47-group structure. Both Po and
P3 CTOss section data are tested. A fixed source is uniformly distributed in the water with a
uniform source spectrum.
Figure 5-1. Uniform medium and source test model.
We ran this model with an S6 quadrature set. As a reference, we also simulated the problem
with the PENTRAN code with the same setup (without acceleration, and with diamond-
differencing scheme only). The calculated scalar fluxes and the relative difference with
PENTRAN for the 3 groups are shown in Figures 5-2, 5-3 and 5-4.
-4.5E-05
-- -6E-05
-5 5E-05
-6E-05
-6 5E-05
-7E-05
-7 5E-05
-8 E-05
N- -8 5E-065
-. -9 E-05
s :::::: 9.5 E -0 5
A B -r
Figure 5-2. Group 1 calculation result. A) Flux. B) Relative difference with PENTRAN.
Grp2
--1 E-05
--1 5 E-05
- 2E-05j
--2. 5E-05
- 3E-05
- 35 E-05
- 4E-05
- 4, E-06
- 5E-05
- 5.5E-05
--6 E-05
BE.5 E-05
78 E-05
I
*!: 'cl
Figure 5-3. Group 2 calculation result. A) Flux. B) Relative difference with PENTRAN.
Grp3
S-2 E-05
S-3.5E-05
-4 E-05
S-4.5E-05
-5 E-05
-5.5 E-05
-6 E-05
-6 5E-05
-7 E-05
i
r-
'r
r
_ ~
1
Figure 5-4. Group 3 calculation result. A) Flux. B) Relative difference with PENTRAN.
As shown in Figures 5-2 to 5-4, TITAN yields the same solution as PENTRAN since the
relative difference (magnitude order of 10- ) is less than the flux tolerance (10-4). It is also worth
noting that relative difference is symmetric, and the larger difference generally occurs around the
corners and edges of the water box, where the scalar fluxes are lower than the center. A test on
code scalability and stability is also performed on a similar problem, in which we keep the same
fine mesh size, but only one coarse mesh for the whole box. TITAN provides the same solution
on the derived model with the similar memory requirement and running time.
As the first testing problem, this benchmark demonstrates that the basic algorithms in the
SN solver are correct. The simple setup of this model is designed to eliminate possible
complicated numerical effects on the SN solver. For example, no spatial or angular proj sections
are required in this model, since no mismatch exists between coarse meshes in either spatial or
angular domain. As a result, the convergence speed for this model is relatively fast (within
seconds), with only 5 or 6 within-group loops required for all the three groups.
Benchmark 2 A Simplified CT Model
A simplified computational tomography (CT) device model is built to test the hybrid
methodology and algorithm. A general CT device is shown in Figure 5-5.
Brral DETECTORS
~~ROTATES AltICaD
~~PATIENTIN
~/X SWCHvITH
Figure 5-5. Computational tomography (CT) scan device.
In a general CT device, the directional gamma rays emitted from the X-ray tube (source)
enter the human body (target) on the center. Some of the gamma particles could be scattered or
absorbed in the target. The uncollided gamma particles, carrying some information about the
attenuation coefficients on different parts of the target, can be recorded by the detector array on
the other side to form a proj section image. Proj sections from different angles, acquired by rotating
the source and detector array, can be used to reconstruct the target cross section image. In our
simplified CT model, we only consider a center slice of a CT device without the target. A 2-D
meshing plot of the simplified CT model is shown in Figure 5-6.
20
0 10 20 30 40 50 60 70
Source XDetectors
Figure 5-6. A simplified CT model.
In the simplified CT model, the photon source and an array of detectors are located on the
left and right side of a slice of the whole CT device, respectively, and the target obj ect is
removed from the center. Our goal is to calculate the scalar fluxes of the 20 fine meshes along y
direction in the detector region (i.e., red region at the right hand side of Figure 5-6). The
relatively large air region between the source and detector usually causes serious ray-effects
when the SN method is used. In order to overcome the ray-effects, The SN algorithm requires
finer discretization grids in both spatial and angular domains. Alternatively, a process called
'smearing' can be used to resolve the discretization grid mismatch in spatial and angular domain
by carefully choosing the mesh size along the discrete ordinates. In this test, we use the ordinate
splitting technique as a ray-effect remedy. And the TITAN solutions with different solvers are
compared with the MCNP5 reference calculation.31
Monte Carlo Model Description
Figure 5-7 shows the geometry for the Monte Carlo MCNP5 model, which is built exactly
as the deterministic model shown in Figure 5-6.
Figure 5-7. MCNP model of the simplified CT device.
We use MCNP5 code in multigroup mode,32 So that we can apply the same cross section
data as used in the deterministic calculations. A mesh tally is used to evaluate the 20 fine-mesh
fluxes in the detector region.
Deterministic Model Description
Figure 5-8 shows the SN solver model, which is composed of 7 coarse meshes with 14,000
fine meshes.
4
2N
201
Figure 5-8. SN solver meshing scheme for the CT model.
Here, we use Hyve coarse meshes in the air region to resolve the ray effect. The average
fluxes for the 20 detector Eine meshes are extracted after the calculation.
Figure 5-9 shows the hybrid solver model with 3 coarse meshes and 3,000 Eine meshes.
40
0 0 20
Figure 5-9. Hybrid model meshing for the CT model.
In the hybrid model, we apply characteristics solver in the air region (coarse mesh #2), and
the SN solver in both the source and detector regions (coarse meshes #1&3). The number of fine
meshes in the hybrid model is much less than the one in the SN model.
Comparison and Analysis of Results
A number of cases are tested for the simplified CT problem. In the first set of cases (Cases
2 and 3), we apply the SN Solver only to solve the problem, and try to alleviate the ray effect by
increasing the SN order. Due to the relatively large distance between the source and the detectors,
and the relatively small size of the detector fine mesh, very high order of quadrature set is
required to eliminate the ray-effects if no other ray effect remedy techniques are applied. This
approach to reduce the ray-effect is not efficient, because the memory requirement is roughly
proportional to square of the SN order. Figure 5-10 shows the results for an Sloo case and an S200
case compared with the MCNP reference case.
i3 4
Cjsr 3. Sr Pr..Tr 521:11:
0 5 10 15 20 25
2.500E-03
2.000E-03
1.500E-03
1.000E-03
5.000E-04
0.000E+00
Figure 5-10. SN simulation results without ordinate splitting.
The ray-effect is obvious in Case 2 with Sloo. Note that in most real problems, SN order
usually can not reach as high as 100 due to the memory limitation. However, since this
simplified model is relatively small with about 14,000 fine meshes, and one group cross section
structure, we are able to apply an S200 PN-TN quadrature set (shown in Figure 5-11A) for Case 2,
in which the ray-effects are significantly reduced.
4 04
Figure 5-11. Quadrature sets used in the CT benchmark. A) PN.TN S200. B) Biased PN.TN S20
In the second set of test cases (Cases 4 and 5), the ordinate splitting technique is applied as
a remedy for elimination of ray-effects. In this model, obviously particles streaming along the
directions close to x axis will contribute the most for the detector fluxes. Therefore, we use a PN-
__ __Y.-C--
-*-D ,ii a
Figure 5-12. Hybrid and SN simulation results with ordinate splitting.
Both cases show a good agreement with the MCNP reference case without ray-effects. It is
worth noting that in the hybrid model, as discussed in the last section, the number of fine meshes
is reduced by a factor of ~5 comparing to the SN model. The run times and error norms as
compared to the MCNP reference case are presented in Table 5-1.
Table 5-1. CT model run time and error norm comparison with the MCNP reference case.
Case Run Time ('! .r ifnr(')
Descriptions Run Time (sec) Err 2-norm Ermfnm
number Comparison Y """
MCNP ref, nps=2e8, rel.err. <0.01 3510 1.0 0.000E+00 0.00%
2 SN PN-TN SI,,, (10,200)" 441.3 7.9 2. 182E-02 5.86%
3 SN PN-TN See,, (40,400)* 1755.8 2.0 2.655E-03 2.41%
4 SN PN-TU S2CI 111.2 111.2 (207)'( 71.4 sec 49.1 2.820E-03 2.09%/
5 Hybrid PV-TN Szot,11.2 t11.2 (207)* 14.1 sec 248.9) 7.510E-03 3.28%
Error 2-norm measures the overall error for the 20 points
2 Error inf-norm represents the maxim local relative error
STotal number of directions
TN S20 quadrature set with the local PN-TN splitting technique on two directions close to the x
axis. both with a splitting order of 11 as shown in Figure 5-11B. The hybrid approach is tested in
Case 5. Figure 5-12 shows the results for the SN solver case and the hybrid case, both compared
with the MCNP reference case.
2.500E-03
2.000E-03
1.500E-03
1.000E-03
5.000E-04
0.000E+00
For the MCNP reference case, we use 200 million particles to yield a relative flux error of
less than 1% for all 20 meshes. Here, we use the infinity-norm and 2-norm to measure the maxim
local relative error and the overall error for the 20 points respectively. All the deterministic cases
show a good agreement with the Monte Carlo reference case and with less computation time.
The hybrid approach (Case 5) is about 5 times faster than the SN solver only case (Case 4), since
in the hybrid model, we use about 5 time less fine meshes than in the SN model. This benchmark
demonstrates that for problems with a large region of low scattering medium, the hybrid
approach can achieve the same level of accuracy as the SN method with much fewer Eine meshes
and thereby significantly lower computation cost.
Benchmark 3 Kobayashi 3-D Problems with Void Ducts
This benchmark consists of three problems with simple geometries and void regions.33
Furthermore, each problem includes two cases: zero-scattering and 50% scattering. We tested all
the three problems with the zero-scattering case. And each problem model is composed of three
regions:
Region 1: Source (no scattering).
Region 2: Void.
Region 3: Pure absorber.
We present the calculation results of our code and the comparison with the analytical
solution provided by the benchmark. Note we use uniform meshing for all the three problems:
each coarse mesh with a size of 10x10x10 cm3, and each Eine mesh with a size of lxlxl cm3.
And the point-wise fluxes in the benchmark are compared with the averaged fluxes calculated
over corresponding coarse mesh.
Problem 1: Shield with Square Void
As shown in Figure 5-13, this box-in-box problem is composed of three cubes: 10x10x10
cm3 Source box in the corner, 50x50x50 cm3 air box, and 100x100x100 cm3 pure absorber box.
10 50 100
Figure 5-13. Kobayashi Problem 1 box-in-box layout.
We consider three cases:
Case 1: MOC solver applied in Region 2 (void), Regions 1 and 3 with SN solver.
Case 2: MOC solver in Region 2&3 (void and pure absorber). SN solver in Region 1.
Case 3: SN solver in all three regions.
Tables 5-2 to 5-4 compare the results of Case I with different quadrature sets for the three
point sets. We also calculate the ratios to analytical solutions.
Table 5-2. Kobayashi Problem 1 Point A set flux results for Case 1.
Point 1A
5,5,5
5,15,,5
5,25,5
5,35,5
5,45,5
5,55,5
5,65,5
5,75,5
5,85,5
5,95,5
ErrNorm
Analytical
5.95659E+00
1.37185E+00
5.00871E-01
2.52429E-01
1.50260E-01
5.95286E-02
1.52283E-02
4.17689E-03
1.18533E-03
3.46846E-04
(Err2Norm ErrlNorm)
Case 1 (S24)
5.94515E+00
1.44872E+00
5.01333E-01
2.48688E-01
1 .45 821E-01
6.16731E-02
1 .5 600 1E-02
4.26493E-03
1.16145E-03
3.13078E-04
1.8232E-02
Ratio
0.9981
1.0560
1.0009
0.9852
0.9705
1.0360
1.0244
1.0211
0.9799
0.9026
9.736%
Case 1 (S30)
5.94414E+00
1.44446E+00
5.00703E-01
2.49114E-01
1.46590E-01
6.21947E-02
1.56733E-02
4.16728E-03
1.18505E-03
3.45040E-04
6.0117E-03
Ratio
0.9979
1.0529
0.9997
0.9869
0.9756
1.0448
1.0292
0.9977
0.9998
0.9948
5.293%
Table 5-4. Kobayashi Problem 1 Point C set flux results for Case 1.
Point IC Analytical Case 1 (S24) Ratio Case 1 (S30) Ratio
5,55,5 5.95286E-02 6.29408E-02 1.0573 6.18784E-02 1.0395
15,55,5 5.50247E-02 6.00183E-02 1.0908 5.95864E-02 1.0829
25,55,5 4.80754E-02 5.14090E-02 1.0693 5.16984E-02 1.0754
35,55,5 3.96765E-02 4.24917E-02 1.0710 4.33243E-02 1.0919
45,55,5 3.16366E-02 3.44892E-02 1.0902 3.48761E-02 1.1024
55,55,5 2.35303E-02 2.15000E-02 0.9137 2.14425E-02 0.9113
65,55,5 5.83721E-03 6.37570E-03 1.0923 6.26243E-03 1.0728
75,55,5 1.56731E-03 1.5 9919E-03 1.0203 1.66064E-03 1.0595
85,55,5 4.53113E-04 4.36921E-04 0.9643 4.82881E-04 1.0657
95,55,5 1.37079E-04 1.46529E-04 1.0689 1.41297E-04 1.0308
ErrNorm (Err2Norm ErrlNorm) 4.7278E-02 9.225% 4.9872E-02 10.240%
In Table 5-3, point (55, 55, 55) has a relative error of 20%, which is largest error among all
points, because it is located in the coarse mesh on the interface between the absorber region and
the air region. The transport solver may encounter difficulties in resolving the highly angular
dependent flux on the interface. Another difficult point (95, 95, 95) is located on the far corner
away from the source, where the ray-effect may be severer than the regions closer to the source.
The S30 CASe shows no significant improvement as compared to the S24 CASe, which may indicate
that we need to apply finer meshes to take advantage of a higher order quadrature set. Tables 5-5
to 5-7 compare the results for Case 2 with an S24 quadrature set for the three point sets.
Table 5-3. Kobavashi Problem 1 Point B set flux results for Case 1.
Point 1B
5,5,5
15,15,15
25,25,25
35,35,35
45,45,45
55,55,55
65,65,65
75,75,75
85,85,85
95,95,95
ErrNorm
Analytical
5.95659E+00
4.70754E-01
1.69968E-01
8.68334E-02
5.25132E-02
1.33378E-02
1.45867E-03
1.75364E-04
2.24607E-05
3.01032E-06
(Err2Norm ErrlNorm)
Case 1 (S24)
5.94515E+00
4.81175E-01
1.70050E-01
8.73159E-02
5.12734E-02
1.08504E-02
1.50095E-03
1.99741E-04
2.42707E-05
2.67405E-06
9.0705E-02
Ratio
0.9981
1.0221
1.0005
1.0056
0.9764
0.8135
1.0290
1.1390
1.0806
0.8883
18.649%
Case 1 (S30)
5.94414E+00
4.79594E-01
1.70665E-01
8.67251E-02
5.29735E-02
1.04048E-02
1.29943E-03
1.78873E-04
2.55221E-05
3.53673E-06
1.3184E-01
Ratio
0.9979
1.0188
1.0041
0.9988
1.0088
0.7801
0.8908
1.0200
1.1363
1.1749
21.990%
Analytical
5.95286E-02
5.50247E-02
4.80754E-02
3.96765E-02
3.16366E-02
2.35303E-02
5.83721E-03
1.56731E-03
4.53113E-04
1.37079E-04
(Err2Norm ErrlNorm)
Tables 5-8 to 5-10 compare the results of Case 3 for different quadrature sets along
different lines to analytical solutions.
I
Kobayashi Problem 1
Table 5-5. Kobayashi Problem 1
Point A set flux results for Case 2.
Case 2 (S24) Ratio
5.94515E+00 0.9981
1.44872E+00 1.0560
5.01333E-01 1.0009
2.48688E-01 0.9852
1.45821E-01 0.9705
6.12631E-02 1.0291
1.55573E-02 1.0216
4.25971E-03 1.0198
1.16837E-03 0.9857
3.18352E-04 0.9178
1.3822E-02 8.215%
Point 1A Analytical
5,5,5 5.95659E+00
5,15,,5 1.37185E+00
5,25,5 5.00871E-01
5,35,5 2.52429E-01
5,45,5 1.50260E-01
5,55,5 5.95286E-02
5,65,5 1.52283E-02
5,75,5 4.17689E-03
5,85,5 1.18533E-03
5,95,5 3.46846E-04
ErrNorm (Err2Norm ErrlNorm)
6.
Table 5-(
Point 1B
5,5,5
15,15,15
25,25,25
35,35,35
45,45,45
55,55,55
65,65,65
75,75,75
85,85,85
95,95,95
ErrNorm
Table 5-
Point IC
5,55,5
15,55,5
25,55,5
35,55,5
45,55,5
55,55,5
65,55,5
75,55,5
85,55,5
95,55,5
ErrNorm
Point B set flux
Case 2 (S24)
5.94515E+00
4.81175E-01
1 .70050OE-0 1
8.73159E-02
5.12734E-02
1.06986E-02
1.45221E-03
1.90111E-04
2.29690E-05
2.51840E-06
1.0662E-01
Point C set flux
Case 2 (S24)
6.24240E-02
5.97140E-02
5.11524E-02
4.22937E-02
3.43198E-02
2.13553E-02
6.35753E-03
1.59983E-03
4.42516E-04
1.45681E-04
4.3423E-02
results for Case 2.
Ratio
0.9981
1.0221
1.0005
1.0056
0.9764
0.8021
0.9956
1.0841
1.0226
0.8366
19.787%
results for Case 2.
Ratio
1.0486
1.0852
1.0640
1.0660
1.0848
0.9076
1.0891
1.0207
0.9766
1.0628
9.243%
Analytical
5.95659E+00
4.70754E-01
1.69968E-01
8.68334E-02
5.25132E-02
1.33378E-02
1.45867E-03
1.75364E-04
2.24607E-05
3.01032E-06
(Err2Norm ErrlNorm)
7.
Kobavashi Problem 1
Ratio
0.9981
1.0542
1.0031
0.9914
0.9823
1.0466
1.0393
1.0333
0.9814
0.9103
8.965%
Case 3 (S34)
5.94319E+00
1.44694E+00
5.03886E-01
2.51595E-01
1.48909E-01
6.32288E-02
1.60293E-02
4.29219E-03
1.20356E-03
3.54708E-04
1.0191E-02
Ratio
0.9978
1.0547
1.0060
0.9967
0.9910
1.0622
1.0526
1.0276
1.0154
1.0227
6.216%
Table 5-9. Kobayashi Problem 1 Point B set flux results for Case 3.
Point 1B Analytical Case 3 (S24) Ratio Case 3 (S34) Ratio
5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94319E+00 0.9978
15,15,15 4.70754E-01 4.80788E-01 1.0213 4.78621E-01 1.0167
25,25,25 1.69968E-01 1 .7005 9E-0 1 1.0005 1 .7 1342E-0 1 1.0081
35,35,35 8.68334E-02 8.75903E-02 1.0087 8.73758E-02 1.0062
45,45,45 5.25132E-02 5.12572E-02 0.9761 5 .24423E-02 0.9986
55,55,55 1.33378E-02 1.09012E-02 0.8173 1.09080E-02 0.8178
65,65,65 1.45867E-03 1.52321E-03 1.0442 1.37740E-03 0.9443
75,75,75 1.75364E-04 2.04277E-04 1.1649 1.66625E-04 0.9502
85,85,85 2.24607E-05 2.50787E-05 1.1166 2.04334E-05 0.9097
95,95,95 3.01032E-06 2.80145E-06 0.9306 2.73143E-06 0.9074
ErrNorm (Err2Norm ErrlNorm) 8.9359E-02 18.268% 7.6500E-02 18.217%
Table 5-10. Kobayashi Problem 1 Point C set flux results for Case 3.
Point IC Analytical Case 3 (S24) Ratio Case 3 (S34) Ratio
5,55,5 5.95286E-02 6.28676E-02 1.0561 6.35577E-02 1.0677
15,55,5 5.50247E-02 5.94963E-02 1.0813 5.94504E-02 1.0804
25,55,5 4.80754E-02 5.16771E-02 1.0749 5.19020E-02 1.0796
35,55,5 3.96765E-02 4.25678E-02 1.0729 4.31326E-02 1.0871
45,55,5 3.16366E-02 3.45271E-02 1.0914 3.47598E-02 1.0987
55,55,5 2.35303E-02 2.14367E-02 0.9110 2.15454E-02 0.9156
65,55,5 5.83721E-03 6.35281E-03 1.0883 6.21321E-03 1.0644
75,55,5 1.56731E-03 1.58707E-03 1.0126 1.64677E-03 1.0507
85,55,5 4.53113E-04 4.34709E-04 0.9594 4.74924E-04 1.0481
95,55,5 1.37079E-04 1.47770E-04 1.0780 1.45363E-04 1.0604
ErrNorm (Err2Norm ErrlNorm) 4.825 6E-02 9.137% 4.9324E-02 9.872%
The above results for Cases 2 and 3 show a similar agreement with the analytical solution
as Case 1, with a largest relative error about 20% on the interface point. Unlike the previous CT
model benchmark, the three point sets in this benchmark cover most of the difficult positions
Table 5-8. Kobayashi Problem 1 Point A set flux results for Case 3
Point 1A Analytical
5,5,5 5.95659E+00
5,15,,5 1.37185E+00
5,25,5 5.00871E-01
5,35,5 2.52429E-01
5,45,5 1.50260E-01
5,55,5 5.95286E-02
5,65,5 1.52283E-02
5,75,5 4.17689E-03
5,85,5 1.18533E-03
5,95,5 3.46846E-04
ErrNorm (Err2Norm ErrlNorm)
Case 3 (S24)
5.94515E+00
1.44622E+00
5.02432E-01
2.50261E-01
1.47601E-01
6.23020E-02
1.58269E-02
4.31608E-03
1.16330E-03
3.15751E-04
1.7566E-02
throughout the model, while in the CT model, we are only interested in the detector region with a
high resolution. It seems that the hybrid approach is more desirable in problems like the previous
benchmark. For this benchmark, the hybrid algorithm performs roughly as efficient as the SN
method for the 1 million mesh model. However, the computation costs are different for the three
cases as listed in Table 5-11.
Table 5-11. CPU time and memory requirement for SN and hybrid methods (1 million meshes
and S24 mOdel).
Solver CPU time Memory
Case #.
Reg. 1 (source) Reg. 2 (air) Reg. 3 (absorber) (sec) (Gigabyte)
1SN MOC SN 690 2.7
2 SN MOC MOC 267 4.3
3 SN SN SN 753 2.5
The characteristics solver is faster, but requires more memory to store the geometry
information. The SN solver is slower, but has a less memory requirement. The tradeoff between
memory and CPU time is always a coding concern, which is reflected in this problem. The CPU
time for Case 3 is reduced by a factor of ~2.8, however, requires about 1.7 time more memory.
It seems that Case 3 is preferred if memory requirement is affordable and/or the speed is the
maj or concern for the user. For simplicity, in the following Problem 2 and 3 calculations, only
the SN solver results are provided.
Problem 2: Shield with Void Duct
Figure 5-14 shows the first z level of the problem layout. The blue region is the source
region, the green region is the void duct, and the rest of the model is filled with a pure absorber,
which is Region 3.
Table 5-12.
Kobavashi Problem 2 Point A set flux results for Case 3.
Point 2A
5,5,5
5,15,,5
5,25,5
5,35,5
5,45,5
5,55,5
5,65,5
5,75,5
5,85,5
5,95,5
ErrNorm
Analytical
5.95659E+00
1.37185E+00
5.00871E-01
2.52429E-01
1.5 0260E-01
9.91726E-02
7.01791E-02
5.22062E-02
4.03188E-02
3.20574E-02
(Err2Norm ErrlNorm)
Case 3 (S24)
5.94515E+00
1.44797E+00
5.03502E-01
2.51243E-01
1.48549E-01
9.71016E-02
6.79254E-02
5.17088E-02
3.91599E-02
2.85735E-02
2.0340E-02
Ratio
0.9981
1.0555
1.0053
0.9953
0.9886
0.9791
0.9679
0.9905
0.9713
0.8913
10.868%
Case 3 (S30)
5.94414E+00
1.44793E+00
5.04210E-01
2.51994E-01
1.49214E-01
9.80078E-02
6.90949E-02
5.03751E-02
3.91877E-02
3.20167E-02
5 .4047E-03
Ratio
0.9979
1.0555
1.0067
0.9983
0.9930
0.9883
0.9846
0.9649
0.9719
0.9987
5.546%
Table 5-13. Kobayashi Problem 2 Point B set flux results for Case 3.
Point 2B Analytical Case 3 (S24) Ratio case SN (S30) Ratio
5,95,5 3.20574E-02 2.85735E-02 0.8913 3.20167E-02 0.9987
15,95,5 1.70541E-03 8.85805E-043 0.5194 1.49781E-033 0.8783
25,95,5 1.40557E-04 1.79639E-04 1.2781 1.53422E-04 1.0915
35,95,5 3.27058E-05 3.17893E-05 0.9720 3.3 9511E-05 1.0381
45,95,5 1.08505E-05 9.20428E-06 0.8483 1.12324E-05 1.0352
55,95,5 4.14132E-06 4.72351E-06 1.1406 4.32799E-06 1.0451
ErrNorm (Err2Norm ErrlNorm) 9.6633E-01 48.059% 3.0605E-02 12.173%
Results are calculated by averaging the corresponding fine mesh(s), instead of coarse mesh.
Our calculation shows a good agreement with the analytical solution on most of points,
except point (15 95 5), which is located on the far side interface between Regions 2 and 3.
Figure 5-14. Kobayashi Problem 2 first z level model layout.
The SN solver calculation results are listed in Tables 5-12 and 5-13.
Problem 3: Shield with Dogleg Void Duct
Figure 5-15 shows the layout of the void duct in the model. The rest of the model is filled
with pure absorber. The SN calculation results are listed in Tables 5-14 to 5-16.
Figure 5-15. Kobayashi Problem 3 void duct layout.
Table 5-14.
Kobayashi Problem 3 Point A set flux results for Case 3.
Point 3A
5,5,5
5,15,,5
5,25,5
5,35,5
5,45,5
5,55,5
5,65,5
5,75,5
5,85,5
5,95,5
ErrNorm
Analytical
5.95659E+00
1.37185E+00
5.00871E-01
2.52429E-01
1.5 0260E-01
9.91726E-02
4.22623E-02
1.14703E-02
3.24662E-03
9.48324E-04
(Err2Norm ErrlNorm)
Case 3 (S24)
5.94515E+00
1.44797E+00
5.03502E-01
2.51243E-01
1.48549E-01
9.71016E-02
4.37756E-02
1.20425E-02
3.34282E-03
8.95157E-04
1.1213E-02
Ratio
0.9981
1.0555
1.0053
0.9953
0.9886
0.9791
1.0358
1.0499
1.0296
0.9439
5.606%
Case 3 (S3,)
5.94414E+00
1.44793E+00
5.04210E-01
2.51994E-01
1.49214E-01
9.80078E-02
4.46212E-02
1.17776E-02
3.32867E-03
9.94322E-04
9.2256E-03
Ratio
0.9998
1.0000
1.0014
1.0030
1.0045
1.0093
1.0193
0.9780
0.9958
1.1108
5.582%
Table 5-15. Kobayashi Problem 3 Point B set flux results for Case 3.
Point 3B Analytical Case 3 (S24) Ratio Case 3 (s3,,) Ratio
5,55,5 9.91726E-02 9.71016E-02 0.9791 9.80078E-02 0.9883
15,55,5 2.45041E-02 2.66812E-02* 1.0888 2.61306E-02* 1.0664
25,55,5 4.54447E-03 4.84126E-03 1.0653 4.91017E-03 1.0805
35,55,5 1.42960E-03 1.46750E-03 1.0265 1.48483E-03 1.0386
45,55,5 2.64846E-04 3.00417E-04* 1.1343 2.88298E-04* 1.0885
55,55,5 9. 14210E-05 9.58897E-05 1.0489 9.55481E-05 1.0451
ErrNorm (Err2Norm ErrlNorm) 2.7730E-02 13.431% 1.9429E-02 8.855%
Results are calculated by averaging the corresponding fine mesh(s), instead of coarse mesh.
Table 5-16. Kobayashi Problem 3 Point C set flux results for Case 3.
Point 3C Analytical Case 3 (S24) Ratio Case 3 (S30) ratio
5,95,35 3.27058E-05 3.46102E-05 1.0582 3.16989E-05 0.9692
15,95,35 2.68415E-05 3 .04241E-05 1.1335 2.88384E-05 1.0744
25,95,35 1.70019E-05 1.61464E-05 0.9497 1.86621E-05 1.0976
35,95,35 3.37981E-05 2.62570E-05 0.7769 2.38136E-05 0.7046
45,95,35 6.04893E-06 5.30795E-063 0.8775 4.85885E-063 0.8033
55,95,35 3.36460E-06 3.43148E-06 1.0199 4.00289E-06 1.1897
ErrNorm (Err2Norm ErrlNorm) 1.2205E-01 22.312% 2.7493E-01 29.542%
Results are calculated by averaging the corresponding fine mesh(s), instead of coarse mesh.
Problem 3 seems to be the most difficult one among the three Kobayashi problems, since
particles tend to streaming along the dogleg void duct. As expected, the worst point (45, 95, 35)
is located on the interface of the far end of the duct.
Analysis of Results
For the three problems, our calculation results show a relatively good agreement with the
analytical solutions for most of the points. The characteristics solver also provides similar results
as the SN solver for problem 1. Figures 5.16-18 show the normalized flux calculation results for
the SN solver for the three problems (PN-TN S24 for Problem 1, PN-TN S30 for Problems 2&3).
0 2 4 6 8 10 12
***** + s
0 2 4 6 8 10 12
O 2 4 6 8 10 12
Figure 5-16. Relative fluxes for Kobayashi Problem 1. A) Point set A. B) Point set B. C) Point
set C
~ -rrrr~r
+ +
0 2 4 6 8 10 1
*
0 2 4 6 8 10 12
Figure 5-17. Relative fluxes for Kobayashi Problem 2. A) Point set A. B) Point set B.
** *,
0 2 4 6 8
0 12 34 56 7
C
Figure 5-18. Relative fluxes for Kobayashi Problem 3. A) Point set A. B) Point set B. C) Point
set C
Figures 5-16 to 5-18 show that points with relatively large errors typically occur on the
interface between the void region and the pure absorber region due to the highly directional
particle streaming on the interface. Since no scattering exists in the model and the source is
located in the corner, ray-effects could be very severe in this 3D model. Therefore, it is difficult
for an SN code without any ray-effect remedies to calculate all the point sets with only one
calculation.34
Benchmark 4 3-D C5G7 MOX Fuel Assembly Benchmark
We tested the k-effective calculation ability of the TITAN code on the extended 3-D C5G7
MOX benchmark.35, 36 TITAN categorizes transport problems into four types: fixed source
problems with only down-scattering, fixed source with up-scattering, criticality with down-
scattering, and criticality with up-scattering. Details on the four kernels are discussed in Chapter
4. This benchmark falls in the fourth category with the reflective boundary condition, which is
numerically the most difficult type. The size of the model also presents a challenge for a serial
non-lattice transport code as TITAN.
Model Description
The C5G7 MOX reactor is a proposed design for this benchmark, which has 2 by 2
assemblies (2 MOX assemblies and 2 UO2 assemblies). Each assembly is composed of 17xl7
fuel pins. And the four fuel assemblies are surrounded by moderator as shown in Figure 5-19.
br tiilr 19< B.C. Vacuum 3 1
UO, VID >. i
E 3 :~_LMOX UU. o
MOX
Moderator
UO2 MOX
A ~Vacuum B.C. BRtleceted B.C.
Figure 5-19. C5G7 MOX reactor layout. A) x-y plane. B) Unrodded configuration.
89
Axially the fuel region of the reactor can be divided equally into three segments as shown
in Figure 5-18B. And control rods can be inserted into different depth of the core. Three control
rod configurations are used in the extended version of this benchmark:36
*Unrodded.
*Rodded A.
*Rodded B.
In the unrodded case, the control rods only reach the moderator region on the top of the
core (grey area in Figure 5-18B). In the other two cases, control rods in the MOX and UO2
assemblies reach different positions in the core.
Several models with different disctetization grids are tested. Only the SN solver is used in
the calculations. The finest grid model we used has about 3 million meshes (12 :levels) with a
Slo quadrature set. This model requires ~1.8Gig memory. Based on the calculation results, the k-
effective is relatively insensitive to the grid size, although the pin-power distribution does
improve slightly with finer discretization grid. Figure 5-20 shows the meshing scheme for the
2x2 fuel assemblies and an individual fuel pin.
A ii B111111
Figure 5-20 3-D C57MXmoe.A ou ulasebis )ulpn
We use 14xl4 fine meshes to represent each fuel pin in this four z-level model, which
leads to a Eine mesh size: 0.09x0.09xl4.28 cm3. The mesh size along : axis is much larger than x-
y size, because Einer meshing is required to represent the round shape of the fuel pin in the
Cartesian geometry. A minimum four z-levels are required to represent the different control rod
configurations. It is necessary to add more z-levels to resolve the axial flux shape because of
different control rod configurations. However, the tests indicate that k-effective is more sensitive
to the x-y size than the z mesh size. Here we only reported the four z-level S6 mOdel calculation
results due to our computation resource limitation. The model has about one million fine meshes.
Note that the multigroup cross section data and the reference solutions (acquired by Monte Carlo
calculations) are provided with the benchmark.
Pin Power Calculation Results
The Monte Carlo reference solution provides the pin power distribution for the three slices
in the reactor core region. In the TITAN model, each fuel pin is composed of 14xl4 fine meshes.
Since the power is proportional to the fission rate, a special subroutine is developed to evaluate
the pin power by summating the fission rates for all the 14xl4 fine meshes and for all the seven
energy groups. Then, the output can be imported to the EXCEL template provided with the
benchmark specification. The differences between user calculation results and the reference are
automatically evaluated by the template. The pin power results calculated by TITAN for the
unrodded case are compared with the reference solution in Table 5-17.
Table 5-17. Pin power calculation results for the unrodded case
Z Slice
#1
Ref.
1.108
Z Slice
#1
User
1.148
Z Slice
#2
Ref.
0.882
Z Slice
#2
User
0.884
Z Slice
#3
Ref.
0.491
Z Slice
#3
User
0.449
Overall
Ref.
2.481
Overall
User
2.481
Specific Pin Power Data
Maximum Pin Power
Percent Error (associated
68% MC)
Distribution Percent Error
Results
Maximum Error (associated
68% MC)
AVG Error
RMS Error
MRE Error
Number of Accurate Fuel
Pin Powers
Number of Fuel Pins Within
68VoMC
Number of Fuel Pins Within
95VoMC
Number of Fuel Pins Within
99% MC
Number of Fuel Pins Within
99.9% MC
Total Number of Fuel Pins
Average Pin Power In Each
Assembly
UO2-1 Power
MOX Power
UO2-2 Power
UO2-1 Power Percent Error
MOX Power Percent Error
UO2-2 Power Percent Error
0.090 3.563 0.100 0.244 0.130 -8.449 0.060 0.007
0.220
0.164
0.171
0.062
4.673
3.340
3.381
1.496
0.320
0.183
0.190
0.055
1.803
0.421
0.536
0.140
0.130
0.245
0.255
0.042
8.449
7.069
7.096
1.445
0.192
0.109
0.114
0.093
1.395
0.268
0.354
0.200
371 0 371 146 371
0 371 147
518 0 518
540 0 540
518 0 518 257
334 540 0 540 336
544
545
219.04
94.53
62.12
0.082
0.061
0.043
0
545
226.70
97.38
64.55
3.498
3.017
3.920
544
545
174.24
75.25
49.45
0.073
0.054
0.038
387
545
173.79
75.10
49.65
-0.258
-0.193
0.404
544
545
97.93
42.92
27.82
0.055
0.041
0.029
0
545
90.54
39.95
25.89
-7.554
-6.911
-6.936
544 398
545 545
491.21
212.70
139.39
0.123
0.092
0.065
491.03
212.44
140.09
-0.038
-0.122
0.506
The format of Table 5-17 is provided by the benchmark template. In the unrodded case,
control rods are inserted to the moderator region on the top of the reactor core. The TITAN
results show a relatively good agreement with the reference solution for the overall pin power
distribution (power summation of the three axial segments). However, large differences exist if
we compare different segments, especially Slices #1 and #3. The error could be attributed to the
large mesh size along the z axis and the lower order of the quadrature set. Similar error pattern
also occurs in the rodded A and B cases as provided in Tables 5-18 and 5-19.
Table 5-18. Pin power calculation results for the rodded A case.
Z Slice
#1
Ref.
1.197
Z Slice
#1
User
1.211
Z Slice
#2
Ref.
0.832
Z Slice
#2
Uer
0.826
Z Slice
#3
Ref.
0.304
Z Slice
#3
User
0.321
Overall
Ref.
2.253
Overall
User
2.274
Specific Pin Power Data
Maximum Pin Power
Percent Error (associated
68% MC)
Distribution Percent Error
Results
Maximum Error (associated
68% MC)
AVG Error
RMS Error
MRE Error
Number of Accurate Fuel
Pin Powers
Number of Fuel Pins Within
68% MC
Number of Fuel Pins Within
95% MC
Number of Fuel Pins Within
99% MC
Number of Fuel Pins Within
99.9% MC
Total Number of Fuel Pins
Average Pin Power In Each
Assembly
UO2-1 Power
MOX Power
UO2-2 Power
UO2-1 Power Percent Error
MOX Power Percent Error
UO2-2 Power Percent Error
0.080 1.145 0.100 -0.696 0.200 5.518 0.059 0.919
0.100
0.157
0.163
0.066
1.625
0.691
0.819
0.388
0.250
0.180
0.186
0.056
1.877
0.760
0.860
0.297
0.330
0.260
0.266
0.037
7.044
3.922
4.251
0.582
0.149
0.108
0.111
0.094
1.701
0.714
0.803
0.690
371 87 371 60 371
11 371 14
518 163 518
540 200 540
113 518 13 518 43
160 540 18 540 70
544
545
237.41
104.48
69.80
0.087
0.065
0.047
237
545
240.06
104.67
70.51
1.118
0.182
1.012
544
545
167.51
78.01
53.39
0.071
0.056
0.040
216
545
165.79
77.42
53.18
-1.029
-0.747
-0.382
544
545
56.26
39.23
28.21
0.040
0.040
0.029
25
545
58.89
38.27
26.85
4.674
-2.447
-4.817
544 104
545 545
461.18
221.71
151.39
0.119
0.094
0.068
464.74
220.36
150.54
0.772
-0.610
-0.565
In the rodded A case, control rods are inserted to the Slice 3
in one assembly. Slice 3 is the
top slice in the reactor core region, which has the least power contribution among the 3 slices.
Slice 3 has the largest percentage error. The maximum error associated 68% Monte Carlo
reference is about 7% for Slice 3, while it is about 2% for the other two slices. The overall
assembly power errors are less than 1% for both UO2 and MOX assembly.
Table 5-19. Pin power calculation results for the rodded B case.
Z Slice Z Slice Z Slice Z Slice Z Slice Z Slice
#1 #1 #2 #2 #3 #3 Overall Overall
Specific Pin Power Data Ref. User Ref. Uer Ref. User Ref. User
Maximum Pin Power 1.200 1.167 0.554 0.585 0.217 0.205 1.835 1.818
Percent Error (associated
68% MC) 0.090 -2.779 0.150 5.603 0.240 -5.657 0.083 -0.890
Distribution Percent Error
Results
Maximum Error (associated
68% MC) 0.090 3.438 0.140 6.689 0.220 15.558 0.071 1.715
AVG Error 0.146 1.313 0.181 2.713 0.285 4.407 0.105 0.710
RMS Error 0.150 1.557 0.184 3.231 0.290 5.714 0.108 0.823
MRE Error 0.073 0.916 0.055 0.971 0.034 0.577 0.098 0.727
Number of Accurate Fuel
Pin Powers
Number of Fuel Pins Within
68VoMC 371 36 371 4 371 31 371 37
Number of Fuel Pins Within
95VoMC 518 75 518 6 518 51 518 78
Number of Fuel Pins Within
99% MC 540 93 540 20 540 70 540 109
Number of Fuel Pins Within
99.9% MC 544 121 544 35 544 88 544 137
Total Number of Fuel Pins 545 545 545 545 545 545 545 545
Average Pin Power In Each
Assembly
UO2-1 Power 247.75 241.84 106.56 112.38 41.12 37.45 395.43 391.67
MOX Power 125.78 124.15 81.41 82.93 29.42 30.42 236.62 237.50
UO2-2 Power 91.64 91.97 65.02 66.40 30.68 30.95 187.34 189.33
UO2-1 Power Percent Error 0.091 -2.385 0.056 5.460 0.035 -8.924 0.112 -0.951
MOX Power Percent Error 0.073 -1.300 0.058 1.875 0.034 3.379 0.100 0.374
UO2-2 Power Percent Error 0.055 0.364 0.046 2.124 0.032 0.899 0.078 1.062
The axial flux profile becomes more and more difficult to resolve from the unrodded case
to the rodded B case, as the control rods insert deeper in the reactor core with different
configurations. As a result, one can observe the overall pin power accuracy worsens slightly
from Table 5-17 to 5-19. This is expected, considering we only use the minimum 4 z-levels
model with the diamond differencing scheme, and a relatively lower quadrature order.
Eigenvalue Comparison
The eigenvalues for the three cases are listed in Table 5-20. The tolerance used in TITAN
input file for keff is 1.0OE-05. Similar to the pin power error, the keff error increases as the control
rods insert further into the core.
Table 5-20. Eigenvalues for three cases of C5G7 MOX benchmark problems.
Case Ref. % Error (68% MC) User Difference (pcm).
Unrodded 1.143080 0.0026 1.13911 169
Rodded A 1.128060 0.0027 1.12600 206
Rodded B 1.077770 0.0028 1.07415 362
Analysis of Results
Since the SN solver in the TITAN code is designed only for Cartesian geometry. We had to
use an 'unusual' meshing scheme in this benchmark: the z mesh size is about 158 times larger
than the x or y mesh size. Such imbalanced meshing could be valid only for problems in which
the axial flux changes very slowly comparing with radical flux profie. Our computer hardware
limitation is another reason why we use a reduced meshing scheme (about 1 million fine
meshes). TITAN is still serial code. And we need to fit the whole problem onto one machine. It
takes about 10 hours to run the 4 z-level S6 mOdel on an AMD Opteron 242 CPU (1.6MHz,
1024k cache) with about 323M memory requirement. The calculation result is reasonable
considering the meshing scheme we used.
Specially designed lattice transport codes for reactor neutronics could be more efficient for
this benchmark. However, TITAN has the potential to increase the efficiency in eigenvalue
problems with some power iteration acceleration techniques implemented. Figure 5-21 shows the
eigenvalue convergence pattern for the rodded A case.
Keff3 0 0 60 7
1trto 14
111ur 5-1 Eievlecnegneptenfo h oddAcniuai
The rodded A case takes more than 2,000 inner iterations and about 60 outer iterations. As
shown in Figure 5-20, the k-effective converged relatively fast for the first 20 iterations without
any power acceleration technique applied. The convergence rate is much slower for the rest of
the iterations, although this pattern is generally expected. The output indicates that some
iterations are wasted to converge fluxes with the un-converged fission source, especially with up-
scattering present. We took some intuitive measures in the code to improve the pattern, including
using adaptive flux convergence criterion, adaptive inner loop and outer loop iteration number
limitations, and Aitken extrapolation method.37 I also combined the up-scattering loop and the
power loop into one loop at the beginning, and separated them toward the end. These measures
are optional in TITAN (Appendix D). And they can improve the convergence rate in certain
situations.
CHAPTER 6
FICTITIOUS QUADRATURE
We introduce a special kind of problems that the TITAN code can be applied: the particle
transport problem within a digital medical phantom. To solve a regular transport problem,
modeling of the problem is required as one of the initial tasks. And a meshing scheme need to be
carefully chosen based on the physics of the problem. While in a digital phantom, the source and
material distributions are stored in the format of voxel values as activity (source) and material
attenuation coefficients. Therefore, it is a natural choice to consider one voxel as one fine mesh
in the initial modeling task. In the TITAN code, a module is developed to process the digital
phantom binary Hiles and automatically generate the meshing scheme. Furthermore, since
transport calculations for medical phantoms often involve the simulations of radiation proj section
images, we developed the fictitious quadrature technique to calculate the angular fluxes for
specific directions of interest that may not be available in a regular quadrature set. The
performance of the technique is tested in a digital heart phantom benchmark.
Extra Sweep with Fictitious Quadrature
In the TITAN code, multiple quadrature sets can be used in one problem model. A regular
quadrature is built based on the criteria of conservation of flux moments. Fictitious quadrature is
designed differently from the regular type of quadrature in that its purpose is to calculate only the
angular fluxes for certain directions, not to conserve the flux moments. Therefore, it can not be
used in a regular sweep process since the scattering source and flux moments cannot be properly
handled. However, it can be used after the source iteration process is complete with the
converged flux moments.
Generally, in a transport problem, users' maj or concern is the scalar flux distribution
and/or k-eff: However, in some cases, the angular fluxes in the directions of interest need to be
evaluated. Since the directions are not necessarily included in the problem quadrature sets, the
angular fluxes in these directions usually cannot directly be calculated by the sweep process with
a regular quadrature set. In the TITAN code, we can define the directions of interest in a
fictitious quadrature set, which is used with an extra sweep process only after the source iteration
process is converged with the regular quadrature set(s). The converged flux moments are used to
evaluate the scattering source in the extra sweep with the fictitious quadrature.
S cattenn = (2 + )ca ugXil~ ) con) +2 (1 k)! (pc)
g =11=0 k=1 (l + k)! (6-1)
>1')
Where, upper script (e.s) stands for extra sweep, (fic) for fictitious, (con) for converged.
con), /k (on) ,and X,"") are the converged Qth Order regular, cosine and sine flux moments.
And (~uP" (se)) specifies a direction in the fictitious quadrature set.
Equation 6-1 is similar to Eq. 2-23, except that we use the converged flux moments after
the source iteration process instead of the flux moments from last iteration. And the polar and
azimuthal angles refer to a direction in the fictitious quadrature set. The fixed source or the
fission source can be evaluated the same way as in a regular sweep process. After the total source
is estimated, we can use the extra sweep process to evaluate the angular fluxes in the directions
of the fictitious quadrature.
One also could choose some other methods based on the calculated angular fluxes in the
quadrature directions to evaluate the angular fluxes of interest. For example, the angular
proj section technique in Chapter 3 can be applied with some modifications. We have tried this
approach in the TITAN code. Another method could be to apply the Legendre expansion of the
angular flux based on the converged flux moments. One potential problem with these two
approaches is that their efficiencies are subj ect to the accuracy of the angular fluxes in the
directions of a regular quadrature set. Usually a convergence criterion is set on the scalar flux in
the source iteration scheme. The accuracy of the angular fluxes or higher moments is not always
granted. And further mathematical manipulations on the angular fluxes or higher moments could
introduce more secondary inaccuracies. One advantage of the fictitious quadrature technique
over the secondary approaches is that the angular fluxes of interest are calculated directly from a
sweep process. And since the sweep process can be considered as a simulation procedure to the
physical particle transport phenomenon in certain directions, some physics of the model along
the interested directions (e.g. Eixed source and scattering) are taken into account in the evaluation
process. Thereby, the extra sweep with the fictitious quadrature has more potential to provide an
accurate estimation on the interested angular fluxes.
Implementation of Fictitious Quadrature
It is straightforward to implement the fictitious quadrature technique, since all the
formulations used in a regular sweep can be applied in the extra sweep. However, due to the
special design of the fictitious quadrature, some modifications on the regular sweep are required
to effectively complete an extra sweep.
Extra Sweep Procedure
The extra sweep starts upon the completion of the source iteration process. The fictitious
quadrature is built as an initialization task before the source iteration starts. Fictitious quadrature
sets can be treated as a regular user-defined quadrature set in the initialization process, except
that any direction regardless of its octant can be defined in the quadrature input fie, and these
directions can be arbitrarily chosen. Note that in a regular user-defined quadrature set, only
directions in the first octant are defined, and directions in other octants are determined by
symmetry. As a result, the extra sweep is performed only along specific directions defined in the
first octant. The extra sweep procedure can be illustrated by Figure 6-1.
Source Iteration Completion
Initialize fictitious 1.Ralcto n.Fu
quadrature set
2. Initialize Boundary flux for group g
Group Iteration 3. Recalculate group g in-moments
g=1,2, ... G
4. Group g extra sweep
-5. Output group g boundary flux
Figure 6-1. Extra sweep procedure with fictitious quadrature.
As shown in Figure 6-1, we start the extra sweep by reallocating the angular flux array
based on the fictitious quadrature set. Since the values of angular fluxes in the regular quadrature
sets will be lost after the memory reallocation, any task which requires the calculated angular
fluxes need to be completed before the extra sweep. At the beginning of the sweep for group g,
we allocate a new array for the boundary angular fluxes, which will be deallocated after the
group g sweep. The original boundary fluxes calculated from regular sweep remain untouched
during the extra sweep, because an angular projection from the regular quadrature to the
fictitious quadrature could be employed on the boundaries if reflective boundary condition is
used. We apply the same scattering-in moment approach discussed in Chapter 5 in the extra
sweep as well. Note that the scattering-in moments are calculated based on the converged flux
moments from regular sweeps, and they are only used for evaluation of the scattering source in
an extra sweep. Also note that the step to calculate flux moments in a regular sweep is removed
in the extra sweep procedure.
Implementation Concerns
We developed a new set of subroutines to complete the extra sweep. Most of these new
routines are on layer 3 or 4, including the angular proj section module, the coarse mesh sweep
routine, and the differencing scheme routine. Although these subroutines share the similar tasks
as their counterparts in the regular sweep, some modifications are required due to the following
concerns:
* Iteration structure.
* Direction singularity.
* Solver compatibility.
Iteration structure
The iteration architecture in a regular sweep for group g is built on the following order
(from outer to inner): Octant loop, coarse mesh loop, direction loop, Eine mesh loop. However
the characteristics of the fictitious quadrature require that the extra sweep to follow a different
order: direction loop, coarse mesh loop, Eine mesh loop. This structure change affects most of
routines on layer 3 and 4, since all the directions in the same octant are handled as a group in the
regular sweep, while in an extra sweep, each direction need to be treated individually. For
example, the coarse mesh or Eine mesh sweep order is assigned individually for each direction
instead by octant. Another modification is made to allow negative directional coordinates in the
user-defined Eictitious quadrature set.
Direction singularity
A regular quadrature set usually avoids directions along an axis or perpendicular to an axis.
Zero directional cosine or sine occurs for these directions. This singularity could cause some
potential problems in the sweep process. For example, in the differencing scheme discussed in
Chapter 2, normally a small perturbation in one boundary incoming angular flux can cast some
effect on all the three outgoing fluxes, since the three components of the incoming angular flux
along x, y and z axes are all positive definite or all zeros. For a singular direction, however, this
is not always true. For example, an incoming angular flux along the x axis only has only one
positive x component. Therefore, while calculating the outgoing fluxes, a differencing scheme
need to take measures to treat a singular incoming angular flux.
Unfortunately, singular directions often happen to be the interested directions in a fictitious
quadrature set. A series of modifications have been made to keep the extra sweep subroutines
singularity safe, including the differencing scheme, the fine mesh sweep procedure, and the
angular proj section routine.
Solver compatibility
The two-solver structure of the TITAN code causes another dimensional difficulty in the
implementation of the fictitious quadrature set. The technique is originally designed for the SN
solver only. Later the compatibility to the characteristics solver is achieved.
Heart Phantom Benchmark
Originally, we developed the fictitious quadrature technique to calculate the boundary
angular fluxes for a single photon emission computed tomography (SPECT) benchmark. In
SPECT, a small amount of photon radiation source is deposited in the target organ with some
nuclear medicine intake. The source distribution in the organ can be reconstructed with the
proj section images. The 3-D source distribution image can be used to diagnose some malfunctions
in the organ. Dozens of proj section images from different angles are required to reconstruct the
source distribution to achieve a good resolution. In medical physics, SPECT simulation usually
is performed with the Monte Carlo approach. In this benchmark, our goal is to simulate the
proj section images of a body phantom with a deterministic transport calculation.
Model Description
We applied the TITAN code on a digital heart phantom generated by the NCAT code.38
NCAT can provide various cardiac-torso phantoms with support of the heart motion. Users can
specify the amount of activity deposited in different organs. The phantom we use in this
benchmark is the first frame of the heart motion cycle. The phantom contains two binary file: the
attenuation file and the activity file. The attenuation file records the linear attenuation
coefficients for each voxel. And the radiation activity in each voxel is stored in the activity file.
A "ii"'t x B~t='1~
Figre -2.Heat hanom ode. A Trso B)Orgns
Fiur -2sow temaeia isriuio f h pano. h sz o tephnomi
matuerials Hare used nthis model. as listed in Table6-.Thmaeildntesndier
attenuation coefficients are given in the output files of the NCAT code. The TITAN code also
can process the phantom attenuation binary file, and automatically generate a material list based
on the different attenuation values in the file.
Table 6-1. Materials list in the heart phantom model.
Mat. Number Mat. name Density (g/cm3) Linear Attenuation Coefficients
(1/pixel) or (0.3 125cm/pixel)
1 Air 1.00E-06
2 Body (Muscle) 1.02 0.0469
3 Dry Spine Bone 1.22 0.0520
4 Dry Rib Bone 1.79 0.0653
5 Lung 0.30 0.0135
Figure 6-3 shows the source activity distribution. The radiation source is deposited only in
the heart, with 75 unit activity in the myocardium (heart muscular substance) and 2 unit activity
in the heart chambers (blood pool).
3 ,1.
i.
Q:f- '
.c-----
_
i.IS--
r 3
;
ti!
I_
+a: ---
n ~-
~---
-"zt-- _,--------'-~'
--
..
i.
Figure 6-3. Activity distribution in the phantom model. A) Heart. B) Heart cross section view.
Photon Cross Section for the Phantom Model
The CEPXS package39 is used to generate the cross section data for this benchmark. The
group structure is decided based on the gamma decay energy of Tc-99m (~140kev), which is
widely used in the area of cardiac nuclear medicine. We also assume a typical 7% energy
resolution for the Nal detectors used in the SPECT camera. Therefore, the width of the first
group is about 10keV with a mid-range energy of 140keV. The rest of the group structure can be
arbitrary chosen, since only the angular flux for the first group is required in this benchmark. In
Table 6-2, we present a four-group structure with only down scattering.
Table 6-2. Group structure of cross section data for the heart phantom benchmark.
Group Number Upper Energy Bound (keV) Lower Energy Bound (keV)
1 145 135
2 135 100
3 100 50
4 501
An ideal SPECT camera takes proj section images only from the uncollided photons.
Therefore, only the first group angular fluxes on the boundaries are required to simulate the
proj sections images. To deliver the cross section data, CEPXS also requires the weight percentage
for each element in a mixture and its density. For the five materials listed in Table 6-1, the body
and lung materials (mat. # 2 and 5) can be considered as water. And we assume the bone
materials are composed of 22% water and 78% calcium. Detailed material compositions and
densities are provided in Table 6-3.
Table 6-3. Material densities and compositions used in CEPXS.
Mat. # Name Density (g/cm3) COmposition (element : weight fraction)
1 Air 1.00E-06 N :0.78 O: 0.22
2 Body (Muscle) 1.02 H: 0.111 O: 0.889
3 Dry Spine Bone 1.22 H: 0.024 O: 0.196 Ca: 0.78
4 Dry Rib Bone 1.79 H: 0.024 O: 0.196 Ca: 0.78
5 Lung 0.30 H :0.111 O: 0.889
Cross section data sets with Legendre order of 0 and 3 are generated based on the group
structure (Table 6-2) and mixture composition (Table 6-3). Note that deterministic calculation
results for the lower groups carry some information about the phantom. They might be useful to
improve the quality of the reconstructed phantom image.
Performance of Fictitious Quadrature Technique
We demonstrate the TITAN code's performance on this benchmark by simulating the four
proj section images along the directions normal to the four boundaries parallel to the : axis. Four
directions are defined in the fictitious quadrature specification file:
Table 6-4. Directions in the fictitious quadrature set for the heart phantom benchmark.
Direction Number rl 5 Description
1 1.0 0 0 Normal to the left boundary
2 -1.0 0 0 Normal to the right boundary
3 0 1.0 0 Normal to the back boundary
4 0 -1.0 0 Normal to the front boundary
The four directions in Table 6-4 are singularity directions. The angular fluxes along the
four directions on the corresponding model boundaries are computed with an extra sweep after
the source iteration process is completed. Assuming a perfect 128xl28 collimator array adjacent
to the body (i.e. all other photons are blocked except those along the interested directions), the
angular flux distribution can be used to simulate the proj section images taken by the SPECT
camera. More directions can be added in the fictitious quadrature to simulate proj section images
from other angles. Since the phantom model has 128xl28xl28 fine meshes, all the four
simulated images have 128xl28 pixels. We simulated several cases with different SN orders (Ss
and Slo) order and PN order (Po and P3) with the SN solver. The output images are similar. We
also performed a Monte Carlo reference calculation with the SIMIND code.40 SIMIND is a
Monte Carlo code used in the nuclear medicine discipline to generate SPECT proj section images.
SIMIND uses about 8 minutes (2 min/proj section) on a 1.5GHz Pentium 4 processor. While
it takes about 4 minutes for TITAN to compute the boundary angular fluxes for the first group on
an AMD Opteron 242 CPU (1.6MHz, 1024k cache), which is about twice faster than the Pentium
CPU. Figure 6-4 compares the globally normalized images calculated by TITAN (Ss and Po) and
SIMIND. And Figure 6-5 compares the individually normalized images.
Al) TITAN: front A2) TITAN: left
A3) TITAN: back A4) TITAN: right
B l) SIMIND: front B2) SIMIND: left B3) SIMIND: back B4) SIMIND: right
Figure 6-4. Globally normalized projection images calculated by TITAN and SIMIND.
Al) TITAN: front
A2) TITAN: left
A3) TITAN: back
A4) TITAN: right
B l) SIMIND: front B2) SIMIND: left B3) SIMIND: back B4) SIMIND: right
Figure 6-5. Individually normalized projection images calculated by TITAN and SIMIND.
107
In Figure 6-4, the four images (front, left, back, and right) are normalized together. It
provides a valid intensity comparison between the four images, among which the right proj section
is the weakest, since it has the longest distance to the heart. In Figure 6-5, the four images are
normalized individually. It shows a clearer view on the difference between SIMIND and TITAN
calculation results. By visual comparison, it seems that the TITAN computed images have a
higher contrast ratio. For a better understanding the amount difference between the results, Table
6-5 provides the overall differences for the voxels above 90% intensity, which are mostly located
in the heart region.
Table 6-5. TITAN calculation errors relative to the SIMIND simulation.
Images Max. Error 2-norm Error
Front 18.89% 3.711E-03
Left 11.29% 1.349E-03
Back 41.92% 6.882E-03
Right 40.22% 8.950E-03
As expected, larger differences are observed in the back and right proj sections that are
farther from the heart as compared to left and front proj sections. Further, the 2-norm of the results
is very low, indicating the maximum errors occur at small fraction of voxels. The differences
could be attributed to the following:
* In SIMIND simulation, we specified a parallel collimator and Nal detector. The effects,
including particle reaction in collimator septa and detector efficiency, are not considered in
the TITAN code.
* SIMIND uses an equal number of particles (i.e., 767,555) to generate all the four
proj section images, while they are located at significantly different distances from the
hearth. Hence, it is expected that the back and right images exhibit larger relative errors.
In order to resolve this important issue, it is essential to determine the pixel-wise statistical
uncertainty map in SIMIND.
* TITAN uses the group cross section Eile generated by CEPXS. While the continuous
energy cross section data built in SIMIND is tuned to the human body materials and
SPECT simulation. Some errors could be due to the cross section data.
Particle transport problems for SPECT traditionally are simulated by the Monte Carlo
approach. Although it is still difficult to perform a strict comparison with the Monte Carlo
simulation by SIMIND due to the reasons discussed in the previous section, the preliminary
results of the TITAN calculation show a reasonably good agreement with the reference. One
potential advantage of deterministic method over the Monte Carlo approach is the reduced
computation time when simulation of a large number of proj section images is required. In a
SIMIND simulation, the CPU time is proportional to the number of projection images. While the
computational cost for TITAN is mostly dedicated to the calculation of the flux moments. After
the flux moments are converged, an extra sweep can compute a projection image with much less
cost. Furthermore, flux moments can be stored after the transport calculation. And proj section
image simulations for different angles can be processed in parallel using the same stored flux
moments. Therefore, TITAN could be much faster for simulation to a large number of proj section
images.
The usage of the fictitious quadrature is not limited to SPECT simulations. The technique
is a relatively reliable approach to evaluate the angular fluxes in interested directions. However,
currently extra sweep with fictitious quadrature can be applied only for problems with vacuum
boundary condition. And although multiple regular quadrature sets can be defined in TITAN,
only one fictitious quadrature is allowed in one problem model.
CHAPTER 7
PENTRAN INTEGRATION AND LIMITATION STUDIES OF THE CHARACTERISTICS
SOLVER
The coarse mesh/fine mesh scheme in the multi-block framework of the TITAN code is the
same as the one used in the PENTRAN code. The block-oriented SN solver and characteristics
solver are developed based on the meshing scheme. We incorporated a modified version of the
characteristics solver into the parallel engine of the PENTRAN code. In this chapter, the
implementation of characteristics solver into PENTRAN is discussed. The performance of the
integrated characteristics solver is tested on the simplified CT model benchmark with different
parallel decomposition schemes. Finally, the limitations of the characteristics solver in TITAN
are examined.
Implementation of the Characteristics Solver in PENTRAN
The data structure difference between PENTRAN and TITAN leads to some modifications
on the characteristics solver in order to complete the integration. PENTRAN' s data structure is
tuned to the parallel environment. The maj or data arrays, including angular flux, flux moment,
etc., are allocated locally. Since TITAN is still serial code, one major challenge is to seamlessly
integrate the serial characteristics solver into the parallel engine.
In PENTRAN, based on the number of Eine meshes within a coarse mesh, a memory-
tuning procedure is used to group the coarse meshes into two categories: medium and large
coarse meshes. While TITAN' s obj ect-oriented programming paradigm allows each coarse mesh
to be treated individually. The structure of the angular flux array is built on the loop architecture
of the source iteration scheme. The dimensions for energy group, coarse mesh, direction octant
in the angular flux array are treated as parent obj ects of the Eine mesh flux. PENTRAN also
stores all the boundary fluxes for each fine mesh, and the boundary flux for each coarse mesh is
stored implicitly with the fine mesh boundary arrays. In TITAN, both coarse mesh and fine mesh
boundary fluxes are treated explicitly by the frontline style sweep procedure, and only the front
line fluxes are dynamically stored. Some differences of the memory structure between the two
codes are listed in Table 7-1.
Table 7-1. Memory structure differences between PENTAN and TITAN
Array name PENTRAN TITAN
Angular flux Two category, locally Coarse mesh individually
Flux moment Two category, locally Coarse mesh individually
Fine mesh boundary flux Stored Not stored, front-line style sweep
Coarse mesh boundary flux Stored Not stored, front-line style sweep
We decided to keep the memory structure untouched in PENTRAN while integrating the
characteristics solver. Thereby, instead of reallocating arrays, new arrays are allocated in
PENTRAN when it is necessary, and de-allocated when they are not needed any more. Table 7-2
compares the characteristics solver in the modified version of PENTRAN (PENTRAN-CM) and
TITAN.
Table 7-2. Comparison of the characteristics solver in PENTAN-CM and TITAN
PENTRAN-CM TITAN
Ray-tracing On the fly Pre-calculated
Geometry information Not stored Stored
Bilinear interpolation Employed Employed
Coarse mesh material Void Void, low-scattering medium,
pure absorber
Projection compatibility Not completely compatible Compatible with angular and
with Taylor projection spatial projection
In the TITAN code, the ray-tracing along the quadrature directions are performed as an
initial task. And the calculated geometry information, such as intersection points, path lengths,
and bilinear interpolation weights, are stored and can be accessed directly in the sweep process.
Depending on the meshing and quadrature set, a relatively large amount of memory is required to
store the geometry information. At the cost of memory, the characteristics solver can sweep the
coarse meshes much faster. In the PENTRAN-CM code, the geometry information is not stored.
The ray-tracing procedure is performed on the fly within every sweep. This approach is CPU-
intensive. However, it reduces the memory requirement. This approach is also suitable to the
PENTRAN' s coarse mesh data structure, thereby, requiring minimal programming changes. For
compatibility reasons, currently, the characteristics solver in PENTRAN-CM can only be used in
void regions. Note that PENTRAN is fully parallelized in the three domains (energy, angle, and
space) of the phase space," while TITAN is a serial code. However, in the PENTRAN-CM
implementation, we take full advantage of the parallel engine, such that the characteristics solver
module can be distributed to different processors to complete the assigned tasks. The individual
tasks for each processor can be transport calculations for a subset of energy groups, octants,
and/or coarse meshes specified by a decomposition scheme."
Benchmarking of PENTRAN-CM
We tested the performance of the characteristics solver in PENTRAN-CM using the
simplified CT benchmark discussed in Chapter 5. Some measures are taken in meshing, cross
section and quadrature set, so that we can make a fair and valid comparison within the
PENTRAN parallel engine.
Meshing, Cross Section and Quadrature Set
We recall that two meshing schemes are used in the original benchmark: the 7-coarse-
mesh model (for the SN solver shown in Figure 5-8) and the 3 coarse mesh model (for the hybrid
solver shown in Figure 5-9). Both models are tested in this PENTRAN-CM benchmarking. A
two-group cross section data file is used to test the parallel decomposition in the energy domain.
The one-group data in the original benchmark is listed in Tables 7-3.
Table 7-3. One group cross section used in the CT benchmark with TITAN.
Material # C, UC C, <
1 (air) 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07
2 (source) 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07
3 (detector) 2.03430E-02 0.0E+00 3.88343E-01 2.6038 7E-01
4 (Water) 7.96423E-04 0.0E+00 1.48783E-01 1.23481E-01
Materials #1 and #2 are the same material. They are represented separately because the
problem can be modeled more easily this way. Material #4 is the characteristics coarse mesh
material. By changing the group constants of material #4, we can further examine the scattering
ratio limitation of the characteristics solver. Table 7-4 lists the two-group cross section data used
in PENTRAN-CM for group parallel decomposition.
Table 7-4. Two group cross section used in the CT benchmark with PENTRAN-CM.
Mat # Grp # Ca UCr Cslg a s a
1 1 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07 0.0E+00
1 2 3.17640E-08 0.0E+00 8.79200E-07 7.75720E-07 1.17630E-07
2 1 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07 0.0E+00
2 2 3.17640E-08 0.0E+00 8.79200E-07 7.75720E-07 1.17630E-07
3 1 2.03430E-02 0.0E+00 3.88343E-01 2.60387E-01 0.0E+00
3 2 1. 083 05E-01 0.0OE+00 5.48045E-01 3.7 8060E-01 1.07613E-01
4 1 7.96423E-04 0.0E+00 1.48783E-01 1.23481E-01 0.0E+00
4 2 5.29416E-03 0.0OE+00 1.80640E-01 1.60378E-01 2.45063E-02
The two-group cross section data is mixed by the GIP code with the Sailor96 library with
only down-scattering.30 And the first group constants in Table 7-4 are the same as the one-group
cross section data in Table 7-3. Therefore, the first group fluxes at the detectors remain the same
regardless of the number of groups. We can compare the detector responses calculated by
PENTRAN-CM with the TITAN results.
In the original CT model, we also applied the PN-TN ordinate splitting technique. In
PENTRAN, only the rectangular ordinate splitting technique is available. In order to use the
same quadrature, the PN-TN S20 quadrature set with two PN-TN splitting directions (Figure 5-11)
is extracted from the TITAN code, and used as a user-defined quadrature set in PENTRAN. A
minor modification in the quadrature routine of PENTRAN is made to process the split
directions. In this quadrature set, there are total 207 directions in each octant.
Benchmark Results and Analysis
A number of cases are tested with the characteristics solver in a parallel environment as
listed in Table 7-5. Note that in PENTRAN, the characteristics solver is included in PENTRAN' s
adaptive differencing strategy as Option 5.15 (i.e. the differencing variable ndmeth=5).
Table 7-5 compares the first 10 detector responses calculated by PENTRAN-CM with
TITAN. Note that the TITAN results are extracted from Table 5-1 for Case 5. The other 10
detectors are symmetric, thereby, they have the same responses as the first 10 detectors.
Table 7-5. Characteristics solver calculated detector response by PENTRAN-CM and TITAN.
Detector # Case la in Table 7-3 Case 5 in Table 5-1 Difference
1 1.345E-03 1.345E-03 4.20E-07
2 1.474E-03 1.475E-03 3.60E-07
3 1.510E-03 1.510E-03 5.40E-07
4 1.579E-03 1.579E-03 -7.00E-08
5 2.095E-03 2.094E-03 -6.00E-07
6 2.123E-03 2.123E-03 -1.80E-07
7 2.131E-03 2.132E-03 9.50E-07
8 2.146E-03 2.146E-03 -4.50E-07
9 2.155E-03 2.155E-03 -5.90E-07
10 2.152E-03 2.152E-03 -4.00E-07
Table 7-5 shows the difference between the two cases is in the order of 10- which is much
lower than the scalar flux convergence tolerance 10-4. Therefore, the characteristics solver
produces the same calculation results within the machine truncation error in both TITAN and
PENTRAN-CM. Table 7-6 compares the CPU time of PENTRAN-CM for a number of cases
with different parallelization decomposition schemes. Note that the detector responses cases are
almost the same as the results in Table 7-5. This also demonstrates the accuracy of PENTRAN' s
parallel engine for different parallelization decompositions schemes.
Table 7-6. Characteristics solver performance in PENTRAN parallel environment.
Case # of # of Decomposition factor (deempv') Differencing Scheme (ndmeth ) CPU
# CM CPU Angular Group Spatial First Middle Last Time
coarse coarse coarse (sec)
mesh mesh mesh
la 3 16 8 2 1 -2 5 -2 7.7
lb 7 16 8 2 1 -2 -2 -2 33.3
2a 3 8 8 1 1 -2 5 -2 10.2
2b 7 8 8 1 1 -2 -2 -2 43.5
3 3 12 2 2 3 -2 5 -2 23.0
4a 3 1 1 1 1 -2 5 -2 64.0
4b 7 1 1 1 1 -2 5 -2 330.0
5a 3 1 Serial Run -2 5 -2 61.4
5b 3 1 Serial Run -2 -2 -2 323.0
SPENTRAN parallel decomposition variable.
2 PENTRAN differencing scheme variable, ndmeth=--2 corresponds to the Directional Theta-Weighted
scheme, and ndmeth=--5 corresponds to the characteristics solver.
Cases la and lb use 16 processors with an angular decomposition factor of 8, an energy
group decomposition factor of 2, and a spatial decomposition factor of 1. In Case la, we use the
characteristics solver by setting ndmeth=5 for coarse mesh #2. Case lb applies the SN solver
only, and uses a total of 7 coarse meshes in order to overcome the ray-effects. The solutions for
both cases are accurate comparing to the solution of Cases 4 and 5 in Table 5-1 respectively
(compared in Table 7-5). An acceleration factor of about 4.3 is achieved with the characteristics
solver comparing to the SN solver, which is slightly lower than in TITAN code.
We can draw the same conclusion based on other cases. Cases 2a, 2b, and 3 use 8 and 12
processors respectively. Cases 4a and 4b are parallel runs, although only one processor is used.
Case Sa and 5b provide the results for serial version of PENTRAN. It takes about 61.4 second
with the characteristics solver, while about 323 seconds for the SN solver with the refined
meshing. This result shows that the characteristics solver is more efficient than the SN solver in
void regions in term of CPU time. In PENTRAN-CM, ray-tracing procedure is performed on the
fly. In TITAN, the characteristics solver can be faster than the SN solver even with the same
meshing, since ray-tracing information is pre-calculated and stored.
Investigation on the Limitations of Characteristics Solver
Thus far, we have benchmarked the characteristics solver in TITAN with the CT model
and Kobayashi problems. We also integrated the solver into the PENTRAN code, and tested the
on-the-fly mode of the solver in a parallel environment. The hybrid approach with the
characteristics solver shows an excellent performance on the benchmarks. However, the
limitations of the solver and its sensitivity related to meshing and quadrature order are not fully
addressed. In this section, we further analyze the characteristics solver based on its memory
requirement, factors that affect accuracy, and possible improvement approaches.
Memory Usage
In the storage module of the characteristics solver, we use an array of user-defined type,
called GEOSET in the TITAN code, to store the coarse mesh geometry information for the
characteristics solver. The size of the GEOSET array equals to the product of the number of fine
meshes on the coarse mesh boundaries and the number of directions in the quadrature set for the
coarse mesh. Therefore, every characteristic ray in the coarse mesh requires a GEOSET obj ect,
which specifies five variables for the ray:
* Fine mesh index i at the incoming boundary (2 byte integer).
* Fine mesh index j at the incoming boundary (2 byte integer).
* Bilinear weight s on the incoming boundary (4 byte real).
* Bilinear weight t on the incoming boundary (4 byte real).
* Track length I of the ray (4 byte real).
These five variables, which are calculated by the ray-tracing routine before the source
iteration process starts, represent all the required geometry information for a characteristic ray, if
we consider the coarse mesh as one region. The pair (i, j) is used to locate the four fine meshes
on the incoming boundary for the bilinear interpolation procedure. The pair (s, t) is the bilinear
weights as defined in Eq. 2.20. And I is the track length across the coarse mesh.
If we consider a four-byte real number as one memory unit, each GEOSET can be stored
with 4 memory units. Note here we store the (i, j) pair as 2-byte integers, instead of the regular 4-
byte integers. So the pair can be considered as one memory unit. The amount of memory
required by the GEOSET can be very large with Eine spatial meshing and high order of
quadrature set. In certain cases, it can be even larger the SN solver. For example, for a coarse
mesh with 10x10x10 Eine meshes and with the same quadrature, the SN solver requires 1000 x
number of direction memory units to store the angular flux. While the characteristics solver
needs 10x10x6 x number of direction x 4 memroy units. The characteristics solver needs about
twice amount of memory as the SN solver. This is demonstrated in Table 5-11 with the
Kobayashi benchmark problems.
The bilinear interpolation procedure requires at least 2x2 meshing on a boundary. On the
other hand, because we use 2-byte integer to store the fine-mesh index in a GEOSET, the number
of boundary Eine meshes is limited to 255x255 for the characteristics solver, which is more than
enough for most problems. We further discuss the mesh size limitation in the next section.
Limitation on the Spatial Discretization
A deterministic solver does not suffer from the statistical uncertainties as in the Monte
Carlo approach. However, since in a deterministic method, the phase space has to be discretized,
the solution accuracy is affected by mesh/grid size. Generally speaking, finer grid size (i.e. Einer
energy group structure, higher order quadrature set, and smaller spatial meshing) should lead to a
more accurate solution at a higher computational cost. It is difficult to set up some universal
criteria on how to decide the optimistic grid size, since it depends on both the algorithms and the
individual problem model. Generally, a good understanding of the physics of the problem can
provide some guidelines in the process of modeling. For example, for a zero-moment SN solver
with the diamond differencing scheme, it is recommended to keep the mesh size under the
material mean free path.
2-D meshing on the coarse mesh boundaries
In the characteristics solver, we integrate the transport equation along the characteristic
rays. A 2-D meshing scheme is required on the coarse mesh boundaries. Generally, the 2-D
meshing scheme is subject to the spatial discretization requirement for a deterministic solver.
Furthermore, we need to consider two maj or factors to determine the mesh size on the coarse
mesh boundaries for the characteristics solver.
* Angular flux distribution fluctuation on the coarse mesh boundaries.
* Angular flux resolution requirement on coarse mesh boundaries for the model.
The first factor is introduced with the bilinear interpolation procedure, which assumes a
linear angular flux distribution on the local four Eine meshes surrounding the intersection point of
each ray with the incoming boundary. With a relatively flat incoming boundary flux distribution,
larger Eine mesh size can be used while preserving the accuracy of the bilinear interpolation. In
an SN coarse mesh, we specify the number of fine meshes (i, j, and k) along x, y, and z axes. In
the characteristics solver, we still use the three integers to define the meshing on each boundary.
For example, the two x-y boundaries have i x j fine meshes. With this meshing scheme, the
bilinear interpolation can keep consistency on the incoming and outgoing boundaries for
directions in different octants. More discussion on the accuracy of the bilinear interpolation was
given in Chapter 2. The second factor can be illustrated with the simplified CT model as shown
in Figure 7-1.
200
~00
Figure 7-1. Characteristics coarse mesh boundary meshing based on flux resolution requirement.
Figure 7-1 shows the meshing scheme of the second coarse mesh in which the
characteristics solver is used. We use 20x10 meshing on the two y-z boundaries, while only 2x2
meshing is applied on the other four boundaries. Our goal is to calculate the detector responses
on the right side of the coarse mesh. Therefore, it is required to apply hier meshing on the y-z
boundaries. We can use much coarser meshing on the other four vacuum boundaries because
these boundary fluxes cannot affect the detector responses. Note that here we choose 2x2
meshing, which is the minimum requirement on meshing for the characteristics solver by the bi-
linear interpolation procedure.
We also investigated the impact of the 2-D meshing on two y-z boundaries. The original
hybrid model uses 20x10 meshing on y-z boundaries of coarse mesh #2. Figure 7-2 examines the
detector response errors as compared to the reference MCNP case by using different number of z
Eine meshes. Case 1 is the MCNP reference case. In Case 2a to 6a, the characteristics solver is
used in coarse mesh #2 with 5, 8, 9, 10, and 12 z Eine meshes. The error curve moves up closer to
the reference solution as increasing the number of z fine meshes from 5 to 8. It indicates that the
characteristics solver provides more accurate solution with Einer discretization grid.
4.00%
300%
-~u *-c ~ase 1 mcnp ref
-m-ase 2a Ray zfm=5
2.00% -case 3a Ray zfm=8
1.00% -case 4a Ray zfm=9
-a-case 5a Ray zfm=10
0.00%- e-* -g -e-ase 6a Ray zfm=12
-1.00%-
-2.00%-
-3.00%-
-4.00%-
-5.00%
Figure 7-2. Detector response relative errors with different number of z fine meshes for the
characteristics solver.
Figure 7-3 shows the relative errors for the SN solver with different z meshing on the same
coarse mesh. Note that for the SN solver, the fine mesh size along x axis is 1cm. Case 2b to 5b
use the SN solver in coarse mesh #2 with 5, 8, 9 and 10 z fine meshes.
3.00%
2.00%-
1.00%-
-+-ase 1 mcnp ref
0.00% +-* -m-case 2b Sn zfm=5
I.' :i' case 3b Sn zfm=8
-1.00% -case 4b Sn zfm=9
-m-case 5b Sn zfm=10
-2.00%-
-3.00%-
-4.00%
Figure 7-3. Detector response relative errors with different number of z fine meshes for the SN
solver.
All the curves in either Figure 7-2 or Figure 7-3 follow a similar trend. One can observe a
jump when increasing zfm~n=9 (Case 4a) to zfmn=10 (Case 4b) f or the characteristics solver (zfmn is
the number of fine mesh along z). It seems that the solutions by the characteristics solver is
affected by the z fine meshing more sensitively than the SN solver. Table 7-7 provides the maxim
percentage and 2-norm errors for all the cases.
Table 7-7. Error comparison with different z meshing.
Number of z Characteristics Error Error SN solver Error Maxim
fine meshes solver cases 1-norm 2-norm cases 2-norm error
5 Case 2a 1.3103E-02 4.536% Case 2b 3.3245E-03 2.360%
8 Case 3a 6.7115E-03 3.622% Case 3b 3.3309E-03 2.786%
9 Case 4a 3.1872E-03 2.771% Case 4b 2.6947E-03 2.285%
10 Case 5a 7.5098E-03 3.280% Case 5b 2.8202E-03 2.092%
12 Case 6a 2.1630E-03 2.515% 3.3245E-03 2.360%
Case Sa and Case 5b are the models used in the CT benchmark discussed in Chapter 5.
Table 7-7 indicates that one can acquire a relatively accurate solution with different zfmn numbers
around 10, which demonstrates the stability of the hybrid algorithm. We further investigated the
effects of y mesh size. Figure 7-4 shows the detector response sensitivity to the number of fine
meshes along y axis (yfmn) for the SN and characteristics solver.
3 OO%-
-*case 1 mcnp ref
2 OO% 4J -' ( -mcase 5a Ray yfm=20
case 5b Sn yfm=20
1 OO% -case 7a Ray yfm=4
-m-case 7b Sn yfm=40
-2 OO%-
-3 OO%
Figure 7-4. Detector response sensitivity to the fine mesh number along y axis.
The curves are acquired by using the same z fine mesh number (zfmn=10), but different y
fine mesh numbers and solvers. Cases Sa and 7a use the characteristics solver with yfmn=20 and
yfmn=40, respectively, While the SN solver is used in Cases 5b and 7b. The two SN curves follow
a similar trend. And as expected, the solution for Case 7b (yfmn=40) is more accurate than Case
5b (yfmn=20). The solutions with yfmn=20 and yfmn=40 are nearly identical for the characteristics
solver. This indicates the yfmn=20 meshing scheme is fine enough for the characteristics solver to
evaluate the 20 detector responses.
Coarse mesh size limitation for the characteristics solver
We further investigated the effects of the coarse mesh size on the accuracy of the
characteristics solver. Since we consider the coarse mesh as one region, the limitation on the path
length of the characteristic rays across the coarse mesh is the maj or factor in determining the
coarse mesh size. The characteristics solver integrates the LBE along the rays with the
assumption that the scattering source is constant throughout the coarse mesh in one sweep (flat
source region). If the material for the coarse mesh is void or pure absorber, such assumption is
valid because the scattering source is always zero. For example, in the CT model, we can use a
large coarse mesh size with the characteristics solver in the air region. In materials other than
void or pure absorber, the scattering ratio of the material is the maj or factor on the size limitation
of the coarse mesh. With scattering collision increasing, we have to reduce the coarse mesh size
to maintain the flat source assumption.
We examined this effect by changing the material cross section data in the CT model.
Here, we use the SN model as the reference, since we already validate the SN solver on this
model. The MCNP model requires much longer CPU time without variance reduction to achieve
a good statistical confidence, because it is more difficult for the detectors to score a particle
when increasing the total cross section in the 'air' region. For the SN model, here we use yfmn=20
and zfi=5. Therefore, all the SN coarse meshes are filled with 1xlx1 cm3 f1ne meshes. Note that
the SN solution is not necessarily as accurate as the MCNP reference we used in the original CT
benchmark. However, it can provide a valid approximate reference solution for the purpose of
this benchmark.
The accuracy of the characteristics solver in void regions is already demonstrated with the
original CT benchmark. Here we further examined the performance of the solver in pure
absorber regions. Figure 7-5 shows the detector response difference for the SN and characteristics
solvers with pure absorber in the 'air region' (cross sections for material in coarse mesh #2 are:
cr, = 1.48783E-01 r, = 0.0 ).
3 3000E-07
3 1000E-07
2 9000E-07
2 7000E-07
2 5000E-07 i-o-- ll~ .. ...t l..: Ili
2 3000E-07
2 1000E-07
1 9000E-07
1 7000E-07
1 5000E-07
0 5 10 15 20 25
Figure 7-5. Detector response comparison between SN and characteristics solver in pure
absorber media.
The characteristics solution (characteristics coarse mesh meshing scheme: yfmn=20, z~i=5,
and x/in=2) shows a relatively close agreement with the SN solution (maxim difference 1. 52%).
This demonstrates that the characteristics solver is accurate in pure absorber media. Figure 7-5
shows that the characteristics solver is less sensitive to the ray-effects, which is also
demonstrated in the original CT benchmark for void regions.
0 5 10 15 20 25
We further changed the cross section data with different scattering ratios while fixing
a, = 1.48783E-02 Figure 7-6 shows the difference between the SN and characteristics
solutions for four different scattering ratios (as / a,=0.05, 0.08, 0. 10, and 0.20). Note here the
characteristics coarse mesh size is 59cm along x axis with meshing scheme: yfmn=20, zfmn=5, and
xfm= 2.
95E-04
90E-04
85E-04
8 0E-04
7 5E-04
70E-04
6 5E-04
60E-04
55E-04
50E-04
95E-04
90E-04
85E-04
8 0E-04
7 5E-04
70E-04
6 5E-04
60E-04
55E-04
50E-04
0 am-,,, ar...
-m-- 1- 5 3. art.-,,,
*' '"
0 5 10 15 20 25
95E-04
90E-04
8 5E-04
8 OE-04
7 5E-04
7 0E-04
6 5E-04
6 0E-04
55E-04
5 0E-04
1 1E-03
1 OE-03
9 OE-04
8 0E-04
7 OE-04
6 OE-04
5 OE-04
a-- 1-
"'C '- 1 11
0 5 10 15 20 25
0I 5 10 15 20 25 1 D
Figure 7-6. Detector response comparison between SN and characteristics solver in media with
different scattering ratio. A) ratio=0.05 B) ratio=0.08 C) ratio=0.10 D) ratio=0.20
By comparing the SN solutions in Figures 7-6A and 7-6B, one can observe that the detector
responses increase very slightly when increasing the scattering ration from 0.01 to 0.2. This is
because that the detector responses are mainly dictated by the magnitude of the total cross
section, which remains the same for all cases. Figure 7-6 also shows that the characteristics
solver tends to over-estimate the solution with higher scattering ratio. This can be attributed to
the flat source assumption in Eq. 2-16, and it can be explained as follows. The scalar flux in the
coarse mesh approximately decreases exponentially from the source region to the detector region
(# = e-o"). Here the scattering source is the only contributing source term, since no fixed source
is present in the characteristics coarse mesh. With the flat source assumption, the scattering
source is calculated by multiplying the coarse-mesh averaged flux and the scattering cross
sections, and it remains the same in the coarse mesh within each iteration. As a result, the
scattering source is over-estimated as x close to the detector region, resulting in the
overestimation of the outgoing angular fluxes for the coarse mesh, which leads to a higher
detector response. The source term's contribution to the outgoing angular fluxes increases with
the scattering ratio. Therefore, Figure 7-6 shows that the detector responses are overestimated
further with higher scattering ratios.
In order to correct this overestimation (i.e. allow the flat source assumption to be
applicable), it is necessary to decrease the length of the characteristic ray, or decrease the size of
the coarse mesh along the axis of interest. Figure 7-7 compares the characteristics results with
different coarse mesh sizes of 46 cm, 36 cm, and 32 cm to the SN reference solution. For this test,
we use a scattering ratio of 0.2.
9 OE-04-
85E-04-
7 5E-04-
7 OE-04-
6~- 5E-0 -
6 OE-04-
5 5E-04-
O 5 10 15 20 25
Figure 7-7. Characteristics solutions with different coarse mesh size along x axis.
As expected the characteristics solution approaches to the SN solution as the coarse mesh
size decreases. The relative errors and CPU time for all the characteristics cases in Figures 7-6
and 7-7 are given in Table 7-8.
Table 7-8. Characteristics solution relative difference to SN solution with different scattering
ratios and coarse mesh size.
Case # Coarse Total cross Scattering Scattering Error Error CPU
mesh size section (cm ') cross section ratio 2-norm 1-norm Time
along x Ratio
(cm, mfp) (SN/Ray)
159 (0.87*) 1.48783E-02 7.43915E-04 0.05 1.2761E-03 1.393% 3.13
2 59 (0.87) 1.48783E-02 1.19026E-03 0.08 1.1499E-02 3.336% 3.10
3 59 (0.87) 1.48783E-02 1.48783E-03 0.10 2.6466E-02 4.646% 3.08
4 59 (0.87) 1.48783E-02 2.97566E-03 0.20 1.9858E-01 11.677% 3.12
5 46 (0.68) 1.48783E-02 2.97566E-03 0.20 8.1535E-03 3.131% 1.88
6 36 (0.54) 1.48783E-02 2.97566E-03 0.20 1.1328E-03 1.330% 1.41
7 32 (0.48) 1.48783E-02 2.97566E-03 0.20 8.0144E-04 1.353% 1.28
8 27 (0.40) 1.48783E-02 3.71958E-03 0.25 7.5987E-03 3.026% 1.13
9 22 (0.32) 1.48783E-02 4.46349E-03 0.30 2.5802E-03 2.223% 1.01
10 17 (0.25) 1.48783E-02 5.95132E-03 0.40 1.6210E-03 1.487% 0.91
Values in the parentheses are in unit of mean free path (mfp).
Based on the results in Table 7-8, we conclude that the characteristics solver can provide
an accurate solution by reducing size of the coarse mesh with higher scattering ratio. For the
cases with scattering ratio of 0.20 (Cases 4 to 7), the limitation on the distance along x is about
36 cm, which is about half of the mean free path for the material (~70 cm). Generally, the
accuracy of the solver depends on both the scattering ratio and mean free path of the material.
Table 7-8 also indicates that the product of scattering ratio and the mean free path, which is
coarse mesh size in unit of scattering mean free path, should be around 0. 1 or less. It is
recommended that the characteristics solver is used for materials with a scattering ration less
than 0.20, because with higher scattering ratio, users need to further refine the coarse mesh size.
And The CPU time comparison in Table 7-8 indicates that the characteristics solver generates
less computational benefits as decreasing the mesh size as shown in Cases 7 to 10. In these four
cases, we keep the coarse mesh size in unit of scattering mean free path close to 0. 1, and the
CPU time ratio decreases gradually. As in Case 10, the SN becomes faster than the characteristics
solver.
Possible Improvements and Extendibility of the Characteristics Solver
The meshing scheme on the characteristics coarse mesh boundaries are limited by the bi-
linear interpolation. And the size and scattering ratio limitations are due to the flat source
assumption. Therefore, we could further study on these two aspects to improve the accuracy of
the characteristics solver. The bi-linear interpolation assumes that the average flux happens on
the center of a fine mesh. We could develop a new interpolation scheme on the incoming
boundaries, which addresses the fact that the point flux actually should be the averaged flux on
the fine mesh area or the cross sectional area of the ray. Instead of assuming a flat scattering
source throughout the region, we could use some higher scheme, for example, linear source
assumption, to represent the source term more accurately. Investigations on these two aspects
will continue.
In summary, the characteristics solver is efficient and accurate in void, pure absorber
regions. For low-scattering medium with scattering ratio less than 0.20, as a conservative
guideline, the size of the characteristics coarse mesh should be equal or less than half of the
mean free path. For higher scattering ratio materials, in which the characteristics solver is not
recommend, the coarse mesh size should be less than tenth of the scattering mean free path.
CHAPTER 8
CONCLUSIONS AND FUTURE WORK
Conclusions
We developed a hybrid algorithm to solve the LBE for realistic 3-D problems, especially
for the problems containing large regions of low scattering media, where the traditional SN
method might become inefficient. A ray-tracing solver is designed and integrated in the TITAN
code along with an SN solver. Both solvers are written under the paradigm of obj ect-oriented
programming with the block-oriented feature. And they are built on the framework of a multi-
block discretization grid (coarse/fine meshing scheme and block-localized angular quadrature).
Both solvers are well-tuned in terms of memory management and CPU efficiency.
The main features of the TITAN code are:
* Integrated SN and characteristics solvers.
* Shared scattering source kernel allowing arbitrary order anisotropic scattering.
* Backward ray-tracing.
* Block-oriented data structure allowing localized quadrature sets and solvers.
* Layered code structure.
* Level-symmetric and PN-TN quadrature sets.
* Incorporation of two ordinate splitting techniques (rectangular and local PN-TN) for the two
type of quadrature sets.
* Fast and memory-efficient spatial and angular proj sections on the interfaces of coarse
meshes by using sparse proj section matrix.
* 'Frontline-style' interface flux handling.
* An efficient algorithm for calculation of the scattering source and the within-group
scattering with a modified scattering kernel.
* A binary I/O library to visualize and post-process data with TECPLOT.
*Extra Sweep technique with the fictitious quadrature technique for calculations of angular
fluxes along arbitrary directions.
We tested the performance of the TITAN code with a number of benchmark problems. For
applications in the Hield of nuclear engineering, TITAN is used to solve the Kobayashi
benchmark, which is a set of difficult shielding problems, and the 3-D C5G7 MOX benchmark,
which is a k-effective problem without homogenization for a MOX/UO2-fueled reactor with
different control rod layouts. For applications in the medical physics Hield, we tested the code on
the CT device model, which is difficult for deterministic codes to solve due to ray-effects, and
the SPECT phantom model, in which transport simulation is commonly performed only by the
Monte Carlo approach. The fictitious quadrature technique we developed for the SPECT model
can be very useful for other medical applications as well. The benchmark results demonstrate not
only the accuracy and efficiency of the code, but also the scalability of the code. For example, in
the CT model, the memory usage still keeps proportional to the quadrature order while
increasing to S200. And in the SPECT model, we are able to use the SN solver in one coarse-mesh
with about two million fine meshes.
Future Work
TITAN provides a code base for future development with its excellent extendibility. There
are still several areas where the code can be further enhanced.
Acceleration Techniques
The loop structure of the code is composed of power iteration loop, upper-scattering loop,
energy group loop, within-group loop, octant loop, coarse mesh loop, direction loop, and fine
mesh/ray loop. Various acceleration techniques can be applied on the power iteration level and
the within-group loop. These acceleration techniques aim to accelerate the convergence of the
fission source or the within-group scattering source. Generally, they can be applied in both SN
and MOC. Coarse mesh rebalancing (CMR) and coarse mesh finite difference (CMFD) are
widely used acceleration techniques,41 which accelerate the within-group loop by forcing the
particle balance in each coarse mesh for each loop. Another physical approach is the synthetic
method,42 in which one can use some lower order methods like diffusion method to acquire a
better estimation of scattering source in-between within-group loops. Some numerical
approaches, such as multi-grid method,43, 44 and pre-conditioned sub space proj section iteration
method,45, 46 can also be applied. However, the general numerical iteration techniques usually
need to be modified here, since in SN codes, we usually do not build up the matrix A in a liner
iteration system x=Ax b due to memory limitation.
Currently, there are two source iteration kernels in the TITAN code. The default kernel is
the S1 kernel, in which the flux moments are updated after angular fluxes are calculated within
each sweep. While the S2 kernel subroutines updates the flux moments immediately after the
angular flux is calculated for each direction. The relationship between S2 and S1 is similar to the
one between Gauss-Seidel and Jacobi iteration methods. Numerically, S2 kernel has a faster
convergence speed than the S1 kernel in most cases without much additional computation cost.
However, it could cause numerical instability problems in some extreme cases. And it is not
preferable for code parallelization in the angular domain. Therefore, currently we choose the S1
kernel as the default option. Another set of kernel subroutines can be added with the flux
moments updated after each octant is processed. This process is numerically similar to the red-
black block or multi-cyclic iteration schemes.
In the future, higher order iteration schemes should be implemented. Krylov subspace
proj section iteration pre-conditioned by CMFD would be a good acceleration combination. The
acceleration subroutines can be inserted into Figure 3-1 around Subroutines L2.7 and/or L3.5.
Parallelization
We can parallelize the TITAN code by using MPI and/or OpenMP. One essential part of
code parallelization work is the loop parallelization. In Figure 3-1, we could break up the coarse
mesh loop and octant loop into a distributed computing environment by using MPI. OpenMP can
be used to parallelize the direction loop in a shared memory environment. Other parallel
algorithms can be applied.47, 48 An MPI and OpenMP hybrid approach can take advantage of the
cost-efficient cluster hardware, as well as multi-CPU nodes and dual-core CPUs. Furthermore,
Code parallelization can benefit from the multi-block framework, since each coarse mesh in the
framework can be treated relatively independently..
Improvements on Characteristics Solver
The TITAN code considers a characteristics coarse mesh as one region, which is sufficient
in this work, since the characteristics solver is only designed for low scattering media. Some
multi-regions data structures already are in place in the code. A more efficient ray-tracer is
required for a multi-region solver.
Other Enhancements
Proj section techniques need to be tested in more problems, since the efficiency and
accuracy of the proj sections are essential under the multi-block framework. It is worth noting that
the multi-block framework can assemble other types of solvers besides SN and MOC. For
example, some non-Cartesian meshing schemes can be implemented in a coarse mesh with a
potential finite element solver.
With above proposed future work, we consider the code still under development. We hope
in the future our community can build an online open-source forum for deterministic
calculations, where users and developers can freely share source codes and ideas.
APPENDIX A
SCATTERING KERNEL IN LINEAR BOLTZMANN EQUATION
Introduction
In the discretized form of the linear Boltzmann equation (Eq. 2-1), the scattering kernel is
the most complicated term. In this appendix, we will prove the following formulation:
(2+1o-~gss 'If P)g,1(F)+ (1k k~(p1) (A-1)
g =1 ~l1 k=1 (1 + k).
[E ,~,t(rF)cos(kp) + ##k,g,t,(r)sin( kp)]}
In Eq. A-1, the discretization in energy domain can be easily separated with the
discretization in the angular domain. The energy and spatial dependency of the scattering source
on the right hand side. Since the $dE' i [ conversion can be achieved straightforwardly by
g'=1
the multigroup approximation, here our main focus is on the conversion of IdA' 4i or
412 l'=1
simplicity, we drop the energy group index (g' and g) and spatial dependency (T) in the flux
moment terms and the cross section moment term. Furthermore, instead of an infinitive Legendre
expansion order, we assume a maxim expansion order ofL. With above simplifications, we can
rewrite the formulation to be proved:
(A-2)
(21+)o-~iflP) +2 (1-k(~)! Ck l COs(kp~)+ ;1 sin(kp)]}
l=1 k=1 (l+ k)!
From now on, we also use the following denotations:
Where 6 is the polar angle with x axis, cp is azimuthal angle on the y-z plane, and
pu = cos(0), pu'= cos(8'). The integration over the unit sphere becomes
Idn' = 2d~P ldp = 4x In some references, for simplicity one can also use
di 1' 1. However, we found it is not necessary to make such assumption, and it
could cause some confusion in the spherical harmonic expansion. So here we still respect the
mathematical fact that the overall solid angle is 4xn. Note that with or without this assumption, the
formulation of Eq. A-2 should remain the same.
In order to prove Eq. A-2, we need to expand the angular flux and the cross section into a
series of Legendre polynomials in the angular domain, respectively. In this appendix, we provide
such an expansion for both the angular flux and cross section. By substitute the two expansion
series into the left hand of Eq. A-1, we can evaluate the new terms, and finally prove the
scattering kernel formulation.
Spherical Harmonic Expansion of the Angular Flux
In this section, we also demonstrate how and why the cosine and sine flux moments are
defined. A smooth function defined on the surface of a unit sphere, such as the angular flux
/(~1') = 1C(pu',y7'), can be expanded by the spherical harmonic function.49, 50
The general form of the spherical harmonic function Y,'(pu', 9') is defined by:
(2n+1) (n-m)!
Y,," (p', cp')= -P," (u' )- e""' (A-5)
4xi (n + m)!
Where P'(pu') is the associated Legendre polynomial. The coefficient a'" is given by:
2x dul+1-UJ~I(I -)Yn(l ~
a"= Inn d~plld dp(pp)Y,"(p,9l) nm ,(u ") A6
2x +1 (2n + 1) (n m)! A6
Where Y,'"(pu, ) is the complex conjugate of Y("(, p) .
The angular flux expansion defined by Eq. A-4 should be a real value. So we expect the
imaginary part of Eq. A-4 is zero. In order to prove this, we rewrite Eq. A-4 as following:
W~p',9')= C fa'",c'"(','= ~jr Y,'(p',Q') + [[arYO'Y,C'"('9)a,'Y'(p '] (A-7)
n=0)m=-n n=() m=1
Based on Eq. A.5, we have:
Y," (p'l,9')= 4,P(pu') (A-8)
By applying the following identity of the spherical harmonic function,49' 51
Y,- "' ( ', yp ') =(- 1) Y,," ( U ', Y ') (A-9)
The coefficient a-'" can be evaluated as:
2=j- + (2n+1) (n+m)l
-1 4xi (n m)!
(2n +1) (n +m)! Cz= +1 (nm1
=~~~~~~ dpd Pp )-(-) "(pu) e""V
4xi (n m)! I (n + m)!
(A-10)
(n- m)! (2n +1) (n +m)! Nx #+
= (-1)m pd "p )"p "
(n + m)! 4xi (n m)! JO -1~~d~ uii~~"0)e"
(2n+1) (n-m)! N #1
= (-1)m pyp ~,"p e"
= (-1)'"a'"
Note in Eq. A-10, we also apply the following identity of the associated Legendre
po ynomial.4
(n nt)
p, "'(p)>= (-1)m .( ("(u)
According Eqs. A-9 and A-10, the last term in Eq. A-7 can be rewritten as:
(A-11)
(a"'Y"(pu',9')
a,"Y "'(p'l, 9')= (-1)"3,' ~-~m(-1)"' Y," (p'1,9')
We substitute Eq. A-12 back to Eq. A-7,
(A-12)
n=0n=-n
m n
~ r O rrO (iu',~71) + ~ r m -iim 011,~71) + a, Y,-"
~ dr r ~ Ilr r
L ~-'n-n LL-'n-n "
n=O m=l
m n
(a, 1' (p, p' + ["'Y' (p',p') (a"Y(" p', ')]}
(A-13)
= :(a~,} '(',Q')+2 aiRe[a"'Y("(,p', )]}
n=0 n;=1
Here we denote the real part of a"'Y,'(p',p') as Re[a,',Y,'(p', p')]. As we expected, the
angular flux is always a real value according Eq. A-13. Now we can further calculate the two
terms in Eq. A-13 based on Eqs. A-5 and A-6. The second term is:
Re[a"'Y,'(p'l, ')]
=~ ~ ~ ~~~P Re[{l dp dp~~)'f()(cos(mp)
=R[JJo ~,drv(ri)ji-1L~~ 4xi (n + nz)!
{ z )~( m) '"~(pu')(cos(n 9') +i sin(n 9'))}]
4n (n + nz)!
isin(nap))}
(A-14)
= ~- ~L "(pnl n-) i') cos(n;9~i' i~~')] dp] dp(pi, V)- ~(f"() cos(n 9)+~
4xi (n + nz)! a-
(2n+ 1) (n n)( 21 +
~~~-p,"(p') sin(mr~p') dp l dp~ wci, a) g;"(p) sin(n p)
4xi (n + n)! J -
And the first term is:
a ~ 2 +1 (2n+1) d~l~U 3 2n+1
-1 4x 4x(A-15)
(2n + 1) 9,1=% r+l
If we define the regular flux moment, cosine moment and sine moment as follows.
1 r22x +1
=~ ) dp1 dy(pu, )- 4,(pU), (A-16)
4xi JO J-1
F,= l'dw dp yip, 9) f,"'(p) cosmp, (A-17)
"':, = 1 dp1dpyr~p,9)-fu"'(p)-sinin;9), (A-18)
We can rewrite Eqs. A-14 and A-15 as follows.
(n nz)
Re[a"'Y ("(p, p')]= (2n +1)- [pf"(pu')cos(mp')("',, + pn"(p1) sin(m p')("' i (A-19)
an, Y,(pl,9') = (2n +1)P, (p1) (-0
By substituting Eqs. A-19 and A-20 into Eq. A-13, finally we derive the expansion
formulation for the angular flux.
n~o n;=>(A-21)
=i~(2N~l" (n -m n)
= f(2n+1)/Y {f(p)( +2f [f'(p1) cos(m p')("',, + pn'"(p') sin(n p')n "')]
One may notice that Eq. A-21 looks similar to Eq. A-4, which is the formulation we need
to prove. However, further derivations are still required to reach Eq. A-4. After the integration,
pu' and 9' disappear on the right hand side of Eq. A-4. And only pu and 9 dependencies are left.
At this point, Eq. A-21 is only a function of pu' and 9'. Here we intentionally use n and na as the
index, so that we can distinguish them with I and k, which we will use in the next section while
expanding the cross section term.
The flux moment formulations, Eqs. A-16 to A-18, are equivalent to Eqs. 2-2 to 2-4 we
discussed in Chapter 2. Note a 42n factor is used in these formulations.
Scattering Cross Section Expansion and the Spherical Harmonic Addition Theorem
The cross section term in Eq. A-2 can be written as follows.
Since the cross section only depends on the scattering angle. With the notations in Eq. A-3,
we can derive the formulation for p,, = O'-0Z .
0'= cos(8')i + sin(8') cos(p7') j + sin(8') sin(7')k~ (A-23)
0z = cos(0)i + sin(0) cos(p) j + sin(0) sin(p)k (A-24)
p, = O'-D2 = cos(0) cos(8') + sin(0) sin(8') cos(6p 9) (A-2.5)
With Eq. A-25, we can apply the spherical harmonic addition theorem.49
4 (p) =((u4 (t')+ 2 (k )Pk (p )[COs(kp~) cos(kpI) + sin(kp) sin(kp')] (A-26)
k=1 (1 + k)!
Now we can expand Eq. A-22 with the Legendre polynomial.
S21+ 1
;~ 4xi '
=i 21 + 1 o 1 Fk(~)p~(U)(-
lo 4xi k=1 (1+k)!
[cos(kp) cos(kp') + sin(kp) sin(kp')])
Note we use the 42n factor in Eq. A-27, because usually we assume o, is the total
scattering cross section. So in case of isotropic scattering, the differential cross section becomes
4xi
Formulation of the Scattering Kernel
So far we have expanded the angular flux with the spherical harmonic function, and the
scattering cross section with the Legendre polynomial. In this section, we multiply the two terms
together and complete the angular integration. Eventually Eq. A-2 is derived.
We begin with rewriting the two expansion formulations (Eqs. A-21 and A-27) and
limiting the expansion order to L.
(n m)1(A-28)
~(2n+1)( {P /(p'nu)A+2i:( P s!(p') [cos(m p')("m, + sin(m p')(" ]}
n=o m= (n + m)!
21 + 1
a~~o, (90) = i{(u)(F~(u') +
(A-29)
2i (1 k(~)! k k U)[COs(kp~) cos(k')+ sin(kp) sin(kp')]}
k=1 (1 + k)!
When we evaluate %~dap' ~dpij', P',')-a,(p-. p') using Eqs. A-28 and A-29, all the tr
and cp terms can be moved out the integration, and obviously a lot of multiplication terms will
appear. Most of the terms become zero. Among the zero terms, some of them are erased by the
orthogonal property of Legendre polynomials, others are scratched off by the facts that:
dp~l~'cos(m7') = 0 and $dyp'sin(mrp')= 0 for m=1, 2.. (A-30)
We will identify these terms step by step. Here, we refer to the term P,(pu')A in Eq. A-28,
and the term (F~(u)P,(u') in Eq. A-29 as 'the first part' of the respective equation, and the
summation term over m or k in both equations as 'the second part'. Now we can apply the
orthogonal property of the regular Legendre polynomials.
2 1 23 4x3
lip)jdp'd'P,(pu')-lIF~(p')= F 1(pu)4 -2xi =1)F(pu)4 (A-31)
-1 21 +1 21 +1
1 l' =n
Where 3 =
Therefore, all the first part multiplication terms become zeros except for those n=1. Now
we consider the first part of Eq. A-28 multiplied by the second part of Eq. A-29 (the summation
term over nt). One can observe that these terms become zeros because of Eq. A-30. Similarly, the
terms, acquired by multiplying the second part of Eq. A-28 with the first part of Eq. A-29,
become zeros as well.
So far the terms we have not covered are the multiplications of the second parts from both
Eqs. A-28 and A-29. A common mistake one might make is to assume
-,idp'l~k )r(U)P=n; 6~,~n~kn The assumption is very convenient here. Unfortunately, such
strict orthogonal relationship for the associated Legendre polynomials can not hold for arbitrary
1, k, n, and nt. However, a relaxed version is always true.49
I 2 (1 +nz)!
dp'l"'(p)P,"(p')= ,,,(A-32)
21+ 1 (1- n)!
In order to apply Eq. A-32, we need to notice the facts that:
J:di'cos(mcp')cos(kpi') dp' sin(mp') sin(kp')= 0 tnien,k=1, 2... (A-33)
$~dp' cos(my'sn; 9') si=k' Xd7' sin(n;9') cos(ka') = 0 forn, k=1, 2.. (A-34)
By using Eqs. A-33 and A-34, we are able to remove all the terms except the terms of
cos(kp')cos(n;9') and sin(kp') sin(n;9') with k=na. Then, we can apply Eq. A-32 on all the
remaining terms. In the end, we can conclude that only the terms with k=n; andlI=n will survive
among all the second part multiplication terms.
Based on the above explanations, we can write the scattering kernel with all the remaining
terms by combining Eqs. A-31 to A-34. Finally, we have proved Eq. A-2.
Li (l + 1)2 4~i x 1- ) 2 (1 + k)!
~ 1"saP,()#,--- +4 [(( k~)? ----- x (A-35)
; 4xi 21+ 1 k=1 (1 + k)! 21+ 1 (1- k)!
Plk flu C,1 COs( kp)+ fasi Jn Yk )]
=i(1 (l+)os,if(irp);l+2~Pk (1-k)!Pkfl) 0COs( kp)+ ##;lsin( kp)])
l=1 k=1 (1 + k)!
Summary
The energy dependency and its integration can be introduced back into Eq. A-3 5. And we
acquire the multigroup form of the scattering kernel. In the TITAN code, we apply the
scattering-in moment form by switching the summation over the group and Legendre order (Eq.
4-1). The switching seems meaningless mathematically. However, it can generate significant
benefits in the coding practice. Further discussions on the scattering-in moment form are already
given in Chapter 4.
In Eq. A-3 5, the direction (pu, ) which is the particle moving direction after a scattering
reaction, is not required to be one of the directions in a quadrature set, although this happens to
be true in the sweep process with a regular quadrature set. Mathematically, (pu, 9) can be an
arbitrary direction in Eq. A-3 5. We take advantage of this fact in the fictitious quadrature
technique we developed in Chapter 6, and also the ordinate splitting technique in Chapter 2. It is
not evident to claim that the scattering source evaluated by Eq. A-3 5 on regular quadrature
directions has a higher accuracy than on an arbitrary direction. Nevertheless, the flux moments
are always calculated with a regular quadrature set to conserve the integration in Eqs. A-16 to
A-18.
Finally, it is worth noting that we choose x axis as the polar axis in all the derivations,
which means 12 is the cosine between CZ and 2. The choice of polar axis does not alternate the
formulation of the scattering kernel. However, the values of some terms in Eq. A-3 5 are affected
by the choice of the polar axis, except for a Level-Symmetric quadrature set, in which all term
values remain the same because of the rotation invariance property. In other quadrature types,
e.g. the Legendre-Chebyshev quadrature, a number of terms in Eq. A-3 5 change with different
polar axes. For example, if we choose the y as the polar axis instead of x, we can build a
relationship between the two systems.
pY = sin(8(X)) cos( )'
rp'" = atan2[sin(8'"x)) sin(r, :), cos(8'"x) (-6
Where atan2 is the extended inverse tangent function, which is available in most math
libraries with various languages. Obviously, Eq. A-36 affects all the terms depending on (pu, )
in Eq. A-3 5, including the flux moments, Legendre polynomial values, cosine' s and sine' s.
However, the overall scattering source should remain the same even with all these changed
terms, because physically the scattering source should not be affected by the choice of polar axis.
Mathematically, one might be able to demonstrate this statement by substituting Eq. A-36 into
Eq. A-3 5 and Eqs. A-16 to A-18. In reality, we can only expand the scattering kernel to a limited
order. In the TITAN code, originally we chose the : axis as the polar axis, later We changed it to
the x axis. The results are almost the same for the first benchmark problem discussed in Chapter.
5. It would be interesting to further investigate the effect of different choices of polar axis on the
scattering kernel.
APPENDIX B
NUMERICAL QUADRATURE ON UNIT SPHERE SURFACE
Introduction
In the process of solving the linear Boltzmann equation, flux moments need to be
evaluated in order to calculate the angular-dependent scattering source term. Flux moment (Eqs.
2-2 to 2-4), by its mathematical nature, is nothing but an integration of a function defined on a
unit sphere surface. The function is the angular flux multiplied by a corresponding regular or
associated Legendre polynomial. Flux moments become angular independent after the
integration over the surface of a unit sphere. The exact distribution of the angular flux on the unit
sphere is unknown. However, we can evaluate function values of the angular flux by the sweep
process at a given number of points (' discrete ordinates') on the unit sphere. Positions and
associated weights of these points are prescribed by a quadrature set. Then, the flux moments can
be simply calculated by a summation of the function values multiplied the associated weights.
Quadrature is a simple but powerful numerical integration technique. For example, a
Gaussian quadrature with an order of N, can acquire the exact value of the integration of any
polynomial up to order of 2N-1 defined within [-1, 1]. In our case, the integration domain is
the surface of a unit sphere. Thereby, we need to build a quadrature to evaluate a double
integration. Mathematically, a good quadrature of a given order always tends to conserve the
integration to the highest order. However, the property of symmetry of a quadrature generally
plays a significant role in a physical problem. For example, in a problem with reflective
boundaries, we obviously hope all reflected directions of a given direction are also in the
quadrature set. Therefore, we often build a quadrature on the balance between keeping symmetry
and conserving higher order integration. For example, the level-symmetric quadrature with an
order of N can conserve moments only up to the Nth order, but with an excellent symmetry
property of rotation invariance. The Legendre-Chebyshev quadrature can conserve moments up
to the 2N-1, but rotation invariance is slightly disturbed.
In this appendix, we prove that the Legendre-Chebyshev quadrature is the best choice in
regards to conserving higher moments. Through the discussion of the procedure, hopefully we
can cast some insights on how a quadrature is built on the balance of simple mathematics and
physics for transport calculations.
General Quadrature Theorem
The popular Gaussian quadrature is built on the orthogonal Legendre polynomial, which is
defined on [-1, 1] with a weighting function w(x)=1. In general, we can consider
(p,, (x) | n > 0 I as the orthogonaIl polynomIIials definedC on (a, b) withI a weighting lfunto ofI~IVIV
w(x) > 0 for a < x < b According to the orthogonality property, we have:
w xp,(x)p,, (~d =( (B-1)
a 7,Y, m =n
Where y, = wnN(x)[p,,(x)] dx. We also denote that p,,(x) = A,,x" +--- an a, = A n
An
the integral of a function f(x) can be represented by an n 'th quadrature formula:
I( f) = w~(x) f (x~dx af w,(,,) f ( ,,I,(f) (B-2)
J-1
For a give~n nulmber of nodes, wei choose the nodep positions {xt~n) and weights (won) in
hoping that we can conserve Eq. B-2 as accurate as possible for any f(x). Mathematically, if we
assume f(x) is a polynomial, this means that the positions and weights of the nodes can hold the
integration exactly as the true value to the highest order of the polynomial. In this sense, the
nodes and weights can by calculated with Theorem B-1,37 which is the fundamental guide for
building the Legendre-Chebyshev quadrature.
Theorem B-1:
For each n > 1, there is unique numerical integration formula of degree of precision 2n-1,
Assuming f(x) is 2n times continuously differentiable on [a b], the formulafor I,V() and its error
is given by
w(x) f(xdv w;, f (,) + f""9) B-3)
For some a < r < b The nodes Cxr) are the zeros of 9,, (x) and the weights fit, are given by:
a,Y,
w, = j=1..,n(B -4)
Legendre-Chebyshev Quadrature on Unit Sphere
Theorem B-1 lays the foundation for building a quadrature set for one-dimensional
integration. In order to apply the theorem for a function defined on a unit sphere, we need to
separate the two-dimensional integration of the angular flux into two one-dimensional
integration.
In general, we consider f (p, 9) is a real smooth function defined on a unit sphere surface,
where pu, 1 < u < 1, is the cosine of the polar angle, and 9, ai < p < +zi is the azimuthal
angle. We need to estimate:
I d (,9)= dp d f (pu, ) (B-5)
First we define a function of g(pu) :
I= Ida f (u,9) = dp g(pu) (B-7)
4 x J
The integration defined by Eq. B-7 can be estimated by a Gaussian quadrature, since the
weighting function isw(x) = 1. Based on Theorem B-1, we choose the quadrature nodes (pu, J as
the roots of the N'th Legendre polynomial.
PN (pI)= 0 (B-8)
Note we usually choose Nas an even integer, so that the roots are symmetrically
distributed on the axis. The weights (wl Jcan be rca~lcla~ted by Eq. B-4. Next we~ needT to
determine the function values of { g/ ,)\ }r .I g\, itself is an integration over a unit circle defined
by Eq. B-6. And it can be estimated by another quadrature, in which we still prefer that the
quadrature nodes are symmetrically distributed on the four quadrant of a unit circle. Thereby, we
separate the integration defined by Eq. B-6 into two parts:
Now we can consider only the integration over the first half of the unit circle, since nodes
on the other half of the circle are decided by symmetry. We denote g(p) = f (p,, 9) and
r = cos(p) The first part of Eq. B-9 can be rewritten as:
Sdp7 f (p,,7 = ) dp7 g(U)) = 11 g(arccos(l7)) = ~ hd) B-0
-dy
Note here dp = d arccos(r) = .And we denote h(r) = g(arccos(r)) .
J1-r
In Eq. B-10, w(r) = is the weighting function for Chebyshev polynomial
T(x) = cos(n arccos(x)) Thereby, we are required to choose the Chebyshev quadrature to
evaluate the integration defined by B-10, so that we can precisely estimate the integration if
h(r) is a polynomial up to the order of 2n-1. Usually, we choose an even integer for n, because
we can keep the symmetry on the top half of the unit circle. Figure B-1 shows the roots of T4 1)
on the unit circle.
zs I, -Y
FigZ3 Z2.Ceyhvrot N=)o aui ice
Fiurthroe B-1. Chebyshey roots (N=4) orlylcae on ah unit circle. n hyar qal
weighted by Eq. B-4.
By combining Eqs. B-7 and B-10, the Legendre-Chebyshev quadrature can be built on a
unit sphere. However, some physical concerns on symmetry still need to be addressed. Normally,
we require the directions in one octant form a 'triangle-shaped' ordering as shown in Figure 2-8
in Chapter 2. And all directions in the other seven octants are decided by symmetry. The
'triangle-shaped' distribution is required to keep the property of 'rotation invariance'. For
example, in the level-symmetric quadrature, number of directions per level increases by one
from one level to the next. And the choice of the polar axis (x, y, or z) does not affect the
distribution of the directions because the directions are perfectly symmetrical. In the Legendre-
Chebyshev quadrature, we can not keep this 'perfect symmetry' because its priority is to
conserve higher moments over rotation invariance. However, we can still keep some 'slightly
disturbed symmetry' of rotation invariance by employing the same 'triangle-shaped' direction
ordering.
The procedure to build a Legendre-Chebyshev Slo quadrature in the first octant can be
explained as follows: We choose the Hyve positive roots ofPlo(x) as the level positions. There is
only one direction on the top level. And its position on the level circle is decided by the positive
root of T2(x). On the second level, the two positive roots of T4(x) become the quadrature node
positions. The third level node positions are chosen by the three roots of T6(x), and so on. On the
bottom level, Hyve directions are to be defined, which are the positive roots of T~o(x). These Hyve
level nodes form a triangle-shaped distribution in the first octant. The final layout of the nodes
has a quite similar look as the level symmetry quadrature of Slo. Figure 2-10A shows the
difference of direction distribution between the level-symmetric and Legendre-Chebyshev
quadrature with an order of 10.
Newton's Method to Find Pn(x) Roots
In the Legendre-Chebyshev quadrature, the roots of Legendre and Chebyshev polynomials
are essential to locate the positions of the quadrature nodes. Chebyshey roots are easy to Eind
since they are uniformly distributed on the unit circle as shown in Figure B-1.
2i -1 2i -1
$.=cr.~co(~)ox ci2n 2n
For a Legendre polynomial f(x)=PN(x), we apply a variant of Newton' s method to find all
the positive zeros { xi) in an inct-lreasng rder asI follows.~
Step 1: Set initial guess x,=0 for the first (smallest) positive root xl.
Step 2: For i=1, 2, ... N, repeat step 3-5, where N, an even integer, is the polynomial rank.
Step 3: Use Newton's method to find root x,.
f(x)
Step 4: Set f (x) =
( x xI )
Step 5: Set initial guess x,= x, for next root x, .~
Step 6: Stop
In Step 3 of the above algorithm, the polynomial f(x) and its derivative can be defined as
follows.
f(x)= P()(B-12)
i(x -x,,)
n =1
d P, ( x) _dP, ( x) 1P, ( x ) 1
(x< x2) (x< x2) (x< x2) ",1 x ,B1
dP, ( x)
P, (x) m ~ ,, x-x
Then we can apply the following iterative formulation ofNewton's method to find root x,
f (x,) P (x, )
x, = xI = x (B-14)
f'(X,>dxN (X, .ii x x,1
In Eq. B-14, P, (x) and P, (x) can be estimated by the recurrence relations of Legendre
polynomial defined in Eqs. B-15 and B-16.
(n +1)P,, (x)- (2n +1)xP, (x)+ nP,_z(x) = 0 (B-15)
(1- x2 P (x) -nxP, (x)+ nP _(x) = (n +1)xP, (x)- (n +1)P,, (x) (B-16)
So far we have set up the layout of the directions on the unit sphere by finding roots of
Pn(x) and T,(x). We will further discuss the node weights in the next section.
Positivity of Weights
Another physical concern is the positivity of the node weights. Level-symmetric
quadrature is limited to the order of 20, because negative weights occur beyond order 20. In the
Legendre-Chebyshev quadrature, the weight for node i is calculated by the product of polar
weight (level weight) and azimuthal weight.
w, = P, T (B-17)
Both the polar weight w, and azimuthal weight wT are calculated by Eq. B-4 with
Legendre and Chebyshev polynomials, respectively. First we evaluate the terms in Eq. B-4 for
azimuthal weights by applying some Chebyshev polynomial properties.
A = 2"lu a -" -2 and 7 = (B-18)
A 2
T~ (xl) = ., and T,, (x, ) = (-1)" sin(P, ) (B-19)
We can substitute Eqs. B-18 and B-19 into Eq. B-4.
-a7 x
w, =" -(B-20)
T, (x, ) T ~(x, ) n
So the Chebyshey nodes are equally weighted. In the TITAN code, we normalize the
azimuthal weights on the same level to one. So we simply use normalized weights.
w, = (B-21)
Where n is level number. Next we can evaluate the level weights by applying some
properties of Legendre polynomial given in Eq. B-22.
(2n)! A,,, [2(n +1)] (2n)! 2n +1 2
A ->~ a = =~and 7 = (B-22)
"2"(n!)2 nAn 2" '[(n+1)!]2 2"(n!)2 n~ 2n +1
By substituting Eq. B-22 into Eq. B-4, and applying the recurrence property of Eq. B-16,
we can rewrite Eq. B-4 as follows.
r = a 7 2 2(1 xx2) (-3
P,(x)P(x)(n + 1)P,'(x,)P,, (x,) (n + 1)2 P ,(x, )12
Note in deriving Eq. B-23, we also apply Pn (xl) = 0. Since 0 < xl <1, wT defined by Eq.
B-23 is positive definite. Therefore, unlike the level-symmetric quadrature, the Legendre-
Chebyshev quadrature weights are always positive. Furthermore, we can prove that the sum of
the weights iwev = 2, because of the follow-ing identity of Legendre polynomial.
i; 1 2 = 1 (B-24)
1 (n + 1) 2 [P ,(x, )12
In the Legendre-Chebyshev quadrature, we always choose n as an even integer. The roots
and weights are symmetrical regarding to x=0. We can apply Eqs. B-17, B-21 and B-24 to
calculate the total weight for all directions in the first octant.
N/2 n N/2 n N/2
C w- = Cwn W Cw n [w [-= w~ = 1 (B-25)
n=1 k=1 n=1 k=1 Y n=1
As the level-symmetric quadrature, all the directions in other octants are determined by
applying symmetry to the ones in the first octant. We can conclude that the sum of the Legendre-
Chebyshev quadrature weights in one octant is equal to one as in the level-symmetric quadrature.
Conclusion
We have proved two very desirable properties of the Legendre-Chebyshev quadrature for
transport calculations. First, it can conserve integration up to 2N-1 order. Second, the weights are
always positive for any order of the quadrature. However, we do lose some symmetry of rotation
invariance. On the other hand, the level symmetry quadrature keeps the perfect symmetry of
rotation invariance at the cost of only Nth order accuracy and an order limitation of 20. These
two quadrature types reflect the trade-off while pursuing mathematical accuracy and physical
symmetry.
In the TITAN code, a quadrature set can be further biased by physical concerns. We can
apply the ordinate splitting technique (Chapter 2) on some directions with more 'physical
importance'. We also developed the fictitious quadrature technique (Chapter 5), which is
designed for calculating the angular fluxes in the directions with more 'physical interests'.
APPENDIX C
IS FORTRAN 90/95 BETTER THAN C++ FOR SCIENTIFIC COMPUTING?
On Nov. 18, 2004, the international FORTRAN standards committee (WG5) published the
FORTRAN 2003 standard under the identification of ISO/IEC 1539-1:2004(E), which is
considered a maj or revision of the previous FORTRAN 95 standard. Among many new features
in the 2003 standard are: derived type enhancements, obj ect-oriented programming (OOP)
support, data manipulation enhancements, and interoperability with the C programming
language. The standard adopts some features of C++ and other modern languages and moves
FORTRAN closer to C++, while trying to keep and enhance the advantages of FORTRAN in
scientific computing. Some of the new features, widely applied in other languages, could play an
important role in scientific programming.
The performance of scientific computer codes has significantly benefited from the fast-
advancing computing technology in terms of processor speed, memory limit, and the concept of
parallel computing. More benefits can be obtained with the use of the new FORTRAN 2003 and
newer compliers. However, the new language features need to be accepted and utilized by the
scientific computing community. Although now no complier can fully support the new standard,
a few compiler vendors are working on the implementation of FORTRAN 2003 in their compiler
products gradually. Among them are the Intel FORTRAN Compiler (IFC), formerly Compaq
Visual FORTRAN compiler (CVF), and Portland Group FORTRAN complier (PGF90). TITAN
uses some FORTRAN 2003 features, which are mainly related to OOP and derived type
enhancements. And it is originally compiled by IFC 8.1 and PGF90 6.1 in both WINDOWS and
LINUX/UNIX with the same source files. As in April 2007, IFC v9.2 and PGF90 v7.0 are
available in both operating systems.
The performance comparison between FORTRAN 77/90/95 and C/C++ has been discussed
for years. C++ and its compilers evolve significantly over the years with a much larger user base.
More and more scientific programmers consider C++ as one of their language choices. In the
nuclear engineering field, however, FORTRAN still remains the first choice for two reasons.
First, data abstraction penalty associated with language features such as OOP could undermine
the performance of a scientific computing code. These new features are not always desirable or
necessary in scientific computing as in computer applications because of the associated
overheads. Codes can be ugly in human eyes, but very desirable in machines' viewpoint. Second,
FORTRAN is traditionally widely used in our community with a large code base. It is not
practical to rewrite the legacy codes in C/C++ or even with a newer FORTRAN standard.
It is difficult to provide a clear direct answer to the question which language is better for
scientific computing, since the results can be affected by the individual coding practice and the
compiler choice. C++ has a much richer feature set than FORTRAN. However, in scientific
computing, one maj or concern of language choice is the array handling. Here we only provide an
individual investigation on this aspect by comparing the C++ 'vector' class template with its
FORTRAN counterpart.
We wrote two small Monte Carlo codes with the particle splitting/rouletting technique in
FORTRAN and C++. The two codes follow the same logic with the same data structure. A
particle object is defined with particle position and direction by a class in the C++ code, and a
user-defined type structure in the FORTRAN code. An array of particle obj ects, called particle
bank, is created by vector class template in the C++ code, and by defining an allocatable array in
the FORTRAN code. We compiled the two codes with Intel Fortran compiler and Inter C++
compiler. The running times for both codes are compared in Table C-1.
Table C-1. Run time comparison of the sample FORTRAN and C++ codes.
Number of Particles Run time of the FORTRAN Code Run time of the C++ code
10 Million 7 sec 7 sec
100 Million 67 sec 64 sec
According to Table C-1, there is no significant performance difference between the two
codes. However, it is worth noting that the size of the particle bank is required to be pre-defined
in the FORTRAN code to avoid memory overflow. While the C++ vector class template
provides a build-in mechanism to adjust the memory buffer after the last element of the vector.
User can 'push' any number of particles into the bank without worrying memory overflow. It is
safe to say that this mechanism in C++ vector template is very efficient, since even with this
overhead, the C++ code still maintains the same level performance as the FORTRAN code, at
least for the relatively small size array in our code. In handling very large size array, FORTRAN
could have some advantages over C++, since it provides some build-in vector operation on
arrays.
The key to a scientific computing code is always the algorithms and the physics
underneath it. However, the paradigm of the code does make a difference on performance. If
some desirable and crucial features are not available in FORTRAN, we should not hesitate to
choose C++ or other languages.
/* C++ source code for comparing performance with FORTRAN*/
/* shielding with variance reduction 1-D slab */
/* Geometry splitting and roulette */
/* Oct. 2005 Author :yice at ufl edu */
#include
#include
#include
#include
using std::string;
using std::cin;
using std::cout;
using std::endl;
using std::vector;
/* for RN generator */
long int rn = 119; /* seed */
/* GGL RN generator */
const int64 a = 16807; /* a=7^'5 */
const int64 c = 0; /* c =0 */
const int64 M=2147483647; /* M=2^31-1 */
class cParticle
public:
/* initial values */
cParticle(): x(0), w(1.0), reg(1), mu(1.0) { }
float x; /* position */
float w; /* weight */
mnt reg; /* region num */
float mu; /* direction cosine */
/* track one particle inside */
int TrackOne();
/* rn generator */
float MyRng();
const double sigma td=10;
const double sigma~st=0.2;
/* num_cell : num of regions with diff. importance */
int num cell = 6;
/* bon: region boundaries */
vector bon;
/* region importance */
vector imp;
/* w~counter: weight counter ; w~square: square sum (for R) ;
w~one: sum of the weights of each starting particle and its children
w xxx[0] : absorbed
w xxx[1] : back-scattered
w xxx[2] : transmitted
w xxx[3] : killed by rouletting */
vector w counter;
vector w_square;
vector w one;
/* particle bank */
vector bank;
/* current partile being followed */
cParticle one;
const float Pi=2*asin(1.0);
/* ******************************************** */
int main()
int i3j,k;
int tot_part=10000000;
int tot tracked=0;
float size_cell=sigma t d/num cell;
/* Initialize varibles */
/* erase counter */
for( i = 0; i <4; ++i)
counter. push back(0. 0) ; /* w counter=0 */
w~square.push back(0. 0) ;
w~one.pushback(0. 0);
/* imp and bod */
imp.push back(0); /* left outside imp=0 */
imp.push back(1); /* region 1 imp=1 */
bon.push back(0);
for( i = 1; i
imp.push back(imp[i]*2) ; /* imp=1,2,4, 8,16 ..*/
bon.push back(bon[i-1]+size_cell); /* bon=0,2,4,6,8,10 */
// imp.push back(1.0) ;
imp.push back(0.0); /* right outside imp=0 */
for (i = 0; i < tot_part ; ++i)
/* initial particle */
one.x=0; one.w=1.0; one.reg=1; one.mu=1.0;
/* push it into bank */
bank.push back(one);
while( !bank.empty() )
one=bank.back(); /* get the last particle in bank */
bank.pop back(); /* pop the last one out of bank */
++tot tracked; /* count tot particle tracked */
// j-bank.size();
k= TrackOne();
w_one [k] =w_one [k] +one. w;
w~counter[k]= w~counter[k] + one.w;
// cout << k=" << k << tot tracked=" << tot tracked <
// cout << w=" << w~counter[k] <
for(j=0; j<4; ++j) {
w~square[j]=w~square[j] + w~onelj]*w~onelij];
w~onelj]=0.0;
cout << tracked= << tot tracked << endl;
cout << transmitted prob.= << w~counter[2]/tot~part << endl;
cout << relative. err. = << sqrt( w~square[2]/pow(w~counter[2],2)-1 .0/tot_part) << endl;
return 0;
/* ************************************************* */
/* track one particle inside a 1-D multi-region shield */
int TrackOne()
float eta, r, mu0, phi,ir;
int k;
while (one.reg >0 && one.reg < num_cell+1 )
eta=MyRng();
r=-log(eta);
one.x=one.x+r~one.mu;
while ( one.x >= bon[one.reg-1] && one.x <= bon[one.reg])
eta=MyRng();
if (eta <= sigma_st)
{ /* scattered */
mu0 = 2*MyRng() 1;
phi = 2*Pi*MyRng();
one.mu = one.mu~mu0 + sqrt(1 -pow(one.mu,2))* sqrt(1-pow(mu0,2))* cos(phi);
r=-log(MyRng() );
one.x=one.x + r one.mu ;
else /* absorbed */
return 0; /*absorbed */
} /* end if eta */
} /* end while loop one.x */
/* cross the right region boundary */
if (one.x > bon[one.reg] )
/* to move forward one region */
one.x=bon[one.reg++];
ir=imp [one. reg]/imp[one. reg-1];
/* cross the left region boundary */
if (one.x < bon[one.reg-1] )
/* to move backward one region */
one.x=bon[--one.reg];
ir=imp[one.reg]/imp[one.reg+1]i;
/* splitting and rouletting */
k=int(ir);
if (ir > 1) /* splitting */
one.w= one.w/ir;
for (int j =1 ; j
bank.push back(one);
if (MyRng() < ir-k )bank.pushl back(one);
} I~llj~ L~ /ILI~*endl ifir greater than 1 */
if (ir < 1 && ir > 0) /* rouletting */
if (MyRng() < ir) {
one.w= one.w/ir;
else {
return 3; /* killed by rouletting */
} /* end if ir less than 1 */
} /* end,;1 wie lop oere */
if (one.reg <1 )
return 1; /* back scattered */
else
return 2; /* transmitted */
} /* end if one.reg */
/* RN generator */
float MyRng()
rn=(a~rm + c)%M;
return 1.*rn/M;
!/* FORTRAN 90 source code for comparing performance with C++*/
!/* shielding with variance reduction 1-D slab */
!/* Geometry splitting and roulette */
module mRNG
integer : : x=119
integer*8 : : a=16807
integer*8 : : M=2 8**31-1
end module
module paraset1
type tParticle
real x
real w
integer reg
real mu
end type tParticle
integer : : banksize=100
type(tParticle), dimension(:), allocatable : : bank
type(tParticle) one
integer : : top=0
end module
module paraset2
real :: sigma~td=10
real :: sigma~st=0.2
! num_cell : num of regions with diff. importance
integer :: num cell= 6
! bon: region boundaries
real dimension(:), allocatable : : bon
! region importance
real, dimension(:) allocatable : : imp
! w~counter: weight counter ; w~square: square sum (for R) ;
!w~one: sum of the weights of each starting particle and its children
!w xxx[0] : absorbed
!w xxx[1] : back-scattered
!w xxx[2] : transmitted
!w xxx[3] : killed by rouletting */
real :: w~counter(0:3)=0
real :: w~square(0:3)=0
real :: w_one(0:3)=0
real pi
end module
program shield
use paraset1
use paraset2
use DFPORT
integer i,k
real eta
integer tot_part,tot~tracked
real size cell
real s1,s2
sl=secnds(0.0)
pi=2*asin(1.0)
tot_part=1000000
tot tracked=0
size_cell=sigma td/num cell
! /* Initialize varibles */
! /* erase counter */
w counter=0
w~square=0
w one=0
!/* imp and bod */
allocate ( imp(0: num_cell+1), bon(0: num_cell) )
imp(0)=0 !left outside
imp(1)=1 !/* region 1 imp=1 */
bon(0)=0
do i = 1, num cell
imp(i+1)=imp(i)*2 /* imp=1,2,4,8,16 ..*/
bon(i)=bon(i-1)+size_cell /* bon=0,2,4,6,8,10 */
! imp (i+1)= 1
enddo
imp(num_cell+1)=0
! /* right outside imp=0 */
allocate (bank(banksize))
top=0
loop_part :do i = 1, tot part
!/* initial particle */
one%x=0
one%w=1.0O
one%reg=1
one%mu=1.0
! /* push it into bank */
top=top+1
bank(top)= one
do while( top .ne. O )
one=bank(top) /* get the last particle in bank */
top-top-1 /* pop the last one out of bank */
tot~tracked=tot~tracked+ 1 /* count tot particle tracked */
call TrackOne(k);
w_one(k)=w~one(k)+one%w
w_counter(k)= w~counter(k) + one%w
enddo
do j=0, 3
w~squaredj)=w~squaredj) + w~onedj)**2
w~onedj)=0.0;
enddo
enddo loop_part
write(*,"('tracked=', IO)") tot tracked
write(*,"('transmitted prob. =', ES12.5)") w~counter(2)/tot_part
write(*,"('relative err. = ', ES12.5)" ) &
sqrt( w~square(2)/(w~counter(2)**"2-1.0/tot_part) )
write(*,'("run time=", fl0.3, "sec" ) ') secnds(sl)
end program
subroutine TrackOne(flag)
use paraset1
use paraset2
integer flag
real eta, r, mu0, phi,ir,temp;
integer k
while~reg : do while (one%reg .gt. O .and. One%reg .It. num_cell+1 )
call MyRng(eta)
r=-log(eta)
one%x-one"/x + r~one%mu
while~xr : do while ( one%x .ge. bon(one%reg-1) .and. One%x .1e. bon(one%reg) )
call MyRng(eta)
if (eta .1e. sigma~st) then /* scattered */
call MyRng(eta)
mu0 = 2*eta 1
call MyRng(eta)
phi = 2*Pi~eta
one%mu = one%mu~mu0 + sqrt(1-one%mu**2)*sqrt(1-mu0**2)*cos(phi)
call MyRng(eta)
r=-log(eta)
one%x-one"/x + r one%mu
else /* absorbed */
flag=0
return
endif
enddo while xr
! /* cross the right region boundary */
if (one%x .gt. bon(one%reg)) then
! /* to move forward one region */
one%x=bon(one.reg)
one%reg=one%reg+1
ir=imp(one%reg)/imp(one%reg-1i)
endif
! /* cross the left region boundary */
if (one.x < bon(one%reg-1) ) then
!/* to move backward one region */
one%reg=one%reg-1
one%x-bon(one"/reg)
ir=imp(one%reg)/imp(one%reg+1i)
endif
! /* splitting and rouletting */
k=int(ir)
if (ir .gt. 1) then !/* splitting */
one%w = one%w/ir
do j=1, k-1
top=top+1
bank(top)= one
enddo
call MyRng(eta)
if ( eta .It. ir-k ) then
top=top+1
bank(top)= one
endif
endif
if (ir .lt. 1 .and. ir .gt. 0) then !/* rouletting */
one%w = one%w/ir
call MyRng(eta)
if ( eta .gt. ir) then
flag=3
return
endif
endif
enddo while~reg
if (one%reg .It. ) then
flag=1
return /* back scattered */
else
flag=2
return /* transmitted */
end if
end subroutine
subroutine MyRng(rn)
use mRNG
real rn
!x=int( mod(a~x,M), 4)
x=mod(a~x,M)
rn=1.0*x/M
return
end subroutine
APPENDIX D
TITAN I/O FILE FORMAT
TITAN Input Files
The TITAN code is developed based on the code base of PENMSH Express,29 which is a
mesh generator I wrote for generating PENTRAN input deck. PENMSH Express, or PENMSH
XP, follows a similar input syntax with PENMSH.28 Therefore, TITAN inherits most of the
PENMSH input fie format. Table D-1 lists the input Hiles of the TITAN code.
Table D-1. TITAN input fie list.
File # File Name Description Memo
1penmsh.inp Meshing parameters Required
2 prbname#.inp Meshing per z level Required
3 prbname.src Fixed source grid Optional
4 prbname.spc Source spectrum Optional
5 prbname.chi Fission spectrum Optional
6 prbname.mba Material balance Optional
7 bonphora.inp General input parameters Required
8 prbname.xs Cross section data Required
Input Hiles #1 to #6 are general PENMSH input files, which define model geometries and
source specifications. We use 'prbnamne' to denote different problem names. General meshing
parameters are specified in the 'penmsh.inp', including number of z levels, z-level boundaries,
etc. Geometries on each z level are specified in a separate Eile (Input file 2). For example,
prbnamnel.inp, prbnamne2. inp,.... These input files can describe various geometries with the
'overlay' feature. Figure D-1 shows the geometries generated by a sample z-level input file. The
fixed source grid can be defined in the 'prbnamne.src' file. prbnamne.spc and 'prbnamne.chi'
specify the source and fission spectrum, respectively. And 'prbnamne.mba' is used to check the
model material balance. More details on Input files #1 to #6 can be found in the manuals of
PENMSH and PENMSH XP. And we will further discuss input file #7 in the next section.
I bonphora.inp: TITAN input file to define transport parameters
#0 Section 0: Global varibles
2 0 /# of quadrature, global DS id
#1 Section 1: Quadrature sets
/quad 1 Pn-Tn
/ Quadrature id order, num of split directions
1 20 2
1spilited directions
46 47 /direction index
11 11 1splitted order
1 1 1splitted id : 1- pn-tn splitting
2 2 /# of directions on the top level
/quad 2 level symmetric
/ Quadrature id order,num of split directions
0 20 1
1spilited directions ids
37 /direction index
8 1splitted order
0 /splitted id : 0- rectangular splitting
0 / not used
#2 Section 2: Coarse mesh specifications
/Solver id
0 10
/qudra_id
121
/Diff scheme
112
Figure D-2. A sample bonphora.inp input fie.
Figure D-1. A 3 by 3 coarse mesh model on one z level.
Bonphora.inp Input File
Input fie 7 (bonphora.inp) is special fie used by TITAN only, which specifies parameters
for transport calculations, such as the quadrature set, differencing scheme, solver, etc. The Eile
supports as many as 4 sections. The following is a sample bonphora.inp fie.
Section 0 is dedicated to specify two parameters: total number of quadrature sets used in
the model, and the global differencing scheme id number, which define the differencing scheme
for all coarse meshes if the number is a positive integer (id=1, diamond with zero fix-up; id=2,
Directional Theta-Weighted). If zero is given as the global differencing scheme id, an additional
card is required to specify an individual differencing scheme for each coarse mesh.
Section 1 is used to define all the quadrature sets used in the model. In this sample input
file, two quadrature set are specified. The first one is a PN-TN quadrature (quadrature id=1) with
an order of 20. The PN-TN splitting technique is applied on two directions in the quadrature set
(direction index number: 46 and 47). The second one is a level-symmetric quadrature set with
rectangular splitting on Direction 37.
Section 2 specifies the parameters for each coarse mesh. In this sample file, the SN solver
will be used for coarse meshes #1 and #3 (solver id=0). Coarse mesh #2 uses the characteristics
solver. Quadrature set #1 specified in Section 1 is applied in coarse meshes #1 and #3.
Quadrature set #2 is used in coarse mesh #2.
Another section can be used to specify the iteration number limitations and tolerances,
especially for eigenvalue problems. The following is the input file for the C5G7 MOX
benchmark problem.
I bonphora.inp: TITAN input file to define transport
parameters
#0 Section 0: Global varibles
1 1 /# of quadrature, global DS id
0600
#3 Iteration parameters
/tolout ,tolin
1.0e-5 1.0e-3
louter,inner
-50 10
Irkdef
Figure D-3. C5G7 MOX benchmark problem bonphora.inp input file.
In this model, an S6 leVel-Symmetric quadrature is used with the diamond differencing
scheme. The SN solver is applied on all the coarse meshes (SN solver is the default solver).
Section 3 specifies some iteration parameters. The outer iteration tolerance is 1.0E-05 (variable
tolout). And the inner iteration tolerance is 1.0E-03(variable tolin). Note if tolin is less than zero,
the adaptive inner loop tolerance control will be engaged. The iteration number limitations are
defined in the next card. The outer and inner iteration limits are 50 and 10 respectively. Negative
numbers means the limitations are adaptive. The last card defines the initial guess of eigenvalue.
Aitken extrapolation37 is used on k-effective if users specify a negative initial guess.
TITAN can automatically convert a digital phantom into a transport calculation model. We
use this feature for the SPECT benchmark problem. The input file format is slightly different for
a medical phantom model. Details can be found in the PENMSH XP manual.
TITAN Output Files and TECPLOT Visualization
Table D-2 list the maj or output files of the TITAN code. The first file is an optional output,
which contains a generated PENTRAN inputd deck. The second output file is a report of material
balance check. The third file, bonphora.log, is the input processing log. And the solver log is
stored in file prbnamnesolver. log, which records all the iteration output.
Table D-2. TITAN output file list.
File # File Name Description
1Prbname out.f90
2 prbname~out.mba Material balance tables
3 Bonphora.log Processing log file
4 Prbname_solver.log Solver log file
5 prbname~mix.plt. TECPLOT binary file, contains all the calculation data
6 prbname.mer TECPLOT macro file
The last two files are used for visualization of the calculation results with the TECPLOT
software. A TECPLOT I/O library is developed and included in the TITAN code. The library,
composed of about 15 subroutines and modules, can generate TECPLOT binary data files as
many as necessary simultaneously. Some other TITAN output files, including the quadrature
data file and the optional boundary angular flux files when a fictitious quadrature set is used, are
also generated by this library. The last file in Table D-2, prbnamne.mcr, is a macro file, which can
be loaded by TECPLOT, to help organize the data in prbnamnemix.plt.
TECPLOT also provides an IO library (without source codes) for users to generate their
own binary data files. However, for practical reasons, here we wrote our own version of
TECPLOT IO library, which is optimized for our purpose. TECPLOT is an excellent
visualization tool. However, it is a commercial software package. We consider migrating to the
widely used visualization toolkit (VTK) platform, which is an open source library for scientific
visualization. A number of front end software packages (e.g. PARAVIEW) are freely available
to visualize the VTK format data file.
TITAN Command Line Option
The common command line option is '-i' option, which specifies the directories where the
input files are located. The default input directory is the current one.
[home/user/]# bonphora -i test
The above command line reads input decks from the /home/user/test directory. Other
command options can be found in the PENMSH XP manual. Users can add their own modules
and subroutines to extract the interested data from the calculation results. All the post-processing
subroutines are called from a container subroutine named Nirvana. The user- defined post-
processing routines can be triggered with a command line option with slight modification of the
code. For example, the option '-mox will trigger the C5G7 MOX post-processing subroutines.
These subroutines are used to calculate the fuel pin powers based on the converged scalar fluxes.
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BIOGRAPHICAL SKETCH
I was born in 1973 in Anshan, China. I went to Tsinghua University in 1992 and got my
bachelor' s degree in nuclear engineering in 1997. I continued on to the graduate school at
Tsinghua, and graduated with a master's degree in nuclear engineering in 2000. The same year, I
went to Penn State University to pursue a doctoral degree. In 2001, I followed Dr. Haghighat
to the University of Florida.
The goal of my study was to develop a hybrid algorithm to solve the LBE
efficiently in low-scattering media and to enhance the efficiencies of the PENTRAN code in
medical applications. I started to write a 3-D SN kernel in April 2005 from the PENMSH XP
code base, which is a mesh generator I wrote for preparing PENTRAN input deck. The 3-D SN
code is originally designed as a test platform for the hybrid algorithm. By the summer of 2005, I
completed the initial versions of both the SN and characteristics solvers. In the summer, I
dedicated most of the time to the University of Florida Training Reactor (UFTR) fuel conversion
proj ect. After that summer, I continued to work on the code and implemented a number of
techniques, including PN-TN quadrature set, PN-TN ordinate splitting, and proj section techniques.
By April 2006, the framework of the code is completed. In the second half of 2006, I worked on
the integration of characteristics solver into PENTRAN. In the first quarter of 2007, the fictitious
quadrature technique is developed for the heart phantom benchmark. And some studies on the
limitations of the hybrid algorithms are performed.
PAGE 1
1 HYBRID DISCRETE ORDINATES AND CHAR ACTERISTICS METHOD FOR SOLVING THE LINEAR BOLTZMANN EQUATION By CE YI 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 Ce Yi
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3 To my Mom, Dad and Brother
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4 ACKNOWLEDGMENTS I thank m y advisor Dr. Alireza Haghighat for hi s instructions and guidance. Without his support, the accomplishment of this research work would have not been possible. And I am grateful to Dr. Glenn Sjoden for his insightful suggestions. His PENTRAN code manual has served as a constant source of knowledge throug hout my studies. I also wish to express my gratitude to other committee members, Dr. David Gilland, Dr. John Wagner, Dr. Jayadeep Gopalakrishnan, and Dr. Shari Mosk ow, for their help and support. I would like to gratef ully acknowledge Mr. Benoit Dionne, Mr. Mike Wenner, and other members of the transport theory group at Univer sity of Florida for their support, especially Benoit for his understanding of this work. The discussions with him on various topics inspired me finding ways to improve the perf ormance of my transport code.
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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ...........................................................................................................................8 LIST OF FIGURES .......................................................................................................................10 ABSTRACT ...................................................................................................................... .............13 CHAP TER 1 INTRODUCTION .................................................................................................................. 15 Overview ...................................................................................................................... ...........15 Linear Boltzmann Equation (LBE) .........................................................................................15 Numerical Methods to Solve the LBE .................................................................................... 18 Discrete Ordinates Method ..............................................................................................18 Method of Characteristics (MOC) ...................................................................................19 Ray-Effects in Low Scattering Region ...................................................................................20 Hybrid Approach ....................................................................................................................21 2 THEORY AND ALGORITHMS ...........................................................................................23 Multi-Block Framework Overview ........................................................................................ 23 Discrete Ordinates Formulations ............................................................................................24 Source Iteration Process .........................................................................................................26 Differencing Scheme ..............................................................................................................27 Characteristics Formulations .................................................................................................. 30 Block-Oriented Characteristics Solver ...................................................................................33 Backward Ray-Tracing Procedure .................................................................................. 34 Advantage of Backward Ray-Tracing .............................................................................36 Ray Tracer .......................................................................................................................37 Interpolation on the Incoming Surface ............................................................................ 38 Quadrature Set ................................................................................................................ ........40 Level-symmetric Quadrature ...........................................................................................43 Legendre-Chebyshev Quadrature .................................................................................... 45 Rectangular and PN-TN Ordinate Splitting ...................................................................... 46 3 PROJECTIONS ON THE INTERFACE OF COARSE MESHES ........................................ 49 Angular Projection ............................................................................................................ ......49 Spatial Projection ....................................................................................................................53 Projection Matrix ............................................................................................................. .......55 4 CODE STRUTURE ................................................................................................................ 56
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6 Block Structure ............................................................................................................... ........56 Processing Block .............................................................................................................. ......57 First Level Routines: Source Iteration Scheme ............................................................... 59 Second Level Routines: Sweeping on Coarse Mesh Level ............................................. 61 Third Level Routines: Sweeping on Fine Mesh Level .................................................... 63 Data Structure and Initialization Subroutines .........................................................................65 Coarse and Fine Mesh Interface Flux Handling ..................................................................... 66 5 BENCHMARKING ................................................................................................................ 70 Benchmark 1 A Uniform Medium and Source Problem ...................................................... 70 Benchmark 2 A Simplified CT Model ................................................................................. 73 Monte Carlo Model Description ...................................................................................... 75 Deterministic Model Description .................................................................................... 75 Comparison and Analysis of Results ............................................................................... 76 Benchmark 3 Kobayashi 3-D Problem s with Void Ducts .................................................... 79 Problem 1: Shield with Square Void ............................................................................... 80 Problem 2: Shield with Void Duct .................................................................................. 84 Problem 3: Shield with Dogleg Void Duct ...................................................................... 86 Analysis of Results ..........................................................................................................87 Benchmark 4 3-D C5G7 MOX Fuel Assem bly Benchmark ................................................ 89 Model Description ...........................................................................................................89 Pin Power Calculation Results ........................................................................................ 91 Eigenvalue Comparison ................................................................................................... 94 Analysis of Results ..........................................................................................................95 6 FICTITIOUS QUADRATURE .............................................................................................. 97 Extra Sweep with Fictitious Quadrature .................................................................................97 Implementation of Fictitious Quadrature ................................................................................ 99 Extra Sweep Procedure ....................................................................................................99 Implementation Concerns .............................................................................................. 101 Iteration structure ...................................................................................................101 Direction singularity ...............................................................................................101 Solver compatibility ...............................................................................................102 Heart Phantom Benchmark ...................................................................................................102 Model Description .........................................................................................................103 Photon Cross Section for the Phantom Model ..............................................................104 Performance of Fictitious Quadrature Technique ......................................................... 106 7 PENTRAN INTEGRATION AND LIMITATION STUDIES OF THE CHARATERISTICS SOLVER ............................................................................................110 Implementation of the Characteristics Solver in PENTRAN ............................................... 110 Benchmarking of PENTRAN-CM .......................................................................................112 Meshing, Cross Section and Quadrature Set ................................................................. 112 Benchmark Results and Analysis .................................................................................. 114
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7 Investigation on the Limitations of Characteristics Solver ...................................................116 Memory Usage ..............................................................................................................116 Limitation on the Spatial Discretization ........................................................................ 117 2-D meshing on the coarse mesh boundaries .........................................................118 Coarse mesh size limitation for the characteristics solver ..................................... 122 Possible Improvements and Extendibility of the Characteristics Solver ....................... 127 8 CONCLUSIONS AND FU TURE WORK ........................................................................... 128 Conclusions ...........................................................................................................................128 Future Work ..........................................................................................................................129 Acceleration Techniques ...............................................................................................129 Parallelization ............................................................................................................... .131 Improvements on Characteristics Solver ....................................................................... 131 Other Enhancements ...................................................................................................... 131 APPENDIX A SCATTERING KERNEL IN LINEAR BOLTZMANN EQUATION ................................ 132 B NUMERICAL QUADRATURE ON UNIT SPHERE SURFACE ......................................142 C IS FORTRAN 90/95 BETTER THAN C++ FOR SCIENTIFIC COMPUTING? ...............152 D TITAN I/O FILE FORMAT ................................................................................................. 166 LIST OF REFERENCES .............................................................................................................171 BIOGRAPHICAL SKETCH .......................................................................................................175
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8 LIST OF TABLES Table page 5-1 CT model run time and error norm co mparison with the MCNP reference case .............. 78 5-2 Kobayashi problem 1 point A set flux results for case 1 ................................................... 80 5-3 Kobayashi problem 1 point B set flux results for case 1 ................................................... 81 5-4 Kobayashi problem 1 point C set flux results for case 1 ................................................... 81 5-5 Kobayashi problem 1 point A set flux results for case 2 ................................................... 82 5-6 Kobayashi problem 1 point B set flux results for case 2 ................................................... 82 5-7 Kobayashi problem 1 point C set flux results for case 2 ................................................... 82 5-8 Kobayashi problem 1 point A set flux results for case 3 ................................................... 83 5-9 Kobayashi problem 1 point B set flux results for case 3 ................................................... 83 5-10 Kobayashi problem 1 point C set flux results for case 3 ................................................... 83 5-11 CPU time and memory requirement for SN and hybrid methods ....................................... 84 5-12 Kobayashi problem 2 point A set flux results for case 3 ................................................... 85 5-13 Kobayashi problem 2 point B set flux results for case 3 ................................................... 85 5-14 Kobayashi problem 3 point A set flux results for case 3 ................................................... 86 5-15 Kobayashi problem 3 point B set flux results for case 3 ................................................... 86 5-16 Kobayashi problem 3 point C set flux results for case 3 ................................................... 87 5-17 Pin power calculation re sults for the unrodded case .......................................................... 92 5-18 Pin power calculation resu lts for the rodded A case .......................................................... 93 5-19 Pin power calculation resu lts for the rodded B case .......................................................... 94 5-20 Eigenvalues for three cases of C5G7 MOX benchmark problems .................................... 95 6-1 Materials list in th e heart phantom model ........................................................................ 104 6-2 Group structure of cross section data for the heart phantom benchmark ........................ 105 6-3 Material densities and co mpositions used in CEPXS ...................................................... 105
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9 6-4 Directions in the fictitious quadrature set for the heart phantom benchmark .................. 106 6-5 TITAN calculation errors rela tive to the SIMIND simulation ......................................... 108 7-1 Memory structure differences between PENTAN and TITAN ....................................... 111 7-2 Comparison of the characteristic s solver in PENTAN-CM and TITAN ......................... 111 7-3 One group cross section used in the CT benchmark with TITAN ................................... 112 7-4 Two group cross section used in th e CT benchmark with PENTRAN-CM .................... 113 7-5 Characteristics solver calculated de tector response by PENTRAN-CM and TITAN .....114 7-6 Characteristics solver performance in PENTRAN parallel environment ........................115 7-7 Error comparison with different z meshing ..................................................................... 121 7-8 Characteristics solution relative difference to SN solution with different scattering ratios and coarse mesh size .............................................................................................. 126 C-1 Run time comparison of the sample FORTRAN and C++ codes ................................... 154 D-1 TITAN input file list ........................................................................................................166 D-2 TITAN output file list ......................................................................................................169
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10 LIST OF FIGURES Figure page 1-1 Angular flux formulation of th e integral transport equation .............................................. 20 2-1 Coarse mesh/fine mesh meshing scheme ........................................................................... 23 2-2 Differencing scheme on one fine mesh .............................................................................. 27 2-3 Schematic of characteristic rays in a coar se mesh using the char acteristics method ......... 31 2-4 A coarse mesh with characteristics solver assigned .......................................................... 34 2-5 Characteristic rays for one fi ne mesh on one outgoing surface ......................................... 35 2-6 Bilinear interpolation for the incoming flux ...................................................................... 38 2-7 Schematic of the S10 level-symmetric quadrature set in one octant ...................................43 2-8 PN-TN quadrature of order 10 ............................................................................................. 45 2-9 Ordinate splitting technique ...............................................................................................46 3-1 Angular projection .............................................................................................................49 3-2 Theta weighting scheme in angular domain. ..................................................................... 50 3-3 Mismatched fine-meshing schemes on the interface of two adjacent coarse meshes ........ 53 4-1 Code structure flowchart ....................................................................................................58 4-2 Pseudo-code of the source iteration scheme ......................................................................59 4-3 Pseudo-code of the coarse mesh sweep process ................................................................62 4-4 Pseudo-code of the fine mesh sweep process .................................................................... 63 4-5 Frontline interface flux handling ....................................................................................... 67 5-1 Uniform medium a nd source test model ............................................................................ 71 5-2 Group 1 calculation result ..................................................................................................71 5-3 Group 2 calculation result ..................................................................................................72 5-4 Group 3 calculation result ..................................................................................................72 5-5 Computational tomography (CT) scan device ................................................................... 73
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11 5-6 A simplified CT model ..................................................................................................... .74 5-7 MCNP model of the simplified CT device ........................................................................ 75 5-8 SN solver meshing scheme for the CT model .................................................................... 75 5-9 Hybrid model meshing for the CT model .......................................................................... 76 5-10 SN simulation results without ordinate splitting ................................................................. 77 5-11 Quadrature sets used in the CT benchmark ....................................................................... 77 5-12 Hybrid and SN simulation results with ordinate splitting ...................................................78 5-13 Kobayashi Problem 1 box-in-box layout ........................................................................... 80 5-14 Kobayashi Problem 2 first z level model layout ................................................................85 5-15 Kobayashi Problem 3 void duct layout .............................................................................. 86 5-16 Relative fluxes for Kobayashi problem 1 .......................................................................... 87 5-17 Relative fluxes for Kobayashi problem 2 .......................................................................... 88 5-18 Relative fluxes for Kobayashi problem 3 .......................................................................... 88 5-19 C5G7 MOX reactor layout .................................................................................................89 5-20 3-D C5G7 MOX model ..................................................................................................... 90 5-21 Eigenvalue convergence pattern for the rodded A configuration ......................................95 6-1 Extra sweep procedure w ith fictitious quadrature ........................................................... 100 6-2 Heart phant om m odel ....................................................................................................... 103 6-3 Activity distribution in the phantom model ..................................................................... 104 6-4 Globally normalized projection imag es calculated by TITAN and SIMIND .................. 107 6-5 Individually normalized projection images calculated by TITAN and SIMIND ............107 7-1 Characteristics coarse mesh boundary me shing based on flux resolution requirement ... 119 7-2 Detector response relative erro rs with different number of z fine meshes for the characteristics solver ........................................................................................................120 7-3 Detector response relative erro rs with different number of z fine meshes for the SN solver ........................................................................................................................ ........120
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12 7-4 Detector response sensitivity to the f ine mesh number along y axis ...............................121 7-5 Detector response comparison between SN and characteristics solv er in pure absorber media ......................................................................................................................... .......123 7-6 Detector response comparison between SN and characteristics so lver in media with different scattering ratio ...................................................................................................124 7-7 Characteristics solutions with different coarse mesh size along x axis ...........................125 B-1 Chebyshev roots (N =4) on a unit circle .......................................................................... 146 D-1 A 3 by 3 coarse mesh model on one z level .....................................................................167 D-2 A sample bonphora.inp input file ....................................................................................167 D-3 C5G7 MOX benchmark problem bonphora.inp input file .............................................. 168
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13 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 HYBRID DISCRETE ORDINATES AND CHAR ACTERISTICS METHOD FOR SOLVING THE LINEAR BOLTZMANN EQUATION By Ce Yi August 2007 Chair: Alireza Haghighat Major: Nuclear Engineering Sciences With the ability of computer hardware and software increasing rapidly, deterministic methods to solve the linear Boltzmann equation (LBE) have attracted some attention for computational applications in both the nuclear engineering an d medical physics fields. Among various deterministic methods, the discrete ordinates method (SN) and the method of characteristics (MOC) are two of th e most widely used methods. The SN method is the traditional approach to solve the LBE for its stability and efficiency. While the MOC has some advantages in treating complicated geometri es. However, in 3-D problems re quiring a dense discretization grid in phase space (i.e., a large number of spat ial meshes, directions, or energy groups), both methods could suffer from the need for larg e amounts of memory and computation time. In our study, we developed a new hybrid algor ithm by combing the two methods into one code, TITAN. The hybrid approach is specif ically designed for application to problems containing low scattering regions. A new serial 3-D time-independe nt transport code has been developed. Under the hybrid appr oach, the preferred method can be applied in different regions (blocks) within the same problem model. Since the characteristics method is numerically more efficient in low scattering media, the hybrid approach uses a block-oriented characteristics solver in low scattering regions, and a block-oriented SN solver in the remainde r of the physical model.
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14 In the TITAN code, a physical problem model is divided into a number of coarse meshes (blocks) in Cartesian geometry. Either the characteristics solver or the SN solver can be chosen to solve the LBE within a coarse mesh. A coarse mesh can be filled with fine meshes or characteristic rays depending on the solver assigned to the coarse mesh. Furthermore, with its object-oriented programming paradigm and layere d code structure, TITAN allows different individual spatial meshing schemes and angular quadrature sets for each coarse mesh. Two quadrature types (level-symmetric and Legendre-Chebyshev quadrat ure) along with the ordinate splitting techniques (rec tangular splitting and PN-TN splitting) are implemented. In the SN solver, we apply a memory-efficient front -line style paradigm to handle the fine mesh interface fluxes. In the characteristics solver, we have devel oped a novel backward ray-tracing approach, in which a bi-linear interpolation procedure is used on the incoming boundaries of a coarse mesh. A CPU-efficient scattering kernel is shared in both solvers within the source iteration scheme. Angular and spatial projection techniques are developed to tran sfer the angular fluxes on the interfaces of coarse meshes with different discretization grids. The performance of the hybrid algorithm is te sted in a number of benchmark problems in both nuclear engineering and medical physics fi elds. Among them are the Kobayashi benchmark problems and a computational tomography (CT) device model. We also developed an extra sweep procedure with the ficti tious quadrature technique to calculate angular fluxes along directions of interest. The technique is appl ied in a single photon emission computed tomography (SPECT) phantom model to simulate the SPECT projection images. The accuracy and efficiency of the TITAN code are demonstrated in these be nchmarks along with its scalability. A modified version of the characteristics solver is integrated in the PENTRAN code and tested within the parallel engine of PENTRAN. The limitations on the hybrid algorithm are also studied.
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15 CHAPTER 1 INTRODUCTION Overview The linear Boltzm ann equation (LBE) (also calle d neutron transport equation) describes the behavior of neutral particles in a system (e.g. a nuclear reactor, a radiological medical device). LBE is derived based on the physics of pa rticle balance in a phase space composed of energy, spatial and angular domains. By solving th e LBE, we can acquire some insights into the characteristics of the system. In this work, we developed a hybrid transport algorithm to solve the LBE, specifically for application to problem s containing regions of low scattering. A new deterministic transport code (TITAN) has been developed based on the new hybrid approach. The code, over 16,000 lines at present, is written in FORTRAN 95 with some language extensions of object-oriented features (par t of the FORTRAN 2003 standard). TITAN is benchmarked for several problems. Linear Boltzmann Equation The original Boltzm ann equation is derived for molecular dynamics of sufficiently dilute gas, in which only binary interaction is considered.1 collisiont f tvrf v m F v t )(),,() ( (1-1) Where, v = the velocity of gas molecules. r = the position of gas molecules. vdrdtvrf ),, (= the expected number of gas molecules in phase spacedrdv F = external force on molecules. m = mass of molecules. Since only binary collision is c onsidered, the collision term on the right side of Eq.1-1 can be written as:
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16 )],,(),,(),',(),',([ ) ( )(1 1 1 1 1tvrftvrftvrftvrf vvvvdvd t fcollision (1-2) Where '1v and 'v are the velocities prior to collision, ) (1vv represents the probability of two molecular collision, and the),',(),',(1tvrftvrf and ),,(),,(1tvrftvrf terms represent the gain and loss of molecules in the ph ase space, respectively. Note that the gain and loss terms are quadratic. Thereby, the original Boltzmann equation (Eq. 1-1) is nonlinear. In order to solve the equa tion numerically, one has to linearize the equation first, which could introduce some system errors to the physics of the real problem to be solved. Usually Monte Carlo approach is used to solve gas dynamics problems without solving the equation directly. Fortunately, the neutron, or in general, ne utral particle trans port phenomenon, can be simulated with the linear form of the Boltzmann equation. The linear form is valid, because neutron-neutron interactions are negligible co mpared to neutron-nucleus interactions. For example, in a typical reactor, the neutron number density is usually ~15 orders of magnitude less than the number density of surrounding medium. In such systems, only the neutron-nucleus interaction is considered, and the medium rema ins unchanged within the time scope of neutron transport. This assumption reduces the collision te rm on the right side of Eq.1-1 from quadratic to a linear term. Further simplif ication can be achieved by assuming 0 F which is true in most situations, because neutrons or neutral particles are not affected by electric or magnetic field, and gravitational force is negligible because of the negligible weight neutr ons and zero weight of gamma rays. With these simplifications to Eq.1-1 the neutron transport equation, or the linear Boltzmann equation (LBE), can be written as:
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17 ), ,,(),' ,',()',(' 4 ),' ,',() ,',(' ), ,,( ), ,,(),( ), ,,(10 4 0 4tErStErErddE tEr EErddE tEr tErEr t tEr vf s t (1-3) Where v is the speed of neutron, ), ,,(), ,,( tErnvtEr is the angular flux, ), ,,( tErn is expected number of neut rons in the phase space of dEdrd t ,s and f are total, scattering and fission cr oss sections of the nuclei in the medium, respectively, )( E is the fission spectrum, and ), ,,( tErS is the independent source. The time-independent linear Boltzmann equation can be written as:2, 3 ) ,,(),' ,',()',(' 4 )' ,',() ,',(' ) ,,( ) ,,(),(0 4 0 4 ErStErErddE Er EErddE Er ErErf s t (1-4) Equation 1-4 is the fundamental equation we are to solve with our code. It represents two basic types of problems. In operator form, they are: Fixed source problem: fixSH Eigenvalue problem: F k H 1 Where the transport operator H and the fission operator F are defined as: ) ,',(' '),( 4 0 EErddEEr Hs E t (1-5) )',(' 4 )(4 0ErddE E Ff (1-6)
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18 And k is the eigenvalue of the system. The fixe d source problem is also referred to as shielding problem and the eigenvalue problem often is called criticality problem in the area of reactor physics. Numerical Methods to Solve the LBE In the past fifty years, numerous numerica l methods have been developed to solve the transport equation. Two of the mo st widely used methods are: Discrete ordinates method (SN). Method of characteristics (MOC). These methods are often referred to as deterministic methods, as opposed to the Monte Carlo method, in the sense that they are to solve the LBE or its derived formulations directly by numerical methods. To numerically solve a differen tial equation, it is requ ired to discretize the equation in its phase space. In the LBE, the angular flux ) ,,( Er is defined in a phase space composed of three domains: spatial, energy a nd angular domain. In deterministic methods, generally, the energy domain variable is discretized using the multigroup approximation.4 The angular domain variables are discretized us ing the numerical quadrature technique.2 And in the spatial domain, different methods may take thei r individual approaches in various geometry systems. For example, one can divide space into structured or unstructured meshes (SN) with finite differencing or finite element approach, or arbitrary-shaped ma terial regions (MOC). Discrete Ordinates Method The SN method was first introduced by Carlson in to the nuclear engineering field in 1958.5 It has been one of the dominant deterministic me thods for its efficiency and numerical stability. In the SN method, Eq. 1-4 is the fundamental equati on to solve. And the angular flux is only calculated in a number of discrete directions. In other words, if we consider the angular flux as a
PAGE 19
19 function defined on the surface of a unit sphere, the SN method evaluates f unction values at discrete points on the surface, which are carefully chosen by a quadrature set in order to conserve the flux moments. In the spatial domain, numeri cal differencing schemes are required in the SN method to evaluate the streaming term. Method of Characteristics (MOC) Recently with the advancements in computi ng hardware, MOC has drawn more and more attentions in both the nuclear engineering and medical physics communities.6, 7 A number of 2D/3D MOC codes 8, 9 have been developed for reactor physics and medical applications. Among advantages of the MOC, its ability to treat arbitrary geomet rical bodies is an attractive feature, especially for medical applications, in which the Monte Carlo approach is still dominant. MOC usually uses the same quadrature technique as the SN method to accomplish angular discretization. It solves the LB E along parallel straight lines (referred to as the characteristic rays) instead of discretized meshes as in SN method. The angular flux al ong a characteristic ray can be described by the formulation of the integral transport equation: ) ,( ) ,( 0 ) ,( 00 4 ) ,( 00) ,, ( ) ,, ( )' ,', () ,', (' ) ,', ()', (' 4 )( ) ,,( lrr lrr R fix lrr R s lrr R fE E E EeERr eElrdlS eElr EElrddEdl eElrElrdEdl E Er (1-7) Where l t EElrdllrr0), '(') ,( is the optical path length along the characteristic ray for particles with energy of E Figure 1-1 illustrates the terms in Eq. 1-6, which is the fundamental formulation for the MOC.
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20 Figure 1-1. Angular flux formulation of the integral transport equation. The streaming term in the LBE disappears in Eq. 1-6 because of the integration over the characteristic ray. Therefore, as one benefit, differencing schemes are not required in the MOC. However, MOC requires a sufficien t number of rays in order to adequately cover the spatial domain. The main disadvantage of the MOC is the need of a la rge amount of memory to store the geometry information for the characteris tic rays. Since the 3-D MOC could be very expensive,10 some synthesis methods, coupled 2D MOC with 1-D nodal/transport method,11 have been developed based on the fact that in mo st reactor system, flux pr ofile changes relatively slowly along z axis, comparing to rapidl y changing profile over the x-y plane. Ray-Effects in Low Scattering Region One numerical difficulty for the determinis tic methods is the so-called ray-effects,12 especially in the SN method with structured meshing in a low scattering medium, where the uncollided flux is dominant. As the distance betw een a localized source and a region of interest increases, the number of discrete ordinates that intersect each di stant spatial mesh is reduced, resulting in unphysical oscillations of the scalar flux. Generally, the meshing and the quadrature O r Rr lr ) ,, ( ERr ) ,, (Elr ) ,,( Er l R
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21 set in the SN method should remain consistent. Otherwise, in a system where spatial and angular domains are tightly coupled, the mismatch between discretization grids in the two domains may cause the ray-effects. The ray-effects can be alleviated naturally by increasing isotropic scattering or fission, since fission is always considered isotropic and an isotropic infl uence tends to flatten the flux in the angular domain. However, the ray-effect s become worse in low scattering medium or a highly absorbing medium, where the flux is usually highly angular dependent. Therefore, particle transport problems in low-scattering media often present a difficulty for deterministic methods. Hybrid Approach Both the SN and characteristics methods have been studied intensively, and utilized into many codes. The goal of this work is to solve the LBE efficiently by taking a hybrid SN and characteristics approach for pr oblems containing low scattering re gions. Such problems are very common in medical physics applications and in many shielding problems. Both methods numerically solve the LBE by discretizing the angular flux in the spatial, angular and energy domains. Howeve r, they solve different formul ations of the LBE, which in return leads to different spatial disc retization approaches In the general SN method, the resolution and accuracy of flux distribution depends on the mesh size and the differencing scheme. In the characteristics method, the resolu tion of flux distribution depends on the sizes of flat source regions. And the accuracy of the flux for each region relies on the densities of characteristic rays. Although the two methods use different discretization methods in the spatial domain, the same discretization approaches (ene rgy group and discrete qu adrature set) can be used in both methods in the ener gy and angular domains. Therefor e, it is possible to combine both methods into one code.
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22 The SN method and the MOC are two of most ef ficient techniques to solve the LBE. However, in 3-D problems requiring a dense gr id in phase space discretization (i.e. a large number of spatial meshes, directions, or ener gy groups), both techniques could suffer from the need for large amounts of memory and computa tion time. In this work, we developed a new transport code (TITAN) with a hybrid discrete or dinates and characteristic method, specifically for application to problems containing regions of low scattering. In this hybrid approach, different methods can be applied to solve the LB E for a given spatial block (coarse mesh) in a physical model. The hybrid approach can take advantages of both methods by applying the preferred method in different regions (block s) based on the problem physics. Since the characteristics method is numerical ly more efficient in low scatte ring media, the hybrid approach uses a block-oriented characteristics solver in low scattering regions, and uses a block-oriented SN solver in the remainder of the physical model.
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23 CHAPTER 2 THEORY AND ALGORITHMS Multi-Block Framework Overview To numerically solve the LBE with a determ inistic method, discretization schemes are required in the energy, angular and spatial domai ns. Once the discretization grid is built in the phase space, one can evaluate the angular flux on each node by sweeping the grid in a specific order repeatedly via an iteration scheme (e .g., the source iteration scheme) until solution convergence is achieved. The hybrid method is built on a multi-block spatial meshing scheme, which is also used in the PENTRAN code.13 The meshing scheme divides the whole problem model into coarse meshes (blocks) in the Cartesian geometry. And each coarse mesh is further filled with uniform fine meshes or characteristic rays depending on which solver is assigned to the coarse mesh. Figure 2-1 shows the multi-block framework of the hybrid approach. Figure 2-1. Coarse mesh/fine mesh meshing scheme. The multi-block framework leads to an importa nt feature of the hybrid code: both the SN and characteristics solvers are coarse-mesh-orie nted. They are designed to solve the transport
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24 equation on the scope of a coarse mesh. A coarse mesh can be considered as a relatively independent coding unit with its own spatial discre tization grid (fine meshes or characteristic rays) and angular discretization grid (quadrature set). Users ca n assign either solver to each coarse mesh. We provide the formulations for the block-oriented SN and characteristics solvers, and demonstrate the two solvers on the multi-block framework. We also discuss the angular quadrature sets used in the TITAN code al ong with the ordinate splitting technique. Discrete Ordinates Formulations Here, we apply the multigroup theory4 to discretize the LBE in the energy domain. And we rewrite Eq. 1-5 in the Cartesian geometry as:3 ,', ', '10 1 ', ', '' ,( )(,,,,)(,,)(,,,,) ()! (21)(,,){()(,,)2 () ()! [(,,)cos()(,,)sin()]} (,,)gg g GL l k sggl lgl l gl k kk Cgl Sgl g fg g oxyzxyzxyz xyz lk lxyzPxyz P lk xyzkxyzk xyz k fix 0g '1(,,) or S(,,,,)G gxyzxyz (2-1) Where, ,and are the x, y and z direction cosines for the discrete ordinates, are the polar and azimuthal angles, respectively. ( ) or ( ) specifies a discrete ordinate, wherecos(), =sin()cos(), sin()sin() ) ( lP is the th Legendre polynomial (for l=1, L where L is Legendre expansion order). And )(k lP is the th, thk associated Legendre polynomial, (,,,,)gxyz is the group g angular flux (for g=1, G where G is the number of groups) at the position of (,,) x yzand in the direction of (,) lg,' is the th Legendre scalar flux moment for group 'g. ',(,,)k Cgl x yz is th, thk cosine associated Legendre scalar flux
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25 moment for group 'g, and ',(,,)k Sgl x yz is th, thk sine associated Legendre scalar flux moment for group 'gat the position of (,,) x yz. These flux moments are defined as: 1 1 2 0 ,')',',,,( 2 )'( 2 ),,( zyx d P d zyxg l lg (2-2) 1 1 2 0 ,')',',,,()'cos( 2 )'( 2 ),,( zyxk d P d zyxg k l lg k C (2-3) 1 1 2 0 ,')',',,,()'sin( 2 )'( 2 ),,( zyxk d P d zyxg k l lg k S (2-4) And other variables are: g : total group macroscopic cross section gsg : th moment of the macroscopic differe ntial scattering cr oss section fromgg '. g : group fission spectrum k0: criticality eigenvalue fg : group fission production ),,,,(zyxSfix g: external source on the position of (x,y,z) and in the direction of ) ,( We can make several observations on Eq. 2-1. First, obviously it accomplishes the discretization in the energy domain by util izing the multigroup theory. As a result, ) ,,( Er becomes (,,,,)gxyz Secondly in the angular domain, no further discretization is required, since we solve for the angular flux in a number of discrete directions of (,) 1,nnnN where N is the total number of directions. The discrete directions are carefully chosen by the quadrature set so that we can conserve the integral quantitie s such as scalar fluxes. Thirdly, if we compare Eqs. 1-5 and 2-1, the most challenging term is the scattering term, in which we convert the integrations over energy and angul ar domain into numerical su mmations for energy groups and Legendre expansion terms. Derivations of the sca ttering kernel are given in Appendix A. It is important to note that in Eq. 2-1, the scatteri ng kernel, as well as th e fission term, does not explicitly depend on the angular flux, but on the flux moments. The relationships between the
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26 angular flux and the flux mo ments are defined by Eqs. 2-2 to 24. Finally the streaming term in Eq. 1-5 becomes a differential term in Cartesian geometry. In order to numerically evaluate the differentials, differencing sc heme is required in the SN method. Source Iteration Process Since the terms on the right hand side of Eq. 2-1, including scatte ring term, fission term and fix-source term, are not explicitly dependent on the angular flux, we can further simplify Eq. 2-1 by combining all the source terms into one source term. ( )(,,,,)(,,)(,,,,)(,,,,)ggggxyzxyzxyzQxyz xyz (2-5) where or g scatteringfissionfixQSSS s catteringS f issionS and f ixS represent the three terms on the right hand side of Eq. 2-1 respectively. Eq 2-5 can be viewed as a numerical iteration equation, which usually is called source iteration scheme (SI).2 In this iteration process, g Q is calculated from previous iteration results. Therefor e, we can solve Eq. 2-5 for the angular flux by taking g Q as a constant. Flux moments can be evalua ted by Eqs. 2-2 to 24 with the latest angular flux, then we can us e the flux moments to update g Q for the next iteration. This process is repeated until the 0th flux moment is converged under some convergence criterion. The iteration process for each group ( g ) can be illustrated as follows: Step 1: Solve Eq. 2-5 for angular flux (,,,,)gxyz Step 2: Evaluate flux moments based on Eqs. 2-2 to 2-4. Step 3: Update the scattering source. Step 4: Repeat the process from Step 1, until ()(1) (1)max(||)ii gg i gtolerence
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27 In Step 1, g is calculated for every fine mesh along a given direction, which is referred to as one direction sweep. After sweeps for ever y direction are completed, flux moments can be updated in Step 2. The group iteration (g=1, G ) needs to repeat only once for fixed source problems with only down-scatteri ng, because the scattering source for the current group only depends on the converged upper group flux mo ments. The summation over groups in the scattering term can be reduced to 1 '1 g g instead of '1 G g However, for problems with upscattering, an outer iteration is required since the scattering source is coupled with lower energy groups. For eigenvalue problems, another outer l oop is necessary so that the fission source and keffective can be updated in between two successive outer iterations. Differencing Scheme From Eq. 1-5 to Eq. 2-1 to Eq. 2-5, we are finally one step away to numerically solving the LBE, which is the evaluation of the differenc ing (streaming) term in Eq. 2-5 by various differencing schemes.14 As shown in Figures 2-2, Eq. 2-5 applies on a spatial domain of a fine mesh with the sizes of and z xy on three axes. y out x y z y in z out t z in A x in x out Figure 2-2. Differencing scheme on one fine mesh.15
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28 Here, we solve for the average flux on the fine mesh. ()1 (,,,,)n g ijk g nn ijk xyzdxdydzxyzu V (2-6) Where i j k are the fine mesh indices, g is the group index, and n is the direction index. ijkVxyz is the volume of the fine mesh. Now, we can finally complete the discretizations on all three domains in the phase space. To calculate () n g ijk we integrate Eq. 2-5 over the fine mesh volume ijkV () () 00 () () 00 () () 00 () () 000 000(,,)(0,,) (,,)(,0,) (,,)(,,0) (,,)(,,)yz nn ngg xz nn ngg xy nn ngg xyz xyz nn ijk g gdydzxyzyz dxdzxyzxz dxdyxyzxy dxdydzxyzdxdydzQxyz (2-7) We assume cross sections are constant inside the fine mesh. In a similar way as Eq. 2-6, we define the fluxes on the three incoming boundaries and the three outgoing boundaries as: () in () out () y in () y out () z in () z out1 (0,,) 1 (,,) 1 (,0,) 1 (,,) 1 (,,0) 1 (,n xg yz n xg yz n g xz n g xz n g xy n g xydydzyz yz dydzxyz yz dxdzxz xz dxdzxyz xz dxdyxy xy dxdyxy xy ,) z (2-8) And the angular source for the fine mesh can be defined as:
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29 ()1 (,,,,)n g ijk g nn ijk xyzQd xd yd z Qxyzu V (2-9) We can divide both sides of Eq. 2-7 by ijkV then substitute Eqs. 2-6, 2-8 and 2-9 into Eq. 2-7, and obtain Eq. 2-10. ()() out iny outy in outz in()()()nn nnn x x z ijkgijkgijkQ xyz (2-10) In Eq. 2-10, the three incoming fluxes (iny inz in, and x ) can be obtained from the fine-mesh boundary conditions at the three in coming surfaces. Therefore, to calculate () n g ijk and the three outgoing fluxes, we need three additional equations, which are provided by the differencing scheme. One of the simplest sche mes is the linear diamond (LD) differencing expressed by: () x out x in () y out y in () z out z in2 2 2n g ijk n g ijk n g ijk (2-11) When moving in positive directions (as s hown in Figure 2-2), we may eliminate the outgoing fluxes in Eq. 2-10 by us ing Eq. 2-11 to obtain Eq. 2-12. () in y in z in ()222 222n nnn x gijk n gijk nnn ijkQ xyz xyz (2-12) The original LBE (Eq. 1-5) finally reduces to a set of linear equations of Eqs. 2-11 and 212. Note that the incoming surfaces change for different directions. The fine mesh sweeping order is decided by the octant number of the dire ction. The same principle is also applied to coarse meshes: we always try to calculate th e outgoing fluxes by solving the LBE based on the incoming fluxes. In this sweeping process, the outgoing fluxes will be the incoming flux for the
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30 next adjacent fine/coarse mesh along the directio n. If the incoming or outgoing boundaries of the fine/coarse mesh are aligned with the model boundaries, model boundary conditions are applied. However, for the coarse mesh sweep, flux proj ections are required on the interface of two adjacent coarse meshes if the two coarse meshes use different spatial and angular discrtetization grids. The projection techniques are discussed in Chapter 3. In Eq. 2-12, the terms of n x n y and nz are always positive, since we always sweep fine meshes along the direction defined by the direction cosines(,,)nnn i.e.,n and x either both are positive, or both are negative. The incoming fluxes, ()n g ijkQ and ijk are positive with their physical meaning. As a result, ()n g ijk is always positive. Howeve r, the outgoing fluxes calculated by Eq. 2-11 of the linear diamond di fferencing scheme could be ne gative, which conflicts with its physical meaning. In order to a void negative fluxes, flux zero fix-up14 is usually applied in the diamond differencing scheme. Furthermore, the diamond differencing scheme introduces artificial oscillations in certain contiditions.16 For this reason, and to facilitate increasing accuracy with adaptive differencing, more advanced differencing schemes17, 18, such as DTW19, EDW20, and EDI21 are implemented in the PENTRAN code. Currently, the diamond and DTW differencing schemes are applied in the TITAN code. Characteristics Formulations Now we further discuss the formulations for the MOC used in the TITAN code. MOC solves the transport equation for the angular fl ux along characteristic ra ys with region-wise discretization grid (i.e. coarse mesh) in the sp atial domain. Since a region can be any shape, MOC has the ability to treat the geometry of a model exactly. Si milar to the coarse/fine mesh sweep process in the SN method, in the MOC, we still calc ulate the outgoing flux based on the
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31 incoming flux for each region, and the outgoing fl ux will be the incoming flux for the next adjacent region. In the angular domain, we perf orm this sweeping process for a number of directions chosen by a quadrature set. Within one region, we assume constant cross sections and calculate the average flux for the region by fill ing the region with char acteristic rays along the directions in a quadrature set. Figure 2-3 shows the parallel char acteristic rays along direction n in a square region i Figure 2-3. Schematic of character istic rays in a coarse mesh us ing the characteristics method. For a given ray of k with a path length of inks we solve the transport equation for () 0 g ink inklls which is the angular flux for group g along direction n, at position l along ray k in region i We denote)0(gink in gink and )(inkgink out ginks The transport equation along ray k can be written as: gin ginkgi gink nQl l )()( (2-13) Where or g inscatteringfissionfixQSSS is the total angular source in region i along direction n for group g. We assume a constant angular source for each ray in region i along direction n. The Sink Region i in g ink n Incoming Boundary Outgoing Boundary out g ink N on fine mesh centers on the incoming boundary Fine mesh centers on the outgoing boundary
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32 streaming term in Eq. 2-13 can be viewed as flux gradients proj ection along direction n, which is the directional derivative of the angular flux. Therefore, Eq. 2-13 can be rewritten as: gin ginkgi ginkQl dl ld )( )( (2-14) Where, l is the path length. Eq. 2-14 can be solv ed analytically if we know the incoming flux (0)in ginkgink as a boundary condition. )1( )(l gi gin l in gink ginkgi gie Q el (2-15) The outgoing flux can be calculated as follows. () (1)giink giinkss gin outin ginkginkinkgink giQ see (2-16) In order to calculate the av erage angular flux in region i first we use Eqs. 2-15 and 2-16 to evaluate the average angular flux fo r each parallel ray along direction n, which is given by: giink gink gi gin giink out gink in gink gi gin l gi gin l in gink s ink gink s ink ginks Q s Q e Q edl s ldl sgi gi ink ink 1 1 )( 1 0 0 (2-17) Where out gink in gink gink. Then, we evaluate the average angular flux for region i by summation of average angular fluxes for all the parallel rays along direction n with a weighting factor of inkink inksAV where inkA is the width (in 2-D) or th e cross sectional area (in 3-D) which ray ( i,n,k ) represents. The average angular flux along direction n is expressed by: k inkink gi k inkink gi gin k inkink k giink gink gi gin inkink k inkink k inkink gin ginsA A Q sA s Q sA sA sA (2-18)
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33 Note that the volume (in 3-D) or the area (2-D) for region i can be represented as k inkink k ink isAVV, if inkA is small enough. Since inkA represents the distance between two adjacent parallel rays, denser rays are required to cover region i as inkA decreases. Therefore, in order to get an accurate region-av eraged angular flux with Eq. 2-18, two conditions are necessary: Region i is small, or flux change s slowly over the region. Rays are dense enough to cover the region. Note that similar conditions are required in the SN method in the sense of spatial domain discretization approach. Generally, in the SN method finer meshes are required to get a more accurate flux distribution. The source iteration scheme can be app lied to the MOC similarly as in the SN method. Eqs. 2-16 and 2-18, as Eqs. 2-11 and 2-12 in the SN method, are the fundamental equations for Step 1 (the sweep process) in the s ource iteration scheme, except that the fine-mesh-averaged angular flux in the SN method becomes region-averaged angular flux in the MOC. Block-Oriented Characteristics Solver The block-oriented characteristics solver is different from the general MOC approach, in the sense that we only apply the solver on an individual block within the multi-block framework. For a characteristics coarse mesh, we build uni form fine meshing on th e boundaries, and draw the characteristic rays from the fine mesh cente rs along quadrature direct ions. We consider the characteristics coarse mesh as one region. And the coarse mesh space is covered with characteristic rays. The boundary fluxes with uniform fine meshing grid are used to communicate with adjacent blocks, since coarse meshes are coupled on th eir interfaces in the sweep process.
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34 Backward Ray-Tracing Procedure Figure 2-4 shows a typical coarse mesh with 55 fine meshes on the 6 surfaces. Note that fine meshing is only applied on the surfaces of a co arse mesh to which the characteristics solver is assigned. The same coarse-mes h volume could be divided into 555 fine meshes if the SN solver is assigned. Figure 2-4. A coarse mesh with characteristics solver assigned. Now we can demonstrate how we set up rays in a coarse mesh shown in Figure 2-4. In the sweep process, our goal is to calculate the outgoing flux based on the incoming flux. In Figure 2-4, the front surface becomes one of the three ou tgoing surfaces for the di rections in four of eight octants in a quadrature set. For the other fo ur octants, it becomes one of the three incoming surfaces. For demonstration purposes, we assume the front surface in Figure 2-4 is one of the outgoing surfaces. Now we need to calculate the outgoing angular flux for each fine mesh on the surface for each direction in the f our octants. Figure 2-5 shows the characteristic rays associated with the center fine mesh on the front surface.
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35 Figure 2-5. Characteristic rays for one fine mesh on one outgoing surface. As shown in Figure 2-5, we draw 12 rays b ackward from the center of one fine mesh (located on the front surface) to the incoming surf aces across the coarse mesh. The four different color rays in Figure 2-5 represent the directions in four octants. Since the intersection positions are not necessarily at the centers of fine me shes on the incoming boundary, an interpolation scheme is required to calculate the incoming fl uxes at the intersecti on positions based on the known incoming fluxes at the fine-mesh centers. Here, we consider an S4 quadrature set which provides three directions per octant For directions in 4 of the 8 octants, the front surface is one of the three outgoing surfaces. Therefore, 12 rays for each fine mesh on the front surfaces are required. The overall characteristic ray density to cover the coarse mesh depends on both the fine mesh grid densities on the outgoi ng boundaries and the number of di rections in the quadrature set. Figure 2-3 also illustrates the characteristic ray drawing pro cedure in 2-D. The green dots on the outgoing boundary in Figure 2-3 are located on the centers of the fine meshes. While the red dots, which represent the in tersection points on the incoming boundary, are off-centered.
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36 Advantage of Backward Ray-Tracing In the characteristic ray drawing procedure, we could choose a forward approach: drawing the characteristic rays from the fine mesh centers on the incoming boundary to the outgoing boundary. The outgoing boundary will experience rays inters ecting its fine meshes in a scattered manner. After the outgoing angular fluxes are calculated, an interpolation procedure is required to project the scattered outgo ing flux onto the fine mesh centers. In a ray drawing procedure, we can always choose a fine mesh center, either on the incoming boundary or on the outgoing boundary, as one node of each characteristic ray to avoid interpolations on that boundary. The other node of the ray will be scattered onto the other boundary, on which interpolations are required rega rdless since we are in terested in the fluxes only on the centers of the fine mesh grid. An interpolation procedure on the incoming boundary needs to evaluate the angular flux at the inco ming node of each characteristic ray based on the known incoming fluxes at the structured fine mesh centers. On the other hand, an interpolation procedure on the outgoing boundary needs to evaluate the outgoing flux at th e center of each fine mesh based on the calculated fluxes at the scat tered outgoing nodes of the rays. The difference between the two choices is: on th e incoming boundary, the interpol ation procedure is carried on from structured data points (incoming fluxes on the fine mesh cen ters) to scattered data points (incoming fluxes for the rays), while on the out going boundary, the procedure is carried on from scattered data points (outgoing fluxes from the rays) to structured data points (outgoing fluxes on the fine mesh centers). In the block-oriented characteristics approach we choose to fix the interpolations on the incoming boundary, because it is numer ically more accurate and efficient to interpolate scattered points from structured points than the other way around. For interpol ations on the outgoing boundary, the scattered outgoing nodes of the rays are the known base points. These scattered
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37 points could be too few, or t oo badly non-uniformly scattered on the boundary, to complete a relatively accurate interpolation to evaluate the flux on the center of every fine mesh. For interpolations on the incoming bound ary, the structured, uniformly distributed fine mesh center fluxes are the known data points. Four closest fine mesh centers to any scattered point can always be found to complete a bi-liner interpol ation. Clearly an interpolation procedure on the incoming boundary is a better choice. The backward ray-tracing facilitates the integration of the block-oriented solvers. Ray Tracer In order to calculate the outgoi ng flux by using Eq. 2-16, we need to evaluate the incoming flux, which is located on the othe r end of the rays on the incomi ng surfaces. The incoming flux is known from the boundary conditions if the incomi ng surface is part of the model boundaries, or from the outgoing flux for the adjacent coarse me sh in the coarse mesh sweep process. We assume these fine-mesh-averaged incoming angular fluxes are located on the center of each fine mesh on the incoming surface. However, the inte rsection point on the incoming surface is not necessarily on the center of a fi ne mesh. Therefore, we need to determine the intersection position of the ray with the incoming surface, and to evaluate the flux at the intersection point by some interpolation method from the fi ne-mesh-centered incoming flux array. In a MOC code, a ray tracer subroutine is requi red to calculate the intersection point of a ray with a surface. The coor dinates of the points along a ray can be defined as: 0 0 0 x xt y yt zzt (2-19) Where 000(,,) x yz is the starting point of the ray, t is path length along the ray, and (,,) are the direction cosines. We can substitute Eq. 2-19 into a region boundary surface
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38 function to evaluate the coordinates of the intersection points of the ray with that surface and the path length t (i.e.,inks in Eqs. 2-16 and 2-18). In the MOC, it can be very expensive, in terms of computer memory, to store the geometry informa tion if the number of rays and the number of regions are very large. For this reason, 3-D MO C could be prohibitive for a large model. The block-oriented characteristics so lver considers the whole coarse mesh as one region. Therefore, for Eq. 2-19, the region boundaries become the coarse mesh surfaces. Beca use the characteristics solver is designed for solving the transport equa tion in a low scattering medium, across which we can expect that the angular flux al ong the ray does not change signi ficantly, it is possible to use a relatively large region (i.e. a coarse me sh) for a flat-source MOC formulation. Interpolation on the Incoming Surface Based on the positions of the intersection po ints of rays on the incoming surface of a coarse mesh, we can further evaluate the averag ed flux for each fine mesh by interpolation. As shown in Figure 2-6, points A B C and D denote the closest 4 neighbors to point P which is the intersection point of a characteristic ray across one incomi ng boundary. We need to evaluate the angular flux at point P based on the fluxes at the 4 neighboring points. Figure 2-6. Bilinear interpolation for the incoming flux. A ( -1 -1 ) B (1,-1) C (1, 1) (-1,1) D P(s,t)
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39 For simplification, we assume the c oordinates for the 4 neighbors and point P are A(-1,-1) B(1, -1) C(1, 1) D(-1, 1) and P(s, t) where s, t are evaluated by the ra y tracer. Note that the actual positions of the fine mesh centers and point P are projected into the coordinates shown in Figure 2-6, in which A B C D and P are located at (-1,-1) (1,-1) (1,1) (-1,1) and (s, t) for the interpolation. Two interpolation techniques are applied in the TIT AN code. Either of them can be used to estimate the incoming flux at point P closest neighbor. P is equal to the angular flux at the closes t neighbor. For example, in Figure 2-6 P will be equal to the A under the closest neighbor approach. bilinear interpolation. A bilinear interpolation formulation is applied:22 (1)(1)(1)(1) (,)(1,1) (1,1) 44 (1)(1)(1)(1) (1,1) (1,1) 44 stst st stst (2-20) Where (1,1) A (1,1)B (1,1)C (1,1)B and (,) P st The truncation error indicates the bilinear approach is a second order interpolation. And it should be more accurate than the first approach, which is a first order interpolation. However, we should note that these point-wise angular fluxes are actually averaged values: fine-mesh-centered fluxes (A B C and D ) are the averaged fluxes on the fine meshes, and the ray intersectionpoint flux (P ) is the averaged flux on the cross sectional area (inkA in Eq. 2-18) of the volume the ray represents. An assumption is made that the averaged flux happens at the center of the fine mesh, or at point P of the ray cross section area. This assumption is reasonable if the fine mesh is small. Therefore, our ray solver may require a relatively finer meshing on the coarse mesh
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40 surfaces, which leads to denser rays in the coar se mesh and longer computer time and memory requirements. On the other hand, if the fine me sh is relatively large, the closest neighbor interpolation scheme is not necessarily less accura te than the advanced bi linear interpolation. The most suitable interpolation scheme coul d depend on the problem and its modeling. By default, the bilinear interpolation scheme is used in the TITAN code. In the characteristics solver, the cross sect ional area represented by each ray (defined in Eq. 2-18) can by calculated by the following formulation: )cos(,, jijiSA (2-21) Where Si,j is the fine mesh area on the outgoing boundary, and is the angle between the ray direction and the direction normal to th e boundary. Even with a uniform fine meshing applied on the surfaces of a coarse mesh for the characteristics so lver, rays are not necessarily distributed uniformly within the coarse mesh vol ume, because rays along a certain direction can form different angles with the normal directions of the three incoming surfaces of the coarse mesh. Non-uniform ray distribution could lead to the requirement of denser rays and/or smaller coarse meshes to maintain accuracy of the bi-linear interpolation. Quadrature Set We discussed the formulations for the SN and characteristics solver, respectively. Our focus has been on the Step 1 of the source iter ation scheme, which is to solve the transport equation for the angular flux. For Steps 2 and 3, the formulations are fundamentally the same for both solvers because of the following similarities between two methods: Calculate the angular flux, alt hough with different formulations. Apply the same energy and angular domain discretization approaches. Use the source iteration scheme.
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41 The major difference between the two methods is the discretization method in spatial domain. Both block-oriented so lvers share the same goal to cal culate the outgoing angular fluxes for a block. However, they complete the task w ith different formulations of the original LBE. Now we can further demonstrate Step 3 of th e source iteration scheme. In both methods, we denote the source term in Eq. 2-5 or Eq. 2-15 by: or s catteringfissionfixQSSS (2-22) For simplification, we omit the index for en ergy group, direction, and fine mesh (SN) or region (MOC). In Eq. 2-22, f ixS is known as external source. s catteringS and f issionS can be evaluated from flux moments cal culated from the results of the previous iteration. () (1) ,',, ',, '10 1 ,(1) ,(1) ',, ',()! (21){()2 () ()! [cos()sin()]}GL l ii k scattering sgglxlnglx ln gl k ki ki Cglx nSgl nlk SlPP lk kk (2-23) Where i is the iteration index, g is the energy group index, l and k are the Legendre expansion indices, (,)nn specifies direction n in the quadrature set, (1),(1) ,(1) ',, ',, ', and ikiki g lxCglxSgl are the flux moments calculated from the last iteration, which is indexed by i-1 here, and x is the fine mesh index in the SN formulation, or the region i ndex in the MOC formulation. The scattering kernel defined by Eq. 2-23 can be expanded to an arbitrary Legendre order if the same order of cross section data is provided. The isotropi c fission source and the k-effective can be evaluated by Eqs. 2-24 a nd 2-25 from an outer iteration. () (1) ',',0, (1) '1G g jj f ission fgxgx i gS k (2-24) )1( )( )1()( j fission j fission jjQ Q kk (2-25)
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42 Where <> denotes the integration over the entire phase space. Note that j is the outer iteration index, while in Eq. 2-23 i is the inner iteration index. Scattering source is updated after one sweep is completed for each group, while th e fission source is update d only after all groups are converged based on the previous fission source. Equations 2-23 and 2-24 are the fo rmulations for Step 3 in th e source iteration scheme. For Step 2, we use a quadrature set to evaluate the integral over angular domai n defined in Eqs. 2-2 to 2-4 for flux moments. N n n n k lnn k lS N n n n k lnn k lC N n nlnn lkPw k Pw Pw1 1 1)sin()( 8 1 )cos()( 8 1 )( 8 1 (2-26) Here, for simplification, we drop the indices for energy group and fine mesh or region. Direction n can be specified by (,) where 11 02nn n n or 222(,,) where 1,,1 1nnn nnnnnn In order to preserve symmetries, a quadrature set only specifies direc tions in the first octant (0,,1nnn ), directions in the other octants can by acquire d by changing the signs of n n and/or n For example,(,,)nnn specifies the opposite directi on corresponding to direction (,,)nnn in another octant. Direction (,,)nnn and all its seven corres ponding directions in other octants have the same weight (nw ). Usually, we keep the total weight for all directions in one octant equal to one. These directio ns and the associated weights (nw ) are carefully chosen by a quadrature set, so that we can accurately evalua te the moments of direction cosines and the flux moments defined by Eq. 2-26. Other concerns related to the phys ics of the problems can affect
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43 the choice of the directions too. Further discu ssions are given in Appendix B. Currently, in the TITAN code, we have two types of quadrature sets available: the level-symmetric quadrature5 and the Legendre-Chebyshev quadrature.23 Level-symmetric Quadrature Figure 2-7 shows a level-symmetric quadrature with an order of 10 (S10). We use a point on the unit sphere to repr esent a direction. The xyz coordinates of the point are the three direction cosines of the direction. These directions are ordered with a t riangle shape formation. To generate a quadrature set, we need to find th e direction cosines and the weights for all the directions. Figure 2-7. Schematic of the S10 level-symmetric quadrat ure set in one octant. S10 specifies 15 directions in the first octant on 5 levels. Directions in the other seven octants are chosen to be symmetric to the directions in the first octant. Therefore, the total number of directions on the unit sphere is 158120 for all 8 octants. Generally, for a levelsymmetric quadrature with an order of N we can calculate the number of levels L and total number of directions M in the first octant by: 8 2)(NN M 2 N L (2-27)
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44 To keep a symmetric lay out of the directions, N is always chosen from even numbers. The level-symmetric quadrature set is widely used in the SN codes for its rotation invariance property and preservation of moments. Rotation invarian ce keeps the quadrature directions unchanged after 90 degree rotation along any ax is. In other words, if (,,)nnn is one direction in the first octant of the quadrature se t, any combinations of n n and n such as (,,)nnn or (,,)nnn are also defined in the first octant of the quadrature set. Note that rotation invariance is different from octant symmet ry of the directions, where ) ,,(nnn defines the eight symmetric directions in the eight octants. Rota tion invariance is very desirable in many real problems to keep the symmetry, especially wh en reflective boundary conditions are applied. However, it also places a strict constraint on the choice of the quadrature directions. The symmetry condition requires kji for 2 ,,1 N kji following the same sequence. 2 )31(2 )1( 2/,,2 ,1,,for 2 1 2 1 N C iC N kjii kji (2-28) In Eq. 2-28, only 1 is free of choice. The remaini ng degrees of freedom on direction weights are used to conserve the odd and even moments of and .10 1 111 1111.0 0 for n odd 1 for n even, 1M m m MMM nnn mmmmmm mmm MMM nnn mmmmmm mmmw www wwwnL n (2-29) The directions and their associated weights can be calculated by Eqs. 2-28 and 2-29. Levelsymmetric quadrature only can conser ve moments to an order of maximum L=N/2 because of the
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45 symmetry condition. Another disadvantage of leve l-symmetric quadrature is that Eqs. 2-28 and 2-29 lead to negative weights if N is greater than 20. Nega tive weights are not physical. Therefore, they cannot be used. This means th at the order of Level-Symmetric quadrature is limited to 20. Legendre-Chebyshev Quadrature The Legendre-Chebyshev quadrature,23 also called PN-TN quadrature, aims to conserve moments to a maximum order without the constr aints of the symmetry condition. Figure 2-8 shows a PN-TN S10 quadrature layout. Figure 2-8. PN-TN quadrature of order 10. The Legendre-Chebyshev quadrature c onserves moments to the order of 2L-1 instead of L in the level-symmetric quadrature set ( L=N/2), at the cost of lack of rotation invariance. Moments in Eq. 2-28 cannot be conserved strictly in the PN-TN quadrature.24 Note that Figures 27 and 2-8 share a similar triangle-shaped dire ction layout on the unit sphere, because Eq 2-27 still holds in the PN-TN quadrature. The direction weights are positive definite in the PN-TN quadrature. Therefore, unlike the le vel-symmetric quadrature set, the PN-TN quadrature order is unlimited mathematically, except for the limitation of computer memory limitation. We have derived the procedure on how to build the PN-TN quadrature on the unit sphere. Based on the procedure, it can be shown that the PN-TN quadrature is the best choice in
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46 mathematically conserving higher moments. We also have proved the positivity of weights in PN-TN quadrature. Details of the above derivatio ns are given in Appendix B. To build a PN-TN quadrature set, it is required to find the roots of an even orde r Legendre polynomial. These roots are used as level positions of the quadrature. A modified Newtons method is applied. Details of the algorithm also are given in Appendix B. Rectangular and PN-TN Ordinate Splitting Ordinate splitting is a technique associated with a quadrature set.25 A selected direction in a quadrature set can be further split into a number of directions. The total weight of the split directions is equal to the weight of the original direction in the quadrature. We apply the ordinate splitting techniques to solve proble ms with highly peaked angula r-dependent flux and/or source. Two splitting methods, rectangular splitting and PN-TN splitting are available in the TITAN code. Figure 2-9 depicts the two splitting dir ections for one di rection of an S10 quadrature set. Note that ordinate splitting technique is in dependent of choice of quadrature set type or order, and can be applied to as many directions as necessary. A B Figure 2-9. Ordinate sp litting technique. A) Rect angular splitting. B) PN-TN splitting.
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47 In the rectangular splitting technique, the split directions are uniformly distributed within a box-shape region centered at the orig inal quadrature direction. In the TITAN code, the size of the box can be defined by users. The total number of splitting directions can be calculated from the user-specified splitting order with Eq. 2-30. 2(21) sl (2-30) Where s is the total number of splitting directions, l is the splitting order. Figure 2-9A shows the 25 split directions for a rectangular sp litting with an order of 3. All the splitting directions are equalweighted, defined as 1 s nww s where nw is the weight of the original direction, which remains in the quadrature set after splitting with a reduced weight. The rectangular-shaped layout of the split direction may not be efficient in conserving the moments. We developed the Legendre-Chebyshev (PN-TN) splitting technique based on the regional angular refinement (RAR) technique.26 In the PN-TN splitting, the original direction can be associated with a local area on the unit sphere surface centered on the or iginal direction. And the range of the area can be decided by users as in the rectangular splitting. The technique projects the directions in th e first octant of a regular PN-TN quadrature set w ith an order of 2 l ( l is the splitting order), into th e local area. For a regular PN-TN quadrature, usually there is only one direction on the top level as show n in Figure 2-8. For the local PN-TN quadrature fitted in the splitting technique, users can spec ify the number of directions on the top level. The number of directions on the following levels increases by on e from the previous level, as for a general PNTN quadrature. Therefore, the total number of split directions can be calculated by: (21) 2 tll s (2-31)
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48 Where t is user-specified number of directions on the top level, and l is the splitting order. The weights of the split dire ctions are calculated in th e same way as a general PN-TN quadrature, except that we normalize the total we ight to the original direction we ight, instead of unity as in a general PN-TN quadrature. The split direction we ights is calculated by Eq. 2-32. TSPSnswwww__ (2-32) Where nw is the original weight of the splitting direction, PSw_ and TSw_ are the level weight and the Chebyshev weight, respectively for one split direction in the local PN-TN quadrature. Note that unlike the rectangular spl itting, the original splitting direction is dropped off after splitting in the PN-TN splitting technique. However, the split directions could be more uniformly distributed within th e splitting region than the rectangu lar splitting, since it is formed uniformly on a sphere surface instead of a rectangular region, and also the PN-TN quadrature conserves integrations more accurately than an equal-weighting formulation. In Chapter 5, we will use the splitting techniques on one benchmark problem. At the end of this chapter, we quote a co mment on different dete rministic methodologies by Weinberg and Wigner.27 The comment was made about half a century ago, yet even today, it provides us some insights on this matter. At present, with so much of the practical work of reactor design being done with large digital computers, the arguments in favor of one me thod of approximation rather than another tend to center around the question of how well suited the method is for dig ital computers. Actually, as the computers become larger, the choice between methods b ecomes less and less clear: any method which converges will do if the computer is large enough. This viewpoint certainly has practical merit; however, convenience for a digita l computer is hardly a substitute for intrinsic mathematical beauty or physical relevance. In this respect th e spherical harmonics method is perhaps most satisfying; its firs t order is identical with diffus ion theory, and its higher orders show the deviations from diffusion theory very clearly. Alvin M. Weinberg & Eugene P. Wigner, 1958
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49 CHAPTER 3 PROJECTIONS ON THE INTERFACE OF COARSE MESHES The TITA N code is built on the multi-block framework with the source iteration scheme. Both the block-oriented SN and characteristics solvers can apply an individual quadrature set and fine-meshing scheme on each coarse mesh. Tran sport calculations can benefit from the multiblock framework, which provides users more opti ons on the choices of discretization grids in different regions of a proble m model. However, the benefits are not free in term of computational cost. In Step 1 of the source it eration scheme, while sw eeping across the interface of two coarse meshes, we need to project the angular flux on the interface from one frame to the other, if the two coarse meshes use different quadrature sets and/or fine-meshing schemes. Therefore, angular and spatial projection techni ques are developed to transfer the interface angular fluxes in the coarse-mesh-level sweep process. Angular Projection Angular projection is trigge red by the two adjacent coarse meshes with different quadrature sets. Figure 3-1 shows the layout of directions in two quadrature sets. A B Figure 3-1. Angular project ion. A) Level-symmetric S10 (red) to PN-TN S10 (green). B) S10 to S8.
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50 Figure 3-1A compares the directio ns for the level-symmetric and PN-TN quadrature sets of order 10. Figure 3-1B presents a more general situation of angular projection: from a higher order quadrature to a lower order quadrature, or vice versa. In general, an angular projection from quadrature P to quadrature Q is used to evaluate the angular fluxes for the directions in quadrature Q for each fine mesh on the interface, based on the angular fluxes from quadrature P For each direction n in quadrature Q we search for the closest th ree neighboring directions in quadrature P to n The angular flux for n can be calculated by a m1 weighting scheme, where m is a positive integer, and is the angle between n and one neighbor direction in quadrature P. Note that also represents the shortest distance between n and its neighbor on the surface of a unit sphere. As shown in Figure 3-2, P1, P2, and P3 are the three closest neighbors in quadrature P to n in quadrature Q Figure 3-2. Theta weighting scheme in angular domain. If we consider that the distances between n and the three closest neighbors are 1 2 and 3 respectively, then the angular flux at n can be written as: P1 P2 P3 n 1 2 3
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51 otherwise f ifm A m A m A m i A m Qi j) ( 1 10),,min( ,321 )( 4 321 )(3 2 1 (3-1) Where )( mf is the mth normalization factor and defined as mmm mf321 )(111 Note that we set the angular flux at n equal to the closest neighbors, if the minimum distance is less or equal than 410 radians. The 0th moment (scalar flux) and the first moment (flux current) of the angular flux have to be conserved after an angular projec tion. Therefore, we need to maintain: jj iiQ M j Q P N i Pw w 1 1 (3-2) jjj iiiQ M j QQ P N i PPw wJ 1 1 (3-3) Where, N and M are the total number of directi ons in one octant in quadratures P and Q respectively. iP is the cosine of the angle between the interface normal direction and direction i in quadrature P jQ is the cosine of the angle between the interface normal direction and direction j in quadrature Q And w s are the direction weights. Note that the total weights are set to one for both quadrature sets (11 1 M j Q N i Pj iww). In order to evaluate )( Q j while conserving the scalar flux and the current, we assume jQ is a linear combination of )1(jQ and )2(jQ )2( )1(j j jQ Q Q (3-4) Where, )1(jQ and )2(jQ are calculated with Eq. 3-1 with m=1, 2 respectively. And and are the linear coefficients, which can be evaluated by substituting Eq. 3-4 into Eqs. 3-2 and 3-3.
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52 )() ()() ()1( 1 1 )2( )2( 1 1 )1( 1 )1( )2( 1jj jjj jj jjj jjj jjQ M j Q M j QQQ Q M j Q M j QQQ M j QQQ Q M j Qw w w w w wJ (3-5) )() ()() ()1( 1 1 )2( )2( 1 1 )1( )1( 1 1 )2(jj jjj jj jjj jj jjjQ M j Q M j QQQ Q M j Q M j QQQ Q M j Q M j QQQw w w w wJ w (3-6) Once )1(jQ ,)2(jQ and are evaluated by Eqs. 3-1, 3-5, and 3-6, jQ can be calculated by Eq. 3-4. Under this angular projection scheme, th e scalar flux and the first flux moment remains the same for each fine mesh on the interface before and after the projection. It is also possible to conserve higher moments at additional computational cost. We can always introduce higher order weighting schemes with Eq. 3-1 (e.g. 31, 41), then more terms and coefficients can be added in Eq. 3-4. In order to calcul ate the linear combination coefficients ( etc.), higher moment conservation equations can be introduced besides Eqs. 3-2 and 3-3. Although the scattering source term defined by Eq. 2-23 is calcul ated with all flux moments up to the order of L generally it is not necessary to conserve flux moments with an order higher than one on the interface, since only the 0th and first moments carry physical meanings (scalar flux and flux current), other than just a mathematical term. In the TITAN code, we also apply a negative fi x-up rule to keep the positivity of angular fluxes by relaxing the 0th and/or the first moment conserva tion rule if necessary. The angular projection can be used with any type of the quadrature set. It is also compatible with the ordinate splitting technique. In order to perform a re latively efficient angul ar projection, it is recommended that both projecting and projected quadrature sets have at least three directions per octant (i.e. at least S4). If there is only one dire ction in one octant (i.e. S2), the direction can be
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53 considered as three directions with the same position and only one-third of the original weight, so the above angular projection procedure stil l can be performed without any modifications. Spatial Projection Spatial projection is triggered if the fine -meshing schemes mismatch on the interface of two adjacent coarse meshes. Figure 3-3 shows a projection situation between a 3x3 meshing scheme and a 2x2 meshing scheme. A B Figure 3-3. Mismatched fine-meshing schemes on the interface of two adjacent coarse meshes. A) 3-D layout. B) 2-D layout. In Figure 3-3B, we denote the 3x3 fine meshes on the green surface as g(1,1), g(2,1) g(3,3) the 2x2 fine meshes on the red surface as r(1,1), r(2,1) r(2,2) The average angular fluxes on these fine meshes can be referred to as ) 3,3()1,1()( )( g g and ) 2,2()1,1()( )( r r Assuming a green-to-red projecti on, we need to calculate )2,2()1,1()( )(r r based on )3,3()1,1()( )( g g by an area weighting scheme. Here, we only demonstrate how to calculate
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54 the angular flux on fine mesh r(1,1) The rest of the red meshes can be evaluated based on the same approach. )1,2()1,2()2,2()2,2()2,1()2,1()1,1()1,1( )1,1( )1,2()2,2()2,1()1,1()1,1( )1,1()( )( )( )( 321 3 2 1 g g g g g g g g g g g g g g rf f f f AAAA A A A A (3-7) Where A1, A2, and A3 are the shade areas in Figure 3-3B. Ag(1,1) is the area of fine mesh g(1,1) Since fine meshes are uniformly dist ributed on either surface, we can denote g gAA )1,1 (. Note that 321)1,1( AAAAAgr is the area of fine mesh r(1,1) Therefore, the factor )( gf can be denoted as: r g r g r g r g gA A f A A f A A f A A f3 )( 2 )( 1 )( )()1,2(,)2,2(,)2,1(,)1,1( (3-8) If we assume a red-to-green projection, () ()(1,1)(3,3)gg will be evaluated based on () ()(1,1)(2,2)rr The same area weighting scheme can be applied: g g r g r g r gA AA A A3 )( 3 )( )( )( )()1,2()1,1()1,2( )1,1()1,1( (3-9) The area weighting scheme can conserve the angular flux for each fine mesh, assuming a flat flux distribution within fine meshes. Therefore, the total a ngular flux over the entire interface is conserved automatically. The post re-normaliza tion process described in the angular projection is not necessary in spatial projection. In the TI TAN code, we separate th e 2-D projection to two single 1-D projections in or der to reduce computation cost. For example, a 2-D 4683 projection can be separated as a 63projection along x axis, and an 48projection along y axis, because x and y projections are actually independent of each other. Genera lly, a projection pair, mn and nm require mn 2 memory units to store the geometry meshing factors
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55 ()( gf )( rf ). However, since most of the factors are zeros, we store only th e non-zero factors with a sparse matrix for each projection pa ir. Note that the factors in an mn projection remain the same whether they are applied in an x or y axis projection. Projection Matrix Both angular and spatial proj ections could be expensive in the source iteration scheme, because for every iteration, th ey are performed whenever th e sweep processes cross the interface of two coarse meshes with different an gular or spatial frame. If both projections are required on an interface, we perfor m the angular projection first, then the spatial projection. A projection from coarse mesh A to coarse mesh B on the interface can be described as BABAP (3-10) Where PAB is a projection matrix, which stores all the necessary geometry information on the interface. Since projection matrices are indepe ndent of angular fluxes, they can be calculated and stored before the sweep process starts.
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56 CHAPTER 4 CODE STRUCTURE The fundamental stru cture of the TITAN code is built on th e four steps of the Source Iteration (SI) scheme with the multi-block framework. And the SN and characteristics solver kernels are integrated in Step 1, in which we apply the sweep process to solve the LBE for angular fluxes. Sweep is a process to calculate the outgoing flux from the incoming flux for a coarse mesh, a fine mesh (SN), or a region (characteristics) by simulating the particle transport along certain directions. The fine me sh/region averaged angul ar fluxes are updated during the process. In Step 2, we evaluate the flux moments base d on the angular flux calculated in Step 1 by a numerical quadrature set, then use the flux moments to update the source in Step 3 for next iteration. The itera tion process continues unt il fluxes are converged based on a convergence criterion. In this chapter, first we introduce the overa ll block structure of the code. Then, we further discuss the transport calc ulation block, with some details of several key subroutines. Finally, the front-line style sw eep process is presented. Block Structure The TITAN code is composed of three majo r blocks: input, processing, and output. The input block loads the input decks to initialize the model material and the fixed source distribution, meshing scheme, a nd some control variables. The processing block performs the transport calculation. And the output block handles the calculation results. In this section, we introduce the input and output blocks. The processing block is disc ussed in the next section. The input decks include the cross-section data file, PENMSH-style input files to build up the model geometry,28, 29 and a block-structured input file ( bonphora.inp ), to setup some control variables such as quadrature sets and solvers for each coarse mesh. By default, the
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57 output block writes up the material number, the source intensity and the calculated scalar flux for each fine mesh into a TECPLOT-format binary da ta file. The data in this file is organized by coarse meshes. Each data point/fine mesh is composed of an array of values: xyz coordinates of the center of th e fine mesh, material number a nd fixed source intensity in the fine mesh, and the average scalar flux for each energy group. Comparing to the ASCII format of the TECPLOT data file, the binary file is smaller in size and faster to load by TECPLOT for various plotting. As an op tion, the output block can also prepare the input deck for the PENTRAN code. More details a bout TITAN I/O file format are given in Appendix D. Processing Block The subroutines in the processing block can be roughly arranged in four levels. The lower level routines are called only by the immediate upper level routines. The top level (0th level) routines choose the corr esponding module for different types of problems (shielding or criticality). The first level ro utines setup the source iterati on schemes for all energy groups. The second level routines complete one system sw eep for all the directio ns in the quadrature sets for one group. The third leve l routines only handl e one sweep for all the directions in one octant for one coarse mesh and one group. Finally on the forth level, we apply the SN or MOC formulations discussed in Chapter 2 to calculate the angu lar flux in one fine mesh (SN) or one region (characteristics) Figure 4-1 shows the major subroutines within the four-level code structure. In the followi ng sections, we further discuss some of the routines on each level.
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58 Figure 4-1. Code structure flowchart. Within group loop till flux converged Ray Sn Loop for each direction in one octant Loop for each parallel ray Loop for each direction in one octant Loop for each FM in the sweep order loo p for octant=1, 8 Loo p for each CM in the swee p orde r L0.1 Input Block L0. 2 Processing Block L0.21 TransCal L0.22 UpScaCal L0.23 Ksearch L0.24 Ksearch u p L0.3 Output Block L1.1-3 CreatQuad L0.23 or L0.24 k outer loo p for criticalit y p roblems L0.22 outer loo p for u p scatterin g L0.21 loo p : g rou p =1,num _g r p L1 1 Ini tS n L1.1-4 InitCMflux L1.1-5 InitPro j ection L1.1-1 G etXs L1.1-2 InitInte r L1.2 GetInMntG L1.3 SolverSn_L1_S1 L1 4 UpdateSca Flx L1.5FissionSrc L1.2-1 GetInMnt_Sn L1.2-2 GetInMnt_Ray L2.1 Map Bnd2inter L2.2 CM SweepOrder L2.3 InitCM L2.3-1 InitCM_Sn L2.3-2 InitCM_Ray L2. 4-1 SolverSn_L2_S1 L2. 4-2 SolverRay_L2_S1 L2.5 FreeCM L2.5-2 FreeCM_Ray L2.5-1 FreeCM_Sn L2.6 Map Inter2Bnd L2.7 CalMnt L2.7-1 CalMnt_Sn L2.7-2 CalMnt_Ray L3.1 Angular Projection L3.2 Spatial Projection L3.3 FM SweepOrder L3.4 Map Sys2CM L3.5 Get FmSrc_CMin L3.6 DiffScheme L3.7 Map CM2Sys L3.8 GetZnSrc_CMin L3.9 GetBakFlx L3.12MapCM2Sys L3.10 GetRayAvg L3.11 GetZnAvg 58
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59 On the top level, TITAN has a simple three-block structure: input bl ock, processing block, and output block. In the processi ng block, four kernel subroutin es are available for different types of problems: L0.21 TransCal: fixed source prob lem with only down scattering. L0.22 UpScaCal: fixed source problem with upscattering. L0.23 Ksearch: criticality problem with only down scattering. L0.24 Ksearch_up: criticality problem with upscattering. Based on some parameters from the input block, we choose one of the four subroutines to perform the transport calculation. TransCal provides the fundamental loop structure of the source iteration scheme. Here, we assume that th e source iteration scheme starts from the energy group loop. The other three subroutines require one ( L0.22 and L0.23 ) or two (L0.24 ) additional outer loops besides the f undamental source iteration scheme loop structure ( L0.21 ). They are designed for problems with upscatter ing and/or criticality problems. First Level Routines: Source Iteration Scheme The flowchart on the first level demonstrates the structure of the processing block. The subroutines on this level can be illu strated in the following pseudo-code. Figure 4-2. Pseudo-code of the source iteration scheme. !! Pseudocode: processing block (Trans Cal, UpScaCal, Ksearch, Ksearch_up) Call InitSn Loop outer_k k loop(power iteration) if eigenvalue problem Loop outer_g outer_g loop if upscattering presents For g=1, num_group group loop call GetInMnt_G(g) while (flux not converged) within group loop call SolverSN_L1_S1(g) call UpdateScaFlx(g) end within group loop end group loop end outer_g loop if upscatter ing presents call FissionSrc if k loop presents End outer_k loop
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60 Subroutine L1.1 InitSn is designed to complete the initia lization works before the transport calculation starts. This initiali zation includes loading cross sec tion data, allocating memory for interface fluxes, angular fluxes, and flux moments, and initialization of the quadrature sets and projection matrices. Subroutine L1.2 GetInMnt_G is called at the beginning of each group loop. And it has only one input argument: group index g GetInMnt_G(g) calculates the flux moment summation for all other groups other than group g which we call scattering-inmoments, or in-moments. Inmoments are used to efficiently calculate the scattering source, wh ich is performed in Step 3 of the source iteration scheme. By applying the in -moments, we can rewrite Eq. 2-23 by switching the group and Legendre order expansion. () (1) ,',, ',, '10 1 ,(1) ,(1) ',, ', ,',,', 0' 1 '()! (21){()2 () ()! [cos()sin()]} (21){()[GL l ii k scattering sgglxlnglx ln gl k ki ki Cglx nSgl n LG lnsgglxg lg gglk SlPP lk kk lP (1) (1) ,,,,,, ,(1) ,(1) ,',, ',,,,, ,, 1' 1 ,( ,',, ', 1' 1 '] ()! 2() c o s () [ ] ()! ()! 2() s i n () [ ()!ii lxsgglxglx lG kk i k i ln n sgglxCglxsgglxCglx kg gg lG kk i lnnsgglxSgl kg gglk Pk lk lk Pk lk 1) ,(1) ,,, ,]}ki sgglxSgl (4-1) In Eq. 4-1, the terms of(1) ,',,',, '1 G i s gglxglx g gg ,(1) ,',, ',, '1 G ki s gglxCglx g gg and ,(1) ,',, ', '1 G ki s gglxSgl g gg are defined as zero in-moments, cosine in-moments and sine in-moments. Mathematically, this formulation seems more complicated than Eq. 2-23. However, it is more efficient to evaluate scattering source. The in-moments can be pre-ca lculated before the within-group starts, since they are independent of group g moments, which are the only changing moment terms between
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61 the within-group loops. Therefore, once the in-moments are pre-calculated by the subroutine GetInMnt_G the summation process over all groups in side the within-group loop reduces to a two-term summation: in-moments plus the group g moments. Inside the subroutine GetInMnt_G we calculate the in-moments for all the coarse meshes. If the characteristics solver is a ssigned to a coarse mesh, Subroutine L1.2-2 GetInMnt_ray is called to calculate the in-moments for each region in the coarse mesh. Otherwise, L1.2-1 GetInMnt_Sn is called to calculate the in-moments fo r each fine mesh within the coarse mesh. Subroutine L1.3 Solver_Sn_L1 is the kernel subroutine on this level, which completes one system sweep for a given group g Its structure is illustrated on the next level. Subroutine L1.4 UpdateScaFlx is used to calculate the scalar fluxes fo r the current iteration, and evaluate the maximum difference from the previous iteration. Solver_Sn_L1 and UpdateFlx are the two major subroutines of the within-group loop. They are repeatedly called until the maximum scalar flux difference between two interations satisfies the user-defined c onvergence criterion. L1.5 FissionSrc is called at the end of each k-effective loop (power iterati on) to update the fission source and the k-effective for the next power iteration. The fission source is considered as an isotropic fixed source for all the other inner loops (within-group loop and upscattering loop). Fission source is evaluated for each fine mesh. Then, the k-effective is calculated by using Eq. 225. More advanced formulas derived from pow er iteration accelerati on techniques can be investigated and applied within the scope of this subroutine. Second Level Routines: Sweeping on Coarse Mesh Level The subroutines on this level are called by the kernel subroutine SolverSN_L1_S1 of the first level. Two inner loops, octant loop and coarse mesh loop are constructed in SolverSN_L1_S1. Its structure can be illustra ted in the following pseudo code.
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62 Figure 4-3. Pseudo-code of th e coarse mesh sweep process. Subroutines L2.4-1 SolverRay_L2_S1 and L2.4-2 SolverSn_L2_S1 are the kernel subroutines, which complete the sweep process within the scope of one coarse mesh for directions in one octant and fo r a given group by using either the characteristics solver or the SN solver. The detail structures of the two subr outines are illustrated in the next section. Subroutines L2.1 MapBnd2inter and L2.6 MapInter2Bnd are used in the sweep process on the system level. The sweep process starts from the three incoming boundaries of the model for the directions in a given octant, and ends at the three outgoing bounda ries. At the incoming surfaces, model boundary conditions need to be applied. And if th e outgoing surfaces are reflective or albedo boundaries, the outgoing angular fluxes need to be reflected back as incoming fluxes for directions in another octant. Therefore, at the beginning of the system sweep process, MapBnd2inter is called to map the incoming syst em boundary conditions to a system interface flux array, while at th e end of the sweep process, MapInter2Bnd is called to map the system interface flux back to the model boundary. Subroutine L2.2 SweepOrder_CM initializes the coarse mesh sweep order for directions in a given octant before the coarse mesh loop starts. Subroutines L2.3 InitCM and L2.5 FreeCM are !! Pseudocode: SolverSn_ L1_S1 (group) !group: energy group index For octant=1, 8 octant loop call MapBnd2inter(octant,group) call SweepOrder_cm(octant) for cm_ijk in the sweeping order !coarse mesh loop if (MOC solver is assigned to cm_ijk) call InitCmRay(cm_ijk) call SolverRay_L2_S1(cm_ijk, octant, group) call FreeCmRay(cm_ijk) else call InitCmSn(cm_ijk) call SolverSn_L2_S1(cm _ijk, octant, group) call FreeCmSn (cm_ijk) endif end cm loop call MapInter2Bnd(octant,group) end octant loop call CalMnt(group)
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63 designed to allocate and free memory for the in terface flux array within one coarse mesh. More details about the interface flux array will be discussed later. Both InitCM and FreeCM have two versions corresponding to the characteristics and SN solver kernel. Subroutine L2.7 CalMnt is called after the system sweep completes. The subroutine is used to evaluate the flux moments (source iteration scheme: Step 2) based on the angular fluxes calculated by the system sweep (s ource iteration scheme: Step 1). Third Level Routines: Sweeping on Fine Mesh Level Two sets of routines are built on this lo west level for the characteristics and SN solvers, respectively. Both calculate angular fluxes within the scope of one coarse mesh, one octant, and one group. Their structures can be illu strated by the following pseudo code. Figure 4-4. Pseudo-code of th e fine mesh sweep process. !! Pseudocode: SolverSn_L2_S1 (cm_ijk, octant, group) call Projection_H0 (cm_ijk octant) angular projection call Projection_D0 (cm_ijk octant) spatial projection call SweepOrder_fm(cm_ijk octant) For direc=1, num_direc direction loop within one octant call MapSys2CM(cm_ijk direc) call GetFmSrc_CMin( cm_ijk, octant, direc, group) for fm_ijk in the sweeping order !fine mesh loop call DiffScheme end fine mesh loop call MapCM2Sys(cm_ijk direct) end direction loop !! Pseudocode: SolverRay_L2_S1 (cm_ijk, octant, group) call Projection_H0 (cm_ijk octant) angular projection call Projection_D0 (cm_ijk octant) spatial projection For direc=1, num_direc direction loop within one octant call GetZnSrc_CMin(cm_ijk, octant, direc, group) for each parallel ray ray loop call GetBakFlx call GetRayAvg end ray loop call GetZnAvg call MapCM2Sys(cm_ijk direct) end direction loop
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64 Subroutines L3.1 Projection_H0 and L3.2 Projection_D0 complete angular and spatial projection procedures. The tw o subroutines, called within SolverSn_L2_S1 and Solver_Ray_L2_S1, remap the incoming flux array onto the same frame (in the angular domain and spatial domain) as the current coarse mesh by the projection techniques. Note that here angular projection is performed firs t if both projections are required. For the SN solver, Subroutine L3.3 SweepOrder_fm initializes the fine mesh sweep order for the following fine mesh loop. L3.4 MapSys2CM and L3.7 MapCM2Sys are similar to their counterparts, L2.1 and L2.7, on the second level. However, here we need to map between the system interface flux array and the coarse mesh interface array, instead of between the model boundaries and the system interface flux array. Subroutine L3.5 GetFmSrc_CMin calculates the total source term for each fine mesh before the fine mesh loop starts. Within the fine mesh loop, L3.6 DiffScheme is called to calculate the outgoing flux and fine-mesh-averaged flux based on the incoming flux by a differencing scheme. The diam ond-differencing and directiontheta-weighted differencing19 schemes are implemented. Other differencing sc hemes can be added into this subroutine. The characteristics subroutine set is similar to the SN set with a two-level loop structure: direction loop and parallel ray loop, instead of fine mesh loop in the SN solver. L3.8 GetZnSrc_CMin as its counterpart L3.5 for the SN solver, calculates th e total source term for each zone, instead of each fine mesh. For each parallel ray, L3.9 GetBakFlx evaluates the incoming flux by the bilinear interpolation scheme. L3.10 GetRayAvg calculates the average angular flux for the current ray. After all the parallel ray aver age fluxes are updated, L3.11 GetZnAvg is used to calculate the average flux for the zone/coarse mesh. And the coarse mesh outgoing flux is mapped back onto the system interface flux array.
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65 Data Structure and Initialization Subroutines The 4-level code flowchart, as outlined in the previous section, is built on the data structure, which organizes of the data arrays, such as angular fluxes and flux moments. In the TITAN code, a number of derived data types ar e defined by applying th e paradigm of objectoriented programming (OOP). These user-defined data objects, such as coarse mesh object, quadrature object, and pr ojection objects, are initialized in subroutine L1.1 InitSn at the beginning of transport calculation. In recent year s, OOP has already evolved into one standard paradigm for modern coding language for co mputer applications. While FORTRAN 90/95, designed mainly for scientific computing, gene rally is not considered as an object-based language. However, FORTRAN 90/95 does provide some tools and language extensions to allow users to utilize some concepts of OOP. And th e OOP support is further enhanced in the new FORTRAN 2003 standard. In the TITAN code, coarse mesh is treated as a relatively independe nt object, within which a number of parameters, arrays, and sub-objec t are defined. Among these parameters are Solver_ID Quad_ID Mat_matrix Src_matrix and angular flux and fl ux moment sub-objects. Solver_ID and Quad_ID specify the solver and quadrature se t for the coarse mesh, respectively. Mat_matrix and Src_matrix are the material and source distri butions within the coarse mesh, respectively. And the angular fl ux and moments for the coarse mesh are de fined as sub-objects for each group and octant. They are initialized in subroutine L1.1-4 InitCMflux Quadrature set is another essential object, wh ich contains the direc tion cosine values and the weights associated with the direct ions for each direction in one octant. L1.1-3 CreatQuad generates all the quadrature sets with ordinate splitting used in the model. For the levelsymmetric quadrature, direction cosines and weight s are preset for quadrature order from 2 to 20. For the PN-TN quadrature set, since the quadrature order is not limited to 20 as level-symmetric
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66 quadrature, directions cosine s and weights are pre-calculat ed by a polynomial root-finding subroutine. After one SN or PN-TN quadrature is created, another s ubroutine is called to build up the splitting ordinates on top of the regular quadrature set. As described by Eq. 2-43, the projection matr ix should be pre-calculated in both spatial and angular domain. In the spatial domain, L1.1-5 InitProjection scans all the coarse mesh interfaces and analyzes all the projections on the interfaces of coarse meshes. Since a 2-D projection is defined by two se parated 1-D projections, only a 35 projection matrix is necessary for a projection of3355 The 2-D projection matrix is built implicitly by the 1D component projection matrix. Furt hermore, 1-D projection matrix is always stored in pair, e.g. 35 and53 because they always happen together on the same coarse mesh interface depending the sweeping direction. Note that sinc e the same projection could happen in a number of interfaces, it is not necessary to build one projection matrix for every coarse mesh interface. In such case, only one projection matrix is stored to reduce the memory cost. And a projection ID is assigned to each coarse mesh interface to specif y the associated projection matrix. The angular projection matrix is built in a similar way, but wi th a subroutine to find the three closest neighbor directions in one quadrature set to every direction in the other quadrature set. Afterwards, the three neighboring direction indice s and the distance weights are st ored in an angular projection matrix. Coarse and Fine Mesh Interface Flux Handling In the sweeping process, the fine-mesh inte rface flux propagates along the sweep direction. Instead of storing all the inte rface fluxes for each fine mesh, we only store the fluxes on the propagation frontline. As shown in Figure 4-2, for a 2-D coarse mesh with 4 by 4 fine meshes, two one dimensional interface arrays, Inter_x(:) and Inter_y(:), can be allocated to store the frontline interface flux, bot h with a size of 4.
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67 Figure 4-5. Frontline interface flux handling. At the beginning of the direction n sweep process, Inter_x and Inter_y are assigned to the incoming fluxes at the bottom and left boundary, respectively. This task is completed by subroutine L3.3 MapSys2CM The sweep process starts from FM (1,1) by using Inter_y(1) and Inter_x(1) as incoming fluxes. After the average flux for FM(1,1) is updated, we assign the outgoing flux for FM(1,1) back into Inter_y(1) and Inter_x(1) And the rest of elements of Inter_x and Inter_y remain the same. Therefore, for FM(1,2) Inter_x(1) and Inter_y(2) become the incoming fluxes. Generally speaking, for FM(m,n) Inter_x(m) and Inter_y(n) always store the incoming fluxes before the sweep begins, an d the outgoing fluxes afterwards. For example, after the sweep process updates th e fluxes for the first 6 fine mesh es, the blue line becomes the propagation frontline. At this point, Inter_x stores the interface fl uxes on the horizontal lines along the blue front line, while Inter_y stores all the interface flux on the vertical lines. After all the fine meshes are processed, Inter_x and Inter_y store the outgoing fluxes for the coarse mesh at the top and right boun daries, respectively. Inter_y(:) 1 2 4 3 1 2 3 4 n Inter_x(:)
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68 The front-line approach to handle the fine-m esh interface fluxes can be extended to the sweep process in a 3-D coarse mesh. We use th ree 2-dimentional arrays to store the interface fluxes: Inter_xy(:,:) Inter_xz(:,:) and Inter_yz(:,:) instead of Inter_x(:) and Inter_y(:) in a 2-D coarse mesh. The front-line shown in Figure 4-2 becomes front-sur face in 3-D along x, y and z axes. The front-line approach is memory-efficient co mpared to the straightforward process to store the interface fluxes for all the fine meshes Under this approach, only the interface fluxes on the marching front-line are stored. For the cas e shown in Figure 4-2, the frontline approach only requires 8 memory units, while 40 memory un its are necessary otherwise. For a 3-D coarse mesh with ijk fine meshes, a total of (1)(1)(1) ijkijkijk memory units are required if all the interface fluxes are stored. While the front-line approach only requires ijikjk memory units. Another benefit of the frontline approach is to avoid memory jumps for the fine mesh incoming fl uxes during the sweep process. As shown in Figure 4-2, the interface flux arrays, Inter_x(:) and Inter_y(:) are always accessed sequentially as the frontline marches forward, which is much more efficient than memory jumps, especially when handling large size arrays. The same approach can be applied on the coar se mesh sweep process, in which a coarse mesh is considered as the finest unit. Howeve r, each element of the in terface flux array becomes another array, or an object, instead of a scalar va lue as in the fine mesh sweep process. Here we use another set of object arrays called system interface arrays Inter_xy_cm(:,:), Inter_xz_cm(:,:), and Inter_yz_cm(:,:) which are similar to Inter_xy(:,:), Inter_xz(:,:), and Inter_yz(:,:). They can be considered as an array of arrays, or an ar ray of objects on the system level, which means each element in Inter_xy_cm(:,:) is another array, instead of a s calar value as in a regular array.
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69 Inter_xy_cm(:,:) represents the front-line coarse mesh fluxes on the xy plane in the global sweep process, as Inter_xy(:,:) represents the front-line fine mesh fluxes in a coarse mesh sweep process. The system interface arrays are initialized by Subroutine L1.1-2 InitInter and connected to coarse mesh interface flux arrays by subroutines L3.3 MapSys2CM and L3.7 MapCM2Sys which performs two mapping actions: Mapping one system array element to the corr esponding coarse mesh interface array as the coarse mesh incoming flux before the fine mesh sweep process starts. Mapping the coarse mesh interface array back on to the system array element afterwards as the outgoing flux.
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70 CHAPTER 5 BENCHMARKING We carefully chose a number of benchm ark problems to test the performance of the TITAN code: A uniform medium and fixed source problem, to test the SN solver. A simplified CT model, to test the hybrid appr oach with the ordinates splitting technique. The Kobayashi benchmark, to test both the SN and hybrid formulations. The C5G7 MOX benchmark, to test eigenvalue problems. These benchmark problems are used to examine different aspects of the code. In this chapter, we present the results of the TITAN code on these benchmark problems, and provide some analysis on the results. Benchmark 1 A Uniform Medium and Source Problem This benchmark is a test problem designed to examine the accuracy of the SN solver of the hybrid algorithm. A 15x15x15 cm3 water cube is divided into 3x3x3 coarse meshes of size of 5x5x5 cm. Each coarse mesh is divided by 5x5x5 fine meshes. The entire model, as shown in Figure 5-1, is composed of 15x15x15 fine meshes in 27 coarse meshes. The fine mesh size is 1x1x1 cm3. The vacuum boundary condition is applied on all the six surfaces of the water box. The cross section data is extracted from the SAILOR-96 library by the GIP code.30 We only use the first 3 neutron group cross se ction data from the SAILOR96 47-group structure. Both P0 and P3 cross section data are tested A fixed source is uniformly di stributed in the water with a uniform source spectrum.
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71 Figure 5-1. Uniform medium and source test model. We ran this model with an S6 quadrature set. As a reference, we also simulated the problem with the PENTRAN code with the same se tup (without acceleration, and with diamonddifferencing scheme only). The calculated scal ar fluxes and the relative difference with PENTRAN for the 3 groups are show n in Figures 5-2, 5-3 and 5-4. A B Figure 5-2. Group 1 calculation result. A) Fl ux. B) Relative difference with PENTRAN.
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72 B Figure 5-3. Group 2 calculation result. A) Fl ux. B) Relative difference with PENTRAN. B Figure 5-4. Group 3 calculation result. A) Fl ux. B) Relative difference with PENTRAN. As shown in Figures 5-2 to 5-4, TITAN yiel ds the same solution as PENTRAN since the relative difference (magnitude order of 10-5) is less than the flux tolerance (10-4). It is also worth noting that relative difference is symmetric, and the larger difference generally occurs around the corners and edges of the water box, where the scalar fluxes are lo wer than the center. A test on code scalability and stability is also performed on a similar problem, in which we keep the same
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73 fine mesh size, but only one co arse mesh for the whole box. TITAN provides the same solution on the derived model with the similar memory requirement and running time. As the first testing problem, this benchmark de monstrates that the basic algorithms in the SN solver are correct. The simple setup of th is model is designed to eliminate possible complicated numerical effects on the SN solver. For example, no spatial or angular projections are required in this model, since no mismatch exis ts between coarse meshes in either spatial or angular domain. As a result, the convergence spee d for this model is relatively fast (within seconds), with only 5 or 6 within-group loops required for all the th ree groups. Benchmark 2 A Simplified CT Model A simplified computational tomography (CT) device model is built to test the hybrid methodology and algorithm. A general CT device is shown in Figure 5-5. Figure 5-5. Computational to mography (CT) scan device. In a general CT device, the directional gamma rays emitted from the X-ray tube (source) enter the human body (target) on the center. Some of the gamma particles could be scattered or absorbed in the target. The uncollided gamma particles, carrying some information about the
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74 attenuation coefficients on different parts of the target, can be r ecorded by the detector array on the other side to form a projection image. Projections from different angles, acquired by rotating the source and detector array, can be used to reconstruct the target cross section image. In our simplified CT model, we only consider a center slice of a CT device without the target. A 2-D meshing plot of the simplified CT model is shown in Figure 5-6. Figure 5-6. A simplified CT model. In the simplified CT model, the photon source and an array of detectors are located on the left and right side of a slice of the whole CT device, respectively, a nd the target object is removed from the center. Our goal is to calculate the scalar fluxe s of the 20 fine meshes along y direction in the detector regi on (i.e., red region at the right hand side of Figure 5-6). The relatively large air region between the source and detector usually caus es serious ray-effects when the SN method is used. In order to overcome the ray-effects, The SN algorithm requires finer discretization grids in both spatial and a ngular domains. Alternativ ely, a process called smearing can be used to resolve the discretiza tion grid mismatch in spatial and angular domain by carefully choosing the mesh size along the discrete ordinates. In this test, we use the ordinate splitting technique as a ray-effect remedy. And the TITAN soluti ons with different solvers are compared with the MCNP5 reference calculation.31 Source Detectors Ai r
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75 Monte Carlo Model Description Figure 5-7 shows the geometry for the Monte Ca rlo MCNP5 model, which is built exactly as the deterministic model shown in Figure 5-6. Figure 5-7. MCNP model of the simplified CT device. We use MCNP5 code in multigroup mode,32 so that we can apply the same cross section data as used in the deterministic calculations. A mesh tally is used to evaluate the 20 fine-mesh fluxes in the detector region. Deterministic Model Description Figure 5-8 shows the SN solver model, which is composed of 7 coarse meshes with 14,000 fine meshes. Figure 5-8. SN solver meshing scheme for the CT model.
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76 Here, we use five coarse meshes in the air region to resolve the ray effect. The average fluxes for the 20 detector fine meshes are extracted after the calculation. Figure 5-9 shows the hybrid solv er model with 3 coarse mesh es and 3,000 fine meshes. Figure 5-9. Hybrid model meshing for the CT model. In the hybrid model, we apply characteristics solver in the air region (coarse mesh #2), and the SN solver in both the source and detector regions (coarse mesh es #1&3). The number of fine meshes in the hybrid model is much less than the one in the SN model. Comparison and Analysis of Results A number of cases are tested for the simplified CT problem. In the first set of cases (Cases 2 and 3), we apply the SN Solver only to solve the problem, a nd try to alleviate the ray effect by increasing the SN order. Due to the relatively large dist ance between the source and the detectors, and the relatively small size of the detector fine mesh, very high order of quadrature set is required to eliminate the ray-eff ects if no other ray effect remedy techniques are applied. This approach to reduce the ray-effect is not efficient, because th e memory requirement is roughly proportional to square of the SN order. Figure 5-10 shows the results for an S100 case and an S200 case compared with the MCNP reference case.
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77 0.000E+00 5.000E-04 1.000E-03 1.500E-03 2.000E-03 2.500E-03 0 5 10 15 20 25 Case 1: MCNP ref Case 2: Sn Pn-Tn S100 Case 3: Sn Pn-Tn S200 Figure 5-10. SN simulation results without ordinate splitting. The ray-effect is obvious in Case 2 with S100. Note that in most real problems, SN order usually can not reach as high as 100 due to the memory limitation. However, since this simplified model is relatively small with about 14,000 fine mesh es, and one group cross section structure, we are able to apply an S200 PN-TN quadrature set (shown in Figure 5-11A) for Case 2, in which the ray-effects are significantly reduced. A B Figure 5-11. Quadrature sets us ed in the CT benchmark. A) PN-TN S200. B) Biased PN-TN S20. In the second set of test cases (Cases 4 and 5) the ordinate splitting technique is applied as a remedy for elimination of ray-effects. In this model, obviously particles streaming along the directions close to x axis will contribute the most for the detector fluxes. Therefore, we use a PN-
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78 TN S20 quadrature set with the local PN-TN splitting techniqu e on two directions close to the x axis. both with a splitting order of 11 as shown in Figure 5-11B. The hybrid approach is tested in Case 5. Figure 5-12 shows the results for the SN solver case and the hybrid case, both compared with the MCNP reference case. 0.000E+00 5.000E-04 1.000E-03 1.500E-03 2.000E-03 2.500E-03 0510152025 Case 1: MCNP ref Case 4: Sn Pn-Tn S20 t11.2 t11.2 Case 5: hybrid Pn-Tn S20 t11.2 t11.2 Figure 5-12. Hybrid and SN simulation results with ordinate splitting. Both cases show a good agreement with the MCNP reference case without ray-effects. It is worth noting that in the hybrid model, as discussed in the last section, the number of fine meshes is reduced by a factor of ~5 comparing to the SN model. The run times and error norms as compared to the MCNP reference ca se are presented in Table 5-1. Table 5-1. CT model run time and error norm comparison with the MCNP reference case. Case number Descriptions Run Time (sec) Run Time Comparison Err 2-norm(1) Err inf-norm(2) 1 MCNP ref, nps=2e8, rel.err. <0.01 3510 1.0 0.000E+00 0.00% 2 SN PN-TN S100 (10,200)* 441.3 7.9 2.182E-02 5.86% 3 SN PN-TN S200 (40,400)* 1755.8 2.0 2.655E-03 2.41% 4 SN PN-TN S20 t11.2 t11.2 (207)* 71.4 sec 49.1 2.820E-03 2.09% 5 Hybrid PN-TN S20 t11.2 t11.2 (207)* 14.1 sec 248.9 7.510E-03 3.28% 1 Error 2-norm measures the overall error for the 20 points 2 Error inf-norm represents the maxim local relative error Total number of directions
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79 For the MCNP reference case, we use 200 million particles to yield a relative flux error of less than 1% for all 20 meshes. Here, we use the infinity-norm and 2-norm to measure the maxim local relative error and the overall error for the 20 points respectively. All the deterministic cases show a good agreement with the Monte Carlo refe rence case and with less computation time. The hybrid approach (Case 5) is about 5 times faster than the SN solver only case (Case 4), since in the hybrid model, we use about 5 ti me less fine meshes than in the SN model. This benchmark demonstrates that for problems with a larg e region of low scattering medium, the hybrid approach can achieve the same level of accuracy as the SN method with much fewer fine meshes and thereby significantly lo wer computation cost. Benchmark 3 Kobayashi 3-D Problems with Void Ducts This benchmark consists of three problems with simple geometries and void regions.33 Furthermore, each problem includes two cases: zero -scattering and 50% scattering. We tested all the three problems with the zero-scattering case And each problem model is composed of three regions: Region 1: Source (no scattering). Region 2: Void. Region 3: Pure absorber. We present the calculation results of our code and the comparison with the analytical solution provided by the benchmark. Note we us e uniform meshing for all the three problems: each coarse mesh with a size of 10x10x10 cm3, and each fine mesh with a size of 1x1x1 cm3. And the point-wise fluxes in th e benchmark are compared with the averaged fluxes calculated over corresponding coarse mesh.
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80 Problem 1: Shield with Square Void As shown in Figure 5-13, this box-in-box problem is composed of three cubes: 10x10x10 cm3 source box in the corner, 50x50x50 cm3 air box, and 100x100x100 cm3 pure absorber box. Figure 5-13. Kobayashi Pr oblem 1 box-in-box layout. We consider three cases: Case 1: MOC solver applied in Regi on 2 (void), Regions 1 and 3 with SN solver. Case 2: MOC solver in Region 2& 3 (void and pure absorber). SN solver in Region 1. Case 3: SN solver in all three regions. Tables 5-2 to 5-4 compare the re sults of Case 1 with different quadrature sets for the three point sets. We also calculate the ratios to analytical solutions. Table 5-2. Kobayashi Problem 1 Poin t A set flux results for Case 1. Point 1A Analytical Case 1 (S24) Ratio Case 1 (S30) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94414E+00 0.9979 5,15,,5 1.37185E+00 1.44872E+00 1.0560 1.44446E+00 1.0529 5,25,5 5.00871E-01 5.01333E-01 1.0009 5.00703E-01 0.9997 5,35,5 2.52429E-01 2.48688E-01 0.9852 2.49114E-01 0.9869 5,45,5 1.50260E-01 1.45821E-01 0.9705 1.46590E-01 0.9756 5,55,5 5.95286E-02 6.16731E-02 1.0360 6.21947E-02 1.0448 5,65,5 1.52283E-02 1.56001E-02 1.0244 1.56733E-02 1.0292 5,75,5 4.17689E-03 4.26493E-03 1.0211 4.16728E-03 0.9977 5,85,5 1.18533E-03 1.16145E-03 0.9799 1.18505E-03 0.9998 5,95,5 3.46846E-04 3.13078E-04 0.9026 3.45040E-04 0.9948 ErrNorm (Err2Norm Err1Norm) 1.8232E-02 9.736% 6.0117E-03 5.293% 100 50 10
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81 Table 5-3. Kobayashi Problem 1 Poin t B set flux results for Case 1. Point 1B Analytical Case 1 (S24) Ratio Case 1 (S30) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94414E+00 0.9979 15,15,15 4.70754E-01 4.81175E-01 1.0221 4.79594E-01 1.0188 25,25,25 1.69968E-01 1.70050E-01 1.0005 1.70665E-01 1.0041 35,35,35 8.68334E-02 8.73159E-02 1.0056 8.67251E-02 0.9988 45,45,45 5.25132E-02 5.12734E-02 0.9764 5.29735E-02 1.0088 55,55,55 1.33378E-02 1.08504E-02 0.8135 1.04048E-02 0.7801 65,65,65 1.45867E-03 1.50095E-03 1.0290 1.29943E-03 0.8908 75,75,75 1.75364E-04 1.99741E-04 1.1390 1.78873E-04 1.0200 85,85,85 2.24607E-05 2.42707E-05 1.0806 2.55221E-05 1.1363 95,95,95 3.01032E-06 2.67405E-06 0.8883 3.53673E-06 1.1749 ErrNorm (Err2Norm Err1Norm) 9.0705E-02 18.649%1.3184E-01 21.990% Table 5-4. Kobayashi Problem 1 Poin t C set flux results for Case 1. Point 1C Analytical Case 1 (S24) Ratio Case 1 (S30) Ratio 5,55,5 5.95286E-02 6.29408E-02 1.0573 6.18784E-02 1.0395 15,55,5 5.50247E-02 6.00183E-02 1.0908 5.95864E-02 1.0829 25,55,5 4.80754E-02 5.14090E-02 1.0693 5.16984E-02 1.0754 35,55,5 3.96765E-02 4.24917E-02 1.0710 4.33243E-02 1.0919 45,55,5 3.16366E-02 3.44892E-02 1.0902 3.48761E-02 1.1024 55,55,5 2.35303E-02 2.15000E-02 0.9137 2.14425E-02 0.9113 65,55,5 5.83721E-03 6.37570E-03 1.0923 6.26243E-03 1.0728 75,55,5 1.56731E-03 1.59919E-03 1.0203 1.66064E-03 1.0595 85,55,5 4.53113E-04 4.36921E-04 0.9643 4.82881E-04 1.0657 95,55,5 1.37079E-04 1.46529E-04 1.0689 1.41297E-04 1.0308 ErrNorm (Err2Norm Err1Norm) 4.7278E-02 9.225% 4.9872E-02 10.240% In Table 5-3, point (55, 55, 55) has a relative error of 20%, which is largest error among all points, because it is located in the coarse mesh on the interface between the absorber region and the air region. The transport solv er may encounter difficulties in resolving the highly angular dependent flux on the interface. Another difficult poi nt (95, 95, 95) is located on the far corner away from the source, where the ray-effect may be severer than the regions closer to the source. The S30 case shows no significant improvement as compared to the S24 case, which may indicate that we need to apply finer meshes to take advant age of a higher order quad rature set. Tables 5-5 to 5-7 compare the results for Case 2 with an S24 quadrature set for th e three point sets.
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82 Table 5-5. Kobayashi Problem 1 Poin t A set flux results for Case 2. Point 1A Analytical Case 2 (S24) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5,15,,5 1.37185E+00 1.44872E+00 1.0560 5,25,5 5.00871E-01 5.01333E-01 1.0009 5,35,5 2.52429E-01 2.48688E-01 0.9852 5,45,5 1.50260E-01 1.45821E-01 0.9705 5,55,5 5.95286E-02 6.12631E-02 1.0291 5,65,5 1.52283E-02 1.55573E-02 1.0216 5,75,5 4.17689E-03 4.25971E-03 1.0198 5,85,5 1.18533E-03 1.16837E-03 0.9857 5,95,5 3.46846E-04 3.18352E-04 0.9178 ErrNorm (Err2Norm Err1Norm) 1.3822E-02 8.215% Table 5-6. Kobayashi Problem 1 Poin t B set flux results for Case 2. Point 1B Analytical Case 2 (S24) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 15,15,15 4.70754E-01 4.81175E-01 1.0221 25,25,25 1.69968E-01 1.70050E-01 1.0005 35,35,35 8.68334E-02 8.73159E-02 1.0056 45,45,45 5.25132E-02 5.12734E-02 0.9764 55,55,55 1.33378E-02 1.06986E-02 0.8021 65,65,65 1.45867E-03 1.45221E-03 0.9956 75,75,75 1.75364E-04 1.90111E-04 1.0841 85,85,85 2.24607E-05 2.29690E-05 1.0226 95,95,95 3.01032E-06 2.51840E-06 0.8366 ErrNorm (Err2Norm Err1Norm) 1.0662E-01 19.787% Table 5-7. Kobayashi Problem 1 Poin t C set flux results for Case 2. Point 1C Analytical Case 2 (S24) Ratio 5,55,5 5.95286E-02 6.24240E-02 1.0486 15,55,5 5.50247E-02 5.97140E-02 1.0852 25,55,5 4.80754E-02 5.11524E-02 1.0640 35,55,5 3.96765E-02 4.22937E-02 1.0660 45,55,5 3.16366E-02 3.43198E-02 1.0848 55,55,5 2.35303E-02 2.13553E-02 0.9076 65,55,5 5.83721E-03 6.35753E-03 1.0891 75,55,5 1.56731E-03 1.59983E-03 1.0207 85,55,5 4.53113E-04 4.42516E-04 0.9766 95,55,5 1.37079E-04 1.45681E-04 1.0628 ErrNorm (Err2Norm Err1Norm) 4.3423E-02 9.243% Tables 5-8 to 5-10 compare the results of Case 3 for different quadrature sets along different lines to analytical solutions.
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83 Table 5-8. Kobayashi Problem 1 Poin t A set flux results for Case 3. Point 1A Analytical Case 3 (S24) Ratio Case 3 (S34) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94319E+00 0.9978 5,15,,5 1.37185E+00 1.44622E+00 1.0542 1.44694E+00 1.0547 5,25,5 5.00871E-01 5.02432E-01 1.0031 5.03886E-01 1.0060 5,35,5 2.52429E-01 2.50261E-01 0.9914 2.51595E-01 0.9967 5,45,5 1.50260E-01 1.47601E-01 0.9823 1.48909E-01 0.9910 5,55,5 5.95286E-02 6.23020E-02 1.0466 6.32288E-02 1.0622 5,65,5 1.52283E-02 1.58269E-02 1.0393 1.60293E-02 1.0526 5,75,5 4.17689E-03 4.31608E-03 1.0333 4.29219E-03 1.0276 5,85,5 1.18533E-03 1.16330E-03 0.9814 1.20356E-03 1.0154 5,95,5 3.46846E-04 3.15751E-04 0.9103 3.54708E-04 1.0227 ErrNorm (Err2Norm Err1Norm) 1.7566E-02 8.965% 1.0191E-02 6.216% Table 5-9. Kobayashi Problem 1 Poin t B set flux results for Case 3. Point 1B Analytical Case 3 (S24) Ratio Case 3 (S34) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94319E+00 0.9978 15,15,15 4.70754E-01 4.80788E-01 1.0213 4.78621E-01 1.0167 25,25,25 1.69968E-01 1.70059E-01 1.0005 1.71342E-01 1.0081 35,35,35 8.68334E-02 8.75903E-02 1.0087 8.73758E-02 1.0062 45,45,45 5.25132E-02 5.12572E-02 0.9761 5.24423E-02 0.9986 55,55,55 1.33378E-02 1.09012E-02 0.8173 1.09080E-02 0.8178 65,65,65 1.45867E-03 1.52321E-03 1.0442 1.37740E-03 0.9443 75,75,75 1.75364E-04 2.04277E-04 1.1649 1.66625E-04 0.9502 85,85,85 2.24607E-05 2.50787E-05 1.1166 2.04334E-05 0.9097 95,95,95 3.01032E-06 2.80145E-06 0.9306 2.73143E-06 0.9074 ErrNorm (Err2Norm Err1Norm) 8.9359E-02 18.268% 7.6500E-02 18.217% Table 5-10. Kobayashi Problem 1 Poin t C set flux results for Case 3. Point 1C Analytical Case 3 (S24) Ratio Case 3 (S34) Ratio 5,55,5 5.95286E-02 6.28676E-02 1.0561 6.35577E-02 1.0677 15,55,5 5.50247E-02 5.94963E-02 1.0813 5.94504E-02 1.0804 25,55,5 4.80754E-02 5.16771E-02 1.0749 5.19020E-02 1.0796 35,55,5 3.96765E-02 4.25678E-02 1.0729 4.31326E-02 1.0871 45,55,5 3.16366E-02 3.45271E-02 1.0914 3.47598E-02 1.0987 55,55,5 2.35303E-02 2.14367E-02 0.9110 2.15454E-02 0.9156 65,55,5 5.83721E-03 6.35281E-03 1.0883 6.21321E-03 1.0644 75,55,5 1.56731E-03 1.58707E-03 1.0126 1.64677E-03 1.0507 85,55,5 4.53113E-04 4.34709E-04 0.9594 4.74924E-04 1.0481 95,55,5 1.37079E-04 1.47770E-04 1.0780 1.45363E-04 1.0604 ErrNorm (Err2Norm Err1Norm) 4.8256E-02 9.137% 4.9324E-02 9.872% The above results for Cases 2 and 3 show a si milar agreement with th e analytical solution as Case 1, with a largest rela tive error about 20% on the interf ace point. Unlike the previous CT model benchmark, the three point sets in this benchmark cover most of the difficult positions
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84 throughout the model, while in the CT model, we ar e only interested in the detector region with a high resolution. It seems that the hybrid approach is more desirabl e in problems like the previous benchmark. For this benchmark, the hybrid algorithm performs roughly as efficient as the SN method for the 1 million mesh model. However, th e computation costs are different for the three cases as listed in Table 5-11. Table 5-11. CPU time and memory requirement for SN and hybrid methods (1 million meshes and S24 model). Case # Solver CPU time (sec) Memory (Gigabyte) Reg. 1 (source) Reg. 2 (air) Reg. 3 (absorber) 1 SN MOC SN 690 2.7 2 SN MOC MOC 267 4.3 3 SN SN SN 753 2.5 The characteristics solver is faster, but re quires more memory to store the geometry information. The SN solver is slower, but has a less me mory requirement. The tradeoff between memory and CPU time is always a coding concern, which is reflected in this problem. The CPU time for Case 3 is reduced by a factor of ~2.8 however, requires about 1.7 time more memory. It seems that Case 3 is preferred if memory re quirement is affordable and/or the speed is the major concern for the user. For simplicity, in the following Problem 2 and 3 calculations, only the SN solver results are provided. Problem 2: Shield with Void Duct Figure 5-14 shows the first z le vel of the problem layout. Th e blue region is the source region, the green region is the voi d duct, and the rest of the mode l is filled with a pure absorber, which is Region 3.
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85 Figure 5-14. Kobayashi Problem 2 first z level model layout. The SN solver calculation results are listed in Tables 5-12 and 5-13. Table 5-12. Kobayashi Problem 2 Poin t A set flux results for Case 3. Point 2A Analytical Case 3 (S24) Ratio Case 3 (S30) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94414E+00 0.9979 5,15,,5 1.37185E+00 1.44797E+00 1.0555 1.44793E+00 1.0555 5,25,5 5.00871E-01 5.03502E-01 1.0053 5.04210E-01 1.0067 5,35,5 2.52429E-01 2.51243E-01 0.9953 2.51994E-01 0.9983 5,45,5 1.50260E-01 1.48549E-01 0.9886 1.49214E-01 0.9930 5,55,5 9.91726E-02 9.71016E-02 0.9791 9.80078E-02 0.9883 5,65,5 7.01791E-02 6.79254E-02 0.9679 6.90949E-02 0.9846 5,75,5 5.22062E-02 5.17088E-02 0.9905 5.03751E-02 0.9649 5,85,5 4.03188E-02 3.91599E-02 0.9713 3.91877E-02 0.9719 5,95,5 3.20574E-02 2.85735E-02 0.8913 3.20167E-02 0.9987 ErrNorm (Err2Norm Err1Norm) 2.0340E-02 10.868% 5.4047E-03 5.546% Table 5-13. Kobayashi Problem 2 Poin t B set flux results for Case 3. Point 2B Analytical Case 3 (S24) Ratio case SN (S30) Ratio 5,95,5 3.20574E-02 2.85735E-02 0.8913 3.20167E-02 0.9987 15,95,5 1.70541E-03 8.85805E-04*0.5194 1.49781E-03*0.8783 25,95,5 1.40557E-04 1.79639E-04 1.2781 1.53422E-04 1.0915 35,95,5 3.27058E-05 3.17893E-05 0.9720 3.39511E-05 1.0381 45,95,5 1.08505E-05 9.20428E-06 0.8483 1.12324E-05 1.0352 55,95,5 4.14132E-06 4.72351E-06 1.1406 4.32799E-06 1.0451 ErrNorm (Err2Norm Err1Norm) 9.6633E-01 48.059% 3.0605E-02 12.173% Results are calculated by averaging the corres ponding fine mesh(s), instead of coarse mesh. Our calculation shows a good agreement with the analytical solution on most of points, except point (15 95 5), which is located on the far side interface between Regions 2 and 3.
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86 Problem 3: Shield with Dogleg Void Duct Figure 5-15 shows the layout of the void duct in the model. The rest of the model is filled with pure absorber. The SN calculation results are liste d in Tables 5-14 to 5-16. Figure 5-15. Kobayashi Pr oblem 3 void duct layout. Table 5-14. Kobayashi Problem 3 Poin t A set flux results for Case 3. Point 3A Analytical Case 3 (S24) Ratio Case 3 (S30) Ratio 5,5,5 5.95659E+00 5.94515E+00 0.9981 5.94414E+00 0.9998 5,15,,5 1.37185E+00 1.44797E+00 1.0555 1.44793E+00 1.0000 5,25,5 5.00871E-01 5.03502E-01 1.0053 5.04210E-01 1.0014 5,35,5 2.52429E-01 2.51243E-01 0.9953 2.51994E-01 1.0030 5,45,5 1.50260E-01 1.48549E-01 0.9886 1.49214E-01 1.0045 5,55,5 9.91726E-02 9.71016E-02 0.9791 9.80078E-02 1.0093 5,65,5 4.22623E-02 4.37756E-02 1.0358 4.46212E-02 1.0193 5,75,5 1.14703E-02 1.20425E-02 1.0499 1.17776E-02 0.9780 5,85,5 3.24662E-03 3.34282E-03 1.0296 3.32867E-03 0.9958 5,95,5 9.48324E-04 8.95157E-04 0.9439 9.94322E-04 1.1108 ErrNorm (Err2Norm Err1Norm) 1.1213E-02 5.606% 9.2256E-03 5.582% Table 5-15. Kobayashi Problem 3 Poin t B set flux results for Case 3. Point 3B Analytical Case 3 (S24) Ratio Case 3 (s30) Ratio 5,55,5 9.91726E-02 9.71016E-02 0.9791 9.80078E-02 0.9883 15,55,5 2.45041E-02 2.66812E-02*1.0888 2.61306E-02* 1.0664 25,55,5 4.54447E-03 4.84126E-03 1.0653 4.91017E-03 1.0805 35,55,5 1.42960E-03 1.46750E-03 1.0265 1.48483E-03 1.0386 45,55,5 2.64846E-04 3.00417E-04* 1.1343 2.88298E-04* 1.0885 55,55,5 9.14210E-05 9.58897E-05 1.0489 9.55481E-05 1.0451 ErrNorm (Err2Norm Err1Norm) 2.7730E-02 13.431% 1.9429E-02 8.855% Results are calculated by averaging the corres ponding fine mesh(s), instead of coarse mesh.
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87 Table 5-16. Kobayashi Problem 3 Poin t C set flux results for Case 3. Point 3C Analytical Case 3 (S24) Ratio Case 3 (S30) ratio 5,95,35 3.27058E-05 3.46102E-05 1.0582 3.16989E-05 0.9692 15,95,35 2.68415E-05 3.04241E-05 1.1335 2.88384E-05 1.0744 25,95,35 1.70019E-05 1.61464E-05 0.9497 1.86621E-05 1.0976 35,95,35 3.37981E-05 2.62570E-05 0.7769 2.38136E-05 0.7046 45,95,35 6.04893E-06 5.30795E-06*0.8775 4.85885E-06*0.8033 55,95,35 3.36460E-06 3.43148E-06 1.0199 4.00289E-06 1.1897 ErrNorm (Err2Norm Err1Norm) 1.2205E-01 22.312% 2.7493E-01 29.542% Results are calculated by averaging the corres ponding fine mesh(s), instead of coarse mesh. Problem 3 seems to be the most difficult one among the three Kobaya shi problems, since particles tend to streaming along the dogleg void duc t. As expected, the wo rst point (45, 95, 35) is located on the interface of the far end of the duct. Analysis of Results For the three problems, our calculation results show a relatively good agreement with the analytical solutions for most of the points. The characteristics solv er also provides similar results as the SN solver for problem 1. Figures 5.16-18 show the normalized flux calculation results for the SN solver for the three problems (PN-TN S24 for Problem 1, PN-TN S30 for Problems 2&3). A 0.0 0.5 1.0 1.5 024681012 B 0.0 0.5 1.0 1.5 024681012 C 0.0 0.5 1.0 1.5 024681 01 2 Figure 5-16. Relative fluxes for Kobayashi Problem 1. A) Point set A. B) Point set B. C) Point set C
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88 A 0.0 0.5 1.0 1.5 024681012 B 0.0 0.5 1.0 1.5 02468 Figure 5-17. Relative fluxes for Kobayashi Problem 2. A) Point set A. B) Point set B. A 0.0 0.5 1.0 1.5 024681012 B 0.00 0.50 1.00 1.50 02468 C 0.0 0.5 1.0 1.5 01234567 Figure 5-18. Relative fluxes for Kobayashi Problem 3. A) Point set A. B) Point set B. C) Point set C Figures 5-16 to 5-18 show that points with relatively large errors typically occur on the interface between the void region and the pure absorber region due to the highly directional particle streaming on the interf ace. Since no scattering exists in the model and the source is
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89 located in the corner, ray-effects could be very seve re in this 3D model. Therefore, it is difficult for an SN code without any ray-effect remedies to calculate all the point sets with only one calculation.34 Benchmark 4 3-D C5G7 MOX Fuel Assembly Benchmark We tested the k-effective calculation ability of the TIT AN code on the extended 3-D C5G7 MOX benchmark.35, 36 TITAN categorizes transport proble ms into four types: fixed source problems with only down-scattering, fixed sour ce with up-scattering, criticality with downscattering, and criticality with up-scattering. Deta ils on the four kernels are discussed in Chapter 4. This benchmark falls in the fourth category with the reflective bounda ry condition, which is numerically the most difficult type. The size of th e model also presents a challenge for a serial non-lattice transport code as TITAN. Model Description The C5G7 MOX reactor is a proposed design for this benchmark, which has 2 by 2 assemblies (2 MOX assemblies and 2 UO2 assemblies). Each assembly is composed of 17x17 fuel pins. And the four fuel assemblies are surrounded by moderator as shown in Figure 5-19. A B Figure 5-19. C5G7 MOX reactor layout. A) x-y plane. B) Unrodded configuration.
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90 Axially the fuel region of the reactor can be divided equally into three segments as shown in Figure 5-18B. And control rods can be inserted into different depth of the core. Three control rod configurations are used in the extended version of this benchmark:36 Unrodded. Rodded A. Rodded B. In the unrodded case, the cont rol rods only reach the modera tor region on the top of the core (grey area in Figure 5-18B). In the other two cases, cont rol rods in the MOX and UO2 assemblies reach different positions in the core. Several models with different disctetization grids are tested. Only the SN solver is used in the calculations. The finest grid model we used has about 3 million meshes (12 z levels) with a S10 quadrature set. This model requires ~1.8Gig memory. Based on the calculation results, the keffective is relatively insensitive to the grid size, although the pin-pow er distribution does improve slightly with finer discretization grid Figure 5-20 shows the meshing scheme for the 2x2 fuel assemblies and an individual fuel pin. A B Figure 5-20. 3-D C5G7 MOX model. A) Four fuel assemblies. B) Fuel pin.
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91 We use 14x14 fine meshes to represent each fuel pin in this four z-level model, which leads to a fine mesh size: 0.09x0.09x14.28 cm3. The mesh size along z axis is much larger than xy size, because finer meshing is required to represent the round shape of the fuel pin in the Cartesian geometry. A minimum four z-levels are required to repr esent the different control rod configurations. It is necessary to add more z-levels to resolv e the axial flux shape because of different control rod configurations However, the tests indicate that k-effective is more sensitive to the x-y size than the z mesh size. Here we only reported the four z-level S6 model calculation results due to our computation resource limitation The model has about one million fine meshes. Note that the multigroup cross section data and the reference solutions (acquired by Monte Carlo calculations) are provided with the benchmark. Pin Power Calculation Results The Monte Carlo reference soluti on provides the pin pow er distribution for the three slices in the reactor core region. In the TITAN model, each fuel pin is composed of 14x14 fine meshes. Since the power is proportional to the fission rate a special subroutine is developed to evaluate the pin power by summating the fission rates for a ll the 14x14 fine meshes and for all the seven energy groups. Then, the output can be imported to the EXCEL template provided with the benchmark specification. The differences between user calculation results and the reference are automatically evaluated by the template. The pin power results calculated by TITAN for the unrodded case are compared with the reference so lution in Table 5-17.
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92 Table 5-17. Pin power calcula tion results for the unrodded case Z Slice #1 Z Slice #1 Z Slice #2 Z Slice #2 Z Slice #3 Z Slice #3 Overall Overall Specific Pin Power Data Ref. User Ref. User Ref. User Ref. User Maximum Pin Power 1.108 1.148 0.882 0.884 0.491 0.449 2.481 2.481 Percent Error (associated 68% MC) 0.090 3.563 0.100 0.244 0.130 -8.449 0.060 0.007 Distribution Percent Error Results Maximum Error (associated 68% MC) 0.220 4.673 0.320 1. 803 0.130 8.449 0.192 1.395 AVG Error 0.164 3.340 0.183 0. 421 0.245 7.069 0.109 0.268 RMS Error 0.171 3.381 0.190 0. 536 0.255 7.096 0.114 0.354 MRE Error 0.062 1.496 0.055 0. 140 0.042 1.445 0.093 0.200 Number of Accurate Fuel Pin Powers Number of Fuel Pins Within 68% MC 371 0 371 146 371 0 371 147 Number of Fuel Pins Within 95% MC 518 0 518 278 518 0 518 257 Number of Fuel Pins Within 99% MC 540 0 540 334 540 0 540 336 Number of Fuel Pins Within 99.9% MC 544 0 544 387 544 0 544 398 Total Number of Fuel Pins 545 545 545 545 545 545 545 545 Average Pin Power In Each Assembly UO2-1 Power 219.04 226.70 174.24 173.79 97.93 90.54 491.21 491.03 MOX Power 94.53 97.38 75.25 75.10 42.92 39.95 212.70 212.44 UO2-2 Power 62.12 64.55 49.45 49.65 27.82 25.89 139.39 140.09 UO2-1 Power Percent Error 0.082 3.498 0. 073 -0.258 0.055 -7.554 0.123 -0.038 MOX Power Percent Error 0.061 3.017 0. 054 -0.193 0.041 -6.911 0.092 -0.122 UO2-2 Power Percent Error 0.043 3.920 0.038 0.404 0.029 -6.936 0.065 0.506 The format of Table 5-17 is provided by th e benchmark template. In the unrodded case, control rods are inserted to the moderator re gion on the top of the reactor core. The TITAN results show a relatively good agreement with th e reference solution for the overall pin power distribution (power summa tion of the three axial segments). However, large differences exist if we compare different segments, especially Slices #1 and #3. The error could be attributed to the large mesh size along the z axis and the lower order of the qua drature set. Similar error pattern also occurs in the rodded A and B cases as provided in Tables 5-18 and 5-19.
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93 Table 5-18. Pin power calculati on results for th e rodded A case. Z Slice #1 Z Slice #1 Z Slice #2 Z Slice #2 Z Slice #3 Z Slice #3 Overall Overall Specific Pin Power Data Ref. User Ref. Uer Ref. User Ref. User Maximum Pin Power 1.197 1.211 0.832 0.826 0.304 0.321 2.253 2.274 Percent Error (associated 68% MC) 0.080 1.145 0.100 -0. 696 0.200 5.518 0.059 0.919 Distribution Percent Error Results Maximum Error (associated 68% MC) 0.100 1.625 0.250 1. 877 0.330 7.044 0.149 1.701 AVG Error 0.157 0.691 0.180 0. 760 0.260 3.922 0.108 0.714 RMS Error 0.163 0.819 0.186 0. 860 0.266 4.251 0.111 0.803 MRE Error 0.066 0.388 0.056 0. 297 0.037 0.582 0.094 0.690 Number of Accurate Fuel Pin Powers Number of Fuel Pins Within 68% MC 371 87 371 60 371 11 371 14 Number of Fuel Pins Within 95% MC 518 163 518 113 518 13 518 43 Number of Fuel Pins Within 99% MC 540 200 540 160 540 18 540 70 Number of Fuel Pins Within 99.9% MC 544 237 544 216 544 25 544 104 Total Number of Fuel Pins 545 545 545 545 545 545 545 545 Average Pin Power In Each Assembly UO2-1 Power 237.41 240.06 167.51 165.79 56.26 58.89 461.18 464.74 MOX Power 104.48 104.67 78.01 77.42 39.23 38.27 221.71 220.36 UO2-2 Power 69.80 70.51 53.39 53.18 28.21 26.85 151.39 150.54 UO2-1 Power Percent Error 0.087 1.118 0.071 -1.029 0.040 4.674 0.119 0.772 MOX Power Percent Error 0.065 0.182 0. 056 -0.747 0.040 -2.447 0.094 -0.610 UO2-2 Power Percent Error 0.047 1.012 0. 040 -0.382 0.029 -4.817 0.068 -0.565 In the rodded A case, control rods are inserted to the Slice 3 in one assembly. Slice 3 is the top slice in the reactor core region, which ha s the least power contribution among the 3 slices. Slice 3 has the largest percentage error. The maximum error associated 68% Monte Carlo reference is about 7% for Slice 3, while it is about 2% for the other two slices. The overall assembly power errors are less than 1% for both UO2 and MOX assembly.
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94 Table 5-19. Pin power calculati on results for th e rodded B case. Z Slice #1 Z Slice #1 Z Slice #2 Z Slice #2 Z Slice #3 Z Slice #3 Overall Overall Specific Pin Power Data Ref. User Ref. Uer Ref. User Ref. User Maximum Pin Power 1.200 1.167 0.554 0.585 0.217 0.205 1.835 1.818 Percent Error (associated 68% MC) 0.090 -2.779 0.150 5.603 0.240 -5.657 0.083 -0.890 Distribution Percent Error Results Maximum Error (associated 68% MC) 0.090 3.438 0.140 6. 689 0.220 15.558 0.071 1.715 AVG Error 0.146 1.313 0.181 2. 713 0.285 4.407 0.105 0.710 RMS Error 0.150 1.557 0.184 3. 231 0.290 5.714 0.108 0.823 MRE Error 0.073 0.916 0.055 0. 971 0.034 0.577 0.098 0.727 Number of Accurate Fuel Pin Powers Number of Fuel Pins Within 68% MC 371 36 371 4 371 31 371 37 Number of Fuel Pins Within 95% MC 518 75 518 6 518 51 518 78 Number of Fuel Pins Within 99% MC 540 93 540 20 540 70 540 109 Number of Fuel Pins Within 99.9% MC 544 121 544 35 544 88 544 137 Total Number of Fuel Pins 545 545 545 545 545 545 545 545 Average Pin Power In Each Assembly UO2-1 Power 247.75 241.84 106.56 112.38 41.12 37.45 395.43 391.67 MOX Power 125.78 124.15 81.41 82.93 29.42 30.42 236.62 237.50 UO2-2 Power 91.64 91.97 65.02 66.40 30.68 30.95 187.34 189.33 UO2-1 Power Percent Error 0.091 -2.385 0.056 5.460 0.035 -8.924 0.112 -0.951 MOX Power Percent Error 0.073 -1.300 0.058 1.875 0.034 3.379 0.100 0.374 UO2-2 Power Percent Error 0.055 0.364 0.046 2.124 0.032 0.899 0.078 1.062 The axial flux profile becomes more and more difficult to resolve from the unrodded case to the rodded B case, as the cont rol rods insert deeper in th e reactor core with different configurations. As a result, one can observe the overall pin power accuracy worsens slightly from Table 5-17 to 5-19. This is expected, co nsidering we only use the minimum 4 z-levels model with the diamond differencing scheme, and a relatively lower quadrature order. Eigenvalue Comparison The eigenvalues for the three cases are listed in Table 5-20. The tolerance used in TITAN input file for keff is 1.0E-05. Similar to the pin power error, the keff error increases as the control rods insert further into the core.
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95 Table 5-20. Eigenvalues for three cas es of C5G7 MOX benchmark problems. Case Ref. % Error (68% MC)User Difference (pcm). Unrodded 1.143080 0.0026 1.13911169 Rodded A 1.128060 0.0027 1.12600206 Rodded B 1.077770 0.0028 1.07415362 Analysis of Results Since the SN solver in the TITAN code is designed only for Cartesian geometry. We had to use an unusual meshing scheme in this benchmark: the z mesh size is about 158 times larger than the x or y mesh size. Such imbalanced meshing could be valid only for problems in which the axial flux changes very slowly comparing with radical flux profile. Our computer hardware limitation is another reason why we use a re duced meshing scheme (about 1 million fine meshes). TITAN is still serial code. And we need to fit the whole problem onto one machine. It takes about 10 hours to run the 4 z-level S6 model on an AMD Opteron 242 CPU (1.6MHz, 1024k cache) with about 323M memory requirement. The calculation result is reasonable considering the meshing scheme we used. Specially designed lattice transport codes for re actor neutronics could be more efficient for this benchmark. However, TITAN has the potential to increase the efficiency in eigenvalue problems with some power iteration acceleration techniques implemented. Figure 5-21 shows the eigenvalue convergence pattern for the rodded A case. Keff 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 01 02 03 04 05 06 07 0 Iteration # Figure 5-21. Eigenvalue convergence pa ttern for the rodded A configuration.
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96 The rodded A case takes more than 2,000 inner it erations and about 60 outer iterations. As shown in Figure 5-20, the k-effective converged relatively fast for the first 20 iterations without any power acceleration technique applied. The conve rgence rate is much slower for the rest of the iterations, although this pattern is generally expected. Th e output indicates that some iterations are wasted to converge fluxes with th e un-converged fission sour ce, especially with upscattering present. We took some intuitive measures in the code to improve the pattern, including using adaptive flux convergence cr iterion, adaptive inner loop a nd outer loop iteration number limitations, and Aitken extrapolation method.37 I also combined the upscattering loop and the power loop into one loop at the beginning, and separated them toward the end. These measures are optional in TITAN (Appendix D). And they can improve the converg ence rate in certain situations.
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97 CHAPTER 6 FICTITIOUS QUADRATURE We introduce a special kind of problem s that th e TITAN code can be ap plied: the particle transport problem within a digital medical pha ntom. To solve a regular transport problem, modeling of the problem is required as one of the initial tasks. And a meshing scheme need to be carefully chosen based on the phys ics of the problem. While in a digital phantom, the source and material distributions are stored in the format of voxe l values as activity (source) and material attenuation coefficients. Therefore, it is a natura l choice to consider one voxel as one fine mesh in the initial modeling task. In the TITAN code, a module is developed to process the digital phantom binary files and automatically genera te the meshing scheme. Furthermore, since transport calculations for medical phantoms often involve the simu lations of radiation projection images, we developed the fictitious quadrature technique to cal culate the angular fluxes for specific directions of interest that may not be available in a regular quadrature set. The performance of the technique is tested in a digital heart ph antom benchmark. Extra Sweep with Fictitious Quadrature In the TITAN code, multiple quadrature sets can be used in one problem model. A regular quadrature is built based on the criteria of conservation of flux moments. Fictitious quadrature is designed differently from the regular type of quadrature in that it s purpose is to calculate only the angular fluxes for certain directions, not to cons erve the flux moments. Th erefore, it can not be used in a regular sweep process since the sca ttering source and flux mo ments cannot be properly handled. However, it can be used after the so urce iteration process is complete with the converged flux moments. Generally, in a transport problem, users major concern is the s calar flux distribution and/or k-eff However, in some cases, the angular fluxes in the directions of interest need to be
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98 evaluated. Since the directions are not necessar ily included in the problem quadrature sets, the angular fluxes in these directions usually cannot directly be calc ulated by the sweep process with a regular quadrature set. In the TITAN code, we can define the directions of interest in a fictitious quadrature set, which is used with an extra sweep process only after the source iteration process is converged with the regular quadrature set(s). The converged flux moments are used to evaluate the scattering source in the extra sweep with the ficti tious quadrature. )sin( )cos( )( )!( )!( 2)({)12()( )( ,', )( )( ,', 1 '1 )( 0 )( ,' )( ,', .).( fic n conk, lgS fic n conk, lgC G g l k fic n k l L l con lg fic nllggs se scatteringk k P kl kl P l S (6-1) Where, upper script (e.s) stands for extra sweep, (fic) for fictitious (con) for converged )( ,' con lg, )(, ,', conk lgC, and )(, ,', conk lgSare the converged th order regular, cosine and sine flux moments. And ) ,()()( fic n fic n specifies a direction in the fictitious quadrature set. Equation 6-1 is similar to Eq. 2-23, except th at we use the converged flux moments after the source iteration process instead of the flux moments from last iteration. And the polar and azimuthal angles refer to a direction in the fic titious quadrature set. Th e fixed source or the fission source can be evaluated the same way as in a regular sweep process. After the total source is estimated, we can use the extra sweep process to evaluate the angular fluxes in the directions of the fictitious quadrature. One also could choose some other methods ba sed on the calculated angular fluxes in the quadrature directions to evalua te the angular fluxes of inte rest. For example, the angular projection technique in Chapter 3 can be applied with some modi fications. We have tried this approach in the TITAN code. A nother method could be to apply the Legendre expansion of the angular flux based on the converged flux mome nts. One potential probl em with these two approaches is that their efficiencies are subj ect to the accuracy of the angular fluxes in the
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99 directions of a regular quadrature set. Usually a convergence criter ion is set on the scalar flux in the source iteration scheme. The accuracy of the a ngular fluxes or higher moments is not always granted. And further mathematical manipulations on the angular fluxes or higher moments could introduce more secondary inaccuracies. One advant age of the fictitious quadrature technique over the secondary approaches is that the angular fluxes of interest are calculated directly from a sweep process. And since the sweep process can be considered as a simulation procedure to the physical particle transport phenomenon in certain directions, some physics of the model along the interested directions (e.g. fi xed source and scattering ) are taken into acc ount in the evaluation process. Thereby, the extra sweep with the fictitio us quadrature has more potential to provide an accurate estimation on the in terested angular fluxes. Implementation of Fictitious Quadrature It is straightforward to implement the fict itious quadrature technique, since all the formulations used in a regular sweep can be a pplied in the extra sweep. However, due to the special design of the fictitious quadrature, some modifications on the regular sweep are required to effectively complete an extra sweep. Extra Sweep Procedure The extra sweep starts upon the completion of the source iteration process. The fictitious quadrature is built as an initialization task before the source iteration star ts. Fictitious quadrature sets can be treated as a regular user-defined qua drature set in the initia lization process, except that any direction regardless of its octant can be defined in the quadrature input file, and these directions can be arbitrarily c hosen. Note that in a regular us er-defined quadrature set, only directions in the first octant are defined, and directions in other octants are determined by symmetry. As a result, the extra sweep is perfor med only along specific directions defined in the first octant. The extra sweep procedure can be illustrated by Figure 6-1.
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100 Figure 6-1. Extra sweep procedure with fictitious quadrature. As shown in Figure 6-1, we start the extr a sweep by reallocating the angular flux array based on the fictitious quadrature set. Since the values of angular fluxes in the regular quadrature sets will be lost after the me mory reallocation, any task whic h requires the calculated angular fluxes need to be completed before the extra sweep. At the beginning of the sweep for group g we allocate a new array for the boundary angular fluxes, which will be deallocated after the group g sweep. The original boundary fluxes calcula ted from regular sweep remain untouched during the extra sweep, because an angular proj ection from the regular quadrature to the fictitious quadrature could be employed on th e boundaries if reflective boundary condition is used. We apply the same scattering-in moment approach discussed in Chapter 5 in the extra sweep as well. Note that the scattering-in moments are calcula ted based on the converged flux moments from regular sweeps, and they are only used for evaluation of the scattering source in an extra sweep. Also note that the step to calcu late flux moments in a regular sweep is removed in the extra sweep procedure. Source Iteration Com p letion 1. Reallocation An g Flux 2. Initialize Boundary flux for group g 3. Recalculate group g in-moments Initialize fictitious quadrature set 4. Group g extra sweep 5. Output group g boundary flux Group Iteration g=1,2, G
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101 Implementation Concerns We developed a new set of subroutines to complete the extra sweep. Most of these new routines are on layer 3 or 4, including the angular projection module, the coarse mesh sweep routine, and the differencing scheme routine. Although these subroutines share the similar tasks as their counterparts in the regular sweep, some modifications are requi red due to the following concerns: Iteration structure. Direction singularity. Solver compatibility. Iteration structure The iteration architecture in a regular sweep for group g is built on the following order (from outer to inner): Octant loop, coarse mesh loop, direction loop, fine mesh loop. However the characteristics of the fictitious quadrature require that the extra sweep to follow a different order: direction loop, coarse mesh loop, fine me sh loop. This structure change affects most of routines on layer 3 and 4, since a ll the directions in the same octant are handled as a group in the regular sweep, while in an extra sweep, each di rection need to be treated individually. For example, the coarse mesh or fine mesh sweep order is assigned indivi dually for each direction instead by octant. Another modification is made to allow negative directional coordinates in the user-defined fictitious quadrature set. Direction singularity A regular quadrature set usually avoids directions along an axis or perpendicula r to an axis. Zero directional cosine or sine occurs for these directions. Th is singularity could cause some potential problems in the sweep process. For ex ample, in the differencing scheme discussed in Chapter 2, normally a small perturbation in one boundary incoming angular flux can cast some
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102 effect on all the three outgoing fl uxes, since the three component s of the incoming angular flux along x, y and z axes are all positive defin ite or all zeros. For a singular direction, however, this is not always true. For example, an incoming angular flux along the x axis only has only one positive x component. Therefore, whil e calculating the outgoing fluxes, a differencing scheme need to take measures to treat a singular incoming angular flux. Unfortunately, singular directions often happen to be the interested directions in a fictitious quadrature set. A series of modi fications have been made to keep the extra sweep subroutines singularity safe, including the differencing scheme, the fine mesh sweep procedure, and the angular projection routine. Solver compatibility The two-solver structure of the TITAN code causes another dimensional difficulty in the implementation of the fictitious quadrature set. The technique is originally designed for the SN solver only. Later the compatibility to the characteristics solver is achieved. Heart Phantom Benchmark Originally, we developed the fictitious qua drature technique to calculate the boundary angular fluxes for a single photon emission computed tomography (SPECT) benchmark. In SPECT, a small amount of photon radiation source is deposited in the target organ with some nuclear medicine intake. The s ource distribution in the organ can be reconstructed with the projection images. The 3-D source distribution image can be used to diagnose some malfunctions in the organ. Dozens of projecti on images from different angles are required to reconstruct the source distribution to achieve a good resolution. In medical phys ics, SPECT simulation usually is performed with the Monte Carl o approach. In this benchmark, our goal is to simulate the projection images of a body phantom with a deterministic transport calculation.
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103 Model Description We applied the TITAN code on a digital h eart phantom generate d by the NCAT code.38 NCAT can provide various cardiac-torso phantoms with support of the heart motion. Users can specify the amount of activity deposited in different organs. The phantom we use in this benchmark is the first frame of the heart motion cy cle. The phantom contains two binary file: the attenuation file and the activity file. The attenuation file records the linear attenuation coefficients for each voxel. And the radiation activ ity in each voxel is stored in the activity file. A B Figure 6-2. Heart phantom mode l. A) Torso. B) Organs. Figure 6-2 shows the material distribution of the phantom. The size of the phantom is 40x40x40 cm3 with 128x128x128 voxels. The voxel size is 0.3125x0.3125x0.3125 cm3. In the deterministic transport calculation, we consider the whole phantom as one coarse mesh with 128x128x128 fine meshes. The model has about 2 m illion fine meshes, and each fine mesh represents a voxel in the phant om. The body is surrounded by air. Most of the organs including the heart are considered the same material, excep t for the lungs and the bo nes. A total of five materials are used in this model as listed in Table 6-1. The material densities and linear
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104 attenuation coefficients are given in the output files of the NCAT code. The TITAN code also can process the phantom attenuati on binary file, and automatically generate a material list based on the different attenuation values in the file. Table 6-1. Materials list in the heart phantom model. Mat. Number Mat. name Density (g/cm3) Linear Attenuation Coefficients (1/pixel) or (0.3125cm/pixel) 1 Air 1.00E-06 2 Body (Muscle) 1.02 0.0469 3 Dry Spine Bone 1.22 0.0520 4 Dry Rib Bone 1.79 0.0653 5 Lung 0.30 0.0135 Figure 6-3 shows the source activity distribution. The radiatio n source is deposited only in the heart, with 75 unit activity in the myocardium (heart muscular substance) and 2 unit activity in the heart chambers (blood pool). A B Figure 6-3. Activity dist ribution in the phantom model. A) Heart. B) Heart cross section view. Photon Cross Section for the Phantom Model The CEPXS package39 is used to generate the cross section data for this benchmark. The group structure is decided based on the gamma decay energy of Tc-99m (~140kev), which is
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105 widely used in the area of cardiac nuclear me dicine. We also assume a typical 7% energy resolution for the NaI detectors used in the SPECT camera. Therefore, the width of the first group is about 10keV with a mid-range energy of 140keV. The rest of the group structure can be arbitrary chosen, since only the angu lar flux for the first group is required in this benchmark. In Table 6-2, we present a four-group st ructure with only down scattering. Table 6-2. Group structure of cross section data for the heart phantom benchmark. Group Number Upper Energy Bound (keV) Lower Energy Bound (keV) 1 145 135 2 135 100 3 100 50 4 50 1 An ideal SPECT camera takes projection images only from the uncollided photons. Therefore, only the first group angular fluxes on the boundaries are required to simulate the projections images. To deliver th e cross section data, CEPXS also requires the weight percentage for each element in a mixture and its density. For the five materials listed in Table 6-1, the body and lung materials (mat. # 2 and 5) can be c onsidered as water. And we assume the bone materials are composed of 22% water and 78% calcium. Detailed material compositions and densities are provide d in Table 6-3. Table 6-3. Material densities and compositions used in CEPXS. Mat. # Name Density (g/cm3) Composition (element : weight fraction) 1 Air 1.00E-06 N : 0.78 O: 0.22 2 Body (Muscle) 1.02 H : 0.111 O: 0.889 3 Dry Spine Bone 1.22 H: 0.024 O: 0.196 Ca: 0.78 4 Dry Rib Bone 1.79 H: 0.024 O: 0.196 Ca: 0.78 5 Lung 0.30 H : 0.111 O: 0.889 Cross section data sets with Legendre order of 0 and 3 ar e generated based on the group structure (Table 6-2) and mixtur e composition (Table 6-3). Note that deterministic calculation results for the lower groups carry some informa tion about the phantom. They might be useful to improve the quality of the reconstructed phantom image.
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106 Performance of Fictitious Quadrature Technique We demonstrate the TITAN codes performan ce on this benchmark by simulating the four projection images along the di rections normal to the four boundaries parallel to the z axis. Four directions are defined in the fictitious quadrature specification file: Table 6-4. Directions in the fictitious qua drature set for the heart phantom benchmark. Direction Number Description 1 1.0 0 0 Normal to the left boundary 2 -1.0 0 0 Normal to the right boundary 3 0 1.0 0 Normal to the back boundary 4 0 -1.0 0 Normal to the front boundary The four directions in Table 6-4 are singularity directions. The angular fluxes along the four directions on the correspond ing model boundaries are computed with an extra sweep after the source iteration process is completed. Assuming a perfect 128x128 collimator array adjacent to the body (i.e. all other photons ar e blocked except those along th e interested di rections), the angular flux distribution can be used to simulate the projection images taken by the SPECT camera. More directions can be added in the fict itious quadrature to simulate projection images from other angles. Since the phantom m odel has 128x128x128 fine meshes, all the four simulated images have 128x128 pixels. We si mulated several cases with different SN orders (S8 and S10) order and PN order (P0 and P3) with the SN solver. The output images are similar. We also performed a Monte Carlo referen ce calculation with the SIMIND code.40 SIMIND is a Monte Carlo code used in the nuclear medicine discipline to generate SPECT projection images. SIMIND uses about 8 minutes (2 min/projectio n) on a 1.5GHz Pentium 4 processor. While it takes about 4 minutes for TITAN to compute the boundary angular fluxes for the first group on an AMD Opteron 242 CPU (1.6MHz, 1024k cache), whic h is about twice faster than the Pentium CPU. Figure 6-4 compares the globally normalized images calculated by TITAN (S8 and P0) and SIMIND. And Figure 6-5 compares th e individually normalized images.
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107 A1) TITAN: front A2) TITAN: left A3) TITAN: back A4) TITAN: right B1) SIMIND: front B2) SIMIND: left B3) SIMIND: back B4) SIMIND: right Figure 6-4. Globally normalized projecti on images calculated by TITAN and SIMIND. A1) TITAN: front A2) TITAN: left A3) TITAN: back A4) TITAN: right B1) SIMIND: front B2) SIMIND: left B3) SIMIND: back B4) SIMIND: right Figure 6-5. Individually nor malized projection images cal culated by TITAN and SIMIND.
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108 In Figure 6-4, the four images (front, left, back, and right) are normalized together. It provides a valid intensity comparison between th e four images, among whic h the right projection is the weakest, since it has the longest distance to the heart. In Figure 6-5, the four images are normalized individually. It shows a clearer view on the differe nce between SIMIND and TITAN calculation results. By visual comparison, it s eems that the TITAN computed images have a higher contrast ratio. For a better understanding the amount difference between the results, Table 6-5 provides the overall differences for the voxels above 90% intensity, whic h are mostly located in the heart region. Table 6-5. TITAN calculation errors relative to the SIMIND simulation. Images Max. Error 2-norm Error Front 18.89% 3.711E-03 Left 11.29% 1.349E-03 Back 41.92% 6.882E-03 Right 40.22% 8.950E-03 As expected, larger differences are observed in the back and right projections that are farther from the heart as compared to left and fr ont projections. Further, the 2-norm of the results is very low, indicating the maximum errors o ccur at small fraction of voxels. The differences could be attributed to the following: In SIMIND simulation, we specified a parallel collimator and NaI detector. The effects, including particle reaction in collimator septa and detector efficiency, are not considered in the TITAN code. SIMIND uses an equal number of particles (i.e., 767,555) to gene rate all the four projection images, while they are located at significantly different distances from the hearth. Hence, it is expected that the back and right images exhibit larger relative errors. In order to resolve this important issue, it is essential to determine the pixel-wise statistical uncertainty map in SIMIND. TITAN uses the group cross section file generated by CEPXS. While the continuous energy cross section data built in SIMIND is tuned to the human body materials and SPECT simulation. Some errors could be due to the cross section data.
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109 Particle transport problems for SPECT traditionally are simulated by the Monte Carlo approach. Although it is still difficult to perf orm a strict comparis on with the Monte Carlo simulation by SIMIND due to the reasons discussed in the previous section, the preliminary results of the TITAN calculation show a reas onably good agreement with the reference. One potential advantage of deterministic method over the Monte Carlo appr oach is the reduced computation time when simulation of a large number of projection images is required. In a SIMIND simulation, the CPU time is proportional to the number of projection images. While the computational cost for TITAN is mostly dedicate d to the calculation of the flux moments. After the flux moments are converged, an extra sweep can compute a projection image with much less cost. Furthermore, flux moments can be stored after the trans port calculation. And projection image simulations for different angles can be processed in parallel using the same stored flux moments. Therefore, TITAN could be much faster for simulation to a large numbe r of projection images. The usage of the fictitious quadrature is not limited to SPECT simulations. The technique is a relatively reliable approach to evaluate the angular fluxes in interested directions. However, currently extra sweep with fictitious quadrature can be applied only for problems with vacuum boundary condition. And although multiple regular quadrature sets can be defined in TITAN, only one fictitious quadrature is allowed in one problem model.
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110 CHAPTER 7 PENTRAN INTEGRATION AND LIMITATION STUDIES OF THE CHARATERISTICS SOLVE R The coarse mesh/fine mesh scheme in the multi-block framework of the TITAN code is the same as the one used in the PENTRAN code. The block-oriented SN solver and characteristics solver are developed based on the meshing scheme. We incorporated a m odified version of the characteristics solver into the parallel engine of the PENTRAN code. In this chapter, the implementation of characteristics solver into PENTRAN is discussed. The performance of the integrated characteristics solver is tested on the simplified CT model benchmark with different parallel decomposition schemes. Finally, the limitations of the characteristics solver in TITAN are examined. Implementation of the Characteristics Solver in PENTRAN The data structure difference between PENTRA N and TITAN leads to some modifications on the characteristics solver in order to complete the integration. PENTRANs data structure is tuned to the parallel environm ent. The major data arrays, in cluding angular flux, flux moment, etc., are allocated locally. Since TITAN is still se rial code, one major challenge is to seamlessly integrate the serial characteristics solver into the parallel engine. In PENTRAN, based on the number of fine meshes within a coarse mesh, a memorytuning procedure is used to gr oup the coarse meshes into two categories: medium and large coarse meshes. While TITANs object-oriented pr ogramming paradigm allows each coarse mesh to be treated individually. The structure of the angular flux array is built on the loop architecture of the source iteration scheme. The dimensions for energy group, coarse mesh, direction octant in the angular flux array are treated as parent objects of the fine mesh flux. PENTRAN also stores all the boundary fluxes for each fine mesh, and the boundary flux for each coarse mesh is stored implicitly with the fine mesh boundary arrays. In TITAN, both coarse mesh and fine mesh
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111 boundary fluxes are treated explicitly by the fron tline style sweep procedure, and only the front line fluxes are dynamically stored. Some differen ces of the memory stru cture between the two codes are listed in Table 7-1. Table 7-1. Memory structure diffe rences between PENTAN and TITAN Array name PENTRAN TITAN Angular flux Two category, locally Coarse mesh individually Flux moment Two category, locally Coarse mesh individually Fine mesh boundary flux Stored Not stored, front-line style sweep Coarse mesh boundary flux Stored Not stored, front-line style sweep We decided to keep the memory structure untouched in PENTRAN while integrating the characteristics solver. Thereby, instead of r eallocating arrays, new arrays are allocated in PENTRAN when it is necessary, and de-allocated when they are not needed any more. Table 7-2 compares the characteristics solver in the modified version of PENTRAN (PENTRAN-CM) and TITAN. Table 7-2. Comparison of the characte ristics solver in PENTAN-CM and TITAN PENTRAN-CM TITAN Ray-tracing On the fly Pre-calculated Geometry information Not stored Stored Bilinear interpolation Employed Employed Coarse mesh material Void Void, low-scattering medium, pure absorber Projection compatibility Not completely compatible with Taylor projection Compatible with angular and spatial projection In the TITAN code, the ray-tracing along the quadrature directions are performed as an initial task. And the calculated geometry information, such as intersectio n points, path lengths, and bilinear interpolation weights, are stored and can be accessed directly in the sweep process. Depending on the meshing and quadrature set, a rela tively large amount of memory is required to store the geometry information. At the cost of memory, the characteristic s solver can sweep the coarse meshes much faster. In the PENTRAN-CM code, the geometry information is not stored. The ray-tracing procedure is performed on the fl y within every sweep. This approach is CPU-
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112 intensive. However, it reduces the memory require ment. This approach is also suitable to the PENTRANs coarse mesh data structure, ther eby, requiring minimal programming changes. For compatibility reasons, currently, the characteristic s solver in PENTRAN-CM can only be used in void regions. Note that PENTRAN is fully paralleli zed in the three domains (energy, angle, and space) of the phase space,15 while TITAN is a serial code. However, in the PENTRAN-CM implementation, we take full advantage of the parallel engine, such that the characteristics solver module can be distributed to diffe rent processors to complete th e assigned tasks. The individual tasks for each processor can be transport calculations for a su bset of energy groups, octants, and/or coarse meshes specified by a decomposition scheme.15 Benchmarking of PENTRAN-CM We tested the performance of the character istics solver in PENTRAN-CM using the simplified CT benchmark discussed in Chapter 5. Some measures are taken in meshing, cross section and quadrature set, so that we can make a fair and valid comparison within the PENTRAN parallel engine. Meshing, Cross Section and Quadrature Set We recall that two meshing schemes are used in the original benchmark: the 7-coarsemesh model (for the SN solver shown in Figure 5-8) and the 3 coarse mesh model (for the hybrid solver shown in Figure 5-9). Both models are tested in this PENTRA N-CM benchmarking. A two-group cross section data file is used to test the parallel decomposition in the energy domain. The one-group data in the original benchmark is listed in Tables 7-3. Table 7-3. One group cross section used in the CT benchmark with TITAN. Material # a f t s 1 (air) 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07 2 (source) 4.77840E-09 0.0E+00 7.16860E-07 5.94460E-07 3 (detector) 2.03430E-02 0.0E+00 3.88343E-01 2.60387E-01 4 (Water) 7.96423E-04 0.0E+00 1.48783E-01 1.23481E-01
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113 Materials #1 and #2 are the same material. Th ey are represented separately because the problem can be modeled more easily this way. Ma terial #4 is the characteristics coarse mesh material. By changing the group constants of mate rial #4, we can further examine the scattering ratio limitation of the characteristics solver. Tabl e 7-4 lists the two-group cross section data used in PENTRAN-CM for group parallel decomposition. Table 7-4. Two group cross section used in the CT be nchmark with PENTRAN-CM. Mat # Grp # a f t gs 1, gs2, 1 1 4.77840E-09 0.0E+00 7.16860E-075.94460E-070.0E+00 1 2 3.17640E-08 0.0E+00 8.79200E-077.75720E-071.17630E-07 2 1 4.77840E-09 0.0E+00 7.16860E-075.94460E-070.0E+00 2 2 3.17640E-08 0.0E+00 8.79200E-077.75720E-071.17630E-07 3 1 2.03430E-02 0.0E+00 3.88343E-012.60387E-010.0E+00 3 2 1.08305E-01 0.0E+00 5.48045E-013.78060E-011.07613E-01 4 1 7.96423E-04 0.0E+00 1.48783E-011.23481E-010.0E+00 4 2 5.29416E-03 0.0E+00 1.80640E-011.60378E-012.45063E-02 The two-group cross section data is mixed by the GIP code with the Sailor96 library with only down-scattering.30 And the first group constants in Tabl e 7-4 are the same as the one-group cross section data in Table 7-3. Therefore, the fi rst group fluxes at the detectors remain the same regardless of the number of groups. We can compare the detector re sponses calculated by PENTRAN-CM with the TITAN results. In the original CT model, we also applied the PN-TN ordinate splitting technique. In PENTRAN, only the rectangular or dinate splitting technique is available. In order to use the same quadrature, the PN-TN S20 quadrature set with two PN-TN splitting directions (Figure 5-11) is extracted from the TITAN code, and used as a user-defined quadrat ure set in PENTRAN. A minor modification in the quadr ature routine of PENTRAN is made to process the split directions. In this quadrature set, there are total 207 directions in each octant.
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114 Benchmark Results and Analysis A number of cases are tested with the charac teristics solver in a parallel environment as listed in Table 7-5. Note that in PENTRAN, the characteristics solver is included in PENTRANs adaptive differencing strategy as Option 5.15 (i.e. the differencing variable ndmeth=5 ) Table 7-5 compares the first 10 detector responses calculated by PENTRAN-CM with TITAN. Note that the TITAN results are extrac ted from Table 5-1 for Case 5. The other 10 detectors are symmetric, thereby, they have the same responses as the first 10 detectors. Table 7-5. Characteristics so lver calculated detector res ponse by PENTRAN-CM and TITAN. Detector # Case 1a in Table 7-3 Case 5 in Table 5-1Difference 1 1.345E-03 1.345E-03 4.20E-07 2 1.474E-03 1.475E-03 3.60E-07 3 1.510E-03 1.510E-03 5.40E-07 4 1.579E-03 1.579E-03 -7.00E-08 5 2.095E-03 2.094E-03 -6.00E-07 6 2.123E-03 2.123E-03 -1.80E-07 7 2.131E-03 2.132E-03 9.50E-07 8 2.146E-03 2.146E-03 -4.50E-07 9 2.155E-03 2.155E-03 -5.90E-07 10 2.152E-03 2.152E-03 -4.00E-07 Table 7-5 shows the difference between the two cases is in the order of 10-7, which is much lower than the scalar fl ux convergence tolerance 10-4. Therefore, the characteristics solver produces the same calculation results within th e machine truncation e rror in both TITAN and PENTRAN-CM. Table 7-6 compares the CPU ti me of PENTRAN-CM for a number of cases with different paralleliza tion decomposition schemes. Note that the detector responses cases are almost the same as the results in Table 7-5. Th is also demonstrates the accuracy of PENTRANs parallel engine for di fferent parallelization decompositions schemes.
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115 Table 7-6. Characteristics solver perfo rmance in PENTRAN parallel environment. Case # # of CM # of CPU Decomposition factor (decmpv1) Differencing Scheme (ndmeth2 ) CPU Time (sec) Angular Group Spatial First coarse mesh Middle coarse mesh Last coarse mesh 1a 3 16 8 2 1 -2 5 -2 7.7 1b 7 16 8 2 1 -2 -2 -2 33.3 2a 3 8 8 1 1 -2 5 -2 10.2 2b 7 8 8 1 1 -2 -2 -2 43.5 3 3 12 2 2 3 -2 5 -2 23.0 4a 3 1 1 1 1 -2 5 -2 64.0 4b 7 1 1 1 1 -2 5 -2 330.0 5a 3 1 Serial Run -2 5 -2 61.4 5b 3 1 Serial Run -2 -2 -2 323.0 1 PENTRAN parallel decomposition variable. 2 PENTRAN differencing scheme variable, ndmeth=-2 corresponds to the Directional Theta-Weighted scheme, and ndmeth=-5 corresponds to the characteristics solver. Cases 1a and 1b use 16 processors with an an gular decomposition factor of 8, an energy group decomposition factor of 2, and a spatial decomposition factor of 1. In Case 1a, we use the characteristics solver by setting ndmeth=5 for coarse mesh #2. Case 1b applies the SN solver only, and uses a total of 7 coarse meshes in or der to overcome the ray-effects. The solutions for both cases are accurate comparing to the soluti on of Cases 4 and 5 in Table 5-1 respectively (compared in Table 7-5). An acceleration factor of about 4.3 is achieved with the characteristics solver comparing to the SN solver, which is slightly lower than in TITAN code. We can draw the same conclusion based on other cases. Cases 2a, 2b, and 3 use 8 and 12 processors respectively. Cases 4a and 4b are pa rallel runs, although only one processor is used. Case 5a and 5b provide the resu lts for serial version of PENT RAN. It takes about 61.4 second with the characteristics solver, while about 323 seconds for the SN solver with the refined meshing. This result shows that the characteristics solver is more efficient than the SN solver in void regions in term of CPU time. In PENTRANCM, ray-tracing procedure is performed on the fly. In TITAN, the characteristics solver can be faster than the SN solver even with the same meshing, since ray-tracing informati on is pre-calculated and stored.
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116 Investigation on the Limitation s of Characteristics Solver Thus far, we have benchmarked the characteristics solver in TITAN with the CT model and Kobayashi problems. We also integrated the solver into the PENTRAN code, and tested the on-the-fly mode of the solver in a parallel environment. The hybrid approach with the characteristics solver shows an excellent performance on the benchmarks. However, the limitations of the solver and its sensitivity rela ted to meshing and quadrature order are not fully addressed. In this section, we further analy ze the characteristics solver based on its memory requirement, factors that affect accuracy, and possible improvement approaches. Memory Usage In the storage module of the characteristics so lver, we use an array of user-defined type, called GEOSET in the TITAN code, to store the coarse mesh geometry information for the characteristics solver. The size of the GEOSET array equals to the product of the number of fine meshes on the coarse mesh boundaries and the number of directions in the quadrature set for the coarse mesh. Therefore, every characteristic ra y in the coarse mesh requires a GEOSET object, which specifies five variables for the ray: Fine mesh index i at the incoming boundary (2 byte integer). Fine mesh index j at the incoming boundary (2 byte integer). Bilinear weight s on the incoming boundary (4 byte real). Bilinear weight t on the incoming boundary (4 byte real). Track length l of the ray (4 byte real). These five variables, which are calculated by the ray-tracing rou tine before the source iteration process starts, represent all the required geometry informa tion for a characteristic ray, if we consider the coarse mesh as one region. The pair ( i, j ) is used to locate the four fine meshes
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117 on the incoming boundary for the bilinear in terpolation procedure. The pair ( s, t ) is the bilinear weights as defined in Eq. 2.20. And l is the track length across the coarse mesh. If we consider a four-byte real number as one memory unit, each GEOSET can be stored with 4 memory units. Note here we store the (i, j ) pair as 2-byte integers instead of the regular 4byte integers. So the pair can be considered as one memory unit. The amount of memory required by the GEOSET can be very large wi th fine spatial meshing and high order of quadrature set. In certain cases, it can be even larger the SN solver. For example, for a coarse mesh with 10x10x10 fine meshes and with the same quadrature, the SN solver requires 1000 number of direction memory units to store the angular flux. While the characteristics solver needs 10x10x6 number of direction 4 memroy units. The characteri stics solver needs about twice amount of memory as the SN solver. This is demonstrated in Table 5-11 with the Kobayashi benchmark problems. The bilinear interpolation procedure requires at least 2x2 meshing on a boundary. On the other hand, because we use 2-byte integer to st ore the fine-mesh index in a GEOSET, the number of boundary fine meshes is limited to 255x255 for the characteristics solver, which is more than enough for most problems. We further discuss the mesh size limita tion in the next section. Limitation on the Spatial Discretization A deterministic solver does not suffer from the statistical uncertainties as in the Monte Carlo approach. However, since in a deterministic method, the phase space has to be discretized, the solution accuracy is affected by mesh/grid size. Generally speaki ng, finer grid size (i.e. finer energy group structure, hi gher order quadrature set, and smaller spatial meshing) should lead to a more accurate solution at a higher computational cost. It is difficult to set up some universal criteria on how to decide the optimistic grid si ze, since it depends on both the algorithms and the individual problem model. Generally, a good unde rstanding of the physics of the problem can
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118 provide some guidelines in the process of modeling. For example, for a zero-moment SN solver with the diamond differencing scheme, it is re commended to keep the mesh size under the material mean free path. 2-D meshing on the coarse mesh boundaries In the characteristics solver, we integrate the transport equation al ong the characteristic rays. A 2-D meshing scheme is required on th e coarse mesh boundaries. Generally, the 2-D meshing scheme is subject to the spatial discre tization requirement for a deterministic solver. Furthermore, we need to consider two major f actors to determine the mesh size on the coarse mesh boundaries for the ch aracteristics solver. Angular flux distribution fluctuatio n on the coarse mesh boundaries. Angular flux resolution requirement on co arse mesh boundaries for the model. The first factor is introduced with the bili near interpolation procedure, which assumes a linear angular flux distribution on the local four fine meshes su rrounding the inters ection point of each ray with the incoming boundary. With a relativ ely flat incoming boundary flux distribution, larger fine mesh size can be used while preservi ng the accuracy of the bilin ear interpolation. In an SN coarse mesh, we specify the number of fine meshes ( i j and k) along x, y, and z axes. In the characteristics solver, we still use the three integers to define the meshing on each boundary. For example, the two x-y boundaries have j i fine meshes. With this meshing scheme, the bilinear interpolation can keep consistency on the incoming and outgoing boundaries for directions in different octants. More discussion on the accuracy of the bilinear interpolation was given in Chapter 2. The second factor can be i llustrated with the simplif ied CT model as shown in Figure 7-1.
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119 Figure 7-1. Characteristics co arse mesh boundary meshing base d on flux resolution requirement. Figure 7-1 shows the meshing scheme of the second coarse mesh in which the characteristics solver is used We use 20x10 meshing on the two y-z boundaries, while only 2x2 meshing is applied on the other four boundaries. Our goal is to cal culate the detector responses on the right side of the coarse mesh. Therefore, it is required to apply finer meshing on the y-z boundaries. We can use much coarser meshing on the other four vacuum boundaries because these boundary fluxes cannot affect the detector responses. Note that here we choose 2x2 meshing, which is the minimum requirement on me shing for the characteristics solver by the bilinear interpolat ion procedure. We also investigated the impact of the 2-D meshing on two y-z boundaries. The original hybrid model uses 20x10 meshing on y-z boundaries of coarse mesh #2. Figure 7-2 examines the detector response errors as comp ared to the reference MCNP case by using different number of z fine meshes. Case 1 is the MCNP reference case. In Case 2a to 6a, the characteristics solver is used in coarse mesh #2 with 5, 8, 9, 10, and 12 z fine meshes. The error curve moves up closer to the reference solution as increasing the number of z fine meshes from 5 to 8. It indicates that the characteristics solver provides more accura te solution with fine r discretization grid.
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120 -5.00% -4.00% -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 0510152025 case 1 mcnp ref case 2a Ray zfm=5 case 3a Ray zfm=8 case 4a Ray zfm=9 case 5a Ray zfm=10 case 6a Ray zfm=12 Figure 7-2. Detector re sponse relative errors with different number of z fine meshes for the characteristics solver. Figure 7-3 shows the relative errors for the SN solver with different z meshing on the same coarse mesh. Note that for the SN solver, the fine mesh size along x axis is 1cm Case 2b to 5b use the SN solver in coarse mesh #2 with 5, 8, 9 and 10 z fine meshes. -4.00% -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 0510152025 case 1 mcnp ref case 2b Sn zfm=5 case 3b Sn zfm=8 case 4b Sn zfm=9 case 5b Sn zfm=10 Figure 7-3. Detector re sponse relative errors with different number of z fine meshes for the SN solver.
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121 All the curves in either Figure 7-2 or Figure 7-3 follow a similar trend. One can observe a jump when increasing zfm=9 (Case 4a) to zfm=10 (Case 4b) for the characteristics solver ( zfm is the number of fine mesh along z). It seems that the solutions by the characteristics solver is affected by the z fine meshing more sensitively than the SN solver. Table 7-7 provides the maxim percentage and 2-norm errors for all the cases. Table 7-7. Error comparison with different z meshing. Number of z fine meshes Characteristics solver cases Error 1-norm Error 2-norm SN solver cases Error 2-norm Maxim error 5 Case 2a 1.3103E-02 4.536% Case 2b 3.3245E-03 2.360% 8 Case 3a 6.7115E-03 3.622% Case 3b 3.3309E-03 2.786% 9 Case 4a 3.1872E-03 2.771% Case 4b 2.6947E-03 2.285% 10 Case 5a 7.5098E-03 3.280% Case 5b 2.8202E-03 2.092% 12 Case 6a 2.1630E-03 2.515% 3.3245E-03 2.360% Case 5a and Case 5b are the models used in the CT benchmark discussed in Chapter 5. Table 7-7 indicates that one can acquire a relatively accurate solution with different zfm numbers around 10, which demonstrates the stability of the hybrid algorithm. We further investigated the effects of y mesh size. Figure 7-4 shows the detector response sensitivity to the number of fine meshes along y axis ( yfm ) for the SN and characteristics solver. -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 0 5 10 15 20 25 case 1 mcnp ref case 5a Ray yfm=20 case 5b Sn yfm=20 case 7a Ray yfm=40 case 7b Sn yfm=40 Figure 7-4. Detector re sponse sensitivity to the fine mesh number along y axis.
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122 The curves are acquired by using the same z fine mesh number ( zfm=10), but different y fine mesh numbers and solvers. Cases 5a and 7a use the characteristics solver with yfm=20 and yfm=40, respectively, While the SN solver is used in Cases 5b and 7b. The two SN curves follow a similar trend. And as expected, the solution for Case 7b ( yfm=40) is more accurate than Case 5b ( yfm=20). The solutions with yfm=20 and yfm=40 are nearly identical for the characteristics solver. This indicates the yfm =20 meshing scheme is fine enough for the characteristics solver to evaluate the 20 detector responses. Coarse mesh size limitation for the characteristics solver We further investigated the effects of th e coarse mesh size on the accuracy of the characteristics solver. Since we consider the coar se mesh as one region, the limitation on the path length of the characteristic rays across the coarse mesh is the major factor in determining the coarse mesh size. The characte ristics solver integrates th e LBE along the rays with the assumption that the scattering source is constant throughout the coarse mesh in one sweep (flat source region). If the material for the coarse mesh is void or pure absorb er, such assumption is valid because the scattering source is always zero. For example, in the CT model, we can use a large coarse mesh size with the characteristics so lver in the air region. In materials other than void or pure absorber, the scatteri ng ratio of the material is the major factor on the size limitation of the coarse mesh. With scattering collision increa sing, we have to reduce the coarse mesh size to maintain the flat source assumption. We examined this effect by changing the mate rial cross section data in the CT model. Here, we use the SN model as the reference, since we already validate the SN solver on this model. The MCNP model requires much longer CPU time without variance reduction to achieve a good statistical confidence, because it is more difficult for the detectors to score a particle when increasing the tota l cross section in the air region. For the SN model, here we use yfm=20
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123 and zfm=5. Therefore, all the SN coarse meshes are filled with 1x1x1 cm3 fine meshes. Note that the SN solution is not necessarily as accurate as th e MCNP reference we used in the original CT benchmark. However, it can provide a valid a pproximate reference solution for the purpose of this benchmark. The accuracy of the characteristics solver in voi d regions is already demonstrated with the original CT benchmark. Here we further examined the performance of the solver in pure absorber regions. Figure 7-5 shows the detector response difference for the SN and characteristics solvers with pure absorber in the air region (cross sections for material in coarse mesh #2 are: 0.0 01-1.48783E s t ). 1.5000E-07 1.7000E-07 1.9000E-07 2.1000E-07 2.3000E-07 2.5000E-07 2.7000E-07 2.9000E-07 3.1000E-07 3.3000E-07 0 5 10 15 20 25 Sn pure absorber Ray pure absorber Figure 7-5. Detector re sponse comparison between SN and characteristics solver in pure absorber media. The characteristics solution (character istics coarse mesh meshing scheme: yfm=20, zfm=5, and xfm=2 ) shows a relatively clos e agreement with the SN solution (maxim difference 1.52% ). This demonstrates that the characteristics solver is accurate in pure absorber media. Figure 7-5 shows that the characteristics solver is less sensitive to the ray-effects, which is also demonstrated in the original CT benchmark for void regions.
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124 We further changed the cross section data with different scattering ratios while fixing 02-1.48783E t Figure 7-6 shows the difference between the SN and characteristics solutions for four different scattering ratios (ts /=0.05, 0.08, 0.10, and 0.20). Note here the characteristics coarse mesh size is 59cm along x axis with meshing scheme: yfm=20, zfm=5, and xfm=2 A 5.0E-04 5.5E-04 6.0E-04 6.5E-04 7.0E-04 7.5E-04 8.0E-04 8.5E-04 9.0E-04 9.5E-04 0 5 10 15 20 25 Sn scattering ratio =0.05 Ray scattering ratio=0.05 5.0E-04 5.5E-04 6.0E-04 6.5E-04 7.0E-04 7.5E-04 8.0E-04 8.5E-04 9.0E-04 9.5E-04 0 5 10 15 20 25 Sn scattering ratio =0.08 Ray scattering ratio=0.08 B C 5.0E-04 5.5E-04 6.0E-04 6.5E-04 7.0E-04 7.5E-04 8.0E-04 8.5E-04 9.0E-04 9.5E-04 0 5 10 15 20 25 Sn scattering ratio =0.1 Ray scattering ratio=0.1 5.0E-04 6.0E-04 7.0E-04 8.0E-04 9.0E-04 1.0E-03 1.1E-03 0 510152025 Sn scattering ratio =0.2 Ray scattering ratio=0.2 D Figure 7-6. Detector re sponse comparison between SN and characteristics so lver in media with different scattering ratio. A) ratio=0.05 B) ratio=0.08 C) ratio=0.10 D) ratio=0.20 By comparing the SN solutions in Figures 7-6A and 7-6B one can observe that the detector responses increase very slightly when increasing the scattering ra tion from 0.01 to 0.2. This is because that the detector responses are mainly dictated by the magnitude of the total cross section, which remains the same for all cases. Figure 7-6 also shows that the characteristics
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125 solver tends to over-estimate the solution with highe r scattering ratio. This can be attributed to the flat source assumption in Eq. 2-16, and it can be explained as follows. The scalar flux in the coarse mesh approximately decreases exponentially from the source region to the detector region (xte ). Here the scattering source is the only contributing source term, since no fixed source is present in the characteristics coarse mesh. With the flat source assumption, the scattering source is calculated by multiplying the coarse-m esh averaged flux and the scattering cross sections, and it remains the same in the coarse mesh within each iteration. As a result, the scattering source is over-estimated as x close to the detector region, resulting in the overestimation of the outgoing angular fluxes fo r the coarse mesh, which leads to a higher detector response. The source terms contributio n to the outgoing angular fluxes increases with the scattering ratio. Therefore, Figure 7-6 show s that the detector res ponses are overestimated further with higher scattering ratios. In order to correct this overestimation (i.e. allow the flat source assumption to be applicable), it is necessary to d ecrease the length of the characterist ic ray, or decrease the size of the coarse mesh along the axis of interest. Figure 7-7 compares the characteristics results with different coarse mesh sizes of 46 cm, 36 cm, and 32 cm to the SN reference solution. For this test, we use a scattering ratio of 0.2. 5.0E-04 5.5E-04 6.0E-04 6.5E-04 7.0E-04 7.5E-04 8.0E-04 8.5E-04 9.0E-04 9.5E-04 0 5 10 15 20 25 Sn scattering ratio =0.2 Ray x_size=46.0 Ray x_size=36.0 Ray x_size=32.0 Figure 7-7. Characteristics solutions with different coarse mesh size along x axis.
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126 As expected the characteristics solution approaches to the SN solution as the coarse mesh size decreases. The relative erro rs and CPU time for all the characteristics cases in Figures 7-6 and 7-7 are given in Table 7-8. Table 7-8. Characteristics so lution relative difference to SN solution with different scattering ratios and coarse mesh size. Case # Coarse mesh size along x (cm, mfp) Total cross section (cm-1) Scattering cross section Scattering ratio Error 2-norm Error 1-norm CPU Time Ratio (SN/Ray) 1 59 (0.87*) 1.48783E-02 7.43915E-04 0.05 1.2761E-03 1.393% 3.13 2 59 (0.87) 1.48783E-02 1.19026E-03 0.08 1.1499E-02 3.336% 3.10 3 59 (0.87) 1.48783E-02 1.48783E-03 0.10 2.6466E-02 4.646% 3.08 4 59 (0.87) 1.48783E-02 2.97566E-03 0.20 1.9858E-01 11.677% 3.12 5 46 (0.68) 1.48783E-02 2.97566E-03 0.20 8.1535E-03 3.131% 1.88 6 36 (0.54) 1.48783E-02 2.97566E-03 0.20 1.1328E-03 1.330% 1.41 7 32 (0.48) 1.48783E-02 2.97566E-03 0.20 8.0144E-04 1.353% 1.28 8 27 (0.40) 1.48783E-02 3.71958E-03 0.25 7.5987E-03 3.026% 1.13 9 22 (0.32) 1.48783E-02 4.46349E-03 0.30 2.5802E-03 2.223% 1.01 10 17 (0.25) 1.48783E-02 5.95132E-03 0.40 1.6210E-03 1.487% 0.91 Values in the parentheses are in unit of mean free path (mfp). Based on the results in Table 78, we conclude that the characteristics solver can provide an accurate solution by reducing size of the co arse mesh with higher scattering ratio. For the cases with scattering ratio of 0.20 (Cases 4 to 7), the limitation on the distance along x is about 36 cm, which is about half of the mean free pa th for the material (~70 cm). Generally, the accuracy of the solver depends on both the scattering ratio and m ean free path of the material. Table 7-8 also indicates that th e product of scatteri ng ratio and the mean free path, which is coarse mesh size in unit of s cattering mean free path, should be around 0.1 or less. It is recommended that the characterist ics solver is used for materials with a scattering ration less than 0.20, because with higher scat tering ratio, users need to further refine the coarse mesh size. And The CPU time comparison in Ta ble 7-8 indicates that the char acteristics solver generates
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127 less computational benefits as decreasing the mesh size as shown in Cases 7 to 10. In these four cases, we keep the coarse mesh size in unit of scattering mean free path close to 0.1, and the CPU time ratio decreases gradually. As in Case 10, the SN becomes faster than the characteristics solver. Possible Improvements and Extendibili ty of the Characteristics Solver The meshing scheme on the characteristics co arse mesh boundaries are limited by the bilinear interpolation. And the size and scatteri ng ratio limitations are due to the flat source assumption. Therefore, we could further study on these two aspects to improve the accuracy of the characteristics solver. The bi-linear interp olation assumes that the average flux happens on the center of a fine mesh. We could develop a new interpolation scheme on the incoming boundaries, which addresses the fact that the point flux actually should be the averaged flux on the fine mesh area or the cross sectional area of the ray. Instead of assuming a flat scattering source throughout the region, we could use some higher scheme, for example, linear source assumption, to represent the source term more accurately. Investigations on these two aspects will continue. In summary, the characteristics solver is efficient and accurate in void, pure absorber regions. For low-scattering medium with scattering ratio less th an 0.20, as a conservative guideline, the size of the characte ristics coarse mesh should be e qual or less than half of the mean free path. For higher scattering ratio materials, in which the characteristics solver is not recommend, the coarse mesh size should be less th an tenth of the scattering mean free path.
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128 CHAPTER 8 CONCLUSIONS AND FUTURE WORK Conclusions We developed a hybrid algorithm to solve the LBE for realistic 3-D problems, especially for the problems containing large regions of lo w scattering media, where the traditional SN method might become inefficient. A ray-tracing solver is designed and in tegrated in the TITAN code along with an SN solver. Both solvers are written under the paradigm of object-oriented programming with the block-oriented feature. And they are built on the framework of a multiblock discretization grid (coarse/ fine meshing scheme and blocklocalized angular quadrature). Both solvers are well-tuned in terms of memory management and CPU efficiency. The main features of the TITAN code are: Integrated SN and characteristics solvers. Shared scattering source kernel allowing arbitrary order anisotr opic scattering. Backward ray-tracing. Block-oriented data structure allowing localized quadrature sets and solvers. Layered code structure. Level-symmetric and PN-TN quadrature sets. Incorporation of two ordi nate splitting techniques (rectangular and local PN-TN) for the two type of quadrature sets. Fast and memory-efficient spatial and angular projections on the in terfaces of coarse meshes by using sparse projection matrix. Frontline-style in terface flux handling. An efficient algorithm for calculation of the scattering source and the within-group scattering with a modified scattering kernel. A binary I/O library to visualize a nd post-process data with TECPLOT.
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129 Extra Sweep technique with the fictitious quadrature technique for cal culations of angular fluxes along arbitrary directions. We tested the performance of the TITAN code with a number of benchmark problems. For applications in the field of nuclear engineer ing, TITAN is used to solve the Kobayashi benchmark, which is a set of difficult shieldi ng problems, and the 3-D C5G7 MOX benchmark, which is a k-effective problem without homogenization fo r a MOX/UO2-fueled reactor with different control rod layouts. For applications in the me dical physics field, we tested the code on the CT device model, which is difficult for deterministic codes to solve due to ray-effects, and the SPECT phantom model, in which transport simulation is commonly performed only by the Monte Carlo approach. The ficti tious quadrature technique we developed for the SPECT model can be very useful for other me dical applications as well. The benchmark results demonstrate not only the accuracy and efficiency of the code, but al so the scalability of the code. For example, in the CT model, the memory usage still keep s proportional to the quadrature order while increasing to S200. And in the SPECT model, we are able to use the SN solver in one coarse-mesh with about two million fine meshes. Future Work TITAN provides a code base for future devel opment with its excellent extendibility. There are still several areas where the code can be further enhanced. Acceleration Techniques The loop structure of the code is composed of power iteration loop, upper-scattering loop, energy group loop, within-group loop, octant loop coarse mesh loop, direction loop, and fine mesh/ray loop. Various acceleration techniques can be applied on the power iteration level and the within-group loop. These acceleration tec hniques aim to accelerate the convergence of the fission source or the within-group scattering source. Generally, they can be applied in both SN
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130 and MOC. Coarse mesh rebalancing (CMR) and coarse mesh finite difference (CMFD) are widely used acceleration techniques,41 which accelerate the within-group loop by forcing the particle balance in each coarse mesh for each loop. Another physical approach is the synthetic method,42 in which one can use some lower order methods like diffusion method to acquire a better estimation of scattering source in-b etween within-group loops. Some numerical approaches, such as multi-grid method,43, 44 and pre-conditioned subs pace projection iteration method,45, 46 can also be applied. However, the gene ral numerical iteratio n techniques usually need to be modified here, since in SN codes, we usually do not build up the matrix A in a liner iteration system x=Ax+b due to memory limitation. Currently, there are two source iteration kernel s in the TITAN code. The default kernel is the S1 kernel, in which the flux moments are updated after angular fluxes ar e calculated within each sweep. While the S2 kernel subroutines updates the fl ux moments immediately after the angular flux is calculated for each direction. The relationship between S2 and S1 is similar to the one between Gauss-Seidel and Jaco bi iteration methods. Numerically, S2 kernel has a faster convergence speed than the S1 kernel in most cases without much additional computation cost. However, it could cause numerical instability problems in some extreme cases. And it is not preferable for code para llelization in the angular domain. Therefore, currently we choose the S1 kernel as the default option. Another set of kernel subroutines can be added with the flux moments updated after each octant is processed. This process is numerically similar to the redblack block or multi-cyclic iteration schemes. In the future, higher order iteration scheme s should be implemented. Krylov subspace projection iteration pre-conditioned by CMFD would be a good acceleration combination. The acceleration subroutines can be inse rted into Figure 3-1 around Subroutines L2.7 and/or L3.5.
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131 Parallelization We can parallelize the TITAN code by using MPI and/or OpenMP. One essential part of code parallelization work is the loop parallelizati on. In Figure 3-1, we could break up the coarse mesh loop and octant loop into a distributed co mputing environment by using MPI. OpenMP can be used to parallelize the di rection loop in a shared memory environment. Other parallel algorithms can be applied.47, 48 An MPI and OpenMP hybrid approach can take advantage of the cost-efficient cluster hardware, as well as mu lti-CPU nodes and dual-core CPUs. Furthermore, Code parallelization can benefit from the multi-block framework, since each coarse mesh in the framework can be treated relatively independently.. Improvements on Characteristics Solver The TITAN code considers a characteristics coar se mesh as one region, which is sufficient in this work, since the characteristics solver is only designed for low scattering media. Some multi-regions data structures already are in place in the code. A more efficient ray-tracer is required for a multi-region solver. Other Enhancements Projection techniques need to be tested in more problems, since the efficiency and accuracy of the projections are essential under the multi-block framework. It is worth noting that the multi-block framework can assemble other types of solvers besides SN and MOC. For example, some non-Cartesian meshing schemes can be implemented in a coarse mesh with a potential finite element solver. With above proposed future work, we consider the code still under development. We hope in the future our community can build an online open-source forum for deterministic calculations, where users and developers can freely share source codes and ideas.
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132 APPENDIX A SCATTERING KERNEL IN LINEAR BOLTZMANN EQUATION Introduction In the discretized form of the linear Boltzmann equation (Eq. 2-1), the scattering kernel is the most complicated term. In this appendix, we will prove the following formulation: )]}sin()()cos()([ )( )!( )!( 2)()(){()12( )' ,',() ,',( ',', ,', 1 '11 ,' ,', 4 0 kr kr P kl kl rPr l Er EErddEk lgS k lgC G gl l k k l lgllggs s (A-1) In Eq. A-1, the discretization in energy domain can be easily separated with the discretization in the angular domain. The energy and spatial dependency of the scattering source on the left hand side is represen ted by the flux moment terms ()(,'rk lg )(,',rk lgC and )(,',rk lgS ) on the right hand side. Since the G gdE1' 0 'conversion can be achieve d straightforwardly by the multigroup approximation, here our main focus is on the conversion of 1' 4 dl. For simplicity, we drop the energy group index ( g and g ) and spatial dependency ( r ) in the flux moment terms and the cross secti on moment term. Furthermore, inst ead of an infinitive Legendre expansion order, we assume a maxim expansion order of L With above simplifications, we can rewrite the formulation to be proved: )]}sin()cos([)( )!( )!( 2)({)12( )' () ( 11 4 k k P kl kl Pl dk lS k lC L l l k k l llls s (A-2) From now on, we also use the following denotations:
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133 ),(),( and ) ','()','(' (A-3) Where is the polar angle with x axis, is azimuthal angle on the y-z plane, and )cos( ) 'cos(' The integration over the unit sphere becomes 4 2 0 1 1 4 ddd In some references, for simplicity one can also use 1 22 2 0 1 1 4 dd d. However, we found it is not necessary to make such assumption, and it could cause some confusion in th e spherical harmonic expansion. So here we still respect the mathematical fact that the overall solid angle is 4 Note that with or without this assumption, the formulation of Eq. A-2 should remain the same. In order to prove Eq. A-2, we need to expand the angular flux and the cross section into a series of Legendre polynomials in the angular domain, respectively. In this appendix, we provide such an expansion for both the angular flux and cross section. By subs titute the two expansion series into the left hand of Eq. A-1, we can evaluate the new terms, and finally prove the scattering kernel formulation. Spherical Harmonic Expansion of the Angular Flux In this section, we also demonstrate how a nd why the cosine and si ne flux moments are defined. A smooth function define d on the surface of a unit sphe re, such as the angular flux )','()' ( can be expanded by the spherical harmonic function.49, 50 0)','( )','()' (n n nm m n m nYa (A-4) The general form of the s pherical harmonic function ) ','(m nY is defined by: ')'( )!( )!( 4 )12( )','( im m n m neP mn mnn Y (A-5)
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134 Where ) '(m nP is the associated Legendre polynomial. The coefficient m na is given by: im m n m n m neP mn mnn dd Y dda )( )!( )!( 4 )12( ) ,( ),() ,( 2 0 1 1 2 0 1 1 (A-6) Where ),(m nY is the complex conjugate of ),(m nY The angular flux expansion defi ned by Eq. A-4 should be a real value. So we expect the imaginary part of Eq. A-4 is zero. In order to prove this, we rewrite Eq. A-4 as following: )]}','()','([)','({)','( )','(001 00 m n m n nn n m m n m n nn n nm m n m nYa Ya Ya Ya (A-7) Based on Eq. A.5, we have: )'( 4 12 )','(0 n nP n Y (A-8) By applying the following identity of the spherical harmonic function,49, 51 )','()1()','( m n m m nY Y (A-9) The coefficientm nacan be evaluated as: m n m im m n im m n im m n im m n m n m na eP dd mn mnn eP dd mn mnn mn mn eP mn mn dd mn mnn eP mn mnn dd Ydda )1( )() ,( )!( )!( 4 )12( (-1) )() ,( )!( )!( 4 )12( )!( )!( (-1) )( )!( )!( (-1)) ,( )!( )!( 4 )12( )( )!( )!( 4 )12( ) ,( ),() ,( 2 0 1 1 m 2 0 1 1 m 2 0 1 1 m 2 0 1 1 2 0 1 1 (A-10)
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135 Note in Eq. A-10, we also apply the following identity of the associated Legendre polynomial.49 )( )!( )!( (-1))(m m n m nP mn mn P (A-11) According Eqs. A-9 and A-10, the last te rm in Eq. A-7 can be rewritten as: )','(()','()','()1()1()','( m n m n m n m n m n m m n m m n m nYa Ya Ya Ya (A-12) We substitute Eq. A-12 back to Eq. A-7, 01 00 01 00 001 00})]','(Re[2)','({ }])','(()','([)','({ )]}','()','([)','({)','( )','(n n m m n m n nn n n m m n m n m n m n nn m n m n nn n m m n m n nn n nm m n m nYa Ya Ya Ya Ya Ya Ya Ya Ya (A-13) Here we denote the real part of )','(m n m nYa as )]','(Re[m n m nYa As we expected, the angular flux is always a real value according Eq. A-13. Now we can further calculate the two terms in Eq. A-13 based on Eqs. A-5 and A-6. The second term is: )sin()() ,( )'sin()'( )!( )!( 4 )12( )cos()() ,( )'cos()'( )!( )!( 4 )12( ))}]'sin()')(cos('( )!( )!( 4 )12( { ))}sin()(cos()( )!( )!( 4 )12( ) ,( Re[{ )]','(Re[2 0 1 1 2 0 1 1 2 0 1 1 m P ddm P mn mnn m P ddm P mn mnn mim P mn mnn mim P mn mnn dd Yam n m n m n m n m n m n m n m n (A-14) And the first term is:
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136 )() ,( )'( 4 )12( )}'( 4 12 {)}( 4 )12( ) ,( {)','(2 0 1 1 2 0 1 1 00 n n n n nnP ddP n P n P n dd Ya (A-15) If we define the regular flux moment, cosi ne moment and sine moment as follows. )() ,( 4 12 0 1 1 n nP dd (A-16) )cos()() ,( 4 12 0 1 1 m P ddm n m nC (A-17) )sin()() ,( 4 12 0 1 1 m P ddm n m nS (A-18) We can rewrite Eqs. A-14 and A-15 as follows. ])'sin()'( )'cos()'([ )!( )!( )12()]','(Re[, m nS m n m nC m n m n m nm P m P mn mn n Ya (A-19) nn nnPn Ya )'()12()','(00 (A-20) By substituting Eqs. A-19 and A-20 into Eq. A-13, finally we derive the expansion formulation for the angular flux. }])'sin()'( )'cos()'([ )!( )!( 2)'(){12( })]','(Re[2)','({)','(0 1 01 00 n m nS m n m nC m n n m nn n n m m n m n nnm P m P mn mn Pn Ya Ya (A-21) One may notice that Eq. A-21 looks similar to Eq. A-4, which is the formulation we need to prove. However, further derivations are still required to reach Eq. A4. After the integration, and disappear on the right hand side of Eq. A-4. And only and dependencies are left. At this point, Eq. A-21 is only a function of and Here we intentionally use n and m as the index, so that we can distinguish them with l and k, which we will use in the next section while expanding the cross section term.
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137 The flux moment formulations, Eqs. A-16 to A18, are equivalent to Eqs. 2-2 to 2-4 we discussed in Chapter 2. Note a 4 factor is used in these formulations. Scattering Cross Section Expansion and the Spherical Harmonic Addition Theorem The cross section term in Eq. A-2 can be written as follows. )() () (0 s s s (A-22) Since the cross section only depends on the scattering angle. With the notations in Eq. A-3, we can derive the formulation for 0. k j i )'sin()'sin()'cos()'sin()'cos(' (A-23) k j i )sin()sin()cos()sin()cos( (A-24) )'cos()'sin()sin()'cos()cos( 0 (A-25) With Eq. A-25, we can apply the sp herical harmonic addition theorem.49 l k k l k l ll lkkkk PP kl kl uPuPP1 0)]'sin()sin()'cos())[cos('()( )!( )!( 2)'()()( (A-26) Now we can expand Eq. A-22 with the Legendre polynomial. )]}'sin()sin()'cos()[cos( )'()( )!( )!( 2)'()({ 4 12 )( 4 12 )(01 0 0, 0 kkkk PP kl kl uPuP l P ll l k k l k l llls l lls s (A-27) Note we use the 4 factor in Eq. A-27, because usually we assume 0, s is the total scattering cross section. So in cas e of isotropic scattering, the di fferential cross section becomes 4 )(0 s s
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138 Formulation of the Scattering Kernel So far we have expanded the angular flux w ith the spherical harmonic function, and the scattering cross section with the Legendre polynomia l. In this section, we multiply the two terms together and complete the angular integr ation. Eventually Eq. A-2 is derived. We begin with rewriting the two expansi on formulations (Eqs. A-21 and A-27) and limiting the expansion order to L }])'sin( )')[cos('( )!( )!( 2)'(){12( )','(0 1 L n m nS m nC m n n m nnm m P mn mn Pn (A-28) })]'sin()sin()'cos())[cos('()( )!( )!( 2 )'()({ 4 12 )(1 0 0 l k k l k l L l llls skkkk PP kl kl uPuP l (A-29) When we evaluate 2 0 1 1)'()','(''sdd using Eqs. A-28 and A-29, all the and terms can be moved out the integration, and obviously a lot of multiplication terms will appear. Most of the terms become zero. Among th e zero terms, some of them are erased by the orthogonal property of Legendre polynomials, ot hers are scratched off by the facts that: 2 00)'cos(' md and 2 00)'sin(' md for m=1, 2 (A-30) We will identify these terms step by step. Here, we refer to the term nnP )' ( in Eq. A-28, and the term ) '()( uPuPll in Eq. A-29 as the first part of the respective equation, and the summation term over m or k in both equations as the second part. Now we can apply the orthogonal property of the regular Legendre polynomials.
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139 12 4 )( 12 2 2)()'()'('')(, 2 0 1 1 l P l PPPddPln ll ln nl l n nl (A-31) Where otherwise nlln 0 1, Therefore, all the first part multiplication terms become zeros except for those n=l. Now we consider the first part of Eq. A-28 multiplied by the second part of Eq. A-29 (the summation term over m ). One can observe that these terms become zeros because of Eq. A-30. Similarly, the terms, acquired by multiplying the second part of Eq. A-28 with the first part of Eq. A-29, become zeros as well. So far the terms we have not covered are the multiplications of the second parts from both Eqs. A-28 and A-29. A common mistake one might make is to assume 1 1 ,,)'()'('mknl m n k lCuPPd The assumption is very convenient here. Unfortunately, such strict orthogonal relationship for the associated Legendre polynomials can not hold for arbitrary l k n, and m However, a relaxed version is always true.49 nl m n m lml ml l PPd, 1 1)!( )!( 12 2 )'()'(' (A-32) In order to apply Eq. A-32, we need to notice the facts that: 2 0 2 0 0 )'sin()'sin(')'cos()'cos(' otherwise mk kmdkmd m,k=1, 2 (A-33) 2 0 2 00)'cos()'sin(')'sin()'cos(' kmdkmd for m,k=1, 2 (A-34) By using Eqs. A-33 and A-34, we are able to remove all the terms except the terms of )'cos()'cos( mkand)'sin()'sin( mk with k=m Then, we can apply Eq. A-32 on all the remaining terms. In the end, we can conclude that only the terms with k=m and l=n will survive among all the second part multiplication terms.
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140 Based on the above explanations, we can write the scattering kernel wi th all the remaining terms by combining Eqs. A-31 to A-34. Finally, we have proved Eq. A-2. )]}sin()cos([)( )!( )!( 2)({)12( )]}sin()cos()[( )!( )!( 12 2 ) )!( )!( [(4 12 4 )({ 4 )12( )' () ( 11 11 2 2 4 k k P kl kl Pl k k P kl kl lkl kl l P l dk lS k lC L l l k k l llls k lS k lC k l L l l k llls s (A-35) Summary The energy dependency and its integration can be introduced back into Eq. A-35. And we acquire the multigroup form of the scattering kernel. In the TITAN code, we apply the scattering-in moment form by switching the su mmation over the group and Legendre order (Eq. 4-1). The switching seems meaningless mathematically. However, it can generate significant benefits in the coding practice. Further discussions on the scattering-in moment form are already given in Chapter 4. In Eq. A-35, the direction ),( which is the particle moving direction after a scattering reaction, is not required to be one of the directions in a quadrature set, although this happens to be true in the sweep process with a regular quadrature set. Mathematically,),( can be an arbitrary direction in Eq. A-35. We take advantage of this fact in the fictitious quadrature technique we developed in Chapter 6, and also the ordina te splitting technique in Chapter 2. It is not evident to claim that the scattering source evaluated by Eq. A-35 on regular quadrature directions has a higher accuracy than on an arbitrary direction. Nevertheless, the flux moments are always calculated with a regular quadrature set to conserve the integrations in Eqs. A-16 to A-18.
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141 Finally, it is worth noting that we choose x axis as the polar axis in all the derivations, which means is the cosine between and x The choice of polar axis does not alternate the formulation of the scattering kern el. However, the values of some terms in Eq. A-35 are affected by the choice of the polar axis, except for a Leve l-Symmetric quadrature set, in which all term values remain the same because of the rotation invariance property. In other quadrature types, e.g. the Legendre-Chebyshev quadrature, a number of terms in Eq. A-35 change with different polar axes. For example, if we choose the y as the polar axis instead of x we can build a relationship between the two systems. )]cos( ),sin()[sin(atan2 )cos()sin()( )( )( )( )( )( )( x x x y x x y (A-36) Where atan2 is the extended inverse tangent function, which is available in most math libraries with various languages. Obviously, Eq. A-36 affects all th e terms depending on ) ,( in Eq. A-35, including the flux moments, Le gendre polynomial values, cosines and sines. However, the overall scattering source should re main the same even with all these changed terms, because physically the scattering source should not be affected by the choice of polar axis. Mathematically, one might be able to demonstrate this statement by substituting Eq. A-36 into Eq. A-35 and Eqs. A-16 to A-18. In reality, we can only expand the scatte ring kernel to a limited order. In the TITAN code, originally we chose the z axis as the polar axis, later We changed it to the x axis. The results are almost the same for the first benchmark problem discussed in Chapter. 5. It would be interesting to further investigate th e effect of different choices of polar axis on the scattering kernel.
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142 APPENDIX B NUMERICAL QUADRATURE ON UNIT SPHERE SURFACE Introduction In the process of solving the linear Bo ltzmann equation, flux moments need to be evaluated in order to calculate the angular-dependent scattering source term. Flux moment (Eqs. 2-2 to 2-4), by its mathematical nature, is noth ing but an integration of a function defined on a unit sphere surface. The function is the angul ar flux multiplied by a corresponding regular or associated Legendre polynomial. Flux moments become angular independent after the integration over the surface of a unit sphere. The exact distributi on of the angular flux on the unit sphere is unknown. However, we can evaluate function values of the angular flux by the sweep process at a given number of points (discrete ordinates) on the unit s phere. Positions and associated weights of these points are prescribed by a quadrature set. Then, the flux moments can be simply calculated by a summation of the function values multiplied the associated weights. Quadrature is a simple but powerful numeri cal integration techni que. For example, a Gaussian quadrature with an order of N, can acquire the exact value of the integration of any polynomial up to order of 2N-1 defined within [-1, +1]. In our case, the integration domain is the surface of a unit sphere. Thereby, we need to build a quadrature to evaluate a double integration. Mathematically, a good quadrature of a given order always tends to conserve the integration to the highest order. However, the property of symmetry of a quadrature generally plays a significant role in a physical problem. For example, in a problem with reflective boundaries, we obviously hope all reflected directi ons of a given direction are also in the quadrature set. Therefore, we often build a quadrature on the balance between keeping symmetry and conserving higher order integration. For exam ple, the level-symmetric quadrature with an order of N can conserve moments only up to the Nth order, but with an excellent symmetry
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143 property of rotation invariance. The LegendreChebyshev quadrature can conserve moments up to the 2N-1, but rotation invariance is slightly disturbed. In this appendix, we prove that the Legendr e-Chebyshev quadrature is the best choice in regards to conserving higher moments. Through th e discussion of the procedure, hopefully we can cast some insights on how a quadrature is built on the bala nce of simple mathematics and physics for transport calculations. General Quadrature Theorem The popular Gaussian quadrature is built on the orthogonal Legendre polynomial, which is defined on [-1, +1] with a weighting function w(x)=1. In general, we can consider 0n | )( xn as the orthogonal polynomials defined on (a, b) with a weighting function of bxaxw for 0) ( According to the orthogonality property, we have: nm nm dxxxxwn b a mn 0 )()()( (B-1) Whereb a n ndxxxw2)]()[( We also denote that n n nxAx)( and n n nA A a1. And the integral of a function f(x) can be represented by an nth quadrature formula: )()( )()()(1 ,,fIxfwdxxfxwfIn b a n j njnj (B-2) For a given number of nodes, we choose the node positions {xj,n} and weights {wj,n} in hoping that we can conserve Eq. B2 as accurate as possible for any f(x). Mathematically, if we assume f(x) is a polynomial, this means that the pos itions and weights of the nodes can hold the integration exactly as the true value to the highe st order of the polynomial. In this sense, the nodes and weights can by calculated with Theorem B-1,37 which is the fundamental guide for building the Legendre-Chebyshev quadrature.
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144 Theorem B-1: For each 1 n, there is unique numerical integratio n formula of degree of precision 2n-1, Assuming f(x) is 2n times continuously diffe rentiable on [a b], the formula for In(f) and its error is given by b a n j n n n jjf nA xfwdxxfxw1 )2( 2)( )!2( )( )()( (B-3) For some b a The nodes {xj} are the zeros of )(xn and the weights {wj} are given by: ..., nj xx a wjnjn nn j1 )()('1 (B-4) Legendre-Chebyshev Quadrature on Unit Sphere Theorem B-1 lays the foundation for buildi ng a quadrature set for one-dimensional integration. In order to apply the theorem for a function define d on a unit sphere, we need to separate the two-dimensional integration of the angular flux into two one-dimensional integrations. In general, we consider ),( f is a real smooth function defined on a unit sphere surface, where 11 is the cosine of the polar angle, and is the azimuthal angle. We need to estimate: 4 1 1 2 0),( ),( fdd fdI (B-5) First we define a function of )( g: 2 0),( )(fd g (B-6) 4 1 1)( ),( gd fdI (B-7)
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145 The integration defined by Eq. B-7 can be es timated by a Gaussian quadrature, since the weighting function is1 )( xw. Based on Theorem B-1, we choose the quadrature nodes } {i as the roots of the Nth Legendre polynomial. 0)( iNP (B-8) Note we usually choose N as an even integer, so that the roots are symmetrically distributed on the axis. The weights } {iwcan be calculated by Eq. B-4. Next we need to determine the function values of )} ({ig ) (ig itself is an integration over a unit circle defined by Eq. B-6. And it can be estimated by another quadrature, in which we still prefer that the quadrature nodes are symmetrically distributed on the four quadrant of a unit circle. Thereby, we separate the integration define d by Eq. B-6 into two parts: 2 0 2 0),( ),( ),( )(i i i ifd fd fd g (B-9) Now we can consider only the integration over the first half of the unit circle, since nodes on the other half of th e circle are decided by symmetry. We denote ) ,()( ifg and )cos( The first part of Eq. B-9 can be rewritten as: 1 1 2 1 1 2 0 0)(h -1 ))(arccos(g -1 )(g ),( d d d fdi (B-10) Note here 21 )arccos( d dd And we denote ))(arccos( )( gh In Eq. B-10, 21 1 )( w is the weighting function for Chebyshev polynomial ))arccos( cos()( xnxTn Thereby, we are required to c hoose the Chebyshev quadrature to evaluate the integration defined by B-10, so that we can precisely estimate the integration if
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146 )( h is a polynomial up to the order of 2n-1 Usually, we choose an even integer for n, because we can keep the symmetry on the top half of the unit circle. Figure B1 shows the roots of T4(x) on the unit circle. X Y Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Figure B-1. Chebyshev root s (N =4) on a unit circle. The x coordinates of Z1-Z4 are the roots of T4(x) For an even order Chebyshev polynomial, Z1 and Z2 are symmetric to Z3 and Z4 respectively. Z5-Z8 are intentionally selected to keep symmetry. As a result, Z1-Z8 are symmetrically distributed over the four quadrants. Furthermore, the Chebyshev roots are uniformly located on the unit circle, and they are equally weighted by Eq. B-4. By combining Eqs. B-7 and B-10, the Legendr e-Chebyshev quadrature can be built on a unit sphere. However, some physical concerns on symmetry still need to be addressed. Normally, we require the directions in one octant form a triangle-shaped ordering as shown in Figure 2-8 in Chapter 2. And all directions in the ot her seven octants are decided by symmetry. The
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147 triangle-shaped distribution is required to keep the propert y of rotation invariance. For example, in the level-symmetric quadrature, num ber of directions per level increases by one from one level to the next. And the choice of the polar axis ( x y or z ) does not affect the distribution of the directions b ecause the directions are perfectly symmetrical. In the LegendreChebyshev quadrature, we can not keep this p erfect symmetry because its priority is to conserve higher moments over rotation invariance However, we can still keep some slightly disturbed symmetry of rotation invariance by em ploying the same trian gle-shaped direction ordering. The procedure to build a Legendre-Chebyshev S10 quadrature in the first octant can be explained as follows: We choose the five positive roots of P10(x) as the level positions. There is only one direction on the top leve l. And its position on the level ci rcle is decided by the positive root of T2(x). On the second level, the two positive roots of T4(x) become the quadrature node positions. The third level node positions are chosen by the three roots of T6(x) and so on. On the bottom level, five directions are to be defined, which are the positive roots of T10(x) These five level nodes form a triangle-shape d distribution in the first octant. The final layout of the nodes has a quite similar look as the level symmetry quadrature of S10. Figure 2-10A shows the difference of direction distri bution between the level-symm etric and Legendre-Chebyshev quadrature with an order of 10. Newtons Method to Find Pn(x) Roots In the Legendre-Chebyshev quadrature, the roots of Legendre and Chebyshev polynomials are essential to locate the positions of the quadrature nodes. Chebyshev roots are easy to find since they are uniformly distributed on th e unit circle as shown in Figure B-1. n i n i xxnxTi i n2 12 2 12 cos 0))arccos( cos()( (B-11)
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148 For a Legendre polynomial f(x) = PN(x), we apply a variant of Newtons method to find all the positive zeros {xi} in an increasing order as follows. Step 1: Set initial guess xg=0 for the first (smallest) positive root x1. Step 2: For i=1, 2, N repeat step 3-5, where N an even integer, is the polynomial rank. Step 3: Use Newtons method to find root xi. Step 4: Set )( )( )(ixx xf xf Step 5: Set initial guess xg= xi for next root xi+1 Step 6: Stop In Step 3 of the above algorithm, the polynomial f(x) and its derivative can be defined as follows. 1 1)( )( )(i m m Nxx xP xf (B-12) 1 1 1 1 1 1 1 1 1 11 )()( )( )( 1 )( )( )( 1)( )( )( )('i m i N N i m i i m i N i m i N i m i Nxx xfxf xP dx xdP xx xx xP xx dx xdP xx xP dx d xf (B-13) Then we can apply the following iterative fo rmulation of Newtons method to find root xi 1 )( )( )( )(' )(1 1 i m i iN iN iN i i i iixx xP dx xdP xP x xf xf xx (B-14) In Eq. B-14, ) ( xPNand ) ('xPNcan be estimated by the recu rrence relations of Legendre polynomial defined in Eqs. B-15 and B-16.
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149 0)()()12()()1(1 1 xnPxxPnxPnn n n (B-15) )()1()()1()()()()1(1 1 '2xPnxxPnxnPxnxPxPxn n n n n (B-16) So far we have set up the layout of the dire ctions on the unit sphere by finding roots of Pn(x) and Tn(x). We will further discuss the node we ights in the next section. Positivity of Weights Another physical concern is the positivity of the node weights. Level-symmetric quadrature is limited to the order of 20, because negative weights occur beyond order 20. In the Legendre-Chebyshev quadrature, the weight for node i is calculated by the product of polar weight (level weight) and azimuthal weight. TPiwww (B-17) Both the polar weight wp and azimuthal weight wT are calculated by Eq. B-4 with Legendre and Chebyshev polynomials, respectively. First we evaluate the terms in Eq. B-4 for azimuthal weights by applying some Chebyshev polynomial properties. 2 1 and 2 2n 1 1 n n n n nA A a A (B-18) )sin()1()( and )sin( )1( )(1 1 i i in i i inxT n xT (B-19) We can substitute Eqs. B-18 and B-19 into Eq. B-4. )()('1nxTxT a winin nn T (B-20) So the Chebyshev nodes are equally weight ed. In the TITAN code, we normalize the azimuthal weights on the same level to one. So we simply use normalized weights. n wT1 (B-21)
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150 Where n is level number. Next we can evalua te the level weights by applying some properties of Legendre polynomial given in Eq. B-22. 12 2 and 1 12 )!(2 )!2( ])!1[(2 )]1(2[ )!(2 )!2(2 2 1 1 2 n n n n n n n A A a n n An n n n n n n n (B-22) By substituting Eq. B-22 into Eq. B-4, and applying the recurrence property of Eq. B-16, we can rewrite Eq. B-4 as follows. 2 1 2 2 1 1)]([)1( )1(2 )()(')1( 2 )()('in i inin inin nn TxPn x xPxPnxPxP a w (B-23) Note in deriving Eq. B-23, we also apply 0 )( inxP Since 1 0 ix wT defined by Eq. B-23 is positive definite. Therefore, unlike th e level-symmetric quadrature, the LegendreChebyshev quadrature weights are always positive. Furthermore, we can prove that the sum of the weights 21 n i iw because of the following identity of Legendre polynomial. n i in ixPn x1 2 1 2 21 )]([)1( 1 (B-24) In the Legendre-Chebyshev quadrature, we always choose n as an even integer. The roots and weights are symmetrical regarding to x=0. We can apply Eqs. B-17, B-21 and B-24 to calculate the total weight for all directions in the first octant. 1 12/ 1 1 2/ 1 1 2/ 1 N n P n n k N n P n n k T k i N n P n iw n wwww (B-25) As the level-symmetric quadrature, all the directions in other octants are determined by applying symmetry to the ones in th e first octant. We can conclude that the sum of the LegendreChebyshev quadrature weights in one octant is equal to one as in the level-symmetric quadrature.
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151 Conclusion We have proved two very desirable propertie s of the Legendre-Chebyshev quadrature for transport calculations. First, it can conserve integration up to 2N-1 order. Second, the weights are always positive for any order of the quadrature. However, we do lose some symmetry of rotation invariance. On the other hand, the level symm etry quadrature keeps the perfect symmetry of rotation invariance at the cost of only Nth order accuracy and an or der limitation of 20. These two quadrature types reflect the trade-off while pursuing mathem atical accuracy and physical symmetry. In the TITAN code, a quadrature set can be further biased by physical concerns. We can apply the ordinate splitting technique (Chapter 2) on some di rections with more physical importance. We also developed the fictitious quadrature technique (C hapter 5), which is designed for calculating the angular fluxes in the directions with more physical interests.
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152 APPENDIX C IS FORTRAN 90/95 BETTER THAN C ++ FOR SCIENTIFIC COMPUTING? On Nov. 18, 2004, the in ternational FORT RAN standards committee (WG5) published the FORTRAN 2003 standard under the identificat ion of ISO/IEC 1539-1:2004(E), which is considered a major revision of the previous FORTRAN 95 standard. Am ong many new features in the 2003 standard are: deri ved type enhancements, object-oriented programming (OOP) support, data manipulation enhancements, and interoperability w ith the C programming language. The standard adopts some features of C++ and other modern languages and moves FORTRAN closer to C++, while trying to keep and enhance the advantages of FORTRAN in scientific computing. So me of the new features, widely applie d in other languages, could play an important role in scientific programming. The performance of scientific computer code s has significantly benefited from the fastadvancing computing technology in terms of pro cessor speed, memory limit, and the concept of parallel computing. More benefits can be obt ained with the use of the new FORTRAN 2003 and newer compliers. However, the new language feat ures need to be accepted and utilized by the scientific computing community. Although now no complier can fully support the new standard, a few compiler vendors are working on the impl ementation of FORTRAN 2003 in their compiler products gradually. Among them are the Intel FORTRAN Compiler (IFC), formerly Compaq Visual FORTRAN compiler (CVF), and Port land Group FORTRAN complier (PGF90). TITAN uses some FORTRAN 2003 features, which are mainly related to OOP and derived type enhancements. And it is originally compiled by IFC 8.1 and PGF90 6.1 in both WINDOWS and LINUX/UNIX with the same source files. As in April 2007, IFC v9.2 and PGF90 v7.0 are available in both operating systems.
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153 The performance comparison between FORTRA N 77/90/95 and C/C++ has been discussed for years. C++ and its compilers ev olve significantly over the years with a much larger user base. More and more scientific programmers consider C++ as one of their language choices. In the nuclear engineering field, however, FORTRAN still remains the first choice for two reasons. First, data abstraction penalty associated with language features such as OOP could undermine the performance of a scientific computing code. These new features are not always desirable or necessary in scientific computing as in com puter applications because of the associated overheads. Codes can be ugly in human eyes, but very desirable in machines viewpoint. Second, FORTRAN is traditionally widely used in our community with a large code base. It is not practical to rewrite the legacy codes in C/C ++ or even with a newer FORTRAN standard. It is difficult to provide a clear direct answ er to the question which language is better for scientific computing, since the results can be affected by the individua l coding practice and the compiler choice. C++ has a much richer feature set than FORTRAN. However, in scientific computing, one major concern of language choice is the array handling. He re we only provide an individual investigati on on this aspect by comparing the C ++ vector class template with its FORTRAN counterpart. We wrote two small Monte Carlo codes with th e particle splitting/rouletting technique in FORTRAN and C++. The two codes follow the same logic with the same data structure. A particle object is defined with particle positi on and direction by a class in the C++ code, and a user-defined type structure in the FORTRAN code. An array of particle objects, called particle bank, is created by vector class template in the C++ code, and by defining an allocatable array in the FORTRAN code. We compiled the two codes with Intel Fortran compiler and Inter C++ compiler. The running times for both c odes are compared in Table C-1.
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154 Table C-1. Run time comparison of th e sample FORTRAN and C++ codes. Number of Particles Run time of the FO RTRAN Code Run time of the C++ code 10 Million 7 sec 7 sec 100 Million 67 sec 64 sec According to Table C-1, there is no signifi cant performance difference between the two codes. However, it is worth noting that the size of the particle bank is requ ired to be pre-defined in the FORTRAN code to avoid memory overf low. While the C++ vector class template provides a build-in mechanism to adjust the memory buffer after the last element of the vector. User can push any number of particles into th e bank without worrying me mory overflow. It is safe to say that this mechanism in C++ vector te mplate is very efficient, since even with this overhead, the C++ code still maintains the same level performance as the FORTRAN code, at least for the relatively small size array in our co de. In handling very la rge size array, FORTRAN could have some advantages over C++, since it provides some build-in vector operation on arrays. The key to a scientific computing code is always the algorithms and the physics underneath it. However, the paradigm of the code does make a difference on performance. If some desirable and crucial features are not avai lable in FORTRAN, we should not hesitate to choose C++ or other languages.
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155 /* C++ source code for compar ing performance with FORTRAN*/ /* shielding with variance reduction 1-D slab */ /* Geometry splitting an d roulette */ /* Oct. 2005 Author :yice at ufl edu */ #include #include #include #include using std::string; using std::cin; using std::cout; using std::endl; using std::vector; /* for RN generator */ long int rn = 119; /* seed */ /* GGL RN generator */ const __int64 a = 16807; /* a=7^5 */ const __int64 c = 0; /* c =0 */ const __int64 M=2147483647; /* M=2^31-1 */ class cParticle { public: /* initial values */ cParticle(): x(0), w(1. 0), reg(1), mu(1.0) { } float x; /* position */ float w; /* weight */ int reg; /* region num */ float mu; /* direction cosine */ }; /* track one particle inside */ int TrackOne(); /* rn generator */ float MyRng(); const double sigma_t_d=10; const double sigma_st=0.2; /* num_cell : num of regions with diff. importance */ int num_cell = 6;
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156 /* bon: region boundaries */ vector bon; /* region importance */ vector imp; /* w_counter: weight counter ; w_ square: square sum (for R) ; w_one: sum of the weights of each star ting particle and its children w_xxx[0] : absorbed w_xxx[1] : back-scattered w_xxx[2] : transmitted w_xxx[3] : killed by rouletting */ vector w_counter; vector w_square; vector w_one; /* particle bank */ vector bank; /* current partile being followed */ cParticle one; const float Pi=2*asin(1.0); /* ************************************************ */ int main() { int i,j,k; int tot_part=10000000; int tot_tracked=0; float size_cell=sigma_t_d/num_cell; /* Initialize varibles */ /* erase counter */ for( i = 0; i <4; ++i) { w_counter.push_back(0.0) ; /* w_counter=0 */ w_square.push_back(0.0) ; w_one.push_back(0.0); } /* imp and bod */ imp.push_back(0); /* le ft outside imp=0 */ imp.push_back(1); /* region 1 imp=1 */ bon.push_back(0); for( i = 1; i
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157 { imp.push_back(imp[i]*2) ; /* imp=1,2,4,8,16 ..*/ bon.push_back(bon[i-1]+size_cell); /* bon=0,2,4,6,8,10 */ // imp.push_back(1.0) ; } imp.push_back(0.0); /* right outside imp=0 */ for (i = 0; i < tot_part ; ++i) { /* initial particle */ one.x=0; one.w=1.0; one.reg=1; one.mu=1.0; /* push it into bank */ bank.push_back(one); while( !bank.empty() ) { one=bank.back(); /* get th e last particle in bank */ bank.pop_back(); /* pop the last one out of bank */ ++tot_tracked; /* count tot particle tracked */ // j=bank.size(); k=TrackOne(); w_one[k]=w_one[k]+one.w; w_counter[k]= w_counter[k] + one.w; // cout << k=" << k << tot_tracked=" << tot_tracked <0 && one.reg < num_cell+1 )
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158 { eta=MyRng(); r=-log(eta); one.x=one.x+r*one.mu; while ( one.x >= bon[one.reg-1] && one.x <= bon[one.reg]) { eta=MyRng(); if (eta <= sigma_st) { /* scattered */ mu0 = 2*MyRng() 1; phi = 2*Pi*MyRng(); one.mu = one.mu*mu0 + sqrt(1-pow(one.m u,2))*sqrt(1-pow(mu0,2))*cos(phi); r=-log(MyRng() ); one.x=one.x + r one.mu ; } else /* absorbed */ { return 0; /*absorbed */ } /* end if eta */ } /* end while loop one.x */ /* cross the right region boundary */ if (one.x > bon[one.reg] ) { /* to move foward one region */ one.x=bon[one.reg++]; ir=imp[one.reg]/imp[one.reg-1]; } /* cross the left region boundary */ if (one.x < bon[one.reg-1] ) { /* to move backward one region */ one.x=bon[--one.reg]; ir=imp[one.reg]/imp[one.reg+1]; } /* splitting and rouletting */ k=int(ir); if ( ir > 1) /* splitting */ { one.w=one.w/ir; for (int j=1; j
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159 if ( MyRng() < ir-k ) bank.push_back(one); } /*end if ir greater than 1 */ if ( ir < 1 && ir > 0) /* rouletting */ { if (MyRng() < ir) { one.w=one.w/ir; } else { return 3; /* killed by rouletting */ } } /* end if ir less than 1 */ } /* end while loop one.reg */ if (one.reg <1 ) { return 1; /* back scattered */ } else { return 2; /* transmitted */ } /* end if one.reg */ } /* RN generator */ float MyRng() { rn=(a*rn + c)%M; return 1.0*rn/M; }
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160 !/* FORTRAN 90 source code for co mparing performance with C++*/ !/* shielding with variance reduction 1-D slab */ !/* Geometry splitting and roulette */ module mRNG integer :: x=119 integer*8 :: a=16807 integer*8 :: M=2_8**31-1 end module module paraset1 type tParticle real x real w integer reg real mu end type tParticle integer :: banksize=100 type(tParticle), dimension(:), allocatable :: bank type(tParticle) one integer :: top=0 end module module paraset2 real :: sigma_t_d=10 real :: sigma_st=0.2 num_cell : num of regions with diff. importance integer :: num_cell = 6 bon: region boundaries real dimension(:), allocatable :: bon region importance real, dimension(:) allocatable :: imp w_counter: weight counter ; w_ square: square sum (for R) ; !w_one: sum of the weights of each st arting particle and its children !w_xxx[0] : absorbed !w_xxx[1] : back-scattered !w_xxx[2] : transmitted
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161 !w_xxx[3] : killed by rouletting */ real :: w_counter(0:3)=0 real :: w_square(0:3)=0 real :: w_one(0:3)=0 real pi end module program shield use paraset1 use paraset2 use DFPORT integer i,k real eta integer tot_part,tot_tracked real size_cell real s1,s2 s1=secnds(0.0) pi=2*asin(1.0) tot_part=1000000 tot_tracked=0 size_cell=sigma_t_d/num_cell /* Initialize varibles */ /* erase counter */ w_counter=0 w_square=0 w_one=0 !/* imp and bod */ allocate ( imp(0:num_cell+1), bon(0:num_cell) ) imp(0)=0 !left outside imp(1)=1 !/* region 1 imp=1 */ bon(0)=0 do i = 1, num_cell imp(i+1)=imp(i)*2 /* imp=1,2,4,8,16 ..*/ bon(i)=bon(i-1)+size_cell /* bon=0,2,4,6,8,10 */ !imp(i+1)=1 enddo
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162 imp(num_cell+1)=0 /* right outside imp=0 */ allocate ( bank(banksize) ) top=0 loop_part : do i = 1, tot_part !/* initial particle */ one%x=0 one%w=1.0 one%reg=1 one%mu=1.0 /* push it into bank */ top=top+1 bank(top)=one do while( top .ne. 0 ) one=bank(top) /* get the last particle in bank */ top=top-1 /* pop the last one out of bank */ tot_tracked=tot_tracked+1 /* count tot particle tracked */ call TrackOne(k); w_one(k)=w_one(k)+one%w w_counter(k)= w_counter(k) + one%w enddo do j=0 3 w_square(j)=w_square(j) + w_one(j)**2 w_one(j)=0.0; enddo enddo loop_part write(*,"('tracked=', I0)") tot_tracked write(*,"('transmitted prob. =', ES12.5)") w_counter(2)/tot_part write(*,"('relative err. = ', ES12.5)" ) & sqrt( w_square(2)/(w_counter(2)**2-1.0/tot_part ) ) write(*,'("run time=", f10.3, "sec" ) ') secnds(s1) end program subroutine TrackOne(flag) use paraset1
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163 use paraset2 integer flag real eta, r, mu0, phi,ir,temp; integer k while_reg : do while (one%reg .gt. 0 .and. one%reg .lt. num_cell+1 ) call MyRng(eta) r=-log(eta) one%x=one%x + r*one%mu while_xr : do while ( one%x .ge. bon(one %reg-1) .and. one%x .le. bon(one%reg) ) call MyRng(eta) if (eta .le. sigma_st) then /* scattered */ call MyRng(eta) mu0 = 2*eta 1 call MyRng(eta) phi = 2*Pi*eta one%mu = one%mu*mu0 + sqrt(1-one %mu**2)*sqrt(1-mu0**2)*cos(phi); call MyRng(eta) r=-log(eta) one%x=one%x + r one%mu else /* absorbed */ flag=0 return endif enddo while_xr /* cross the righ t region boundary */ if (one%x .gt. bon(one%reg) ) then /* to move foward one region */ one%x=bon(one.reg) one%reg=one%reg+1 ir=imp(one%reg)/imp(one%reg-1) endif /* cross the left region boundary */
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164 if (one.x < bon(one%reg-1) ) then !/* to move backward one region */ one%reg=one%reg-1 one%x=bon(one%reg) ir=imp(one%reg)/imp(one%reg+1) endif /* splitting and rouletting */ k=int(ir) if ( ir .gt. 1) then !/* splitting */ one%w = one%w/ir do j=1, k-1 top=top+1 bank(top)=one enddo call MyRng(eta) if ( eta .lt. ir-k ) then top=top+1 bank(top)=one endif endif if ( ir .lt. 1 .and. ir .gt. 0) then !/* rouletting */ one%w = one%w/ir call MyRng(eta) if ( eta .gt. ir) then flag=3 return endif endif enddo while_reg if (one%reg .lt. 1 ) then flag=1 return /* back scattered */ else flag=2
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165 return /* transmitted */ end if end subroutine subroutine MyRng(rn) use mRNG real rn !x=int( mod(a*x,M), 4 ) x=mod(a*x,M) rn=1.0*x/M return end subroutine
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166 APPENDIX D TITAN I/O FILE FORMAT TITAN Input Files The TITAN code is developed based on the code base of PENMSH Express,29 which is a mesh generator I wrote for generating PENTRA N input deck. PENMSH Express, or PENMSH XP, follows a similar input syntax with PENMSH.28 Therefore, TITAN inherits most of the PENMSH input file format. Table D-1 lis ts the input files of the TITAN code. Table D-1. TITAN input file list. File # File Name Description Memo 1 penmsh.inp Meshing parameters Required 2 prbname#.inp Meshing per z level Required 3 prbname.src Fixed source grid Optional 4 prbname.spc Source spectrum Optional 5 prbname.chi Fission spectrum Optional 6 prbname.mba Material balance Optional 7 bonphora.inp General input parameters Required 8 prbname.xs Cross section data Required Input files #1 to #6 are general PENMSH input files, which define model geometries and source specifications. We use prbname to denote different problem names. General meshing parameters are specified in the penmsh.inp, including number of z levels, z -level boundaries, etc. Geometries on each z level are specified in a separate file (Input file 2). For example, prbname1.inp, prbname2.inp,. These input files can describe various geometries with the overlay feature. Figure D-1 shows th e geometries generated by a sample z -level input file. The fixed source grid can be defined in the prbname.src file. prbname.spc and prbname.chi specify the source and fission spectrum, respectively. And prbname.mba is used to check the model material balance. More details on Input files #1 to #6 can be found in the manuals of PENMSH and PENMSH XP. And we will further di scuss input file #7 in the next section.
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167 Figure D-1. A 3 by 3 coarse mesh model on one z level. Bonphora.inp Input File Input file 7 ( bonphora.inp) is special file used by TITAN only, which specifies parameters for transport calculations, such as the quadrature set, differencing scheme, solver, etc. The file supports as many as 4 sections. The following is a sample bonphora.inp file. Figure D-2. A sample bonphora.inp input file. / bonphora.inp: TITAN input file to define transport parameters #0 Section 0: Global varibles 2 0 /# of quadrature, global DS id #1 Section 1: Quadrature sets /quad 1 Pn-Tn / Quadrature id order, num of split directions 1 20 2 /spilited directions 46 47 /direction index 11 11 /splitted order 1 1 /splitted id : 1pn-tn splitting 2 2 /# of directions on the top level /quad 2 level symmetric / Quadrature id order,num of split directions 0 20 1 /spilited directions ids 37 /direction index 8 /splitted order 0 /splitted id : 0rectangular splitting 0 / not used #2 Section 2: Coarse mesh specifications /Solver_id 0 1 0 /qudra_id 1 2 1 /Diff scheme 1 1 2
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168 Section 0 is dedicated to specify two paramete rs: total number of quadrature sets used in the model, and the global differencing scheme id number, which define the differencing scheme for all coarse meshes if the number is a positive integer ( id=1 diamond with zero fix-up; id=2 Directional Theta-Weighted). If zero is given as the global diffe rencing scheme id, an additional card is required to specify an individual differencing scheme for each coarse mesh. Section 1 is used to define all the quadrature sets used in the model. In this sample input file, two quadrature set are specified. The first one is a PN-TN quadrature (quadrature id=1) with an order of 20. The PN-TN splitting technique is applied on two directions in the quadrature set (direction index number: 46 and 47). The second one is a level-symmetric quadrature set with rectangular splitting on Direction 37. Section 2 specifies the parameters for each coarse mesh. In this sample file, the SN solver will be used for coarse meshes #1 and #3 (solver id=0). Coarse mesh #2 uses the characteristics solver. Quadrature set #1 specified in Secti on 1 is applied in coarse meshes #1 and #3. Quadrature set #2 is used in coarse mesh #2. Another section can be used to specify the iteration number limitations and tolerances, especially for eigenvalue problems. The follo wing is the input f ile for the C5G7 MOX benchmark problem. Figure D-3. C5G7 MOX benchmark problem bonphora.inp input file. / bonphora.inp: TITAN input file to define transport parameters #0 Section 0: Global varibles 1 1 /# of quadrature, global DS id #1 0 6 0 #3 Iteration parameters /tolout ,tolin 1.0e-5 1.0e-3 /outer,inner -50 10 /rkdef 10
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169 In this model, an S6 level-symmetric quadrature is us ed with the diamond differencing scheme. The SN solver is applied on all the coarse meshes (SN solver is the default solver). Section 3 specifies some iteration parameters. Th e outer iteration tolerance is 1.0E-05 (variable tolout ). And the inner iteration to lerance is 1.0E-03(variable tolin ). Note if tolin is less than zero, the adaptive inner loop to lerance control will be engaged. The iteration number limitations are defined in the next card. The outer and inner iteration limits ar e 50 and 10 respectively. Negative numbers means the limitations are adaptive. The last card defines the init ial guess of eigenvalue. Aitken extrapolation37 is used on k-effective if users specify a negative initial guess. TITAN can automatically convert a digital phantom into a tran sport calculation model. We use this feature for the SPECT benchmark problem. Th e input file format is slightly different for a medical phantom model. Details can be found in the PENMSH XP manual. TITAN Output Files and TECPLOT Visualization Table D-2 list the major output f iles of the TITAN code. The first file is an optional output, which contains a generated PENTRAN inputd deck. The second output file is a report of material balance check. The third file, bonphora.log, is the input processing l og. And the solver log is stored in file prbname_solver.log, which records all the iteration output. Table D-2. TITAN output file list. File # File Name Description 1 Prbname_out.f90 2 prbname_out.mba Material balance tables 3 Bonphora.log Processing log file 4 Prbname_solver.log Solver log file 5 prbname_mix.plt. TECPLOT binary file, contains all the calculation data 6 prbname.mcr TECPLOT macro file The last two files are used for visualizati on of the calculation results with the TECPLOT software. A TECPLOT I/O library is developed and included in the TITAN code. The library, composed of about 15 subroutines and modules, can generate TECPLOT binary data files as
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170 many as necessary simultaneously. Some other TITAN output files, including the quadrature data file and the optional boundary angular flux file s when a fictitious quadrature set is used, are also generated by this library. The last file in Table D-2, prbname.mcr is a macro file, which can be loaded by TECPLOT, to help organize the data in prbname_mix.plt TECPLOT also provides an IO library (without source codes) for users to generate their own binary data files. However, for practical reasons, here we wrot e our own version of TECPLOT IO library, which is optimized for our purpose. TECPLOT is an excellent visualization tool. However, it is a commercial software package. We consider migrating to the widely used visualization toolkit (VTK) platform which is an open source library for scientific visualization. A number of front end software packages (e.g. PARAVIEW) are freely available to visualize the VTK format data file. TITAN Command Line Option The common command line option is -i option, which specifies the directories where the input files are located. The default input directory is the current one. [home/user/]# bonphora i test The above command line reads input decks from the /home/user/test directory. Other command options can be found in the PENMSH XP manual. Users can add their own modules and subroutines to extract the interested data fr om the calculation result s. All the post-processing subroutines are called from a container subroutine named Nirvana The userdefined postprocessing routines can be triggered with a comm and line option with slight modification of the code. For example, the option -mox will trigger the C5G7 MOX post-processing subroutines. These subroutines are used to calculate the fuel pin powers based on the c onverged scalar fluxes.
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171 LIST OF REFERENCES 1. B. V. ALEXEEV, Ge neralized Boltzmann Physical Kinetics, Elsevier Science Publishing (2004). 2. E. E. LEWIS and W. F. MILLER, Computational Method of Neutron Transport John Wiley & Sons, New York (1984). 3. G. I. BELL, and S. GLASSTONE, Nuclear Reactor Theory Robert E. Krieger Publishing, Malabar, FL (1985). 4. J. J. DUDERSTADT and L. J. HAMILTON, Nuclear Reactor Analysis, 1st ed., John Wiley & Sons, New York (1976). 5. B. G. CARLSON and K.D. LATHROP, Dis crete Ordinates Angular Quadrature of the Neutron Transport Equation, LA-3186, Los Alamos Nationa l Laboratory (1965). 6. J. R. ASKEW, A Characteristics Formul ation of the Neutron Transport Equation in Complicated Geometries, AEEW-M1108, United Kingdom Atomic Energy Authority (UKAEA), Winfrith (1972). 7. M. D. BROUGH and C.T. CHUDLEY, Char acteristic Ray Solution of the Transport Equation, Advances in Nuclear Science and Technology Yearbook (1980). 8. S. G. HONG and N. Z. CHO, CRX: A Code for Rectangular and Hexagonal Lattices Based on the Method of Characteristics, Ann. Nucl. Energy, 25, 547 (1998). 9. M. HURISN and T. JEVREMOVIC, AGENT Code Neutron Transport Benchmark Example and Extension to 3D Lattice Geometry, Nuclear Technology and Radiation Protection XX, 10 (2005). 10. R. ROY, Large-Scale 3D Characteri stics Solver: Can the Dream Live On? Proc. Int. Conf. on Mathematics and Computation (M&C 2005) Avignon, France, American Nuclear Society (2005). 11. N. Z. CHO, G. S. LEE, and C. J. PARK Fusion of Method of Characteristics and Nodal Method for 3-D Whole Core Transport Calculation, Trans. Am. Nucl. Soc., 86, 322 (2002). 12. K. D. LATHROP, Remedies for Ray Effects, Nucl. Sci. Eng. 45, 255 (1971). 13. G. E. SJODEN and A. HAGHIGHAT, PENTRAN Parallel Enviroment Neutral Particle TRANsport in 3-D Cartesian Geometry, Proc. Int. Conf. on Ma thematical Methods and Supercomputing for Nuclear Applications (M&C 1997) Saratoga Springs, NY, American Nuclear Society (1997). 14. K. D. LATHROP, Spatial Differencing of th e Transport Equation: Positivity vs. Accuracy, J. Comput. Phys., 4, 475 (1969).
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172 15. G. E. SJODEN and A. HAGHIGHAT, PENTRAN : Parallel Environment Neutral-particle TRANsport SN in 3-D Cartesian Geometry User Guid e Version 9.30c, University of Florida (2004). 16. B. PETROVIC and A. HAGHIGHAT, Ana lysis of Inherent Oscillations in Multidimensional SN Solutions of the Neutron Transport Equation, Nucl. Sci. Eng. 124, 31 (1996). 17. A. M. KIRK, On the Propagation of Rays in Discrete Ordinates, Nucl. Sci. Eng. 132, 155 (1999). 18. W. RHOADES and W. ENGLE, A New Weighted Differen ce Formulation for Discrete Ordinates Calculations, Trans. Am. Nucl. Soc., 27, 776 (1977). 19. B. PETROVIC and A. HAGHIGHAT, N ew Directional Theta-Weighted SN Differencing Scheme, Trans. Am. Nucl. Soc. 73, 195 (1995). 20. G. E. SJODEN and A. HAGHIGHAT, The Exponential Directional Weighted (EDW) Differencing Scheme in 3-D Cartesian Geometry, Proc. Int. Conf. on Mathematical Methods and Supercomputing for Nuclear Applications (M&C 1997) Saratoga Springs, NY, American Nuclear Society (1997). 21. G. E. SJODEN, An Efficient Exponential Dir ectional Iterative Differe ncing Scheme for 3-D SN Computations in XYZ Geometry, Nucl. Sci. Eng. 155, 179 (2007). 22. W. T. VETTERLING and B. P. FLANNERY, Numerical Recipes in C++: the Art of Scientific Computing, Cambridge University Press (2002). 23. B. G. CARLSON, Transport Theory: Discrete Ordinates Quadrature over the Unit Sphere, LA-4554, Los Alamos Nati onal Laboratory (1970). 24. G. LONGONI, Advanced Quadrature Sets, Acceleration and Preconditioning techniques for the Discrete Ordinates Method in Parallel Computing Environments, PhD Thesis, University of Florida (2004). 25. G. LONGONI and A. HAGHIGHAT, Developm ent of New Quadrature Sets with the Ordinate Splitting Technique, Proc. Int. Conf. on Mathemat ical Methods and Supercomputing for Nuclear Applications (M&C 2001), Salt Lake City, UT, American Nuclear Society (2001). 26. G. LONGONI and A. HAGHIGHAT, Developm ent of the Regional Angular Refinement and Its Application to the CT-Scan Device, Trans. Am. Nucl. Soc., 86, 246 (2002). 27. A. M. WEINBERG and E. P. WIGNER, Physical Theory of Neutron Chain Reactors, University of Chicago Press (1958).
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173 28. A. HAGHIGHAT, A Manual of PENMSH Version 5 A Cartesian-Based 3-D Mesh Generator, University of Florida (2004). 29. C. YI, PENMSH XP manual: A Mesh Gene rator to Build PENTRAN Input Deck with Compatibility to PENMSH, University of Florida (2007). 30. J. E. WHITE et al., Bugl e 96: Coupled 47 Neutron, 20 Gamma-ray Group Cross-Section Library Derived from ENDF/B-VI for the LW R Shielding and Pressure Vessel Dosimetry Applications Oak Ridge Na tional Laboratory (1996). 31. X-5 Monte Carlo Team, MCNP-A General Monte Carlo Code for Neutron and Photon Transport, Version 5, Los Alam os National Laboratory (2003). 32. J. C. WAGNER et al., MCN P: Multigroup/Adjoint Capabil ities, Los Alamos National Laboratory (1994). 33. K. KOBAYASHI, N. SUGIMURA, and Y. NAGAYA, -D Radiation Transport Benchmarks for Simple Geometries with Void Regions, OECD/NEA (2000). 34. A. HAGHIGHAT, G. E. SJODEN, and V. KUCUKBOYACI, Effectiv eness of PENTRAN's Unique Numerics for Simulation of the Kobayashi Benchmarks, Prog. Nucl. Energy 39, 191 (2001). 35. E. E. LEWIS et al., Benchmark Specification for Deterministic 2-D/3-D MOX Fuel Assembly Transport Calculations without Sp atial Homogenization (C 5G7 MOX), OECD/NEA (2001). 36. E. E. LEWIS et al., Proposal for Extended C5G7 MOX Benchmark, OECD/NEA (2002). 37. K. ATKINSON, An Introduction to Numerical Analysis, 2nd ed., John Wiley & Sons, New York (1989). 38. W. P. SEGARS, Development and appli cation of the new dynamic NURBS-based cardiactorso (NCAT) phantom, PhD Thesis, University of Nort h Carolina (2001). 39. L. J. LORENCE, J. E. MOREL, and G. D. VALDEZ, User's Guide to CEPXS/ONELD: A One-Dimensional Coupled Electron-Photon Discre te Ordinates Code Package, Sandia National Laboratory (1989). 40. M. LJUNGBERG, S. STRAND, and M. A. KING, The SIMIND Monte Carlo program: Monte Carlo Calculation in Nuclear Medicine, Applications in Diagnostic Imaging 11, 145 (1998). 41. A. YAMAMOTO, Generalized Coarse-M esh Rebalance Method for Acceleration of Neutron Transport Calculations, Nucl. Sci. Eng. 151, 274 (2005).
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174 42. J. S. WARSA, T. A. WARE ING, and J. E. MOREL, Kry lov Iterative Methods and the Degraded Effectiveness of Diffusion Synt hetic Acceleration for Multidimensional SN Calculations in Problems with Material Discontinuities, Nucl. Sci. Eng. 147, 218 (2004). 43. V. KUCUKBOYACI and A. HAGHIGHAT, Angular Multigrid Ac celeration for Parallel SN Method with Application to Shielding Problems, Proc. Int. Conf. on Advances in Reactor Physics and Mathematics and Computation into the Next Millennium (PHYSOR 2000), Pittsburgh, PA, American Nuclear Society (2000). 44. P. NOWAK, E. LARSEN, and W. MARTIN, Multigrid Methods for SN Problems, Trans. Am. Nucl. Soc. 55, 355 (1987). 45. Y. SAAD, Numerical Methods for Large Eigenvalue Problems, John Wiley & Sons, New York (1992). 46. Y. SAAD, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003) 47. A. HAGHIGHAT, M. HUNTER, and R. MATTIS, Iterative Schemes for Parallel SN Algorithms in a Shared Memory Computing Environment, Nucl. Sci. Eng. 121, 103 (1995). 48. A. HAGHIGHAT, G. E. SJODEN, and M. HUNTER, Parallel Algorithms for the Linear Boltzmann Equation Complete Phase Space Decomposition, Society for Industrial and Applied Mathematics (SIAM) Annual Meeting Kansas City, MO (1996). 49. G. ARFKEN, Mathematical Methods for Physicists, Academic Press, New York (1970). 50. L. I. SCHIFF, Quantum Mechanics, McGraw-Hill, New York (1968). 51. E. W. HOBSON, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press (1931).
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175 BIOGRAPHICAL SKETCH I was born in 1973 in Anshan, China. I went to Tsinghua University in 1992 and got my bachelors degree in nuclear e ngineering in 1997. I continued on to the graduate school at Tsinghua, and graduated with a masters degree in nuclear engineering in 2000. The same year, I went to Penn State University to pursue a doctoral degree. In 2001, I followed Dr. Haghighat to the University of Florida. The goal of my study was to develop a hybrid algorithm to solve the LBE efficiently in low-scattering media and to enha nce the efficiencies of the PENTRAN code in medical applications. I started to write a 3-D SN kernel in April 2005 from the PENMSH XP code base, which is a mesh generator I wrot e for preparing PENTRAN input deck. The 3-D SN code is originally designed as a test platform for the hybrid algorithm. By the summer of 2005, I completed the initial versions of both the SN and characteristics solvers. In the summer, I dedicated most of the time to the University of Florida Training Reactor (UFTR) fuel conversion project. After that summer, I continued to wo rk on the code and implemented a number of techniques, including PN-TN quadrature set, PN-TN ordinate splitting, an d projection techniques. By April 2006, the framework of the code is comp leted. In the second half of 2006, I worked on the integration of characteristics solver into PENTRAN. In the firs t quarter of 2007, the fictitious quadrature technique is develope d for the heart phantom benchmark. And some studies on the limitations of the hybrid algorithms are performed.