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Advanced quadrature sets, acceleration and preconditioning techniques for the discrete ordinates method in parallel comp...

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PAGE 1

ADVANCED QUADRATURE SETS, ACCELERATION AND PRECONDITIONING TECHNIQUES FOR THE DISCRETE ORDINATES METHOD IN PARALLEL COMPUTI NG ENVIRONMENTS By GIANLUCA LONGONI 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 2004

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Copyright 2004 by GIANLUCA LONGONI

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I dedicate this research work to Rossana and I thank her for the support and affection demonstrated to me during these years in co llege. This work is dedicated also to my family, and especially to my father Gi ancarlo, who always shared my dreams and encouraged me in pursuing them. “Only he who can see the invi sible, can do the impossible.” By Frank Gaines

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iv ACKNOWLEDGMENTS The accomplishments achieved in this research work would have not been possible without the guidance of a mentor such as Prof. Alireza Haghighat; he has been my inspiration for achieving what nobody else has d one before in the radiation transport area. I wish to thank Prof. Glenn E. Sjoden for his endless help and moral support in my formation as a scientist. I also express my gratitude to Dr. Alan D. George, for providing me with the excellent comput ational facility at the Hi gh-Performance Computing and Simulation Research Lab. I am thankful to Prof. Edward T. Dugan for his useful suggestions and comments, as well as to Dr. Ray G. Gami no from Lockheed Martin KAPL and Dr. Joseph Glover, for being part of my Ph.D. committee. I also thank UFTTG for the interesting conversations regarding radiation transport physics and computer science. I am also grateful to the U.S. DOE Nuclear Education E ngineering Research (NEER) program, the College of Engineer ing, and the Nuclear and Radiological Engineering Department at the University of Florida, for supporting the development of this research work.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES...........................................................................................................xi ABSTRACT.....................................................................................................................xv i CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Overview............................................................................................................1 1.2 The Linear Boltzmann Equation........................................................................1 1.3 Advanced Angular Quadrature Sets for the Discrete Ordinates Method...........2 1.4 Advanced Acceleration Algorithms for the SN Method on Parallel Computing Environments.....................................................................................................5 1.5 The Even-Parity Simplified SN Method.............................................................6 1.6 A New Synthetic Acceleration Algorithm Based on the EP-SSN Method......10 1.7 An Automatic Preconditioning Algorithm for the SN Method: FAST (Flux Acceleration Simplified Transport).................................................................12 1.8 Outline..............................................................................................................13 2 THE DISCRETE ORDINATES METHOD...............................................................14 2.1 Discrete Ordinates Method (SN)......................................................................14 2.1.1 Discretization of the Energy Variable.....................................................14 2.1.2 Discretization of the Angular Variable...................................................17 2.1.3 Discretization of the Spatial Variable.....................................................20 2.1.4 Differencing Schemes.............................................................................21 2.1.4.1 Linear-Diamond Scheme (LD)......................................................22 2.1.4.2 Directional Theta-Weighted Scheme (DTW)...............................23 2.1.4.3 Exponential Directional-Weighted Scheme (EDW).....................24 2.1.5 The Flux Moments..................................................................................25 2.1.6 Boundary Conditions..............................................................................25 2.2 Source Iteration Method..................................................................................25 2.3 Power Iteration Method...................................................................................26 2.4 Acceleration Algorithms for the SN Method....................................................27

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vi 3 ADVANCED QUADRATURE SETS FOR THE SN METHOD..............................29 3.1 Legendre Equal-Weight (PN-EW) Quadrature Set..........................................30 3.2 Legendre-Chebyshev (PN-TN) Quadrature Set.................................................31 3.3 The Regional Angular Refi nement (RAR) Technique....................................33 3.4 Analysis of the Accuracy of the PN-EW and PN-TN Quadrature Sets..............34 3.5 Testing the Effectiveness of the New Quadrature Sets....................................38 3.5.1 Kobayashi Benchmark Problem 3..........................................................38 3.5.2 CT-Scan Device for Medical/Indus trial Imaging Applications..............43 4 DERIVATION OF THE EVEN -PARITY SIMPLIFIED SN EQUATIONS.............47 4.1 Derivation of the Simplified Spherical Harmonics (SPN) Equations...............48 4.2 Derivation of the Even-Parity Simplified SN (EP-SSN) Equations..................51 4.2.1 Boundary Conditions for the EP-SSN Equations....................................55 4.2.2 Fourier Analysis of the EP-SSN Equations.............................................56 4.2.3 A New Formulation of the EP-SSN Equations for Improving the Convergence Rate of the Source Iteration Method.................................59 4.3 Comparison of the P1 Spherical Harmonics and SP1 Equations......................60 5 NUMERICAL METHODS FOR SOLVING THE EP-SSN EQUATIONS...............65 5.1 Discretization of the EP-SSN Equations Using the Finite-Volume Method....65 5.2 Numerical Treatment of the Boundary Conditions..........................................72 5.3 The Compressed Diagon al Storage Method....................................................74 5.4 Coarse Mesh Interface Projection Algorithm..................................................75 5.5 Krylov Subspace Iterative Solvers...................................................................80 5.5.1 The Conjugate Gradient (CG) Method...................................................82 5.5.2 The Bi-Conjugate Gradient Method.......................................................83 5.5.3 Preconditioners for Krylov Subspace Methods......................................84 6 DEVELOPMENT AND BENCHMARKI NG OF THE PENSSn CODE..................86 6.1 Development of the PENSSn (Paral lel Environment Neutral-particle Simplified Sn) Code.........................................................................................87 6.2 Numerical Analysis of Krylov Subspace Methods..........................................92 6.2.1 Coarse Mesh Partitioning of the Model..................................................92 6.2.2 Boundary Conditions..............................................................................95 6.2.3 Material Heterogeneities.........................................................................96 6.2.4 Convergence Behavior of Higher EP-SSN Order Methods.....................97 6.3 Testing the Incomplete Cholesky C onjugate Gradient (ICCG) Algorithm.....99 6.4 Testing the Accuracy of the EP-SSN Method................................................100 6.4.1 Scattering Ratio.....................................................................................100 6.4.2 Spatial Truncation Error.......................................................................103 6.4.3 Low Density Materials..........................................................................104 6.4.4 Material Discontinuities........................................................................108 6.4.5 Anisotropic Scattering..........................................................................111

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vii 6.4.6 Small Light Water Reactor (LWR ) Criticality Benchmark Problem....115 6.4.7 Small Fast Breeder Reactor (F BR) Criticality Benchmark Problem....120 6.4.8 The MOX 2-D Fuel Assembly Benchmark Problem............................124 7 PARALLEL ALGORITHMS FOR SOLVING THE EP-SSN EQUATIONS ON DISTRIBUTED MEMORY ARCHITECTURES...................................................128 7.1 Parallel Algorithms for the PENSSn Code....................................................128 7.2 Domain Decomposition Strategies................................................................130 7.2.1 Angular Domain Decomposition..........................................................130 7.2.2 Spatial Domain Decomposition............................................................131 7.2.3 Hybrid Domain Decomposition............................................................131 7.3 Parallel Performance of the PENSSn Code...................................................132 7.4 Parallel Performance of PENSSn Ap plied to the MOX 2-D Fuel Assembly Benchmark Problem.......................................................................................139 8 DEVELOPMENT OF A NEW SYNTH ETIC ACCELERATION METHOD BASED ON THE EP-SSN EQUATIONS..............................................................................140 8.1 The EP-SSN Synthetic Acceleration Method.................................................141 8.2 Spectral Analysis of the EP-SSN Synthetic Acceleration Method.................145 8.3 Analysis of the Algorithm Stab ility Based on Spatial Mesh Size.................148 8.3.1 Comparison of the EP-SSN Synthetic Acceleratio n with the Simplified Angular Multigrid Method.......................................................................150 8.4 Limitations of the EP-SSN Synthetic Acceleration Method..........................153 9 FAST: FLUX ACCELERATION SI MPLIFIED TRANSPORT PRECONDITIONER BASED ON THE EP-SSN METHOD..................................154 9.1 Development and Implementation of FAST................................................154 9.2 Testing the Performance of the FAST Preconditioning Algorithm.............157 9.2.1 Criticality Eigenvalue Problem.............................................................157 9.2.2 Fixed Source Problem...........................................................................159 9.3 The MOX 3-D Fuel Assembly Benchmark Problem.....................................161 9.3.1 MOX 3-D Unrodded Configuration......................................................162 9.3.2 MOX 3-D Rodded-A Configuration.....................................................165 9.3.3 MOX 3-D Rodded-B Configuration.....................................................167 10 SUMMARY, CONCLUSION, AND FUTURE WORK.........................................171 APPENDIX A EXPANSION OF THE SCATTERING TERM IN SPHERICAL HARMONICS..175 B PERFORMANCE OF THE NEW EP-SSN FORMULATION................................177 LIST OF REFERENCES.................................................................................................180

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viii BIOGRAPHICAL SKETCH...........................................................................................185

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ix LIST OF TABLES Table page 3-1. Even-moments obtained with a PN-EW S30 quadrature set.......................................35 3-2. Even-moments obtained with a PN-TN S30 quadrature set.........................................36 3-3. CPU time and total number of directi ons required for the CT-Scan simulation........45 6-1. Comparison of number of iterations re quired to converge for the CG and Bi-CG algorithms.................................................................................................................93 6-2. Number of Krylov iterati ons required to converge fo r the CG and Bi-CG algorithms with different boundary conditions..........................................................................95 6-3. Number of Krylov iterati ons required to converge fo r the CG and Bi-CG algorithms for heterogeneous the box in a box problem............................................................96 6-4. Number of Krylov iterati ons required to converge fo r the CG and Bi-CG algorithms for the EP-SS8 equations..........................................................................................97 6-5. Number of iterations for the ICCG and CG algorithms.............................................99 6-6. Two groups cross-sectio ns and fission spectrum.....................................................106 6-7. Comparison of keff obtained with the EP-SSN method using DFM versus PENTRAN* S6 (Note that DFM=1.0 implies no cross-sections scaling)...................................106 6-8. Balance tables for the EP-SSN and S16 methods and relative di fferences versus the S16 solution.............................................................................................................111 6-9. Integral boundary leakage for the EP-SSN and S16 methods and relative differences versus the S16 solution............................................................................................111 6-10. Fixed source energy spectrum and energy range....................................................112 6-11. Maximum and minimum re lative differences versus the S8 method for energy group 1 and 2..........................................................................................................113 6-12. Two-group cross-sections for the small LWR problem.........................................116 6-12. Two-group cross-sections for the small LWR problem (Continued).....................116

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x 6-13. Fission spectrum and energy range for the small LWR problem...........................117 6-14. Criticality eigenvalues cal culated with different EP-SSN orders and relative error compared to Monte Carlo predictions....................................................................117 6-15. CRWs estimated with the EP-SSN method.............................................................118 6-16. Criticality eigenvalues for the small FBR model...................................................122 6-17. CRWs estimated with the EP-SSN and Monte Carlo methods...............................122 6-18. Criticality eigenvalues and relative er rors for the MOX 2-D benchmark problem.125 7-1. Data relative to the load imbala nce generated by the Krylov solver........................136 7-2. Parallel performance data obtained on PCPENII Cluster.........................................138 7-3. Parallel performance data obtained on Kappa Cluster.............................................138 7-4. Parallel performance data for the 2D MOX Fuel Assembly Benchmark problem (PCPENII Cluster)..................................................................................................139 8-1. Spectral radius for the di fferent iterative methods...................................................147 8-2. Comparison of the number of inner iteration between EP-SSN synthetic methods and unaccelerated transport...........................................................................................149 9-1. Criticality eigenvalues obtained with the preconditi oned PENTRAN-SSn code for different EP-SSN orders..........................................................................................159 9-2. Results obtained for the MOX 3D in the Unrodded configuration.........................162 9-3. Results obtained for the M OX 3-D Rodded-A configuration..................................165 9-4. Results obtained for the M OX 3-D Rodded-B configuration...................................167 B-1. Performance data for the standard EP-SSN formulation..........................................177 B-2. Performance data for the new EP-SSN formulation.................................................177 B-3. Ratio between Krylov iterati ons and inner iterations..............................................178 B-4. Inner iterations and tim e ratios for different SSN orders..........................................178

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xi LIST OF FIGURES Figure page 2-1. Cartesian space-angle coordinates system in 3-D geometry......................................16 2-2. Point weight arrangement for a S8 level-symmetric quadrature set...........................19 2-3. S20 LQN quadrature set...............................................................................................20 3-1. S28 PN-EW quadrature set...........................................................................................31 3-2. S28 PN-TN quadrature set.............................................................................................33 3-3. PN-TN quadrature set (S16) refined with the RAR technique......................................34 3-4. Configuration of the test problem fo r the validation of the quadrature sets...............37 3-5. Relative difference between the currents Jx and Jz for the test problem.....................37 3.6. Mesh distribution for the Kobayashi be nchmark problem 3: A) Variable mesh distribution; B) Unifor m mesh distribution..............................................................39 3-7. Comparison of S20 quadrature sets in zone 1 at x =5.0 cm and z =5.0 cm...................40 3-8. Comparison of S20 quadrature sets in zone 2 at y =55.0 cm and z =5.0 cm.................40 3-9. Comparison of PN-EW quadrature sets for different SN orders in zone 1 at x =5.0 cm and z =5.0 cm............................................................................................................41 3-10. Comparison of PN-EW quadrature sets for different SN orders in zone 2 at y =55.0 cm and z =5.0 cm.......................................................................................................41 3-11. Comparison of PN-TN quadrature sets for different SN orders in zone 1 at y =5.0 cm and z =5.0 cm............................................................................................................42 3-12. Comparison of PN-TN quadrature sets for different SN orders in zone 2 at y =55.0 cm and z =5.0 cm............................................................................................................42 3-13. Cross-sectional view of the CT-Scan model on the x y plane..................................43 3-14. Scalar flux di stribution on the x y plane obtained with an S20 level-symmetric quadrature set...........................................................................................................44

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xii 3-15. Scalar flux di stribution on the x y plane obtained with an S50 PN-TN quadrature set.44 3-16. Scalar flux di stribution on the x y plane obtained with an S30 PN-TN quadrature set biased with RAR......................................................................................................45 3.17. Comparison of the scalar flux at detector position ( x =72.0 cm)...............................46 5.1. Fine mesh representation on a 3-D Cartesian grid......................................................67 5.2. View of a fine mesh along the x -axis..........................................................................68 5.3. Representation of a coarse mesh interface..................................................................76 5.4. Representation of the interface projection algorithm between two coarse meshes....79 6-1. Description of PENSSn input file...............................................................................88 6-2. Flow-chart of the PENSSn code.................................................................................90 6-2. Flow-chart of the PENSSn code (Continued).............................................................91 6-3. Configuration of the 3-D test problem.......................................................................92 6-4. Convergence behavior of the CG al gorithm for the non-partitioned model...............94 6-5. Heterogeneous configurati on for the 3-D test problem..............................................96 6-6. Distribution of eigenvalues for the EP-SS8 equations................................................98 6-7. Configuration of the 2-D cr iticality eigenvalue problem.........................................101 6-8. Criticality eigenvalues as a function of the scattering ratio (c) for different methods.101 6-9. Relative difference for criticality ei genvalues obtained with different EP-SSN methods compared to the S16 solution (PENTRAN code).....................................102 6-10. Plot of criticality eigenvalu es for different mesh sizes...........................................103 6-11. Plot of the relative difference of the EP-SSN solutions versus transport S16 for different mesh sizes................................................................................................104 6-12. Uranium assembly test problem view on the x y plane..........................................105 6-13. Relative difference of physical quantitie s of interest calculated with EP-SSN method compared to the S6 PENTRAN solution................................................................107 6-14. Convergence behavior of the PENSSn with DFM=100.0 and PENTRAN S6.......108 6-15. Geometric and material configur ation for the 2-D test problem............................109

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xiii 6-16. Scalar flux distributi on at material interface ( y =4.84 cm)......................................109 6-17. Relative difference versus S16 calculations at material interface ( y =4.84 cm).......110 6-18. Fraction of scalar flux values within different ranges of relative difference (R.D.) in energy group 1........................................................................................................112 6-19. Fraction of scalar flux values within different ranges of relative difference (R.D.) in energy group 2........................................................................................................113 6-20. Front view of the relati ve difference between the scalar fluxes obtained with the EP-SS8 and S8 methods in energy group 1.............................................................114 6-21. Rear view of the relative difference be tween the scalar fluxes obtained with the EPSS8 and S8 methods in energy group 1...................................................................115 6-22. Model view on the x y plane. A) view of the model from z =0.0 cm to 15.0 cm, B) view of the model from z =15.0 cm to z =25.0 cm...................................................115 6-23. Model view on the x z plane...................................................................................116 6-24. Normalized scalar flux for case 1, in group 1 along the x -axis at y =2.5 cm and z =7.5 cm...........................................................................................................................11 8 6-25. Scalar flux distributions A) Case 1 energy group 1, B) Case 2 energy group 1, C) Case 1 energy group 2, D) Case 2 energy group 2.................................................119 6-26. View on the x y plane of the small FBR model......................................................120 6-27. View on the x z plane of the small FBR model......................................................121 6-28. Scalar flux distributi on in energy group 1: A) Case 1; B) Case 2..........................123 6-29. Scalar flux distributi on in energy group 4: A) Case 1; B) Case 2..........................123 6-30. Mesh distribution of the MOX 2-D Fuel Assembly Benchmark problem.............124 6-31. Scalar flux distribution for the 2-D MOX Fuel Assembly benchmark problem (EPSS4): A) Energy group 1; B) Energy group 2; C) Energy group 3; D) Energy group 4; E) Energy group 5; F) Ener gy group 6; G) Energy group 7..............................126 6-32. Normalized pin power distribution fo r the 2-D MOX Fuel Assembly benchmark problem (EP-SS4): A) 2-D view; B) 3-D view.......................................................127 7-1. Hybrid decomposition for an EP-SS6 calculation (3 directions) for a system partitioned with 4 coarse meshes on 6 processors..................................................132 7-2. Speed-up obtained by running PENSSn on the Kappa and PCPENII Clusters........134

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xiv 7-3. Parallel efficiency obtained by running PENSSn on the Kappa and PCPENII Clusters...................................................................................................................135 7-4. Angular domain decompos ition based on the automatic load balancing algorithm.137 7-5. Parallel fraction obtained with the PENSSn code....................................................137 8-1. Spectrum of eigenvalues for the Source Iteration and Synthetic Methods based on different SSN orders................................................................................................147 8-2. Number of inner iterations required by each acceleration method as a function of the mesh size................................................................................................................149 8-3. Number of inner iterations as a func tion of the scattering ratio (DZ differencing scheme)...................................................................................................................150 8-4. Number of inner iterations as a functi on of the scattering ratio (DTW differencing scheme)...................................................................................................................151 8-5. Number of inner iterations as a functi on of the scattering ratio (EDW differencing scheme)...................................................................................................................151 8-6. Number of inner it erations for the EP-SS2 synthetic method obtained with DZ, DTW, and EDW differencing schemes.............................................................................152 9-1. Card required in PENTRAN-SS n input deck to initiate SSN preconditioning.........155 9-2. Flow-chart of the PENTRAN-SSn Code System.....................................................156 9-3. Ratio of total number of inner iterati ons required to solv e the problem with preconditioned PENTRAN-SSn and non-preconditioned PENTRAN..................158 9-4. Relative change in flux value in group 1..................................................................160 9-5. Relative change in flux value in group 2..................................................................160 9-6. Behavior of the criticality eigenvalue as a function of the outer iterations..............163 9-7. Convergence behavior of the criticality eigenvalue.................................................164 9-8. Preconditioning and transport calculation phases with relative computation time required...................................................................................................................164 9-9. Behavior of the criticality eigenvalue as a function of the outer iterations..............165 9-10. Convergence behavior of the criticality eigenvalue...............................................166 9-11. Preconditioning and transport calculation phases with relative computation time required...................................................................................................................167

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xv 9-12. Behavior of the criticality eigenvalu e as a function of th e outer iterations............168 9-13. Convergence behavior of the criticality eigenvalue...............................................169 9-14. Preconditioning and transport calculation phases with relative computation time required...................................................................................................................169

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xvi 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 ADVANCED QUADRATURE SETS, ACCELERATION AND PRECONDITIONING TECHNIQUES FOR THE DISCRETE ORDINATES METHOD IN PARALLEL COMPUTI NG ENVIRONMENTS By Gianluca Longoni December 2004 Chair: Alireza Haghighat Major Department: Nuclear and Radiological Engineering In the nuclear science and engineering fiel d, radiation transport calculations play a key-role in the design and optimization of nuclear devices. The linear Boltzmann equation describes the angular, energy and spatia l variations of the particle or radiation distribution. The discre te ordinates method (SN) is the most widely used technique for solving the linear Boltzmann equation. Howeve r, for realistic problems, the memory and computing time require the use of supercomputers. This research is devoted to the development of new formulations for the SN method, especially for highly angular dependent problems, in parallel environments The present research work addresses two main issues affecting the accuracy and performance of SN transport theory methods: quadrature sets and acceleration techniques. New advanced quadrature techniques which a llow for large numbers of angles with a capability for local angular refinement ha ve been developed. These techniques have

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xvii been integrated into the 3-D SN PENTRAN (Parallel Environment Neutral-particle TRANsport) code and applied to highly angul ar dependent problems, such as CT-Scan devices, that are widely used to obtain detailed 3-D images for industrial/medical applications. In addition, the accurate simulation of co re physics and shielding problems with strong heterogeneities and tr ansport effects requires the numerical solution of the transport equation. In genera l, the convergence rate of the solution methods for the transport equation is reduced for large pr oblems with optically thick regions and scattering ratios approaching unity. To remedy this situation, new acceleration algorithms based on the Even-Parity Simplified SN (EP-SSN) method have been developed. A new stand-alone code system, PENSSn (Parallel En vironment Neutral-particle Simplified Sn), has been developed based on the EP-SSN method. The code is designed for parallel computing environments with spatial, angular and hybrid (s patial/angular) domain decomposition strategies. The accuracy and perf ormance of PENSSn has been tested for both criticality eigenvalue and fixed source problems. PENSSn has been coupled as a preconditioner and accelerator for the SN method using the PENTRAN code. This work has cu lminated in the development of the Flux Acceleration Simplified Transport (FAST) preconditioning algorithm, which constitutes a completely automated system for precondi tioning radiation transp ort calculations in parallel computing environments.

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1 CHAPTER 1 INTRODUCTION 1.1 Overview In the nuclear engineering fi eld, particle transport calcul ations play a key-role in the design and optimization of nuclear devi ces. The Linear Boltzmann Equation (LBE) is used to describe the angular, energy and spatial variations of the particle distribution, i.e., the particle angular flux.1 Due to the integro-differential nature of this equation, an analytical solution cannot be obtained, except for very simple problems. For real applications, the LBE must be solved numerically via an iterative process. To solve large, real-world problems, significant memory and computational requirements can be handled using parallel computing environments, en abling memory partitioning and multitasking. The objective of this dissertation is to i nvestigate new techniques for improving the efficiency of the of the SN method for solving problems with highly angular dependent sources and fluxes in parallel environments. In order to achieve this goal, I have investigated two major areas: 1. Advanced quadrature sets for problem s characterized with highly angular dependent fluxes and/or sources. 2. Advanced acceleration/preconditioning algorithms. 1.2 The Linear Boltzmann Equation The LBE is an integro-partial differential equation, which describes the behavior of neutral particle transport. The Boltzmann equation, together with the appropriate boundary conditions, constitutes a mathemati cally well-posed problem having a unique solution. The solution is the distribution of particles throughout the phase space, i.e.,

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2 space, energy, and angle. The distribution of pa rticles is, in general, a function of seven independent variables: three spatial, two angular, one ener gy, and one time variable. The time-independent LBE in its ge neral integro-differential form1 is given by Eq. 1.1. ). ˆ ( ) ( ' 4 ) ˆ ( ) ˆ ˆ ( ' ) ˆ , ( ) ˆ , ( ) ( ) ˆ , ( ˆ04 04 E r E r d dE E E r E E r d dE E r q E r E r E rf s ext t (1.1) In Eq. 1.1, I have defined the following quantities: : Angular flux [particles/cm2/MeV/sterad/sec] r : Particle position in a 3-D space [cm]. E: Particle energy [MeV]. ˆ: Particle direction unit vector. t : Macroscopic total cross-section [1/cm/MeV]. extq: External independent source [particles/cm3/MeV/sterad/sec] s : Macroscopic double-differential scatteri ng cross-section [1/cm/sterad/MeV]. : Fission spectrum [1/MeV]. : Average number of neutrons generated per fission. f : Macroscopic fission cross-section [1/cm/MeV]. 1.3 Advanced Angular Quadrature Sets for the Discrete Or dinates Method The discrete ordinates method (SN) is widely used in nucle ar engineering to obtain a numerical solution of the integro-differen tial form of the Boltz mann transport equation. The method solves the LBE for a discrete number of ordinate s (directions) with associated weights.2 The combination of discrete direc tions and weights is referred to as quadrature set.3 The major drawback of the SN method is the limited number of directions involved, which, in certain situations, may lead to the so called ray-effects, which appear as unphysical flux oscillations. In general, this behavior occurs for problems with highly

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3 angular dependent sources and/or fluxes, or when the source is localized in a small region of space, in low density or highly absorbent media. In the past, several remedies for ray-e ffects have been proposed; the most obvious one is to increase the number of directions of the quadratur e set, or equivalently the SN order. However, this approach can lead to significant memory requirements and longer computational times. Carlson and Lathrop proposed a number of remedies4-5 for rayeffects based on specialized quadrature sets, which satisfy higher order moments of the direction cosines. Remedies based on firstcollision approximati ons have also been investigated.3 The source of particles generated fro m first-collision processes is often less localized than the true source; hence, the flux due to this source is usually less prone to ray-effects than the flux from the original source. If the true source is simple enough, analytic expressions can be obtained for th e uncollided flux and used to produce a first collision source; however, for general sour ces and deep-penetration problems, this method can be very time-consuming. An alternative approach6 is to expand the angular flux in terms of spherical harmonics (PN). The PN method does not suffer from rayeffects, because the angular dependency in th e angular flux is trea ted using continuous polynomial functions. However, the PN method has found limited ap plicability due to its computational intensive requirements. One of the most widely used techniques fo r generating quadrature sets is the levelsymmetric3 (LQN) method; however, the LQN method yields negative weights beyond order S20. In problems with large re gions of air or highly abso rbent materials, higher order (>20) quadrature sets are needed in order to mitigate ray-effects; therefore, it is necessary to develop techniques which allow fo r higher order or biased quadrature sets.

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4 In the past, different techniques have been investigated to remedy this issue. The equal weight quadrature set (EQN), developed by Carlson,7 is characterized by positive weights for any SN order. Other quadrature sets have been derived, by relaxing the constraints imposed by the LQN method;2 for this purpose, the Gauss quadrature formula and Chebyshev polynomials have been used for one-dimensional cylindrical or twodimensional rectangular geometries.4 In a recent study, the uniform positive-weight quadrature sets8 (UEN and UGN) have been derived. The UEN quadrature set is derived by uniformly partitioning the unit sphere into the number of directions defined by the SN order while the UGN quadrature set selects the ordinates along the z-axis as roots of Legendre polynomials. A new biasing technique, named Ordinate Splitting (OS), has been developed9 for the Equal Weight (EW) quadrature set; the OS technique is a method which is suitable to solve problems in which the pa rticle angular flux and/or so urce are peaked along certain directions of the unit sphere. The idea is to se lect a direction of flight of the neutron and split it into a certain number of directions of equal weights, while conserving the original weight. This new biasing technique has been implemented in the PENTRAN code10 and it has been proven very effective for medi cal physics applications such as CT-Scan devices.11-15 In this research work, I have developed new quadrature sets11-12 based on Legendre (PN) and Chebyshev (TN) polynomials. The Legendre-Chebyshev (PN-TN) quadrature set is derived by setting the polar angles equa l to the roots of the Legendre polynomial of order N, and the azimuthal angles are calculate d by finding the roots of the Chebyshev polynomials.

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5 The Legendre Equal-Weight (PN-EW) quadrature set is derived by choosing the polar angles as the roots of the Legendre polynomial of order N, while the azimuthal angles are calculated by equally partitioning a 90 degree angle. The set of directions is then arranged on the octant of the unit sphere similar to the level-symmetric triangular pattern. The main advantage of these new qua drature sets is the absence of negative weights for any SN order, and their superior accuracy compared to other positive schemes such as equal weight quadrature sets. Moreover, I have developed a new refineme nt technique, referred to as Regional Angular Refinement (RAR), which lead s to a biased angular quadrature set.13-15 The RAR technique consists of fitting an auxiliary quadrature set in a regi on of the unit sphere, where refinement is needed. These quadrature sets have been a pplied successfully to large problems, such as a CT-Scan de vice used for medical/industrial imaging9 and a Time-of-Flight (TOF) experiment simulation.16 The benefit of usi ng biased quadrature sets is to achieve accurate solutions for hi ghly angular dependent pr oblems with reduced computational time. 1.4 Advanced Acceleration Algorithms for the SN Method on Parallel Computing Environments Radiation transport calculations for real istic systems involve the solution of the discretized SN equations. A typical 3-D transport m odel requires the discretization of the SN equations in ~300,000 spatial meshes, 47 energy groups, and considering an S8 calculation, a total of 80 direc tions on the unit sphere. Thes e figures yield approximately 1.13 billion unknowns. In terms of computer me mory, this number translates into ~ 9 GBytes of RAM for storage (in single precision) of only the angular fluxes. It is clear that an efficient solution for such a problem is out of the scope of a regular workstation

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6 available with current technology. Therefore, it is necessary to develop new algorithms capable of harnessing the computational capab ilities of supercomputers. Based on this philosophy, in the late 1990s G. Sjoden a nd A. Haghighat have developed a new 3-D parallel radiation transport code: PENTRA N (Parallel Environment Neutral-particle Transport).10 However, besides the size and complex ity of the problem being solved, other aspects come into play, especially when d ealing with criticality eigenvalue problems. Because of the physics of these problems, th e convergence rate of the currently used iterative methods is quite poor For realistic problems, su ch as whole-core reactor calculations performed in a 3-D geometry, the solution of the SN equations may become impractical if proper acceleration methods17 are not employed. The main philosophy behind the novel accel eration algorithms developed in this work is to employ a simplified mathematical model which closely approximates the SN equations, yet can be solved efficiently. The new acceleration/preconditioning algor ithms have been developed during the course of this research in three major phases: 1. Development of the PENSSn code based on the Even-Parity Simplified SN (EPSSN) method. 2. Investigation of a new synthetic accel eration algorithm based on the EP-SSN method. 3. Development of an automated acceleration system for the SN method on parallel environments: FAST (Flux Acceleration Simplified Transport). 1.5 The Even-Parity Simplified SN Method The Even-Parity Simplified SN (EP-SSN) method is developed based on the philosophy considered in the PN and Simplified Spherical Harmonics (SPN) methods.18

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7 The spherical harmonics (PN) approximation to the transport equation is obtained by expanding the angular flux using spherical harmonics fu nctions truncated to order N, where N is an odd number; these functions form a complete basis in the limit of the truncation error. In the limit of N, the exact transport solution is recovered.1 In 3-D geometries, the number of PN equations grows as 21 N The PN equations can be reformulated in a second-order form, cast as 2 / 12 N diffusion-like equations, characterized by an elliptic operator. Howeve r, the number of these equations is very large, and the coupling invol ves not only angular moments, but also mixed spatial derivatives of these moments.6 Because of these issues, to reduce the computing time in the early 1960s, Gelbard et al. proposed the Simplified Spherical Harmonics18 or SPN method. The Gelbard procedure consists of extendi ng the spatial variable to 3-D by substituting the second order derivatives in the 1-D PN equations with the 3-D Lapl acian operator. As a result, coupling of spatial derivatives is eliminated, yielding only ( N+ 1) equations as compared to 21 N Further, since the SPN equations can be reformulat ed as second-order elliptic equations, effective iterative tec hniques such as Krylov subspace19-20 methods, can be employed. The main disadvantage of the SPN equations is that the exact solution to the transport equation is not recovered as N, due to terms that are inherently omitted in replacing a 1-D leakage term with a simplifie d 3-D formulation. However, for idealized systems characterized by homogeneous ma terials and isotropic scattering, the SPN and the SN equations yield the same solution, gi ven the same quadrat ure set and spatial discretization is used for both methods.

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8 Despite this fact, the SPN equations yield improved solutions21-22 compared to the currently used diffusion equation. The theoretical basis for the SPN equations has been provided by many authors using variationa l methods and asymptotic analysis.22 It has been shown that these equations are higher-ord er asymptotic solutions of the transport equation. Moreover, Pomraning has demonstrated that the SPN equations, for odd N are a variational approximation to the one-group even -parity transport equa tion with isotropic scattering in an infini te homogeneous medium. Recently, the SPN formulation has received renewed interest, especially in reactor physics applications. For applications such as the MOX fuel assemblies22-23 or for reactor problems with strong transport effects,24 diffusion theory does not provide accurate results, while the SPN equations improve the accuracy of the solution within a reasonable amount of computation time. Initially, I derived the SP3 equations starting from the 1-D P3 equations;25 however, for developing a general order algorithm, I de rived a general formulation using the evenparity form of the SN transport equation.26 The Even-Parity Simplified SN (EP-SSN) formulation has some interesting properties th at make it suitable to develop algorithms of any arbitrary order. Chapter 4 is dedicated to this issue. To make the method more effective, the convergence properties of the EP-SSN equations were improved by modifying the scattering term; it will be show n that this improved derivation is problem dependent but can reduce the number of iterations significantly. I have developed a general 3-D parallel code,23 PENSSn (Parallel Environment Neutral-particle Simplified SN), based on the EP-SSN equations. The EP-SSN equations are discretized with a finite-volume approach, and the spatial domain is partitioned into

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9 coarse meshes with variable fine grid de nsity in each coarse mesh emulating the PENTRAN grid structure. A projection algorith m is used to couple the coarse meshes based on the values of the even-parity angular fluxes on the interfaces. PENSSn uses iterative solvers based on Krylov subspace19 methods: the Conjugate Gradient (CG) and the Bi-Conjugate Gradient (Bi-CG) solver s. In addition, I have developed a preconditioner based on the Incomp lete Cholesky factorization20 for the CG method. The finite-volume discretization of the EP-SSN equations in a 3-D Cartesian geometry yields sparse matrices with a 7-diagonal sparse st ructure. Therefore, I optimized the memory management of PENSSn by using a Comp ressed Diagonal Storage (CDS) approach, where only the non-zero elements on th e diagonals are stored in memory. The PENSSn code is designed for parallel computing environments with angular, spatial and hybrid (angular/spatia l) domain decomposition algorithms.23 The space decomposition algorithm partitions the 3-D Cart esian space into coarse meshes which are then distributed among the processors while the angular decomposition algorithm partitions the directions among the processo rs. The hybrid decomposition algorithm is a combination of the two algorithms discussed above. Note that the hybrid decomposition combines the benefits of memory partit ioning offered by the spatial decomposition algorithm, and the speed-up offered by the a ngular decomposition algorithm. The code is written in Fortran 90, and for seamless parallelization, the MPI (Message Passing Interface) library27 is used. I have tested the PENSSn code for pr oblems characterized by strong transport effects, and have shown that the impr ovements over the diffusion equation can be significant.26 The solutions obtained with the EP-SSN method are in good agreement with

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10 SN and Monte Carlo methods; however, the co mputation time is si gnificantly lower. Hence, these results indicate that the EP-SSN method is an ideal candidate for the development of an effective acceleration or preconditioning algorithm for radiation transport calculations. 1.6 A New Synthetic Acceleration Algorithm Based on the EP-SSN Method As mentioned earlier in this chapter, the solution of the linear Boltzmann equation is obtained numerically via an iterative pro cess. The most widely used technique to iteratively solve the transport equation is the Source Iteration (SI) method3 or Richardson iteration. The convergence propertie s of this method are related to the spectral radius of the transport operator. It can be shown that for an infinite, homogeneous medium, the spectral radius of the tr ansport operator is dominat ed by the scattering ratio c given by t sc (1.2) where s is the macroscopic scattering cross-sect ion and is the macroscopic total crosssection. Note that Eq. 1.2 can be obtained by Fourier analysis in an infinite homogeneous medium. The asymptotic convergence rate v is defined as ) log( c v (1.3) Eq. 1.3 indicates that for problems with scattering ratios close to unity, the unaccelerated SI method is ineffective, b ecause the asymptotic convergence rate tends toward zero. Hence, the use of an acceleration scheme is necessary. In the past, many acceleration techniques have been proposed25 for solving the steady-state transport equation. The synthe tic methods have emerged as effective techniques to speed-up the converg ence of the SI iterative process.28 In the synthetic

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11 acceleration process a lower-order approxi mation of the transport equation (e.g., diffusion theory) is corrected using the transp ort equation at each iteration. In this way the spectral radius of the accelerated transpor t operator is reduced with consequent speedup of the iteration process. Two categories of synthetic methods have b een investigated so far, the Diffusion Synthetic Acceleration (DSA) and the Tr ansport Synthetic Acceleration (TSA).29-30 The first method has been proven to be very eff ective for 1-D problems and for certain classes of multi-dimensional problems. However, recently it has been shown that for multidimensional problems with significant material heterogeneities, the DSA method fails to converge efficiently.31-32 The same behavior, along with possible divergence, has been reported also for TSA.30 I have developed and tested a ne w synthetic acceleration algorithm33 based on the simplified form of the Even-Parity transport equations (EP-SSN). I tested the EP-SSN algorithm for simple 3-D problems and I conc luded that it is affected by instability problems. These instabilities are similar in na ture to what has been reported for DSA by Warsa, Wareing, and Morel.31-32 The main problems affec ting the stability of the synthetic methods are material heterogeneitie s and the inconsistent discretization of the lower-order operator with the transport operator, which leads to divergence if the mesh size is greater than ~1 mean free path. Moreover, because the synthetic accelera tion method has been implemented into the PENTRAN code, another consideration come s into play. The spatial differencing in the PENTRAN code system is based on an Adaptive Differencing Strategy10 (ADS); with this method the code automatically selects the most appropriate differencing scheme

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12 based on the physics of the problem. Hence, the discretization of the lower-order operator should be consistent with every differencing scheme present in the code. This task is feasible if we consider only the linear-d iamond (LD) differencing scheme; however, the complexity increases if we consider the Directional Theta Weighted (DTW) or the Exponential Directional Weighted (EDW) differencing schemes.34 Moreover, it has been shown that even a fully consistent discre tization of the lower-order operator does not guarantee the convergence of the synthetic method. Due to the issues discussed above, I have developed a system that utilizes the EPSSN method as preconditioner for the SN method. 1.7 An Automatic Preconditioning Algorithm for the SN Method: FAST (Flux Acceleration Simplified Transport) The last phase of the development of an effective accelerati on algorithm for the SN method has culminated in th e development of the FAST system (Flux Acceleration Simplified Transport). The FAST algorithm is based on the kernel of the PENSSn code. The main philosophy followed in the developmen t of the system is completely antithetic to the synthetic acceleration approach. The id ea is to quickly obtain a relatively accurate solution with the EP-SSN method and to use it a precondi tioned initial guess for the SN transport calculation. This approach has the main advantage of decoupling the two methods, hence avoiding all the st ability issues discussed above. The FAST system is currently implemen ted into the PENTRAN-SSn Code System for distributed memory environments, and it is completely automated from a user point of view. Currently, the system has su ccessfully accelerated large 3-D criticality eigenvalue problems, speeding-up the calculations by a factor of 3 to 5 times, and hence reducing significantly the spectral radius.

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13 1.8 Outline The remainder of this dissertation is orga nized as follows. Chapter 2 provides the theory for the discrete ordinates method. Th e discretization of the phase space variables in the transport equation will be discusse d, along with the proper boundary conditions. Chapter 3 discusses the theoretical developm ent of the advanced and biased quadrature sets for the discrete ordinates method. It also presents the application of the new quadrature sets for the simulation of a CT-S can device and for the Kobayashi benchmark problem 3. Chapter 4 discusse s the derivation of the EP-SSN equations. Chapter 5 discusses the numerical methods for the solution of the EP-SSN equations, along with the iterative solvers based on Krylov subspace methods. Chapter 6 addresses the numerics and accuracy of the EP-SSN equations. Chapter 7 presents the parallel algorithms implemented in the PENSSn code for dist ributed memory architectures. Chapter 8 describes the development of a new syntheti c acceleration algorithm based on the EP-SSN method and its limitations. Chapter 9 focuses on the development of the FAST preconditioner; the performance of the algorith m is measured with two test problems and a large 3-D whole-core criticality eigenvalu e calculation. Chapter 10 will draw the conclusions on the objectives accomplished and it will point out some aspects for future development.

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14 CHAPTER 2 THE DISCRETE ORDINATES METHOD In this chapter, the Discrete Ordinates Method (SN) will be discussed in detail. The discretized form of the transport equation is formulated in a 3-D Cartesian geometry. I also address the iterative techniques a nd acceleration methods used to solve the SN transport equations. 2.1 Discrete Ordinates Method (SN) The Discrete Ordinates Method (SN) is widely used to obtai n numerical solutions of the linear Boltzmann equation. In the SN method, all of the inde pendent variables (space, energy and angle) are discre tized as discussed below. 2.1.1 Discretization of the Energy Variable The energy variable of the transport e quation is discretized using the multigroup approach.3 The energy domain is partitioned in to a number of di screte intervals ( g =1… G ), starting with the hi ghest energy particles ( g =1), and ending with the lowest ( g = G ). The particles in energy group g are those with energies between Eg-1 and Eg. The multigroup cross-sections for a generic reaction process x are defined as 1 14 4 ,) ˆ , ( ) ˆ , ( ) ( ) (g g g gE E x E E g xE r d dE E r E r d dE r (2.1) Based on the definition given in Eq. 2.1, the group constants are defined in Eqs. 2.2, 2.3, and 2.4, for the “total,” “fissi on” and “scattering” processes, respectively.

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15 1 14 4 ,) ˆ , ( ) ˆ , ( ) ( ) (g g g gE E t E E g tE r d dE E r E r d dE r (2.2) 1 14 4 ,) ˆ , ( ) ˆ , ( ) ( ) (g g g gE E f E E g fE r d dE E r E r d dE r (2.3) 1 ' 1 '4 4 ,) ˆ , ( ˆ ) ˆ , ( ) ˆ ˆ ( ' ) (g g g gE E s E E gg sE r d dE E r E E r d dE r (2.4) With the group constants defined above, th e multigroup formulation of the transport equation is written as ), ˆ ( ) ( ) ( 1 ) ˆ ( ) ˆ ˆ ( ) ˆ ( ) ( ) ˆ ( ˆ' 1 ' ' 1 4 r q r r k r r d r r re g g G g g f g g gg G g g g g (2.5) for g= 1 G where the angular flux in group g is defined as 1) ˆ , ( ) ˆ (g gE E gE r dE r (2.6) In Eq. 2.5, ) ˆ ( r qe g is the angular dependent fixed source; in general, for criticality eigenvalue problems, this term is set to zero. The scalar flux in Eq. 2.5 is defined as ) ˆ ( ) (4 r d rg g (2.7) In a 3-D Cartesian geometry, the “str eaming” term can be expressed as z y x ˆ (2.8)

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16 where the direction cosines are defined as i ˆ ˆ j ˆ ˆ k ˆ ˆ (2.9) Figure 2-1 shows the Cartesian space-angl e system of coordinates in three dimensions. Figure 2-1. Cartesian space-angle coor dinates system in 3-D geometry. The multigroup transport equation, with the scattering kernel expanded in terms of Legendre polynomials and the angular flux in terms of spherical harmonics is given by Eq. 2.10. The complete derivation of the sc attering kernel expansion in spherical harmonics shown in Eq. 2.10 is given in Appendix A. r x y z i ˆj ˆ ˆ

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17 ) ˆ ( ) , ( ) , ( )]} sin( ) , ( ) cos( ) , ( [ ) ( )! ( )! ( 2 ) , ( ) ( ){ , ( ) 1 2 ( ) , , ( ) , (1 ' 0 ' 1 1 '0 ' G g e g g g f g k g Sl k g Cl l k k l G g L l g l l g g sl g gr q z y x z y x k k z y x k z y x P k l k l z y x P z y x l z y x z y x z y x (2.10) where : direction cosine along the x -axis : direction cosine along the y -axis : direction cosine along the z -axis g : total macroscopic cross-section : azimuthal angle, i.e. arctan ) , , ( z y xg: angular flux in energy group g g g sl' : lth moment of the macroscopic transfer cross-section ) ( lP : lth Legendre Polynomial ) (, g l: lth flux moment ) (k lP : associated lth, kth Legendre Polynomial ) (, k g Cl: cosine associated lth, kth flux moment ) (, k g Sl: sine associated lth, kth flux moment g : group fission spectrum k: criticality eigenvalue g f : fission neutron generation cross-section 2.1.2 Discretization of the Angular Variable The angular variable, in the transport equation is discretized by considering a finite number of directions, and the angular flux is evaluated only along these directions. Each discrete direction can be visualized as a point on the surface of a unit sphere with an associated surface area which mathematically corresponds to the weight of the integration scheme. The combination of the discrete di rections and the corresponding weights is

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18 referred to as quadrature set. In general, quadrature sets should satisfy the following properties:3 The associated weights must be positive and normalized to a constant, usually chosen to be one M m mw10 1. (2.11) The quadrature set is usually chosen to be symmetric over the unit sphere, so the solution will be invariant with respect to a 90-degree axis rotation and reflection. This condition results in the odd-moments of the direction cosines having the following property M m M m M m n m m n m m n m mw w w1110 0 for n odd. (2.12) The quadrature set must lead to accurate values for moments of the angular flux (i.e., scalar flux, currents); this require ment is satisfied by the following conditions on the even-moments of the di rection cosines as follows M m M m M m n m m n m m n m mn w w w1111 1 for n even. (2.13) A widely used method for generating a quadrature set is the level-symmetric technique (LQN). In this technique, the directions are ordered on each octant of the unit sphere along the z -axis ( ) on N /2 distinct levels. The number of directions on each level is equal to 1 2 i N for i =1, N /2. It is worth noting that in 3-D geometries, the total number of directions is M = N ( N +2), where N is the order of the SN method. Considering 12 2 2 k j i and 2 2 N k j i where N refers to the number of levels and i, j, k are the indices of the direction co sines, we derive a formulation for determining the directions as follows ) 1 (2 1 2ii (2.14) where

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19 ) 2 ( ) 3 1 ( 22 1 N, for 2 2 N i and 3 1 02 1 (2.15) In Eq. 2.14 the choice of 1 determines the distribution of directions on the octant. If the value of 1 is small, the ordinates will be clustered near the poles of the sphere; alternatively, if the value of 1 is large, the ordinates will be placed far from the poles. The weight associated to each direction, called a point weight, is then evaluated with another set of equations. Fo r example, in the case of an S8 level-symmetric quadrature set, this condition can be formulated as follows 1 2 12 2w p p (2.16a) 2 3 22 w p p (2.16b) 3 22 w p (2.16c) 4 11w p (2.16d) where p1, p2 and p3 are point weights associated with each direction, and w1, w2, w3, w4 are the weights associated with the levels, as shown in Figure 2-2. Figure 2-2. Point weight arrangement for a S8 level-symmetric quadrature set.

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20 As an example, Figure 2-3 shows the S20 LQN quadrature set for one octant of the unit sphere. Figure 2-3. S20 LQN quadrature set. Note that, this quadrature set is li mited by unphysical negative weights beyond order S20. Therefore, if a higher order quadrature set is needed beyond S20, another formulation has to be developed, which sa tisfies the evenand odd-moments conditions. To address this issue, I have developed new quadrature techniques based on the GaussLegendre quadrature formula a nd on the Chebyshev polynomials. 2.1.3 Discretization of the Spatial Variable The linear Boltzmann equation, given in Eq. 2.10, can be rewritten in an abbreviated form as ) ( ) ( ) (, ,r Q r r z y xg m g m g m m m (2.17) for M m 1 and G g 1 The angle and energy dependence are denoted by the indices m and g respectively. The right hand side of Eq. 2.17 represents the sum of the scattering, fission and external sources.

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21 The spatial domain is partitioned into computational cells, bounded by x1/2, x3/2,…, xI+1/2; y1/2, y3/2,…, yJ+1/2; z1/2, z3/2,…, zK+1/2. The cross-sections are assumed to be constant within each cell and they are denoted byk j i , Eq. 2.17 is then integrated over the cell volumek j i k j iz y x V ,, and then divided by the cell volume to obtain the volumeand surface-averaged fluxes, therefore reducing to ., , , , , , 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 g m k j i g m k j i k j i g m k j i g m k j i k m g m k j i g m k j i j m g m k j i g m k j i i mQ z y x (2.18) In Eq. 2.18, the indices i, j, k represent the cell-center values, while i 1/2, j 1/2, k 1/2 refer to the surface values. 2.1.4 Differencing Schemes For the SN method, different classes of di fferencing schemes are available. Loworder differencing schemes require only the angular fluxes, and the average values at the cell boundaries to be related at the cell average value. Various forms of Weighted Difference (WD) schemes belong to this class. High-order differencing schemes require higher order moments, and may be linear or non-linear. Di scontinuous, characteristic, and exponential schemes are examples of high-order differencing schemes.35 The solution of the SN equations is obtained by marching along the discrete directions generated in each octant of the unit sphere; this process is usually referred to as a transport sweep .3 For each computational cell, the angular fluxes on the three incoming surfaces are already known, from a previous calculation or boundary conditions. The cellcenter fluxes and the fluxes on the three out going surfaces must be calculated, hence

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22 additional relationships are n eeded. The additional relationshi ps are referred to as the “ differencing schemes” The general form of WD schemes can be expressed as g m k j i g m k j i g m k j i g m k j i g m k j ia a, , 2 / 1 , , , , 2 / 1 , , , ,) 1 ( (2.19a) g m k j i g m k j i g m k j i g m k j i g m k j ib b, , 2 / 1 , , , , 2 / 1 , , , , ,) 1 ( (2.19b) g m k j i g m k j i g m k j i g m k j i g m k j ic c, 2 / 1 , , , , 2 / 1 , , , , ,) 1 ( (2.19c) The values ai,j,k,m,g, bi,j,k,m,g, and ci,j,k,m,g are determined based on the type of weighted scheme employed. 2.1.4.1 Linear-Diamond Scheme (LD) In the LD scheme, the cell-average flux is an arithmetic average of any two opposite boundary fluxes; hence the wei ghts are set to constant values a = b = c =1/2. 2 1 2 1 2 1, 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 , , 2 / 1 , , g m k j i g m k j i g m k j i g m k j i g m k j i g m k j i g m k j i (2.20) For example, in the direction0 m ,0 m ,0 m the outgoing fluxes are obtained as follows g m k j i g m k j i g m k j i , , 2 / 1 , , , , 2 / 12 (2.21a) g m k j i g m k j i g m k j i , 2 / 1 , , , , 2 / 1 ,2 (2.21b) g m k j i g m k j i g m k j i , 2 / 1 , , , , 2 / 1 ,2 (2.21c) We can then eliminate the fluxes on the outgoing surfaces in Eq. 2.18 and obtain the center-cell angular flux

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23 k m j m i m k j i k j i g m k j i k m g m k j i j m g m k j i i m g m k j iz y x Q z y x 2 2 2 2 2 2, , , 2 / 1 , , 2 / 1 , , 2 / 1 , ,. (2.22) The outgoing fluxes are then eval uated using Eqs. 2.21a, b, and c. The LD differencing scheme may yield ne gative angular fluxe s in regions where the flux gradient is large, even if the inco ming fluxes and scattering source are positive. In this case, one approach is to set negative fluxes equal to zero, a nd then the cell-average flux is recalculated to preserve the balance of particles. This appro ach is referred to as negative flux fix-up (DZ). The DZ scheme performs better than the LD in practical applications, but the linearity and accuracy of the LD equations is not preserved. 2.1.4.2 Directional Theta-Weighted Scheme (DTW) This scheme uses a direction-based pa rameter to obtain the weighting factors a b and c which are restricted to the range 0.5 and 1.0. The DTW scheme uses a directionbased parameter to obtain an angular flux we ighting factor, which ensures positivity of the angular flux and removes the oscilla tions due to the spatial and angular discretization.36 The DTW average cell angular flux is given by z c y b x a z c y b x a qg m k j i m g m k j i m g m k j i m z in g m k j i m y in g m k j i m x in g m k j i m k j i g m k j i , , , , , , , , , , , , , , , , ,. (2.23) In the DTW scheme, the weights (ai,j,k,m,g, bi,j,k,m,g, ci,j,k,m,g) are restricted to the range between 0.5 and 1.0, approaching second order accuracy when all weights are equal to 0.5, which is in this case is equivalent to the LD scheme.

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24 2.1.4.3 Exponential Directional-Weighted Scheme (EDW) The Exponential Directional Weighted (EDW) differencing scheme34, implemented in PENTRAN, is a predictor-corrector sche me, which utilizes the DTW to predict a solution that is then corrected using an expone ntial fit. The EDW is an inherently positive scheme, and the auxiliary equations deri ved for this method are given in Eq. 2.24. m k m j m i mz P y P x P a z y x / ) ( exp / ) ( exp / ) ( exp ) , (1 1 1. (2.24) The DTW scheme is used to calculate the angular fluxes () ~ needed for the estimation of the coefficientsi ,j and k given in Eqs. 2.25a, b, and c, respectively. A m x in x out i ~ 2 ~ ~ , (2.25a) A m y in y out j ~ 2 ~ ~ , (2.25b) A m z in z out k ~ 2 ~ ~ , (2.25c) where the subscripts in and out refers to the incoming a nd outgoing surface averaged angular fluxes. The cell-average angular flux formulated with the EDW scheme is given in Eq. 2.26. ), ( 1 1 ) 2 exp( 1 ) 2 exp( 1 ) 2 exp(, , z in m y in m x in m A m k m j m i Az y x q (2.26) where is calculated using the coefficien ts given in Eqs. 2.25a, b, and c.37

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25 2.1.5 The Flux Moments The flux moments are obtained from th e angular fluxes using the following formulation M m m l k j i m l k j iP w1 , , ,) ( (2.27) Note that Eq. 2.27 for l=0 yields the scalar flux. Th e Associated Legendre moments are calculated using Eqs. 2.28 and 2.29. ) cos( ) (1 , , , m M m m n l k j i m n l C k j in P w (2.28) ) sin( ) (1 , , , m M m m n l k j i m n l S k j in P w (2.29) 2.1.6 Boundary Conditions Three major boundary conditions can be expressed with a general formula as m s m sr a r , (2.30) where n nm m '. Depending on the value of coefficient a the three boundary conditions are: a =1, reflective boundary condition. a =0, vacuum or non-reentr ant boundary condition. a = albedo boundary condition. 2.2 Source Iteration Method Due to the integro-differential nature of the transport equation, the solution of the multigroup discrete ordinates equations is obt ained by means of an iterative process, named the source iteration .3 The source iteration method c onsists of guessing a source (i.e., in-group scattering source), then sweep ing through the angular, spatial and energy

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26 domains of the discretized system with the appropriate boundary conditions. When the sweep is completed, integral quantities such as scalar flux and fl ux moments are obtained from the angular fluxes, and then a new in-g roup scattering source is calculated, and the iteration process continues until a convergence criterion, shown in Eq. 2.31, is satisfied. Typical tolerances for fixed source calculations range in the order of 1.0e-3 to 1.0e-4. .1 1 i i i (2.31) If fission and/or up-scattering processes are present, outer iterations are performed on the fission and transfer scattering sour ces. Acceleration techniques may be applied between source iterations to speed-up the conve rgence rate by determining a better guess for the flux moments and the source. 2.3 Power Iteration Method Criticality eigenvalue problems are solved using the method of power iteration.38 For this method, it is assumed that the eigenvalue problem has a largest positive eigenvalue, k>0, with an associated fission distribution ) (r F that is nonnegative. Hence by considering k0>0 and 0 ) (0r F as initial guesses, we calculate the eigenvalue at iteration i as follows ) ( ) (1 1r F r d dE r F r d dE k ki i i i (2.32) where ) ( ) ( ) (E r E r r Fi f i is the fission source di stribution at iteration i. The iterative process is continued until the desi red convergence is reached, as shown in Eq. 2.33.

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27 .1 1 k i i i k k k (2.33) Generally, the tolerance required for criticality calculation is 1.0e-4 to 1.0e-6. 2.4 Acceleration Algorithms for the SN Method Many acceleration methods have been proposed to speed-up the convergence of the iterative methods used to solve the steady-state tr ansport equation.17 There are a number of problems where standard non-accelerated ite rative methods converge too slowly to be practical. Most of these problems are char acterized by optically thick regions with scattering ratio near unity. The three major acceleration approaches are the Coarse Mesh Rebalance (CMR), Multigrid, and Synthetic methods. The CMR a pproach is based on the fact that the converged solution must satisfy the particle balance equation.3 By imposing this balance condition on the unconverged solution over coar se regions of the problem domain, it is possible to obtain an iteration procedure th at usually converges more rapidly to the correct solution. However, this method is hi ghly susceptible to the choice of the coarse mesh structure and can be unstable. The multigrid approach has been used to accelerate the SN calculations; the basic principle of the method is to solve the equa tions on a coarse grid and to project the solution onto a finer grid. Different types of mu ltigrid approaches exist, such as “/” Slashcycle, V-Cycle and W-Cycle, and the Simplified Angular Multigrid39 (SAM). In Chapter 8, I will compare the results obtained with the EP-SSN synthetic acceleration and the SAM method. The synthetic acceleration approach is based on using a lower-order operator, generally diffusion theory, as a means to accelerate the numerical solution of the

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28 transport equation.17 In the late 1960s, Gelbard a nd Hageman developed a synthetic acceleration method based on the diffusion and the S4 equations.28 Later, Reed independently derived a simila r synthetic acceleration scheme40 and pointed out some limitations of the method derived by Gelb ard and Hageman. The synthetic method developed by Reed has the advantage of being very effective for small mesh sizes, but it is unstable for mesh sizes greater than ~1 mfp (mean free path). Later, Alcouffe independently derived the Diffusion S ynthetic Acceleration (DSA) method.29 He addressed the issue of stabil ity of the method and derived an unconditionally stable DSA algorithm. Alcouffe pointed out that in order to obtain an unconditionally stable method, the diffusion equation must be derived consis tently from the discretized version of the transport equation. In this way, the consiste ncy between the two operators is preserved. Recently, the DSA method has b een found to be ineffective30-32 for multidimensional problems with strong heterogeneities, even when a consistent discretization of the lowe r order and transport op erators is performed.

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29 CHAPTER 3 ADVANCED QUADRATURE SETS FOR THE SN METHOD This chapter covers the development of advanced quadrature sets for solving the neutron transport equation vi a the Discrete Ordinates (SN) method. The level-symmetric (LQN) quadrature set is the sta ndard quadrature set for SN calculations; although, as discussed in the previous chap ter, this quadrature set is limited to order 20. The Equal Weight (EW) quadrature set ha s been proposed to resolve th e issue of negative weights; this quadrature set is generated by partitioning the unit sphere into M directions, where ) 2 ( N N M and by assigning an equal weight to each direction M wi1 The EW quadrature set yields positive weights for any SN order; however, it does not completely satisfy the even-moment conditions given in Eqs. 2-13. I have developed and tested new qua drature sets based on the Legendre (PN) and Chebyshev (TN) polynomials. In this chapter, I di scuss the Legendre Equal-Weight (PNEW), the Legendre-Chebyshev (PN-TN) quadrature sets, and the Regional Angular Refinement (RAR) technique for local angular refinements. The PN-EW and PN-TN have no limitations on the number of di rections, and the RAR technique is an alternative to the Ordinate Splitting (OS) technique.11 The OS technique has been developed to refine the directions of a standard quadrature set us ing equal-spaced and equal-weight directions, while the RAR utilizes the PN-TN quadrature set over a subset of ordinates. The main difference between the OS and RAR technique s is in the refining methodology; the OS technique focuses on the refinement of each single direction, while the RAR considers a

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30 sector of the unit sphere. Also the biased qua drature set generated with RAR satisfies the conditions on the oddand even-moments of the direction cosines (Eq. 2-12 and 2-13). To examine the effectiveness of the new techniques for angular quadrature generation, each technique has been implemented into the PENTRAN code10 (Parallel Environment Neutral-Particle TRANsport), and utilized fo r a number of problems of practical interest. 3.1 Legendre Equal-Weight (PN-EW) Quadrature Set In order to develop a quadrature se t which is not limited to order S20, I have investigated the Gauss-Le gendre quadrature technique.4 This quadrature set is characterized by the same arrangement of directions as the LQN, but the directions and weights are evaluated differently. Given the SN order for the discrete set of directions, we apply the Gauss-Legendre quadrature formula using the following recursive formulation 1 1) 1 2 ( ) 1 ( j j jjP P j P j for j = 0, N, (3.1) where 1 1 0 ) (1 P, and 1 ) (0 P. (3.2) The -levels or polar angles, along the z-axis are set equal to the roots of Eq. 3.1. The values represent the levels of the quadrature set. Once we have evaluated the ordinates along the z-axis, we obtain the weights associated with each level using the following recursive formulation 2 2) 1 ( 2 id dP wN i i for 2 1N i (3.3)

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31 In order to complete the definition of each discrete direction, the azimuthal angle is evaluated on each level by equally partitioning a 90 degree angle into 2 2 i N angular intervals, where i=1, N/2. Hence, the weight associated with each direction is given by j w pi j i,, for 2 ... 1 N i (3.4) In Eq. 3.4, 1 2 ... 1 i N j is the number of directions with equal weights on the ith level. Figure 3-1 shows the directions and the associated weights for an S28 PN-EW quadrature set on one octant of the unit sphere. Figure 3-1. S28 PN-EW quadrature set. Note that in Figure 3-1, all directions on the same -level have the same weight, as indicated by the color. 3.2 Legendre-Chebyshev (PN-TN) Quadrature Set In the PN-TN quadrature set, similar to PN-EW, we choose the -levels on the z-axis equal to the roots of the Gauss-Legendre quadr ature formula given in Eqs. 3.1 and 3.2; however, the azimuthal angles on each level ar e set equal to the roots of the Chebyshev

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32 (TN) polynomials of the first kind. Chebyshe v polynomials of the first kind are formulated as follows l Tlcos cos (3.5) The Chebyshev polynomials are orthogonal and satisfy the following conditions ) cos( 0 2 / 0 , 0 ) 1 )( ( ) (1 1 2 / 1 2 y k l k l k l y y T y dyTk l (3.6) Again, using the ordering of the LQN quadrature set, we define the azimuthal angles on each level using the following formulation 2 2 1 2 2, i j ij i, (3.7) where 2 0, j i and i=1, N/2. In Eq. 3.7, i is the level number and 1 2 ... 1 i N j. The level and point weights are generated in the same way as for the PN-EW. Note that both PN-EW and PN-TN quadrature sets do not present negative weights for SN orders higher than 20. Figure 3-2 shows the directions and the associated weights for an S28 PN-TN quadrature set on one octant of the unit sphere.

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33 Figure 3-2. S40 PN-TN quadrature set over the unit sphere. 3.3 The Regional Angular Refinement (RAR) Technique The RAR method is developed for solving problems with highly peaked angular fluxes and/or sources. The approach consists of two steps. In the first step, we derive a PN-TN quadrature set of arbitrary order on one octant of the unit sphere, as described in Section 3.2. In the second step, we define the area (angul ar segment) of the octant to be refined along with the order (N’) of the PN’-TN’ quadrature set to be used in this region. The area to be refined is characterized by the polar range ( min, max) and the azimuthal range ( min, max). Generally, the biased region is selected based on the physical properties of the model. For example, if a di rectional source is forward peaked along the x-axis, the quadrature set will be refine d on the pole along the x-axis. The -levels of the PN’-TN’ quadrature set are calculated using Eqs. 3.1 and 3.2; therefore, they are mapped onto the ( min, max) sub-domain using the following formula: 2 2 ~min max min max i i, for i=1, N’/2. (3.8)

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34 Hence, the azimuthal angles are evaluated using the Chebyshev polynomials as follows 2 2 1 2 2min max min max N j Nj i, (3.9) where 2 0, j i, i=1, N’/2 and j=1, N’/2. The number of directions in the refined region is equal to 22 N. The weights in the refined region are renormalized to preserve the overall normalization on the unit sphere. Figure 3-3 shows the RAR t echnique applied to an S16 PN-TN quadrature set. In the biased region, which extends from =0.0 to =0.2 along the z-axis, and from =0 to =10 on the azimuthal plane, an S10 PN-TN quadrature set is fitted. Figure 3-3. PN-TN quadrature set (S16) refined with the RAR technique. 3.4 Analysis of the Accuracy of the PN-EW and PN-TN Quadrature Sets The main advantage of the LQN technique is the fact that it preserves moments of the direction cosines, thereby lead ing to an accurate solution. The PN-EW and PN-TN

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35 quadrature sets attempt to preserve th ese quantities independently along the -, -, and axes. Therefore, I verified the capability of th e new quadrature sets in preserving the even moments of the direction cosines, which are directly related to the accuracy of the quadrature set. Table 3-1 compares the even-m oments of the direc tion cosines calculated with an S30 PN-EW quadrature set with the exact value. Table 3-1. Even-moments obtained with a PN-EW S30 quadrature set. Moment Order (n even) M i n i iw1 M i n i iw1 M i n i iw1 Exact value n1 1 2 0.333333333 0.333333333 0.333333333 0.333333333 4 0.194318587 0.194318587 0.2 0.2 6 0.135907964 0.135907964 0.142857143 0.142857143 8 0.103874123 0.103874123 0.111111111 0.111111111 10 0.083691837 0.083691837 0.090909091 0.090909091 12 0.06983611 0.06983611 0.076923077 0.076923077 14 0.059749561 0.059749561 0.066666667 0.066666667 16 0.052087133 0.052087133 0.058823529 0.058823529 18 0.046074419 0.046074419 0.052631579 0.052631579 20 0.041234348 0.041234348 0.047619048 0.047619048 22 0.037257143 0.037257143 0.043478261 0.043478261 24 0.033932989 0.033932989 0.043478261 0.043478261 26 0.031114781 0.031114781 0.037037037 0.037037037 28 0.028696358 0.028696358 0.034482759 0.034482759 30 0.026599209 0.026599209 0.032258065 0.032258065 As expected, the PN-EW preserves exactly the even-moments conditions along the -axis, while on the -, -axes these conditions are onl y partially preserved; the maximum relative difference between the even-moments calculated with PN-EW and the exact solution is 17.0%. Table 3-2 shows the comparison of the even-moments evaluated with an S30 PN-TN quadrature set and the exact value.

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36 Table 3-2. Even-moments obtained with a PN-TN S30 quadrature set. Moment Order (n even) M i n i iw1 M i n i iw1 M i n i iw1 Exact value n1 1 2 0.333333333 0.333333333 0.333333333 0.333333333 4 0.199999962 0.199999962 0.2 0.2 6 0.142857143 0.142857143 0.142857143 0.142857143 8 0.111111111 0.111111111 0.111111111 0.111111111 10 0.090909091 0.090909091 0.090909091 0.090909091 12 0.076923077 0.076923077 0.076923077 0.076923077 14 0.066666667 0.066666667 0.066666667 0.066666667 16 0.058823529 0.058823529 0.058823529 0.058823529 18 0.052631579 0.052631579 0.052631579 0.052631579 20 0.047619048 0.047619048 0.047619048 0.047619048 22 0.043478261 0.043478261 0.043478261 0.043478261 24 0.043478261 0.043478261 0.043478261 0.043478261 26 0.037037037 0.037037037 0.037037037 0.037037037 28 0.034482759 0.034482759 0.034482759 0.034482759 30 0.032258065 0.032258065 0.032258065 0.032258065 It is clear from Table 3-2 that the PN-TN quadrature set completely satisfies the even-moment conditions. This is possible because both roots of Legendre and Chebyshev polynomials satisfy the even-moment conditions given by Eqs. 2.13. In order to further verify the accuracy of the PN-EW and PN-TN quadrature sets, and to check their accuracy, I used a simple te st problem, consisting of a homogeneous parallelepiped, where an isotropic source is pl aced in its lower left corner as shown in Figure 3-4. Due to the symmetry of the problem, it is expected that the particle currents flowing out of regions A and B (see Figure 3-4) have the same value.

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37 Figure 3-4. Configuration of th e test problem for the valida tion of the quadrature sets. Figure 3-5 shows the relative difference betw een the particle current in regions A and B for the EW, PN-EW, and PN-TN as compared to the LQN technique. 0 2 4 6 8 10 12 8101214161820Sn OrderRelative Difference Jx/Jz (%) EW Pn-Tn Pn-EW Figure 3-5. Relative difference between the currents Jx and Jz for the test problem. Level-symmetric is considered as the re ference because it preserves moments of both azimuthal and polar direction cosines. The PN-TN yields almost perfect symmetry, while the PN-EW and Equal Weight show maximum re lative differences of 4% and 10%, respectively. It is worth noting that the lo ss in accuracy of the EW quadrature set is attributed to the fact that the even-m oment conditions are not satisfied. The PN-EW yields

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38 higher accuracy compared to EW, because the even-moment conditions are satisfied along the z-axis. 3.5 Testing the Effectiveness of the New Quadrature Sets In this section, the effectiveness of the new quadrature sets is examined by simulating two test problems: Kobayashi be nchmark problem 3 and a CT-Scan device for industrial/medical imaging applications. 3.5.1 Kobayashi Benchmark Problem 3 To examine the effectiveness of the new quadrature sets, I have used the first axial slice of the Kobayashi43 3-D benchmark problem 3 with pure absorber. Figures 3-6 show two different mesh distributions: Figure 3-6A is obtained from a previous study42, where an appropriate variable mesh was devel oped; Figure 3-6B shows a uniform mesh distribution that I have deve loped for the current study. The uniform mesh is used in order to separate the effects of the angular discretization from the spatial discretization. The reference semi-analytical solutions are evaluated in two spatial zones shown in Figures 3-6A and 3-6B (zone 1 along y-axis, at every 10.0 cm intervals between 5.0 and 95.0 cm; zone 2 along x-axis, y = 55.0 cm, every 10.0 cm, between 5.0 cm and 55.0 cm).

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39 A B Figure 3.6. Mesh distribution for the Kobayash i benchmark problem 3: A) Variable mesh distribution; B) Unifor m mesh distribution. Figure 3-7 shows the ratio of the calculated to the exact solution (C/E) for the levelsymmetric, PN-EW and PN-TN quadrature sets of order 20 for zone 1. Also, in this figure, I present a solution34 obtained in a previous study which uses the variable mesh distribution shown in Figure 3-6a. In the previous study, by taking advantage of the variable mesh distribution, the solution obtained with the level-symmetric quadrature set presented a maximum relative error of ~6% in zone 1. In the current study, the solution obtained with the level-symmetric quadrature set and uniform spatial mesh yields a maximum relative error of ~10% in zone 1. In zone 1, the PN-EW and PN-TN quadrature sets underest imate the scalar flux by ~51.9% and ~8.5%, respectively, on the last point of zone 1; this is due to the fact that the PN-EW quadrature set has fewer di rections clustered around the y-axis as compared to the PN-TN and level-symmetric quadrature sets.

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40 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 5152535455565758595 Y (cm)C/E Ratio LQn S20 (Uniform mesh) LQn S20 (Variable mesh) Pn-Tn S20 Pn-EW S20 Figure 3-7. Comparison of S20 quadrature sets in zone 1 at x=5.0 cm and z=5.0 cm. Figure 3-8 compares the scalar flux obtain ed in zone 2 of the benchmark problem. While using a uniform spatial mesh, the PN-TN quadrature set yields s lightly better results compared to level-symmetric. 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 51525354555 X (cm)C/E Ratio LQn S20 (Uniform mesh) LQn S20 (Variable mesh) Pn-Tn S20 Pn-EW S20 Figure 3-8. Comparison of S20 quadrature sets in zone 2 at y=55.0 cm and z=5.0 cm. In zone 2, the maximum re lative error obtained with PN-TN is ~18.6%, while for level-symmetric it is ~21.9% using the uniform mesh distribution, and ~6% using variable meshing. However, an error of ~28.2% is observed for the PN-EW quadrature set. Figures 3-9 and 3-10, show the solutions obtained with the PN-EW quadrature set for different SN orders compared to level-symmetric S20, in zone 1 and 2 respectively.

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41 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00 1.10E+00 5152535455565758595 Y (cm)C/E Ratio LQn S20 Pn-EW S20 Pn-EW S22 Pn-EW S24 Pn-EW S26 Figure 3-9. Comparison of PN-EW quadrature sets for different SN orders in zone 1 at x=5.0 cm and z=5.0 cm. 7.00E-01 8.00E-01 9.00E-01 1.00E+00 1.10E+00 1.20E+00 1.30E+00 1.40E+00 51525354555 X (cm)C/E Ratio LQn S20 Pn-EW S20 Pn-EW S22 Pn-EW S24 Pn-EW S26 Figure 3-10. Comparison of PN-EW quadrature sets for different SN orders in zone 2 at y=55.0 cm and z=5.0 cm. In zone 1 (Figure 3-9), the PN-EW is not as accurate as level-symmetric, because fewer directions are clustered near the y-axis; however in zone 1, the solution improves somewhat by increasing the SN order. In zone 2 (Figure 3-10) the PN-EW set yields inaccurate results, with a maximum relative error of ~36% for the S20 case. Figure 3-11 compares the ratios of diffe rent computed solutions to the exact solution; the computed solutions wered obtained with the PN-TN quadrature set for orders S20, S22, S24, S26 and with the S20 level-symmetric quadrature set. It appears that the increase in the quadrature order does not have a noticeable effect in improving the

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42 accuracy. However, this behavior can be attribut ed to the fact we have retained the same spatial mesh discretization. 8.00E-01 8.50E-01 9.00E-01 9.50E-01 1.00E+00 1.05E+00 1.10E+00 1.15E+00 5152535455565758595 Y (cm)C/E Ratio LQn S20 Pn-Tn S20 Pn-Tn S22 Pn-Tn S24 Pn-Tn S26 Figure 3-11. Comparison of PN-TN quadrature sets for different SN orders in zone 1 at y=5.0 cm and z=5.0 cm. In zone 2 (Figure 3-12), th e solution obtained with an S22 PN-TN quadrature set is more accurate than what obtained with level-symmetric. The S22 PN-TN yields a maximum relative error of ~9% compared to ~2 2% from level-symmetric. Again, in zone 2, the accuracy somewhat decreases as the SN order increases, because the spatial mesh is not consistently refined. 7.00E-01 8.00E-01 9.00E-01 1.00E+00 1.10E+00 1.20E+00 1.30E+00 51525354555 X (cm)C/E Ratio LQn S20 Pn-Tn S20 Pn-Tn S22 Pn-Tn S24 Pn-Tn S26 Figure 3-12. Comparison of PN-TN quadrature sets for different SN orders in zone 2 at y=55.0 cm and z=5.0 cm.

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43 3.5.2 CT-Scan Device for Medical/I ndustrial Imaging Applications The model of a CT-Scan device used for me dical/industrial applica tions is used in this section to verify the accuracy and pe rformance of the RAR technique. A CT-Scan device utilizes a collimated x-ray source (fan -beam) to scan an object or a patient. The main components of a CT-Scan device are an x-ray source mounted on a rotating gantry and an array of sensors. The patient is positi oned on a sliding bed that is moved inside the CT-Scan. The mesh distribution for this model, is shown in Figure 3-13. Figure 3-13. Cross-sectional view of the CT-Scan model on the x-y plane. Figure 3-13 shows the simplified PENTRA N model which represents the x-ray directional source (“fan” beam), a large region of air and an array of sensors. The size of this model is 74 cm along the y-axis and 20 cm along the x-axis. The array of detectors is located at 72 cm from the source along the x-axis. The materials are described using one-g roup cross-sections from the 20-group gamma of the BUGLE-96 cross-sections lib rary. The group corresponds to an x-ray source emitting photons in an energy range of 100 KeV to 200 KeV. The cross-sections were prepared using a P3 expansion for the scattering kernel. Because of the presence of large void regi ons and a directional source, the solution of the transport equati on is significantly affected by the ray-effects.3 One remedy is to use

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44 high order quadrature sets with biasing, such as RAR. We compared the solutions obtained with an S50 PN-TN quadrature set. The RAR techni que has been applied to an S30 PN-TN quadrature set; the biased region on the positive octant extends from z=0.0 cm to z=0.3 cm and the azimuthal angle extends from 0.0 to 5.0 degrees. In the biased region an S10 PN-TN quadrature set is used. The PN-TN quadrature set biased with RAR resulted in 142 directions per octant. The unbiased S50 PN-TN quadrature set yielded 325 directions per octant. The S20 level-symmetric quadrature set yielded 55 directions per octant. Figures 3-14, 3-15, and 3-16 show the flux distributions in the x-y plane, obtained with the level-symmetric S20, S50 PN-TN and S30 PN-TN with RAR, respectively. Figure 3-14. Scalar fl ux distribution on the x-y plane obtained with an S20 levelsymmetric quadrature set. Figure 3-15. Scalar fl ux distribution on the x-y plane obtained with an S50 PN-TN quadrature set.

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45 Figure 3-16. Scalar fl ux distribution on the x-y plane obtained with an S30 PN-TN quadrature set biased with RAR. The above results indicate that the le vel-symmetric quadrature set exhibits significant ray-effects, while S50 PN-TN and S30 PN-TN with RAR quadrature sets, yield similar solutions without any ray-effects. The main advantage of using a biased quadrature set is the significant reduction in computational cost and memory requirement. Table 3-3 compares the CPU time and memory requirements for the three calculations presented above. Table 3-3. CPU time and total number of dire ctions required for the CT-Scan simulation. Quadrature Set Directions CPU Time(sec) Memory ratioa Time ratioa S50 PN-TN 2600 166.4 1.0 1.0 S30 PN-TN RAR (S10) 1136 79.4 0.51 0.47 S20 LQN 440 33.3 0.2 0.2 a memory and time ratio are referred to the S50 PN-TN quadrature set. The RAR technique lessens the ray effect in the flux distribution and greatly reduces the computational time by more than a factor of 2 compared to S50. The new quadrature sets biased with the OS rather than the RAR technique have also been examined based on the CT-Scan model.9 Figure 3-17 compares the results of PENTRAN with a reference Monte Carlo solution. For all cases, the first direction of the

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46 lowest level in quadrature set is split in 9 or 25 directions; for example, PN-TN 22-2-55 corresponds to PN-TN S22 with direction 55 split in 9 directions. All the quadrature sets biased with the OS technique yield accurate results within the statistical uncertainty of the Monte Carlo predictions. Due to the significant ray-effects, the level-symmetric S20 quadrature set without ordinate splitting yields poor accuracy. Note that, even by using high order quadrature sets, such as PN-TN S28 (840 directions), the solution at detector position is under predicted by ~21%. 4.00E+08 4.50E+08 5.00E+08 5.50E+08 6.00E+08 6.50E+08 7.00E+08 7.50E+08 8.00E+08 8.50E+08 9.00E+08 02468101214161820 Y-Axis Position (cm)X-Ray Flux Pn-Tn22-2-55 Pn-Tn-20-3-46 MCNP Tally Flux S20-LS-3-46 S20-EW-5-46S20-LS (No OS) Pn-Tn28 (No OS) Figure 3.17. Comparison of the scal ar flux at detector position (x=72.0 cm).

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47 CHAPTER 4 DERIVATION OF THE EVEN -PARITY SIMPLIFIED SN EQUATIONS This chapter presents the initial derivation of the Simplified Spherical Harmonics (SPN) equations starting from the PN equations in 1-D geometry, and it discusses the issues related to the coupling of the SPN moments on the vacuum boundary conditions. Because of this peculiarity, the implementation of the general SPN equations into a computer code proved to be cumbersome. Howeve r, I will present the initial derivation of the SP3 equations, successively implemented in to a new computer code named PENSP3 (Parallel Environment Neutral-particles SP3). To overcome the difficulties rela ted to the coupling of the SPN moments in the vacuum boundary conditions, I adopted a diffe rent formulation based on the Even-Parity Simplified SN (EP-SSN) equations. These equations are derived starting from the 1-D SN equations, and using the same assumpti ons made for the derivation of the SPN equations; however, the main advantage of this formulation is the natural decoupling of the evenparity angular fluxes for the vacuum boundary conditions. Therefore, a Fourier analysis of the EP-SSN equations will follow, along with the derivation of a new formulation to accelerat e the convergence of the source iteration method applied to the EP-SSN equations. This chapter is conc luded with the derivation of the 3-D P1 equations. I will compare the P1 equations with the SP1 equations, and I will describe the assumptions made in the derivation of the SP1 equations and the relation with the spherical harmonics P1 formulation.

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48 4.1 Derivation of the Simplified Spherical Harmonics (SPN) Equations The SPN equations were initially proposed by Gelbard18 in the early 1960s. However, they did not receive much atten tion due to the weak theoretical support. Recently, the theoretical foundations of the SPN equations have been significantly strengthened using a variational analysis approach.21-22 I derive the multigroup SPN equations starting from the 1-D multigroup PN equations and by applying the procedure originally outlined by Gelbard. The multigroup 1-D PN equations are given by ) ( ) ( ) ( ) ( ) ( 1 2 ) ( 1 2 1, , , 1 1x q x x x x x n n x x n ng n g n g g sn g t g n g n (4.1) for n=0, N and g=1, G, where ) ( ) ( ) ( ) ( ) ( 4 1 2 ) (' 1 , ' 0 ,x S x q x x P l x qG g g g ext g n g f g l g g sl L l l g n (4.2) In Eqs. 4.1 and 4.2, I defined the following quantities: ) (,xg t total macroscopic cross-section in group g. ) (,xg g sn Legendre moment of the in-group macr oscopic scattering cross-section of order n. ) (' ,xg g sl Legendre moment of the group transfer macroscopic scatte ring cross-section of order l. ) (,xg n Legendre moment of th e angular flux of order n. ) (,x qg f, fission source. ) (,x Sext g n, Legendre moment of the i nhomogeneous source of order n. G, total number of energy groups and L, order of the Legendre expansion of th e macroscopic scattering cross-section (L
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49 In Eq. 4.1, the angular flux is expand ed in terms of Legendre polynomials as L l g l l gx P l x0 ,) ( ) ( 4 1 2 (4.3) In Eq. 4.1, the term dx dg N 1 is defined to be identically zero when n=N. This assumption closes the PN equations, yielding N+1 equations with N+1 unknowns. The procedure prescribed by Gelb ard to obtain the 3-D SPN equations from the 1-D PN, consists of the following steps: 4. Replace the partial derivative operator in Eq. 4.1 for even n with the divergence operator ) ( 5. Replace the partial derivative operator in Eq. 4.1 for odd n with the gradient operator ) ( By applying this procedure to Eq. 4.1, it reduces to ) ( ) ( ) ( ) ( ) ( 1 2 ) ( 1 2 1, , , 1 1r q r r r r n n r n ng n g n g g sn g t g n g n (4.4a) for n=0,2,…,N-1, g=1, G, and V r ) ( ) ( ) ( ) ( ) ( 1 2 ) ( 1 2 1, , , 1 1r q r r r r n n r n ng n g n g g sn g t g n g n (4.4b) for n=1,3,…,N, g=1, G, and V r The SPN equations can be reformulated in term s of a second-order elliptic operator, if one solves for the odd-parity moments using Eqs. 4.4b i.e., g an g n g n g an g n g an g nr q r n n r n n r, , 1 , 1 ,) ( ) ( 1 2 1 ) ( 1 2 1 1 ) ( (4.5) where ) ( ) (, ,r rg g sn g t g an (4.6) and then substitute Eqs. 4.5 into Eqs. 4.4a to obtain

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50 1 2 1 2 1 1 1 2 1 1 2 1 1 2 1 2 1 3 2 1 1 2 1 1 3 2 2 1 2 1, 1 1 1 1 , 1 2 1 , 1 , 1 2 1 g an g n g an g n g n g n g an g n g an g n g an g n g an g n g anq n n q n n q n n n n n n n n n n n n n n n n (4.7) for n=0,2,…,N-1 and g=1, G. Note that for simplicity, the spatial dependency of has been eliminated. The SPN equations yield a system of (N+1)/2 coupled partial differentia l equations that can be solved using standard iterative methods, such as Gauss-Seidel or Krylov subspace methods. The main disadvantage of this form ulation is that it yi elds Marshak-like vacuum boundary conditions, coupled through the moments. Because of this issue the implementation of this formulation into a computer code becomes cumbersome. However, to study the effectiveness of the SPN method, I developed a 3-D parallel SP3 code,25 PENSP3 (Parallel Environment Neutral-particles SP3). The PENSP3 code is based on Eqs. 4.8a and 4.8b, which are derived assuming isotropic scattering and isotropic inhomogeneous source. 1 21 0 0 ' 0 1 ' 0 0 0 2 , 1r S r r k r r r r r r r F r r F r DG g ext g g g f g g G g g g g g s g g g s g g t g t g (4.8a) 1 5 21 0 0 ' 0 1 ' 0 0 0 2 , 2 2 G g ext g g g f g g G g g g g g s g g a g g t g gr S r r k r r r r r r r r D (4.8b) for g=1, G, where

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51 r r r Fg g g 0 22 r r Dg t g , 13 1, and r r Dg t g , 235 9. (4.9) The SP3 Marshak-like vacuum boundary conditions for Eqs. 4.8a and 4.8b are given by 0 1 2 0 2 1, 2 8 3 ) ( 2 1 d d r r r F n D r Fb g g g g (4.10a) 0 1 3 2 0 2 2 2, 2 5 3 40 3 ) ( 40 21 d d r P r F r n D rb g g g g .(4.10b) Eqs. 4.10a and 4.10b are termed Marshak-like boundary conditions, because in 1-D geometry they reduce to standard Marshak boundary conditions. Implementation of these formulations into a computer code is difficult because of the coupling of the SPN moments. The reflective boundary condition is repres ented by setting the odd-moments equal to zero on the boundary, i.e., 0 ) ( ˆ, b g nr n for n odd, (4.11) where V rb and nˆ is the normal to the surface considered. 4.2 Derivation of the Even-Parity Simplified SN (EP-SSN) Equations I have derived the Even-Parity Simplified SN (EP-SSN) equations starting from the 1-D SN equations given by ) ( ) ( ,m m t m mx Q x x x x for m=1, N, (4.12) where ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) (0 0x q x S P l x x P l x Qf L l l m l L l l sl m l m (4.13) and

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52 ) ( ) ( 1 ) (0x x k x qf f ; m m l N m m lx P w x ) ( 2 1 ) (1 (4.14) A Gauss-Legendre symmetric quadrature set (PN) is considered, where) 1 1 ( m 0 21M m mw, and M=N(N+2). In Eq. 4.13, L is the order of the Legendre expansion for both the macroscopic scattering crosssection and the inhomogeneous source (L
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53 ) ( ) ( l lP P for l odd. (4.19b) Eq. 4.18 can be rewritten as ) ( ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) (1 .. 2 0x q x S x x P l x x x xf L even l l l sl m l E m t O m m (4.20) Similarly, by subtracting Eq. 4.16 from Eq. 4.12, I obtain ) ( ) ( , ) ( ,m m m m t m m mx Q x Q x x x x x x (4.21) and by using the definitions of evenand odd-parity angular fluxes, I obtain ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( 1 2 ) ( ) ( 2 ) ( 20 0 L l l m l m l L l l sl m l m l O m t E m mx S P P l x x P P l x x x x (4.22) Following the use of the Legendre polynomial identities (Eqs. 4.19a and 4.19b), Eq. 4.22 reduces to L odd l l l sl m l O m t E m mx S x x P l x x x x.. 3 1) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( (4.23) Now, the odd-parity angular fluxes are then obtained from Eq. 4.23 as L odd l l l sl m l t E m t m O mx S x x P l x x x x x.. 3 1) ( ) ( ) ( ) ( 1 2 ) ( 1 ) ( ) ( ) ( (4.24) Then, using Eq. 4.24 in Eq. 4.20, I obtain ). ( ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) (.. 3 1 1 .. 2 0 2x q x S x x P l x x x S x x P l x x x x x xf L odd l l l sl m l t m L even l l l sl m l E m t E m t m (4.25)

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54 Finally, the EP-SSN equations in 3-D Cartesian geometry with anisotropic scattering kernel and anisotropic inho mogeneous source of arbitrary order L are obtained by applying the procedure outlined by Gelbard (i.e., substitution of first order partial differential operators with the gradient operator) to Eq. 4.25. ), ( ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) (.. 3 1 1 .. 2 0 2r q r S r r P l r r S r r P l r r r rf L odd l l l sl m l t m L even l l l sl m l E m t E m t m (4.26) for m= 1 N/2 where L odd l l l sl m l t E m t m O mr S r r P l r r r r.. 3 1) ( ) ( ) ( ) ( 1 2 ) ( 1 ) ( ) ( ) ( (4.27) Due to the symmetry of the GaussLegendre quadrature set, the EP-SSN equations only need to be solved on half of the angular domain, e.g. 1 0 The moments of the evenand odd-parity angul ar fluxes are evaluated by ) ( ) ( ) (2 / 1r P w rN m E m m l m l for l even (4.28a) and ) ( ) ( ) (2 / 1r P w rN m O m m l m l for l odd (4.28b) The multigroup form of Eqs. 4.26 and 4.27 with anisotropic scattering and source are written as

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55 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) (, 1 '.. 3 1 , 1 .. 2 0 1 '.. 3 1 ' , 1 1 .. 2 0 ' , , 2r q r S P l r r S P l r r P l r r r P l r r r rg f G g L odd l g l m l g t m L even l g l m l G g L odd l g l g g sl m l g t m G g L even l g l g g sl m l E g m g t E g m g t m (4.29) and G g L odd l g l g l g g sl m l g t E g m g t m O g mr S r r P l r r r r1 '.. 3 1 ' , , ,) ( ) ( ) ( ) ( 1 2 ) ( 1 ) ( ) ( ) ( (4.30) for m= 1 N/2 and g= 1 G 4.2.1 Boundary Conditions for the EP-SSN Equations The boundary conditions for the EP-SSN equations are based on the assumption that the angular flux on the boundary surface is azimuthally symmetric about the surface normal vector. The EP-SSN boundary conditions follow directly from the 1-D even-parity SN boundary conditions. Hence, by considering th e positive half of the angular domain for the EP-SSN equations, the 1-D source boundary condi tion at the right boundary face is given by ) ( ) ( ) (m s m s O m s Ex x x (4.31) and the corresponding condition in 3-D is given by ) ( ) ( ) (m s m s O m s Er r n r (4.32)

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56 Note that in 3-D geometry, the EP-SSN formulation requires the incoming boundary flux to be azimuthally symmetric about the surface normal vector. The 3-D albedo boundary condition is given by ) ( 1 1 ) (m s E m s Or r n (4.33) Note that in Eq. 4.33, the vacuum boundary condition is obtained by setting 0 while the reflective boundary c ondition is obtained by setting 1 Note that the main advantage of the EP-SSN formulation compared to SPN is the decoupling of the even-parity angular fluxes for the vacuum boundary conditions. 4.2.2 Fourier Analysis of the EP-SSN Equations The EP-SSN equations are solved iteratively usin g the source iteration method. This method is based on performing iterative cycles on the scattering source; moreover, the method has a clear physical inte rpretation that allows one to predict classes of problems where it should yield fast convergence. The source iteration method for the EP-SSN equations is defined by ) ( ) (, 1 , ,q q r r Hl g g m s l E g m g m L for m= 1 N/2 and g= 1 G (4.34) where, g m LH, is the EP-SSN leakage plus collision operator, l g g m sq, is the in-group scattering source, and q is a fixed source term that in cludes scattering transfers from energy groups other than g the external source and fission sources. The iterative method begins by assuming a flux guess; then, Eq. 4.34 is solved for 1 ,l E g m and the in-group scattering source is updated. This process cont inues until a certain convergence criterion is satisfied.

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57 The convergence rate of any iterati ve method is characterized by the spectral radius For the source iteration method, in an infinite homogeneous medium, it is well known that the spectral radius is equal to the scattering ratio (c) given by g t g g s gc, (4.35) The scattering ratio is bounded between 0 and 1; hence, a c -ratio approaching one means that the problem will converge slowly, while, oppositely, a c -ratio close to zero, indicates a fast converging problem. Fourier analysis is the tool of choice to analyze th e convergence behavior of iterative methods. For simplicity, I will consider the 1-D EP-SSN equations with isotropic scattering and source, given by ) ( ) ( ) ( ) ( ) ( ) ( ) (, 0 0 2 / 1 2 / 1 2x S x x x x x x x xext l s E l t E l t (4.36) which following division by ) ( xt reduces to ) ( ) ( ) ( ) ( ) ( ) (, 0 2 / 1 2 / 1 2 2x x S x c x x x x xt ext l E l E l t (4.37) where c is the scattering rati o defined by Eq. 4.35. The EP-SSN equations can be rewritten in terms of the error between two consecutive iterations as ) ( ) ( ) ( ) (, 0 2 / 1 2 / 1 2 2x c x x x x xl E l E l t (4.38) where ) ( ) ( ) (2 / 1 2 / 1 2 / 1 x x xE l E l E l (4.39) and

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58 ) ( ) ( ) (1 0 0 0x x xl l l (4.40) The error terms are then expanded in term s of the Fourier modes, considering an infinite homogeneous medium, the Four ier ansatz is defined as follows ) exp( ) ( ) (2 / 1x i f xE l and) exp( ) (, 0x i xl (4.41) where 1 i and By substituting the above relations into Eq. 4.38, I obtain the EP-SSN equations mapped onto the frequency domain, resulting in function ) ( f given by 21 ) ( tc f. (4.42) Therefore, the spectrum of eigenvalues is obtained by observing that the error in the scalar flux at iteration l+1 can be written as 1 0 2 1 0 1 0 2 / 1 1 01 ) exp( ) ( ) exp( ) ( ) ( t E l lc d x i f d x i x d x. (4.43) By performing the integration over the angular variable in Eq. 4.43, the spectrum of eigenvalues is found to be equal to t tc arctan. (4.44) The result obtained in Eq. 4.44 is similar to what is obtained for the SN equations. The spectral radius is found to be equal to c ) ( max However, the convergence behavior of the EP-SSN equations is also affected by the value of the total scattering

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59 cross-section. For optically thin media, where the total cross-section assumes small values, Eq. 4.44 suggests that the convergence should be very fast, and in the limit as 0 t the spectral radius will tend to zero. 4.2.3 A New Formulation of the EP-SSN Equations for Improving the Convergence Rate of the Source Iteration Method As discussed in the previous paragraph, the performance of the source iteration method applied to the EP-SSN equations is similar to the SN equations. However, I have derived a new formulation of the EPSSN equations which reduces the spectral radius for the source iteration method. Appendix B addresses the performance of the new formulation for a criticality eigenvalu e benchmark problem; note that the new formulation is a key aspect for the successf ul implementation of an acceleration method for the SN equations. The main idea behind the new formulation is to remove the in-group component of the scattering kernel for each di rection. In order to reformulate the EPSSN equations, we note that the even-moments in the in-group portion of the scattering kernel can be expanded as follows 1 .. 2 0 2 / 1 , 1 .. 2 0 ,) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2L even l N m E g m m l m g g sl m l L even l g l g g sl m lr P w r P l r r P l (4.45) The term ) (,rE g m is consistently removed fr om the in-group portion of the scattering kernel and from the collision term on the left-hand side of the EPSSN equations.

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60 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) (, 1 '.. 3 1 , 1 .. 2 0 1 '.. 3 1 ' , 1 1 .. 2 0 ' 1 .. 2 0 2 / 1 ' ' , , 2r q r S P l r r S P l r r P l r r r P l r P w r P l r r r rg f G g L odd l g l m l g t m L even l g l m l G g L odd l g l g g sl m l g t m G g g g L even l g l g g sl m l L even l N m m m E g m m l m g g sl m l E g m R g m E g m g t m for m= 1 N/2 and g= 1 G (4.46) Note that in Eq. 4.46, the total cross-secti on is replaced, in the collision term, with a direction-dependent remova l cross-section as follows 1 .. 2 0 2 ,) ( ) ( 1 2 ) ( ) (L even l g g sl m m l g t R g mr w P l r r (4.47) In this new formulation, the main idea is to remove a “degree of freedom” from the iteration process in order to re duce the iterations on the component) (,rE g m This modification leads to a drastic reduction of the spectral radius. 4.3 Comparison of the P1 Spherical Harmonics and SP1 Equations In order to understand the assumptions on which the SPN and the EP-SSN equations are based, it is useful to examine the 3-D P1 spherical harmonics equations.6 The expansion in spherical harmonics of the angular flux can be written as follows N l l m lm lm m lm r m r P l r00) sin( ) ( ) cos( ) ( ) (cos ) 1 2 ( ˆ , (4.48) where 0 and 2 0 In the following discussion, for simplicity I will assume a P1 expansion of the angular flux in spherical harmonics, given by

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61 sin ) ( cos ) ( ) (cos 3 ) ( ) (cos 3 ) ( ) (cos ˆ ,11 11 1 1 10 0 1 00 0 0r r P r P r P r (4.49) By substituting the definitions of the Associated Legendre polynomials6 in Eq. 4.49, I obtain the P1 expansion for the angular flux: sin ) ( cos ) ( sin 3 ) ( 3 ) ( ˆ ,11 11 10 00r r r r r (4.50) where cos The derivation of the SPN equations outlined by Gelbard, assumes implicitly that the angular flux be azimuthally independent, and hence symmetric with respect to the azimuthal variable. By introducing this assumption on the P1 expansion of the angular flux in Eq. 4.50, I obtain d r r r r d r r 2 0 11 11 10 00 2 0sin ) ( cos ) ( sin 3 ) ( 3 ) ( ˆ , ~ (4.51) Therefore, by performing the integration on Eq. 4.51, I obtain ) ( 3 ) ( ~ 10 00r r r (4.52) It is evident that the angular flux obtai ned in Eq. 4.52 is equivalent to the SP1 angular flux where, 00 is the scalar flux and 10 is the total current. The general formulation of the multigroup PN equations6, with anisotropic scattering and source, is obtained by substi tuting Eq. 4.48 into the linear Boltzmann equation and deriving a set of coupled partia l differential equati ons for the moments ) ( rg lm and ) ( rg lm

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62 2 ) 1 2 ( 2 ) )( 1 ( ) 1 )( 2 ( ) ( 2 ) 1 ( 2, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1g lm g lm g l g m l g m l g m l g m l g m l g m l g m l g m l g m l g m lS l y x m l m l y x m l m l y x y x z m l z m l (4.53a) 2 ) 1 2 ( 2 ) )( 1 ( ) 1 )( 2 ( ) ( 2 ) 1 ( 2, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1g lm g lm g l g m l g m l g m l g m l g m l g m l g m l g m l g m l g m lS l x y m l m l x y m l m l x y x y z m l z m l (4.53b) for g= 1 G where g g sl g t g l , Therefore, the P1 equations are obtained by evaluating Eqs. 4.53a and 4.53b for l= 0, 1 and m= 0, 1, as follows (l= 0 m= 0 ) g g g g g gS y x z, 00 00 0 11 11 102 2 2 2 (4.54a) g g g g g gS x y z, 00 00 0 11 11 102 2 2 2 (4.54b) (l= 1 m= 0 ) g g g g g g gS y x z z, 10 10 1 21 21 00 202 6 6 2 4 (4.54c)

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63 g g g g g g gS x y z z, 10 10 1 21 21 00 202 6 6 2 4 (4.54d) (l= 1 m= 1 ) g g g g g g g g g gS y x y x y x z, 11 11 1 22 22 20 20 00 00 212 6 12 2 6 (4.54e) g g g g g g g g g gS x y x y x y z, 11 11 1 22 22 20 20 00 00 212 6 12 2 6 (4.54f) The terms with l> 1 and m> 1 are dropped from Eqs. 4.54c through f, yielding the following relationships g g g g gS z, 1 10 00 1 103 3 1 (4.55a) g g g g gS z, 1 10 00 1 103 3 1 (4.55b) g g g g g gS y x, 1 11 00 00 1 113 3 1 (4.55c) g g g g g gS x y, 1 11 00 00 1 113 3 1 (4.55d) Then, by substituting Eqs. 4.55a, c and d in Eq. 4.54a, I obtain g g g g g g g g g gS S x y y x, 1 00 00 0 00 1 00 1 00 1~ 3 1 3 1 3 1 (4.56) where g g g g g g gS y S x S z S, 1 11 1 11 1 10 13 3 3 ~ (4.56a) Analogously, by using Eqs. 4.55b, c and d in Eq. 4.54b, I obtain

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64 g g g g g g g g g gS S x y y x, 1 00 00 0 00 1 00 1 00 1~ 3 1 3 1 3 1 (4.57) where g g g g g g gS y S x S z S, 1 11 1 11 1 10 13 3 3 ~ (4.57a) Eqs. 4.56 and 4.57 constitute a coupled system of partial differential equations for g 00 andg 00, which must be solved iteratively. Recall that the assumption made in the SPN methodology is that the angular flux is azi muthally symmetric; therefore, to obtain the SP1 equations (Eq. 4.58 or 4.59), terms such asg 00are dropped from Eqs. 4.56 and 4.57, as follows g g g g g g gS S, 1 1 00 00 0 00 13 3 1 (4.58) or g g g g g g gS S, 1 1 0 0 0 0 13 3 1 (4.59) Here, I can also conclude that in the ca se of a homogeneous medium, with isotropic scattering, the P1 and the SP1 equations yield the same so lution, because the azimuthal dependency on the angular flux is removed. Note that this result can also be generalized to the SPN equations.

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65 CHAPTER 5 NUMERICAL METHODS FOR SOLVING THE EP-SSN EQUATIONS This chapter addresses the numerical t echniques utilized to solve the EP-SSN equations; I will describe the discretization of the EP-SSN equations in a 3-D Cartesian geometry using the finite-volume method, along with the matrix operator formulation utilized and the boundary conditions. I w ill also introduce the Compressed Diagonal Storage (CDS) method, which is fundamental for reducing the memory requirements and the computational complexity of the iterative solvers. Further, a new coarse mesh based projection algorithm for elliptic-type partial differential equations will be presented. Finally, I will describe a class of iter ative solvers based on the Krylov subspace methods, such as the Conjugate Gradient (C G) and the Bi-Conjugate Gradient methods (Bi-CG). The CG and Bi-CG methods have b een implemented to so lve the linear systems of equations arising from the finite -volume discretization of the EP-SSN equations. Furthermore, the issue of preconditioning of the CG methodology will be discussed. 5.1 Discretization of the EP-SSN Equations Using the Finite-Volume Method The EP-SSN equations derived in Chapter 4 are discretized using the finite-volume approach. For this purpose, I consider a general volume V in a 3-D Cartesian geometry. The volume V is then partitioned into non-overlapping sub-domains Vj, called coarse meshes Note that, the coarse mesh sub-dom ains are generally defined along the boundaries of material regions. As I will discu ss in Chapter 7, the main purpose of this approach is to partition the pr oblem for parallel processing.

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66 The discretization of the spatial domain is completed by defining a fine-mesh grid onto each coarse mesh. I have derived a formulation of the discretized EP-SSN equations which allows for variable fine mesh density on different regions of the problem; this approach is very effective to generate an effective mesh di stribution, because it allows a finer refinement only in those regions where higher accuracy is needed. The finite-volume discretization of the multigroup EP-SSN equations (Eqs. 4.29) is obtained by performing a triple integration on a finite volume, dxdydz dr as follows vv g f m g ext m g s E g m g t g t mdr r Q r Q r Q dr r r r ) ( ) ( ) ( ) ( ) ( ) (, , , , 2 (5.1) where ) ( ) ( ) ( 1 2 ) ( ) ( ) ( ) ( 1 2 ) (1 '.. 3 1 ' , 1 1 .. 2 0 ' , G g L odd l g l g g sl m l g t m G g L even l g l g g sl m l m g sr r P l r r r P l r Q (5.1a) L odd l g l m l g t m L even l g l m l m g extr S P l r r S P l r Q.. 3 1 , 1 .. 2 0 , ,) ( ) ( 1 2 ) ( ) ( ) ( 1 2 ) ( (5.1b) and ) ( ) ( 1 ) (0 ,r r k r Qg f g f (5.1c) For this derivation, I consid er a central finite-differen ce scheme for generic mesh element with coordinates xi, yj and zk; an example of a fine mesh element and its neighbor points is shown in Figure 5.1.

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67 Figure 5.1. Fine mesh represen tation on a 3-D Cartesian grid. The generic fine mesh element is de fined by the discreti zation step sizes,cx cy and cz along the x -, y and z -axis, respectively. Note that the discretization steps are constant within each coarse mesh; hence, a non-uniform mesh distribut ion is not allowed. The discretization steps are defined as follows c x c x cN L x c y c y cN L y c z c z cN L z and c c c cz y x v (5.2) for c= 1 Ncm where, Ncm is the total number of coarse meshes; c xL ,c yL, andc zL are the dimensions of the coarse mesh ( c ), along the x -, y and z -axis, respectively; and c xN ,c yN, and c zN refer to the number of fine meshes along the x -, y and z -axis, respectively. Note that, Eq. 5.1 is numerically integrated on a generic finite volumecv I will first consider the integration of the el liptic or leakage operator (first term in Eq. 5.1) as follows (i, j, k) (i+1, j, k) (i-1, j, k) (i, j+1, k) (i, j-1, k) (i, j, k+1) (i, j, k-1) x y z

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68 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1, 2 , 2 , 2 , 2 , 2 , 2 , 2 k k j j i i k k j j i iz E g m g t m z E g m g t m c c y E g m g t m y E g m g t m c c x E g m g t m x E g m g t m c c z z E g m g t m y y x xr z r r z r y x r y r r y r z x r x r r x r z y r r dz dy dx (5.3) For simplicity, I will derive the discretized operator along the x -axis; the treatment is analogous along the y and z -axis. Figure 5.2 represents the view of a fine mesh and its neighbor points along the x -axis. Figure 5.2. View of a fine mesh along the x -axis. In Figure 5.2, x represents a generic macrosc opic cross-section (e.g., total, fission, etc.) which is constant within the fine mesh. In Eq. 5.3, I evaluate the right-side and left-side partial derivatives along the x -axis at2 / 1ix xi x i+1xi-1 xi+1/2 xi-1/2 i x 1 ,i x 1 ,i x x cx 2 /cx 2 /cx

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69 2 / ) , ( ) , ( ) , ( ) , (, 2 / 1 ) ( , 2 2 / 1 ) ( c k j i E g m k j i E g m k j i g t m k j i E g mx z y x z y x z y x z y x f (5.4) 2 / ) , ( ) , ( ) , ( ) , (2 / 1 ) ( 1 1 2 2 / 1 ) ( c k j i E g m k j i E g m k j i g t m k j i E g mx z y x z y x z y x z y x f (5.5) In order for the elliptic operator to be defined, the function ) , (,z y xE g m must be continuous along with its first derivative ) , (,z y x fE g mand second derivative, which translates into the fact that the even-parity angular flux belongs to a C2 functional space, or 2 ,) , ( C z y xE g m Therefore, the following relationships hold true ) , ( ) , ( ) , (2 / 1 ) ( 2 / 1 ) ( 2 / 1 k j i E g m k j i E g m k j i E g mz y x z y x z y x (5.6) and ) , ( ) , ( ) , (2 / 1 ) ( 2 / 1 ) ( 2 / 1 k j i E g m k j i E g m k j i E g mz y x f z y x f z y x f (5.7) Therefore, I eliminate the value of) , (2 / 1 ,z y xi E g m in Eqs. 5.4, obtaining the second order, central-finite differencing fo rmula for the even-parity angular flux: g m k j i g m k j i E g m k j i x g m k j i E g m k j i x g m k j i E g m j i id d d d, , , , 1 , , , , , , 1 , , 1 , , 2 / 1 (5.8) and the even-parity current density E g m k j i E g m k j i x g m k j i x g m k j i x g m k j i x g m k j i E g m k j id d d d f, , , , 1 , , 1 , , , , 1 , , , , 2 / 12 (5.9) In Eqs. 5.8 and 5.9, I have defined the pseudo-diffusion coefficients along the x axis, as c g k j i t m x g m k j ix d , , 2 , , c g k j i t m x g m k j ix d , 1 2 , , 1 c g k j i t m x g m k j ix d , 1 2 , , 1 (5.10)

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70 Analogously, the expression for ) , (2 / 1 k j i Ez y x f is obtained as follows E g m k j i E g m k j i x g m k j i x g m k j i x g m k j i x g m k j i E g m k j id d d d f, , 1 , , , , 1 , , , , 1 , , , , 2 / 12 (5.11) The partial derivatives along the y and z -axis are discretized in a similar fashion, yielding the finite-volume discreti zed elliptic operator given by ) ( ) (, 1 , , , , 1 , , , 1 , , 1 , , 1 , , , 1 , , , , 1 , 1 , , 1 , , , 1 , , , , 1 , 1 , 22 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1E g m k j i E g m k j i g m k k E g m k j i E g m k j i g m k k c c E g m k j i E g m k j i g m j j E g m k j i E g m k j i g m j j c c E g m k j i E g m k j i g m i i E g m k j i E g m k j i g m i i c c z z E g m g t m y y x xy x z x z y r r dz dy dxk k j j i i (5.12) where x g m k j i x g m k j i x g m k j i x g m k j i g m i id d d d, , 1 , , , , 1 , , , 1 ,2 x g m k j i x g m k j i x g m k j i x g m k j i g m i id d d d, , 1 , , , , 1 , , , 1 ,2 (5.13a) y g m k j i y g m k j i y g m k j i y g m k j i g m j jd d d d, , 1 , , , , 1 , , , 1 ,2 y g m k j i y g m k j i y g m k j i y g m k j i g m j jd d d d, , 1 , , , , 1 , , , 1 ,2 (5.13b) z g m k j i z g m k j i z g m k j i z g m k j i g m k kd d d d, 1 , , , , 1 , , , , 1 ,2 z g m k j i z g m k j i z g m k j i z g m k j i g m k kd d d d, 1 , , , , 1 , , , , 1 ,2 (5.13c) and c g k j i t m y g m k j iy d , , 2 , , c g k j i t m y g m k j iy d , 1 , 2 , 1 , c g k j i t m y g m k j iy d , 1 , 2 , 1 , (5.14) c g k j i t m z g m k j iz d , , 2 , , c g k j i t m z g m k j iz d 1 , 2 , 1 , c g k j i t m z g m k j iz d 1 , 2 , 1 , (5.15) Finally, by integrating the remaining terms of the EP-SSN equations, I obtain the complete multigroup EP-SSN formulation with anisotropic scattering and source as follows

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71 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2, , 1 '.. 3 1 2 / 1 , 2 / 1 , , , , , 1 '.. 3 1 2 / 1 , , 2 / 1 , , , , , 1 '.. 3 1 , 2 / 1 , 2 / 1 , , , , 1 .. 2 0 , , 1 '.. 3 1 2 / 1 , 2 / 1 , , , , , 1 '.. 3 1 2 / 1 , , 2 / 1 , , , , , 1 '.. 3 1 , 2 / 1 , 2 / 1 , , , , 1 1 .. 2 0 , , , , , , , , 1 , , , , 1 , , , 1 , , 1 , , 1 , , , 1 , , , , 1 , 1 , , 1 , , , 1 , , , , 1 , 1 ,c k j i g f G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c L even l c k j i g l m l G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c G g L odd l k j i g l k j i g l k j i g g sl m l k j i g t m c c G g L even l c k j i g l k j i g g sl m l c E g m k j i k j i g t E g m k j i E g m k j i g m k k E g m k j i E g m k j i g m k k c c E g m k j i E g m k j i g m j j E g m k j i E g m k j i g m j j c c E g m k j i E g m k j i g m i i E g m k j i E g m k j i g m i i c cv Q S S P l y x S S P l z x S S P l z y v S P l P l y x P l z x P l z y v P l v y x z x z y (5.16) for c=1, Ncm, m=1, N/2 L=0, N-1 g=1, G The EP-SSN equations discretized with the fi nite-volume method can be expressed in a matrix operator form characteri zed by a 7-diagonal banded structure.

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72 ,, , , , , , , , , , , , , , , , , g m x g m y g m z g m x g m g m x g m y g m z g m x g m g m x g m y g m y g m x g m g m x g m y g m y g m x g m g m x g m z g m y g m x g m g m x g m z g m y g m x g m g m g m cD L L L U D L L L U D L L U U D L L U U D L U U U D L U U U D A for c= 1 Ncm; m= 1 N/ 2; g= 1 G where ,, 1 , 1 , 1 , 1 , 1 , 1 , g m k k g m k k c c g m j j g m j j c c g m i i g m i i c c x g my x z x z y D g m i i c c x g mz y U, 1 , ,g m i i c c x g mz y L, 1 , g m j j c c y g mz x U, 1 , ,g m j j c c y g mz x L, 1 , g m k k c c z g my x U, 1 , ,g m k k c c z g my x L, 1 , for i= 2 1 c xN j= 2 1 c yN, k= 2 1 c zN c= 1 Ncm, m= 1 N/ 2, g= 1 G 5.2 Numerical Treatment of the Boundary Conditions The boundary conditions for the EP-SSN equations are discreti zed as well using the finite-volume method. In general, the BCs can be prescribed at back (-xb), front (+xb), left (-yb), right (+yb), bottom (-zb), and top (+zb). The reflective boundary conditions are simply derived as follows: -xb) 0, 2 / 1 , O k j g m, +xb) 0, 2 / 1 ,O k j N g mx (5.17a) -yb) 0, 2 / 1 , O k i g m, +yb) 0, 2 / 1 , ,O k N i g my (5.17b) -zb) 02 / 1 , , O j i g m, +zb) 02 / 1 , ,O N j i g mz (5.17c)

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73 The vacuum boundary conditions are obt ained from Eq. 4.32, by setting0 Hence, the vacuum boundary conditions along the x -, y and z -axis are given below: Front side vacuum boundary condition x = +x b ) ( 1 2 1 1 11 '.. 3 1 , 2 / 1 , 2 / 1 , , , , , , , , , , , , 2 / 1 G g L odd l k j N g l k j N g l k j N g g sl m l k j N g t g m N g m N E g m k j N g m N g m N O g m k j Nx x x x x x x x x xS P l a a a a (5.18a) Back side vacuum boundary condition x = -x b ) ( 1 2 1 1 11 '.. 3 1 , 2 / 3 , 2 / 3 , 1 , 1 , , 1 , 1 , , 1 , 1 , 1 , , 2 / 1 G g L odd l k j g l k j g l k j g g sl m l k j g t g m g m E g m k j g m g m O g m k jS P l a a a a (5.18b) Right side vacuum boundary condition y = +y b ) ( 1 2 1 1 11 '.. 3 1 2 / 1 , , 2 / 1 , , , , , , , , , , , , , 2 / 1 G g L odd l k N i g l k N i g l k N i g g sl m l k N i g t g m N g m N E g m k N i g m N g m N O g m k N iy y y y y y y y y yS P l b b b b (5.19a) Left side vacuum boundary condition y = -y b ) ( 1 2 1 1 11 '.. 3 1 2 / 3 , , 2 / 3 , , 1 , , 1 , , 1 , 1 , 1 , 1 , 1 , 2 / 1 G g L odd l k i g l k i g l k i g g sl m l k i g t g m g m E g m k i g m g m O g m k iS P l b b b b (5.19b)

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74 Bottom side vacuum boundary condition z = -z b ) ( 1 2 1 1 11 '.. 3 1 2 / 1 , 2 / 1 , , , , , , , , , , , , 2 / 1 , G g L odd l N j i g l N j i g l N j i g g sl m l N j i g t g m N g m N E g m N j i g m N g m N O g m N j iz z z z z z z z z zS P l c c c c (5.20a) Top side vacuum boundary condition z = +z b ) ( 1 2 1 1 11 '.. 3 1 2 / 3 , 2 / 3 , 1 , 1 , , , 1 , 1 , 1 , , 1 , 1 , 2 / 1 , G g L odd l j i g l j i g l j i g g sl m l j i g t g m g m E g m j i g m g m O g m j iS P l c c c c (5.20b) where k j g t c m g mx a, 1 , , 12 k j N g t c m g m Nx xx a, , , ,2 k i g t c m g my b, 1 , , 12 k N i g t c m g m Ny yy b, , , ,2 1 , , , 12j i g t c m g mz c z zN j i g t c m g m Nz c, , , ,2 and cmN c c x xN N1, cmN c c y yN N1, cmN c c z zN N1. 5.3 The Compressed Diagonal Storage Method Due to the sparse structure of the ma trices involved, I have adopted the Compressed Diagonal Storage (CDS) method in order to efficiently store the matrix operators. The CDS method stores only the non -zero elements of the coefficient matrix and it uses an auxiliary vector to identify th e column position of each element. Due to the

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75 banded structure of the coefficients matrix a mapping algorithm is easily defined for a generic square matrix as follows: J j I i A j i aj i, 1 1 ) (, 3 3 1 ~ ) ( ~ d I i A d i ad i, ) ( d i jcol. (5.21) The algorithm defined in Eq. 5.21, maps th e full structure of the matrix A into a compressed diagonal structure, where for each element on row i there is an associated diagonal index ranging from -3 to 3, with index 0 being the main diagonal, and an auxiliary vector jcol which stores the column position of each element. If we consider a 360x360 full matrix in single precision, with a total of 129600 elements, the memory required for allocating the matrix is roughl y 2.1 MB. However, if the CDS method is used, the total number of non-zero elements to be stored is only 2520, for a total memory requirement of 42 KB, which is a reduction of a factor of 50 compared to the full matrix storage. Moreover, since the CDS method stores only non-ze ro elements, I have also obtained a reduction in the number of ope rations involved in the matrix-vector multiplication algorithms. 5.4 Coarse Mesh Interface Projection Algorithm The partitioning of the spatial domain into non-overlapping coarse meshes leads to a situation in which the EP-SSN equations have to be discretized independently for each coarse mesh. Therefore, each coarse mesh is considered as an independent transport problem; however, to obtain the solution on the whole domai n, an interface projection algorithm has to be used in conjunction with an iterative method. The matrix operators have to be modified on the interfaces in order to couple the equations on each coarse

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76 mesh. For explanatory purposes, consider Fi gure 5.3, which shows the interface region between two coarse meshes. Figure 5.3. Representation of a coarse mesh interface The coordinates xN+1/2 and x1/2’ represent the interface on coarse mesh 1 and 2, respectively. As shown in Figure 5.3, the disc retization of the ellipt ic operator for coarse mesh 1, using the central finite difference me thod, would require the values of the evenparity angular flux at points xN-1, xN, and x1’. Similarly, in coarse mesh 2, the discretization would involve the value of the EP angular flux at points xN, x1’, and x2’. However, the point x1’ is located on coarse mesh 2 and point xN is located on coarse mesh 1; hence this term does not appear explicitly in the matrix operator for both coarse meshes. In order to couple the equations on the inte rface, I have reformul ated the discretized equations by bringing the unknown points on the right side of the equations. The numerical discretization of the EP-SSN equations in coarse mesh 1 would yield xN x1’ Coarsemesh2 xN+1/2 x1/2’ xN-1 x2’ xN-1/2 x3/2’ x1’ xN Coarse mesh 1

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77 ~ , , , , , , , 1 , , 1 , , 1 , 1 N g f N N m g ext N N m g s N N E g m N N g t E g m N E g m N g m N N E g m N E g m g m NQ x Q x Q x x (5.22) where x g m x g m N x g m x g m N g m Nd d d d, 1 , , 1 , , 1 ,2 and x g m N x g m N x g m N x g m N g m N Nd d d d, 1 , , 1 , , 1 ,2 (5.23) The coefficient x g md, 1 depends on the material properties and fine mesh discretization of coarse mesh 2, and it is calculated a prio ri; however, in Eq. 5.22, the termE g m , 1 ~ is unknown, and hence has to be eval uated iteratively by placing it in the source term, as shown in Eq. 5.24. ~, 1 , 1 , , , , , , , 1 , 1 , , 1 , , 1 E g m g m N N g f N N m g ext N N m g s N N E g m N N g t E g m N g m N N E g m N g m N N E g m N g m NQ x Q x Q x x (5.24) A similar equation can be formulated for coarse mesh 2, as follows ~ 1 , 1 1 , 1 1 , 1 1 , 1 1 , , , 1 , 1 , 1 , 2 , 2 1 g f m g ext m g s E g m g t E g m N E g m g m N E g m E g m g mQ x Q x Q x x (5.25) or ~, , , 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , , 1 , 1 , 1 , 2 1 , 2 , 2 1 E g m N g m N g f m g ext m g s E g m g t E g m g m N E g m g m E g m g mQ x Q x Q x x (5.26) where x g m x g m x g m x g m g md d d d, 2 , 1 , 2 , 1 , 2 12 and x g m N x g m x g m N x g m g m Nd d d d, , 1 , , 1 , 12 (5.27)

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78 Therefore, Eq. 5.24 and 5.26 are coupled th rough the value of th e EP angular fluxes E g m , 1 ~ andE g m N , ~ The EP-SSN equations are solved iteratively starting in coarse mesh 1, and assuming an initial guess forE g m , 1 ~ Once the calculation is completed the value of E g m N , ~ in Eq. 5.26, is set equal toE g m N , Hence, once the calculation is completed on coarse mesh 2, the value obtained for E g m , 1 is used in Eq. 5.24, to update the value ofE g m , 1 ~ ; this procedure contin ues until a convergence criterion is satisfied. In a 3-D Cartesian geometry the coupling on the coarse mesh interfaces is achieved exactly as described above; however, in this case the coarse meshes can be discretized with different fine mesh grid densities. Th e variable grid density requires a projection algorithm in order to map the EP angular fl uxes and the pseudo-diffusion coefficients among different grids. As stated earlier in this chapter, the va riable density grid approach is very effective to refine only those regions of the model where a higher accuracy is needed; note that the main constraint on the fi ne mesh grid is the mesh size being smaller than the mean free path for that particular material region. The main philosophy behind the projection algorithm is derived from the multigrid method, where a prolongation/injection operator is used to map a vector onto grids with different discretizations. Figure 5-4 shows the application of the projection algorithm along the y -axis on the interface between two coarse meshes.

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79 Figure 5.4. Representation of the interface projection algorithm between two coarse meshes. For simplicity, I will consider the proj ection of a vector between two coarse meshes, along the y-axis, as shown in Figure 5.4. The fine-to-coarse projection of a vector is obtained by collap sing the values as follows 4 1 1,i iF iF CG w F (5.28) where C iF iFA A w for i =1, 4 (5.29) In Eq. 5.29, iFA andCA are the areas associated w ith the fine-mesh and coarsemesh grid, respectively. Conversely, the coar se-to-fine projection is obtained as follows C F FF w G1 1 1 (5.30a) C F FF w G1 2 2 (5.30b) x z Grid 2 F1C F2F3F4 x z Grid 1 G1F G2F G3F G4F G5F G6F G7F G8F x y z Coarse mesh 1 (Finer grid) Coarse mesh 2 (Coarser grid)

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80 C F FF w G1 3 3 (5.30c) C F FF w G1 4 4 (5.30d) In general, the fine-to-coarse mesh projec tion is obtained with the following formulation ,1FN j jF jF iCG w F (5.31) where .iC jF jFA A w (5.32) The weights in Eq. 5.32 are the ratios of th e areas of the fine meshes intercepted by the coarse meshes on which the values are be ing mapped. Similarly, the coarse-to-fine mesh projection algorithm is defined as follows ,1CN j jC jC iFF w G (5.33) where .jC iF jCA A w (5.34) By using the above formulations, the ev en-parity angular fluxes and the pseudodiffusion coefficients are projected among coarse meshes with different grid densities. Note that the projected pse udo-diffusion coefficients need to be calculated only one time at the beginning of the calculation, while, th e projections for the EP angular fluxes have to be updated at every iteration. 5.5 Krylov Subspace Iterative Solvers Due to the size and sparse structure of the matrix operators obtained from the discretization of the EP-SSN equations, direct solution methods such as LU

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81 decomposition and Gaussian elimination do not perform effectively both in terms of computation time and memory requirements. In contrast, the Krylov subspace iterative methods, such as Conjugate Gradient (CG), ar e specifically designed to efficiently solve large linear systems of equations charact erized by sparse matrix operators. Note that in many engineeri ng applications, the matrix operators resulting from a finite-difference discretization is usually positive-definite and diagonally dominant. These conditions are fundamental in ensuring th e existence of a unique solution. A matrix is positive-definite if it satisfies the following condition 0 x A xT for every vector0 x (5.35) Moreover, a matrix is defined to be diagona lly dominant if the following condition holds true. n i j j ij iia a1, for i= 1, n (5.36) The CG algorithm is based on the fact that the solution of the linear system b x A is equivalent to finding the minimum of a quadratic form given by c x b x A x x fT T 2 1 ) (. (5.37) The minimum of the quadratic form of Eq. 5.37 is evaluated by calculating its gradient as follows ) ( ) ( ) (1x f x x f x x fn (5.38)

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82 The gradient of a function is a ve ctor field, and for a given point x points in the direction of the greatest increase of ) ( x f Because the matrix A is positive-definite, the surface defined by the function ) ( x f presents a paraboloid shape, which ensures the existence of a global minimum. Moreover, the diagonal dominance of the matrix A ensures the existence of a unique solution. By applying Eqs. 5.37 and Eq. 5.38, we derive the formulation for the gradient of the function ) ( x f given by b x A x A x fT 2 1 2 1 ) (. (5.39) If the matrix A is symmetric, Eq. 5.39 reduces to b x A x f ) (. (5.40) Therefore, by setting ) ( x f in Eq. 5.40 equal to zero, we find the initial problem that we wish to solve. 5.5.1 The Conjugate Gradient (CG) Method The CG method is based on finding the minimum of the function ) ( x f using a line search method. The calculation begins by gue ssing a first set of search directions 0d using the residual as follows: 0 0 0x A b r d (5.41) The multiplier for the search directions is calculated as follows i T i i T i id A d r r (5.42) where i is the iteration index. The multiplier is chosen such that the function ) ( x f is minimized along the search direction. Therefore, the solu tion and the residuals are updat ed using Eqs. 5.43 and 5.44.

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83 i i i id x x 1, (5.43) i i i id A r r 1. (5.44) The Gram-Schmidt orthogonalization met hod is used to update the search directions by requiring the residuals to be orthogonal at two cons ecutive iterations. The orthogonalization method consists of calculating the search directions i i i id r d 1 1 1 (5.45) where the coefficients are given by i T i i T i ir r r r 1 1 1 (5.46) Note that Eq. 5.44 indicates that the new residuals are a linear combination of the residual at the previous iteration and id A It follows that the new search directions are produced by a successive application of the matrix operator A on the directions at a previous iteration id. The successive application of the matrix operator A on the search directions id generates a vector space called Krylov subspace, represented by 0 1 0 2 0 0,..., ,d A d A d A d spani i (5.47) This iterative procedure is terminated when the residuals satisfy the following convergence criterion 1ir MAX, (5.48) where is the value of the tolerance, which is usually set to 1.0e-6. 5.5.2 The Bi-Conjugate Gradient Method The Bi-Conjugate Gradient (Bi-CG) has b een developed for solving non-symmetric linear systems. The update relations for the residuals are similar to the CG method;

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84 however, they involve the transpose of the ma trix operator. Hence, the residuals and the search directions are updated with the following equations: i i i iAp r r 1, (5.49a) i T i i ip A r r ~ ~ ~ 1 (5.49b) 1 1 1 i i i ip r p (5.49c) 1 1 1 ~ ~ ~ i i i ip r p (5.49d) where i T i i T i iAp p r r ~ ~ 1 1 and 1 1~ ~ i T i i T i ir r r r. (5.50) 5.5.3 Preconditioners for Krylov Subspace Methods The convergence rate of iterative methods depends on spectral properties of the coefficient matrix. The main philosophy of preconditioning is based on the attempt to transform the linear system into one that preserve the solution, but that has more favorable spectral properties. Th e spectral radius in norm L2 for a symmetric matrix A is defined by 2A A (5.51) The spectral radius so defined, gives an indication of the convergence behavior of the iterative method used. In the ca se of preconditioning, if a matrix M approximates the coefficient matrix A the transformed system b M x A M 1 1 (5.52) has the same solution of the original system b x A but the spectral radius of its coefficient matrix A M 1 is generally smaller than the original system. Various preconditioning techniques include the Jacobi or diagonal scaling, the Incomplete

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85 Cholesky, and the multigrid. The Jacobi preconditioner is the most straightforward preconditioner and it is based on using the main diagonal of the matrix A otherwise. 0 j i if ,i i j ia m (5.53) This method is the least demanding in terms of memory requirements and computation time; however, the method also pres ents limited performance characteristics. I have developed an Incomplete Cholesky preconditioner for the Conjugate Gradient (ICCG) method. This method is well su ited for symmetric definite matrices and it is based on decomposing the matrix A using the Cholesky factorization method.24 Since the matrix is symmetric, only the lower triangular part L is computed, thereby saving half of the operation required for a classic LU decomposition. The preconditioning matrix can be written as follows TLL M (5.54) The elements of the matrix L decomposed with Incomplete Cholesky algorithm are given by 2 / 1 11 11a l For i = 2 to n For j = 1 to i 1 If aij = 0 then lij = 0 else lij = jj j k jk ik ijl l l a /1 1 lij = 2 / 1 1 1 2 j k ik iil a

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86 CHAPTER 6 DEVELOPMENT AND BENCHMARKING OF THE PENSSN CODE In this chapter, I will present the devel opment of the new PENSSn code, and then I will test its numerics and the accuracy. In par ticular, I will address the performance of the Krylov subspace methods, including the CG a nd Bi-CG iterative solvers, along with the Incomplete Cholesky preconditioner for the CG method. The accuracy of the EP-SSN method will be tested for the following parameters Scattering ratio; Spatial truncation error; Low density materials; Material discontinuities; Anisotropic scattering. In addition, I will analyz e the method based on two 3-D criticality benchmark problems proposed by Takeda and Ikeda.43 The first problem invol ves the simulation of the Kyoto University Critical Assembly (KUCA) reactor. This problem is characterized by significant transport effects due to the pres ence of a control rod and a void-like region. The second problem involves the simulation of a small Fast Breeder Reactor (FBR) with a control rod half-inserted into the core. Th e solutions obtained for these two benchmarks will be compared with the Monte Carlo and SN methods. Finally, I will present the resu lts obtained for the OECD/NEA1 MOX 2-D Fuel Assembly Benchmark problem.44 1 OECD/NEA Organisation for Economic Co-operation and Development/Nuclear Energy Agency

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87 6.1 Development of the PENSSn (Parallel Environment Neutral-particle Simplified Sn) Code I have developed a new 3-D radiation tr ansport code, PENSSn, based on the EPSSN formulation. The code development began in 2001 utilizing the Simplified P3 formulation, that led to the development of the PENSP3 (Parallel Environment Neutralparticle SP3) code.25 However, the extension of the SP3 algorithm to an arbitrary order (N) proved to be impractical. Hence, I re directed the work by deriving a 3-D EP-SSN formulation. PENSSn consists of ~10,000 lines of code entirely written in ANSI/FORTRAN-90, using the Message Passing Interface (MPI) libraries for parallelization.27 PENSSn is a standalone code which solves the multigroup EP-SSN equations of arbitrary order with arbitrary anisotropi c scattering expansion. To improve the convergence rate of the Source Iteration met hod, a modified formulation of the EP-SSN equations (see Section 4.2.3) has been inte grated into PENSSn. Currently both fixed source and criticality eigenvalue calculati ons can be performed with upand downscattering processes. The discretized EP-SSN equations are solved using the Krylov subspace methods described in Chapter 5, i.e. CG and Bi-CG. However, in the parallel version of PENSSn, only the Bi-CG algorithm is implemented due to its superior parallel performance and numerical robustness as compared to CG. Angular, spatial and hybrid (spatial/angular) domain d ecomposition algorithms have been developed to achieve full-memory partitioning and multi-tasking. The code is capable of parallel I/O in order to deal effi ciently with large data structures. A complete description of the domain decomposition algorithms is given in Chapter 7.

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88 PENSSn produces balance tables and a co mplete description of the model solved, along with performance and timing data. The code is completed by a parallel data processor, PDATA, which collects the output files produced by differe nt processors and generates a single file for each energy group for plotting or further analysis. Currently, the geometry and material dist ribution are prepared for PENSSn using the PENMSH45 tool in the PENTRAN Code Sy stem. PENSSn requires only one additional input file which is defined as problem_name.psn The PENSSn input file is shown in Figure 6-1. Figure 6-1. Description of PENSSn input file. As shown above, the input file prov ides three groups of information: General PENSSn settings; Parallel Environment settings; Convergence control parameters. The General PENSSn group is used to input the SSN and PN order for the calculation. Note that the SSN order is an even number and also it holds the conditionN NP SS The Parallel Environment group is used to specify the decomposition vector for the parallel environment. Note that the number of coarse meshes has to be divisible by the number of processors specified for the spatia l domain, and also the number of directions has to be divisible by the nu mber of processors specifi ed for the angular domain.

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89 The Convergence control parameters group is used to specify the inner, outer, and Krylov subspace (CG) tolerances. Also th e maximum number of inner, outer, upscattering and Krylov iterations can be specified. PENSSn can be run in parallel or serial m ode; note that for the serial mode version, both CG and Bi-CG algorithms are available. A flow-chart for the PENSSn code is shown in Figure 6-2.

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90 Figure 6-2. Flow-chart of the PENSSn code. Initiate PENSSn Subroutine INPROC Process input files: penmsh.inp problem_name#zlev.inp problem_name.psn Parallel vector is accepted? PENSSn Halting execution. No Subroutine MEMALC Performs memory allocation in parallel environment. Subroutine MAPPING Define the arrays for the 3-D Cartesian geometry; calculates Gauss-Legendre roots and weights. Subroutine DECOMP Define the 3-D Virtual Topology and creates the MPI communicators. A Yes

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91 Figure 6-2. Flow-chart of th e PENSSn code (Continued). Subroutine MATDIST Process cross-sections and material distribution. Subroutine MBUILD EP-SSN matrix operators generation with CDS method. A Problem Type Subroutine PROCSRC Generates source distribution. Subroutine SOLVER Solves the EP-SSN equations with up-/downscattering. Subroutine POWITER Calculates the criticality eigenvalue with the Power Iteration method. PENSSn run completion Subroutine BTABLE: Generates balance table for the system. Subroutine WREPT: Output a file with problem summary ( problem_name.prep ). Subroutine DATAOUT: Parallel I/O of the even-parity angular flux moments (i.e., scalar Fixed source Criticality ei g envalue

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92 6.2 Numerical Analysis of Krylov Subspace Methods In this section, I will present a detailed analysis for the CG and Bi-CG algorithms as applied to problems with different numeri cal properties. In part icular, I will analyze the convergence performan ce of these algorithms in the following cases: Coarse mesh partitioning of the model; Boundary conditions; Material heterogeneities; Higher order EP-SSN methods. 6.2.1 Coarse Mesh Partitioning of the Model In this section, I will study the performance of the iterative solvers when the model is partitioned into coarse meshes. The first test problem consists of a simple symmetric 3D problem shown in Figure 6-3. The pr oblem size is 10.0x10.0x10.0 cm; a uniform distributed source is located within a cube of side 5.0 cm. Vacuum boundary conditions are prescribed for this model on every su rface. The model is characterized by one homogeneous material with one-group cross-sec tions; the total crosssection is equal to 1.0, and the scattering ratio is equal to 0.9. Figure 6-3. Configuration of the 3-D test problem. Coarse mesh 1 S=1.0 n/cm3/s c=0.9 Coarse mesh 2 S=0.0 n/cm3/s c=0.9 Coarse mesh 3 S=0.0 n/cm3/s c=0.9 Coarse mesh 4 S=0.0 n/cm3/s c=0.9 0.0 5.0 10.0 5.0 10.0 x y

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93 The system is discretized with a 1.0 cm uniform mesh along the x -, y and z -axes. The EP-SS2 equation is solved using the CG and Bi-CG algorithms; the convergence criteria for the source iteration and the Krylov methods are 1.0e-5 and 1.0e-6, respectively. The formulation used for the convergence criterion in the source iteration is given in Eqs. 6.1. Source iteration method convergence criterion 5 1 1 ,0 1 e r r ri g m i g m i g m (6.1) Table 6-1 compares the number of iterati ons for the Krylov solvers, CG and Bi-CG in two cases. In the first case, the model is partitioned into coarse meshes (Partitioned model); in the second case, the model is cons idered as a whole and no coarse meshes are specified (Non-partitioned). Table 6-1. Comparison of number of iterati ons required to converge for the CG and BiCG algorithms. Partitioned model Non-partitioned model Method Krylov iterations Inner iterations Krylov iterations Inner iterations Bi-CG 995 58 165 57 CG 1620 58 270 57 An increase of a factor of 6 is observed in the Krylov iteratio ns by partitioning the model into coarse meshes. The coarse mesh pa rtitioned model requires a larger number of iterations to converge, because the values of the angular fluxes on the interfaces of the coarse meshes are calculated iteratively. Notic e that this effect is purely numerical and only related to the Krylov solvers. In fact, I did not observe any signi ficant change in the number of inner iterations, wh ich is exclusively related to the scattering ratio and hence to the physics of the problem.

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94 I calculated the spectral c ondition number with an L2 norm for the partitioned and the non-partitioned system. The sp ectral condition number in L2 norm is defined by ) ( ) ( ) (min max 2A A A k (6.2) where, ) (maxA and ) (minA are the maximum and minimum eigenvalues of the matrix A The spectral condition number gives an i ndication of the convergence behavior of the iterative method. For the CG algorithm the number of ite rations required to reach a relative reduction of (one order of magnitude) in the error is proportional to 2k. For the non-partitioned model, I obtained6 42 k, while for the coarse mesh partitioned model, I obtained0 42 kin each coarse mesh. Figure 6-4 confirms the prediction base d on the spectral condition number; the number of iterations required to reduce the error by one order of magnitude is approximately 2.1. 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 123456789101112 Iteration numberCG Error Figure 6-4. Convergence behavior of the CG algorithm for the non-partitioned model.

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95 Based on these results, I conc lude that the increase in the number of Krylov iterations observed between th e partitioned and non-partitioned models is due to the presence of the coarse mesh interfaces. Moreover, these tests show the superior performance of the Bi-CG algorithm compared to CG; the Bi-CG algorithm requires only ~61% of the CG iterations for both the non-partitioned and partitioned models. 6.2.2 Boundary Conditions The objective of the following test problem is to analyze the effect of different boundary conditions on the convergence behavior of the Krylov solvers. The 3-D test problem used in the previous section ha s been modified by prescribing reflective boundary conditions on the planes at x =0.0, y =0.0 and z =0.0, and vacuum boundary conditions on the planes at x =10.0 cm, y =10.0 cm and z =10.0 cm. The model is partitioned into four coarse meshes, which are discretized with a 1.0 cm uniform mesh. Table 6-2 lists the number of iterations requi red by the Bi-CG and CG method to achieve convergence, along with the sp ectral condition number (2k) calculated for each coarse mesh. Table 6-2. Number of Krylov iterations required to converge for the CG and Bi-CG algorithms with different boundary conditions. Coarse mesh 2k Bi-CG method CG method 1 4.5 399 649 2 4.25 361 595 3 4.25 362 594 4 4.05 338 546 As expected, the number of Krylov iterati ons is higher for coarse meshes with larger condition number. However, the Kryl ov solvers require a different number of iterations per each coarse mesh, due to th e effect of boundary conditions. As will be

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96 discussed in Chapter 7, this situation will have a slight im pact on the performance of the parallel algorithm. 6.2.3 Material Heterogeneities Realistic engineering applic ations are characterized by material heterogeneities; hence, it is important to determine their imp act on the Krylov solver s. For this purpose, I have modified the 3-D test problem described earlier, w ith a heterogeneous material configuration, shown in Figure 6-5. The bounda ry conditions prescribed are the same as for the previous test case. Figure 6-5. Heterogeneous configur ation for the 3-D test problem. Table 6-3 demonstrates that the spectra l condition number is affected by the different material configuration in each co arse mesh, leading to a different number of iterations required by the Krylov solvers for each coarse mesh. Table 6-3. Number of Krylov iterations required to converge for CG and Bi-CG. Coarse mesh 2k Bi-CG method CG method 1 4.5 539 875 2 1.88 302 518 3 1.88 302 518 4 1.86 282 483 Coarse mesh 1 S=1.0 n/cm3/s 0 1 t c=0.9 Coarse mesh 2 S=0.0 n/cm3/s 0 2 t c=0.9 Coarse mesh 3 S=0.0 n/cm3/s 0 2 t c=0.9 Coarse mesh 4 S=0.0 n/cm3/s 0 2 t c=0.9 0.0 5.0 10.0 5.0 10.0 x y

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97 Note that the spectral condition numbers for coarse meshes 2, 3, and 4 are relatively lower than for coarse mesh 1. This is due to th e fact that the materials in coarse meshes 2, 3, and 4 are optically thicker than region 1. The optical thickness directly impacts the condition number. In general, optically thin re gions present matrix op erators with larger spectral condition numbers; conversely, optically thick regions present matrices with smaller spectral condition numbers. 6.2.4 Convergence Behavior of Higher EP-SSN Order Methods In this section, I will anal yze the convergence behavior of the Krylov methods for high EP-SSN order methods. The test problem cons idered is a cube with homogeneous material, and one group cross-sections with c= 0.9. The side of the cube measures 5.0 cm and it is discretized with a 1.0 unifo rm mesh. Vacuum boundary conditions are prescribed on every side of the model; also a uniform distributed source is present in the model, which emits 1.0 particles/cm3/sec. This problem has been solved with the EP-SS8 equations; the convergence criteria prescrib ed for the source iteration and Krylov methods are 1.0e-5 and 1.0e-6, respectivel y. Table 6-4 shows the spectral condition number (2k) as a function of direction for the EP-SS8 equations. For simplicity I have selected the EP-SS8 equations; however the discussion below can be extended to any SSN order. Table 6-4. Number of Krylov iterations required to converge for the CG and Bi-CG algorithms for the EP-SS8 equations. Direction number Direction cosine ( ) 2k Bi-CG method CG method 1 0.1834346 1.34 49 85 2 0.5255324 3.35 73 114 3 0.7966665 5.62 89 144 4 0.9602898 7.11 101 161

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98 Table 6-4 indicates that the spectral cond ition number increases as the direction cosine approaches 1.0; hence, the number of Krylov iterations required to achieve convergence increases as well. This behavi or can be explaine d by considering the definition of condition number (i.e., Eq. 6.2), and by obser ving the distribution of the eigenvalues for the matrix operators on each direction ( ), as shown in Figure 6-6. Figure 6-6. Distribution of eigenvalues for the EP-SS8 equations. The distribution of eigenvalues is cluste red toward the value of 1.0 for smaller values of ; however, as the direction cosine increa ses, the eigenvalue distributions start to drift away from 1.0. Th erefore, based on the definiti on of condition number, those matrices with a distribution of eigenvalues clustered around 1.0 present the smallest condition number, and consequently the Kr ylov method requires fewer iterations to converge.

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99 6.3 Testing the Incomplete Cholesky Co njugate Gradient (ICCG) Algorithm This section presents the numerical tes ting of the Incomplete Cholesky Conjugate Gradient (ICCG) algorithm. In order to op timize the preconditioner for large sparse matrices, I have utilized the Incomplete Chol esky (IC0) no-fill factorization. With this method the Cholesky factorization is comput ed only for non-zero elements of the EP-SSN matrix operators. The test problem utilized is a 3-D cube with a homogeneous material. The side of the cube measures 5.0 cm. The boundary c onditions prescribed, are reflective on the planes along x =0.0 cm, y =0.0 cm, and z =0.0 cm, and vacuum on the planes along x =5.0 cm, y =5.0 cm, and z =5.0 cm. The model is discretized with a 1.0 cm uniform mesh along the x y and z axis. The convergence criteria prescr ibed for the source iteration and ICCG methods are 1.0e-5 and 1.0e-6, respectively. Table 6-5 shows the number of iterations for the ICCG method compared to the non-preconditioned CG algorithm. Table 6-5. Number of iterations for the ICCG and CG algorithms. MethodICCG CG EP-SS4 37 68 EP-SS6 65 120 EP-SS8 91 173 For SSN orders ranging from 4 to 8, the ICCG method yields a reduction of the number of iterations by a factor of two. The main disadvantage of the ICCG met hod is the computation time and memory required to perform the Incomplete Cholesky f actorization of the coefficient matrices; I have observed that for large problems, the factorization phase account for ~30% of the total computation time. In addition, the tim e spent for each Krylov iteration increases because the preconditioning matrix has to be solved as well. However, the ICCG method

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100 yields an overall reduction of the computation time by a factor of ~1.5. In spite of its good performance, the ICCG method has not been implemented in the current algorithms; this decision has been dictated mainly by the large memory required by the ICCG method. 6.4 Testing the Accuracy of the EP-SSN Method In this section, I will test the accuracy of the EP-SSN equations, and I will identify the limitations of the methodology with respec t to scattering ratio, spatial truncation error, low density materials, material disc ontinuities, and anisotr opic scattering order. In conclusion, I will analyze the accuracy of the EP-SSN using two 3-D criticality benchmark problems proposed by Takeda and Ikeda.44 6.4.1 Scattering Ratio The objective of this test is to ca lculate the criticality eigenvalue ( keff) as a function of the scattering ratio ( c ). For this purpose, I will cons ider a simple 2-D criticality eigenvalue problem with0 4 0 0 y x The boundary conditions prescribed are reflective at x =0.0, y =0.0, and vacuum at x =4.0 cm, y =4.0. The model is discretized with a 0.25 cm uniform mesh. The model conf iguration is shown in Figure 6-7.

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101 Figure 6-7. Configuration of the 2-D criticality eigenvalue problem. For this problem, I will progressively modify the scattering ratio in region 2; however, as the scattering rati o decreases, the outer region becomes less diffusive, so it is expected that the diffusion equation will be less accurate compared to higher order SSN methods. Figure 6-8 shows the comparison among the cr iticality eigenvalues obtained with the EP-SSN and S16 methods for different scattering ra tios in a range of 0.6 – 0.99. The S16 transport solutions have been obtai ned with the PENTRAN and DORT codes. Figure 6-8. Criticality eigenvalu es as a function of the scat tering ratio (c) for different methods. 0.0 2.0 4.0 2.0 4.0 x y Region 2 0 1 t cs 0 0 f Region 1 0 1 t 9 0 s 2 0 f

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102 Based on the data presented in Figure 6-8, the EP-SS2 method yields inaccurate results for every scattering ratio. Because of large particle leakage from the system, the EP-SS2 method yields inaccurate results also for scattering ratio greater than 0.9, where the physics is dominated by diffusive processes. However, higher order EP-SSN methods yield accurate results for ev ery scattering ratio, with a ma ximum relative difference of 0.85% compared to the S16 transport calculation. Note also that by increasing the SSN order, the accuracy is not improved as well; this behavior is due to the fact that the EP-SSN formulation does not yield the transport solution as the order increases. Figure 6-9, clearly shows the increased accuracy obtained using higher order EPSSN methods relative to th e diffusion equation. Figure 6-9. Relative difference for criticality eigenvalues obtained with different EP-SSN methods compared to the S16 solution (PENTRAN code). It is interesting to note that the highe st accuracy is achieved with the EP-SS4 method with a relative difference of 0.12% for c =0.99. As will be shown in the next section, this behavior is part ly due to the spatial truncation error. As expected all the EP-

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103 SSN methods with N>2 yield accurate results for every scattering ratio, while the diffusion equation degrades as th e scattering ratio decreases. 6.4.2 Spatial Truncation Error In order to further test the accuracy of the EP-SSN method, I have investigated the effect of the spatial discretization for critic ality eigenvalue calculations. For this purpose I solved the problem presented in Section 6.3. 1 with different fine mesh discretizations. The case considered has scattering ratio in region 2 equal to 0.6 (see Figure 6-6). I have compared the soluti ons obtained with the EP-SSN method with orders ranging from 4 to 10, with an S16 transport calculation obtai ned with the PENTRAN and DORT codes. Figure 6-10 presents the cri ticality eigenvalues obtained with these methods for different fine mesh discretizations. Figure 6-10. Plot of criticality ei genvalues for different mesh sizes. As we can see, the EP-SSN method converges to the transport solutions by increasing the spatial resolution of the problem; however, the EP-SS4 equations yield more accurate results compared to higher order SSN equations, down to a mesh size of 0.25 cm.

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104 Figure 6-11 shows the relative difference of the EP-SSN solutions compared to the S16 transport calculation obtaine d with the PENTRAN code. Figure 6-11. Plot of the rela tive difference of the EP-SSN solutions versus transport S16 for different mesh sizes. The EP-SS4 method yields relative differences of 9.87%, 2.06% and -0.09% for mesh sizes of 1.0, 0.5 and 0.25 cm, respectively. For a mesh size of 0.125 cm, the EP-SS6 method yields the most accurate results, w ith a relative difference of 0.03%. In conclusion, this test problem demonstrates that the spatial discretization has to be refined as the SSN is increased; note that this behavior is similar to what is observed for the SN method. 6.4.3 Low Density Materials The problem considered in this section c onsists of two blocks of uranium dioxide highly enriched at 93.2 %, surrounded by air with an 80% relative humidity. Figure 6-12 shows a view on the x y plane of test problem consid ered. The model extends along the z axis from 0.0 to 12.0 cm, and vacuum boundary conditions are prescribed on all surfaces.

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105 Figure 6-12. Uranium assembly test problem view on the x y plane. The purpose of this test problem is to assess the accuracy of the EP-SSN methodology in the presence of low density medi a such as air gaps. In low density or void-like regions, the particle physics is not dominated by diffusive processes, where elliptic-type mathematical models such as the EP-SSN equations yield acc urate results. In this type of problems the particle behavior is well described by the transport equation which behaves like a hyperbolic wave equation. The major issue affecting the accuracy of the EP-SSN equations is the diffusion coefficient defined by r r Dt m m 2 (6.3) In low density media, the value of the tota l scattering cross-sec tion is usually below 1.0e-3 [1/cm]. Hence, the value of the di ffusion coefficient becomes abnormally large, leading to numerical difficulties and to an underestimation of the leakage term. In order to remedy this situation, I have introduced a density factor multiplier (DFM) in order to scale up the cross-section only in low density regions. The density factor is applied to the 0.0 25.0 x y 20.0 5.0 5.0 20.025.040.0 45.0 Air 80% relative humidity UO2 (93.2%) UO2 (93.2%)

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106 cross-sections of materials for which the tota l cross-sections have a value within 1.0e-3 and 1.0e-7. The value of DFM is chosen such that the scaled cross-sections have a value of ~1.0e-1. This is an ad-hoc treatment, but it has been proven effective in improving the convergence properties and accuracy of the EP-SSN method for problems with low density regions. The two-group cross-sections a nd the fission spectrum are listed in Table 6-6; group 1 spans an energy range between 0.4 and 10.0 MeV, while group 2 spans a range between 0.0 and 0.4 MeV. Table 6-6. Two groups cross-sections and fission spectrum. Material Group (g) a f t 1 g s 2 g s ) ( g UO2 1 6.1902e-02 1.4436e-012.3968e-01 1.5220e-01 0.0 0.896 2 8.6126e-02 1.7309e-014.2551e-01 3.3938e-01 2.5582e-02 0.104 Air 1 3.3372e-06 0.0 1.0115e-04 8.6948e-05 0.0 0.0 2 6.1639e-06 0.0 2.8127e-04 2.7511e-04 1.0868e-05 0.0 For this problem, I calculated physical quantities of interest such as k-effective, leakage, collision and scattering term. Then, I compared these quantities with a transport solution obtained with the PENTRAN code Table 6-7 compares the criticality eigenvalues obtained by the EP-SSN method with different DFM values, with the PENTRAN S6 solution. Table 6-7. Comparison of keff obtained with the EP-SSN method using DFM versus PENTRAN* S6 (Note that DFM=1.0 implies no cross-sections scaling). Method keff (DFM=100.0) Rel. difference vs. S6 keff (DFM=1.0) Rel. difference vs. S6 EP-SS2 0.77621 -18.47% 0.69437 -27.07% EP-SS4 0.90544 -4.90% 0.83097 -12.72% EP-SS6 0.92138 -3.23% 0.8464 -11.10% EP-SS8 0.92436 -2.91% 0.84894 -10.84% EP-SS10 0.92529 -2.82% 0.8496 -10.77% *PENTRAN S6 predicts keff= 0.95211. Table 6-7 indicates that a de nsity factor of 100.0, largely improves the accuracy of keff. However, if DFM is not uti lized, the accuracy of the EP-SSN method is poor for this

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107 problem, and the convergence trend is charact erized by an oscillatory behavior which leads to divergence within the maximum numb er of outer iterations specified (50). Figure 6-13 shows the relative difference between the physical quantities calculated with EP-SSN (DFM=100.0) and S6 PENTRAN methods. -25.0% -20.0% -15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%EP-SS2EP-SS4EP-SS6EP-SS8EP-SS10Relative difference k-effective Leakage term Collision term Scattering term Figure 6-13. Relative difference of physical quan tities of interest calculated with EP-SSN method compared to the S6 PENTRAN solution. For this problem, transport effects are si gnificant due to the large boundary leakage and highly angular behavior due to the low density medium. Hence, higher order EP-SSN methods yield a better angular representation of the particle flux, therefore leading to more accurate results. The EP-SS10 method yields a relative di fference compared to the S6 solution of 2.8%, 1.9%, -1.2% and -0.7% for keff, leakage, collision and sc attering term, respectively. Note also that the density factor (DFM=100. 0) improves the convergence behavior of the EP-SSN method. Figure 6-14 presents the cri ticality eigenvalue relative e rror as a function of the outer iteration number for PENSSn with DFM=100.0 and PENTRAN S6.

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108 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+0212345678910111213141516171819202122232425262728293031323334Outer iterationsRelative error (dk/k) PENTRAN S-6 EP-SS2 EP-SS4 EP-SS6 EP-SS8 EP-SS10 Figure 6-14. Convergence behavior of th e PENSSn with DFM=100.0 and PENTRAN S6. The PENTRAN relative error presents an os cillatory behavior due to the Aitken’s extrapolation method utilized.39 The EP-SS2 relative error presents a sudden drop from 1.0e-4 to 1.0e-5, probably indica ting false convergence. The EP-SSN calculations with N>2 all indicate a rather stable convergence behavior. 6.4.4 Material Discontinuities In this section, I will analyze material discontinuities which may introduce significant angular dependencies on the particle flux at the material interface. The test problem considered is a simple 2-D model made of two heterogeneous regions, with a fixed source. The geometric and material conf iguration for the test problem is shown in Figure 6-15. The test problem is characterized by a steep change in the total cross-section between regions 1 and 2; also, region 2 is de fined as a highly absorbent material. Because of these features the problem pr esents strong transport effects.

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109 Figure 6-15. Geometric and material conf iguration for the 2-D test problem. The solution for this problem is obtained with the EP-SS2, EP-SS4 and PENTRAN S16 methods. Figure 6-16 shows the flux distribution, along the x -axis. 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+010. 16 0.4 7 0.78 1 .09 1. 41 1. 72 2. 03 2.34 2. 66 2. 97 3.2 8 3.59 3 .91 4. 22 4. 53 4. 84 5.16 5 .47 5. 78 6. 09 6. 41 6.72 7 .03 7. 34 7.6 6 7.97 8 .28 8. 59 8. 91 9. 22 9.53 9 .84x (cm)Scalar flux (n/cm^2/s) PENTRAN S-16 EP-SS2 EP-SS4 Figure 6-16. Scalar flux distri bution at material interface ( y =4.84 cm). As indicated by Figure 6-16, the EP-SS4 yields an accurate solution compared to S16; the maximum relative difference between the two methods (15.58%) is found at x =4.84 cm and y =4.84 cm, which is the fine mesh on the corner of region 1. At this mesh 0.0 5.0 10.0 5.0 10.0 x y Region 1 0 1 t 5 0 s 0 1 S Region 2 0 2 t 1 0 s 0 0 S

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110 location, the transport effects due to material transition are significant, resulting in the largest difference between the EP-SS4 and S16 methods. As expected, the EP-SS2 method is accurate in region 1; however, the solution rapidly degrades as we move into region 2 where the transport effe cts are significant. Figure 6-17 presents the rela tive difference for the EP-SS2 and EP-SS4 methods as compared to PENTRAN S16 at the material interface. -100.00% -80.00% -60.00% -40.00% -20.00% 0.00% 20.00% 40.00%0.1 6 0 .47 0.78 1 0 9 1.41 1 7 2 2.03 2 3 4 2.66 2 9 7 3.28 3.5 9 3 .91 4 2 2 4.53 4.8 4 5 .16 5.47 5 7 8 6.0 9 6 .41 6.72 7 0 3 7.34 7 6 6 7.97 8 2 8 8.59 8 9 1 9.22 9.5 3 9 .84x (cm)Relative difference EP-SS2 EP-SS4 Figure 6-17. Relative difference versus S16 calculations at material interface ( y =4.84 cm). Figure 6-17 shows that the EP-SS4 method exhibits a maximum relative difference of ~15.6% at the material interface. This problem clearly shows how higher order EP-SSN methods introduce more transport physics into the solution compared to the diffusion-like equation. The balance table (Table 6-8) demonstrat es that the leakage term is the major component affecting the accuracy of the EP-SSN method for problems with strong transport effects. The EP-SS4 method yields a relative diffe rence of only -1.12% for the

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111 leakage term. Note that the collision and s cattering terms, are relatively well represented by both EP-SS2 and EP-SS4 methods. Table 6-8. Balance tables for the EP-SSN and S16 methods and relative differences versus the S16 solution. Integral system balance Relative difference vs. S16 Method Leakage Collision Scatte r Leakage Collision Scatter EP-SS2 -1.76e-06 -4.61e+01 2.11e+01 -94.62% -0.37% -0.81% EP-SS4 -3.23e-05 -4.65e+01 2.15e+01 -1.12% 0.47% 1.03% S16 -3.27e-05 -4.62e+01 2.12e+01 These findings are further confirmed by observing the integral boundary leakage for different boundary surfaces. Table 6-9 clearly indicates that the predicted leakage rate is underestimated by ~98.7% using the EP-SS2 method, while it is only underestimated by ~2% using the EP-SS4 method. Table 6-9. Integral boundary leakage for the EP-SSN and S16 methods and relative differences versus the S16 solution. Integral boundary leakage Relative difference vs. S16 Method East (+ x ) North(+ y )East (+ x ) North(+ y ) EP-SS2 2.12e-07 2.12e-07 -98.70% -98.70% EP-SS4 1.59e-05 1.59e-05 -1.98% -1.99% S16 1.63e-05 1.63e-05 East (+ x ) refers to the right boundary of the system at x =10.0 cm, while North (+ y ) refers to the top boundary of the system at y =10.0 cm. This is very encouraging becau se it indicates that the EP-SSN methodology could be applicable for shielding problems. 6.4.5 Anisotropic Scattering This section addresses th e accuracy of the EP-SSN equations for problems characterized by anisotropic sc attering. The test problem c onsists of a cylinder with a 20.0 cm radius, extending axially for 30.0 cm, representing a fuel region; the cylinder is then surrounded by water extending from 20.0 cm to 30.0 cm along the x and y -axis, and from 30.0 to 40.0 cm along the z -axis. Reflective boundary conditions are prescribed on

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112 the planes at x =0.0 cm, y =0.0 cm and z =0.0 cm; vacuum boundary conditions are prescribed on the planes at x =30.0 cm, y =30.0 cm and z =40.0 cm. A two-group crosssection set is generated using the first two groups of the BUGLE-96 library with P3 anisotropic scattering order. A uniform fixed s ource is placed in the fuel region, with an energy spectrum given in Table 6-10. Table 6-10. Fixed source ener gy spectrum and energy range. Energy group Group upper boundary (MeV) Group lower boundary (MeV) Fixed source (n/cm3/s) 1 17.3 14.2 4.25838e-5 2 14.2 12.2 1.84253e-4 I have compared the results obtained with the EP-SSN method with a PENTRAN S8 transport solution. The convergence criterion fo r the angular flux has been set to 1.0e-4. I calculated the relative difference between the solutions obtained with the EP-SSN and the S8 methods. Figures 6-18 and 6-19 show the fract ion of scalar flux valu es within different ranges of relative difference (compared to S8) for energy group 1 and 2 respectively. R.D.< 5% 5% < R.D.< 10% 10% < R.D.< 20% 20% < R.D. < 30% EP-SS4 EP-SS6 EP-SS8 66.10 21.50 12.40 0.00 64.10 16.60 18.90 0.40 45.30 29.30 23.20 2.20 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 Figure 6-18. Fraction of scalar flux values with in different ranges of relative difference (R.D.) in energy group 1.

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113 R.D.< 5% 5% < R.D.< 10% 10% < R.D.< 20% 20% < R.D. < 30% EP-SS4 EP-SS6 EP-SS8 78.30 18.20 3.50 0.00 73.30 21.40 5.30 0.00 44.90 44.00 11.10 0.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 Figure 6-19. Fraction of scalar flux values with in different ranges of relative difference (R.D.) in energy group 2. Note that by increasing the SSN order, the number of scalar flux values with relative difference less than 5% increases in bo th groups; this behavior demonstrates that higher order EP-SSN methods improve the accuracy of th e solution, especially for highly angular dependent proble ms. As expected the ac curacy of the EP-SSN method increases for lower energy groups because the probability of leakage decreases and the medium becomes optically thicker. Table 6-11 shows the maximum and minimum relative difference in the scalar flux versus the S8 method2, in energy groups 1 and 2. Table 6-11. Maximum and minimum relative di fferences in the scalar flux versus the S8 method for energy group 1 and 2. Group 1 Group 2 Method MAX MIN MAX MIN EP-SS4 24.42 1.292e-03 17.86 1.379e-04 EP-SS6 21.34 1.401e-04 15.01 6.053e-05 EP-SS8 18.37 4.226e-04 13.74 2.508e-04 2 The MAX and MIN relative difference compared to the S8 method are defined as [MAX|( S8EP-SSn)|/ S8] and MIN[|( S8EP-SSn)|/ S8] respectively.

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114 Note that the EP-SS8 method significantly improves the accuracy yielding a maximum relative difference in the scalar flux of 18.37% and 13.74% in energy groups 1 and 2, respectively. Figures 6-20 and 6-21 show the rela tive difference between the EP-SS8 and S8 flux solutions in group 1. The front view result s, Figure 6-20, indi cate that the largest differences occur on the external surface of the model, where vacuum boundary conditions are specified; as expected the relati ve difference is larger in this region due to the approximate vacuum boundary c onditions derived for the EP-SSN method. The rear view results, shown in Figure 6-21, indicate a noticeable a larger relative difference on the material interface between the fuel regi on and the moderator due to higher order angular dependencies. Figure 6-20. Front view of th e relative difference between th e scalar fluxes obtained with the EP-SS8 and S8 methods in energy group 1.

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115 Figure 6-21. Rear view of the relative differe nce between the scalar fluxes obtained with the EP-SS8 and S8 methods in energy group 1. 6.4.6 Small Light Water Reactor (LWR) Criticality Benchmark Problem A small LWR benchmark problem has been proposed by Takeda and Ikeda and it is one of the 3-D Neutron Transport Benchmarks by OECD/NEA.43 The model represents the core of the Kyoto University Critical Assembly (KUCA) as shown in Figures 6-22 and 6-23. A B Figure 6-22. Model view on the x y plane3. A) view of the model from z =0.0 cm to 15.0 cm, B) view of the model from z =15.0 cm to z =25.0 cm. 3 CR is the abbreviation for Control Rod.

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116 Figure 6-23. Model view on the x z plane. The model is discretized with a 1.0 cm uniform mesh. The core is polyethylene moderated and it consists of 93 w/o enrich ed U-Al alloy and natural uranium metal plates, with a moderation ratio of 1.5. The two-group cross-sections have been modified using the transport cross-section in place of th e total cross-section in order to account for P1 anisotropic scattering. The cross-sectio ns are given in Table 6-12 and the fission spectrum along with energy rang e are given in Table 6-13. Table 6-12. Two-group cross-sect ions for the small LWR problem. Material Group (g)a f t Core 1 8.52709e-03 9.09319e-03 2.23775e-01 2 1.58196e-01 2.90183e-01 1.03864 Reflector 1 4.16392e-04 0.0 2.50367e-01 2 2.02999e-02 0.0 1.64482 CR 1 1.74439e-02 0.0 8.52325e-02 2 1.82224e-01 0.0 2.17460e-01 Void 1 4.65132e-05 0.0 1.28407e-02 2 1.32890e-03 0.0 1.20676e-02 Table 6-12. Two-group cross-sections for the small LWR problem (Continued). Material Group (g) 1 g s 2 g s Core 1 1.92423e-01 0.0 2 8.80439e-01 2.28253e-02 Reflector 1 1.93446e-01 0.0 2 1.62452 5.65042e-02 CR 1 6.77241e-02 0.0 2 3.52358e-02 6.45461e-05

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117 Table 6-12. Two-group cross-sections the small LWR problem (Continued). Material Group (g) 1 g s 2 g s Void 1 1.27700e-02 0.0 2 1.07387e-02 2.40997e-05 Table 6-13. Fission spectrum and ener gy range for the small LWR problem. Group Upper energy boundary (eV) Lower energy boundary (eV) Fission spectrum 1 10.0e7 6.8256e-01 1.0 2 6.8256e-01 1.0e-05 0.0 For this problem, two cases have been c onsidered: in case 1, the control rod is withdrawn from the reactor and it is replaced wi th a void-like region; in case 2 the control rod is completely inserted into the core. This problem is particularly challenging due to the transport effects introduced by the cont rol rod and the void-lik e region. For this problem, I have not modified the cross-sections with the density factor multiplier in order to show the limitations of the EP-SSN method in dealing with this type of medium. Therefore, I have calcula ted the Control Rod Worth44 (CRW) defined by 1 21 1Case eff Case effk k CRW (6.4) The criticality eigenvalues ( keff) calculated with different SSN orders and the error relative to the Monte Carlo predictions are given in Table 6-14 for both cases. Table 6-14. Criticality eigenvalues calculated with different EP-SSN orders and relative error compared to Monte Carlo predictions. Case 1 Case 2 Method keff Error (pcm) keff Error (pcm) Monte Carlo 0.97900.00060.96240.0006EP-SS2 0.92325 -5598.2 0.9288 -3491.3 EP-SS4 0.95266 -2591.0 0.95854 -401.1 EP-SS6 0.95338 -2517.4 0.95931 -321.1 EP-SS8 0.95341 -2514.3 0.95926 -326.3

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118 The EP-SSN method predicts relatively accurate values of the keff for case 2; however, the method under-predi cts the criticality eigenvalue when the control rod is withdrawn in case 1. This behavior is due to the intrinsic limitations of the method in dealing with void-like regions. The CR Ws estimated with different EP-SSN methods and the Monte Carlo method, are given in Table 6-15. Table 6-15. CRWs estimated with the EP-SSN method. Method CRW Monte Carlo 1.66e-20.09e-2 EP-SS2 -6.47e-03 EP-SS4 -6.44e-03 EP-SS6 -6.48e-03 EP-SS8 -6.40e-03 Based on the definition given in Eq. 6. 4, a negative CRW would represent a positive insertion of reactivity by the control rod, which is clearly unphysical. Figure 624 shows the normalized scalar flux4 for case 1, in energy group 1 along the x -axis at y =2.5 cm and z =7.5 cm (i.e., core mid-plane). 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000 .5 0 1.76 3 .0 3 4 .2 9 5.55 6 .8 2 8.08 9.34 10.6 1 11.87 1 3. 1 3 14.3 9 15.66 1 6. 9 2 18.1 8 1 9. 4 5 2 0. 7 1 21.9 7 2 3. 2 4 2 4. 5 0x (cm)Normalized scalar flux (n/cm^2/s) EP-SS8 PENTRAN S-8 Figure 6-24. Normalized scalar fl ux for case 1, in group 1 along the x -axis at y =2.5 cm and z =7.5 cm. 4 The scalar flux is normalized to the maximum value.

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119 The EP-SS8 method underestimates the flux distri bution in the core region (0.0 to 15.0 cm) compared to the S8 solution; this is due to an overestimation of the leakage term in the void-like region. Note that the inte grated leakage term estimated by the EP-SS8 method in the void-like region, is equal to 8.46280e-04 particles/sec while the S8 method yields 6.61564e-05 particles/sec in the same re gion. Hence, due to the underestimation of the scalar flux in the core region, the critical ity eigenvalues obtained in case 1 with the EP-SSN method are also underestimated Figures 6-25 show the EP-SS8 predicted flux distributions for both cas es in both energy groups. Figure 6-25. Scalar flux distri butions. A) Case 1 energy group 1, B) Case 2 energy group 1, C) Case 1 energy group 2, D) Case 2 energy group 2.

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120 Figure 6-25A and C clearly show the flat scalar flux distributi on in the void-like region for both energy groups. In contra st, Figure 6-25D shows a pronounced flux depression in the control rod region in energy group 2. Note, that transport effects are significant in case 2, due to a steep flux gradie nt in the control rod region in the thermal range. However for this case, the EP-SSN method (N>2) yields an accurate solution due to its higher order a ngular representation of the particle flux. 6.4.7 Small Fast Breeder Reactor (FBR ) Criticality Benchmark Problem The small FBR benchmark problem has also been proposed by Takeda and Ikeda and it is part of the OECD/NEA 3-D Neutr on Transport Benchmarks. Views of the model on the x y and x z planes are shown in Figur e 6-26 and 6-27 respectively. Figure 6-26. View on the x y plane of the small FBR model.

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121 Figure 6-27. View on the x z plane of the small FBR model. For this benchmark two cases have been c onsidered: in case 1 the control rod is fully withdrawn from the reactor and the cha nnel is filled with sodium; in case 2 the control rod is half-inserted as shown in Figur e 6-27. Note that the control rod introduces strong transport effects; howev er, no void-like regions are present for this problem, hence the EP-SSN method is expected to yield relati vely accurate solutions. The model is discretized with a 5 cm unifo rm mesh, which is the reference mesh size used in the benchmark; also, four-group cross-secti ons are used in these calculations. The criticality eigenvalues ( keff) obtained for this problem are given in Table 6-16, along with the relative erro r compared to the Monte Carlo reference solutions.

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122 Table 6-16. Criticality eigenvalu es for the small FBR model. Case 1 Control rod withdrawn Ca se 2 Control rod half-inserted Method keff keff 5 (pcm) Method keff keff d (pcm) Monte Carlo 0.97320.0002Monte Carlo 0.95940.0002 EP-SS2 0.96888 -443.90 EP-SS2 0.95467 -493.02 EP-SS4 0.97388 69.87 EP-SS4 0.96017 80.26 EP-SS6 0.97396 78.09 EP-SS6 0.96024 87.55 EP-SS8 0.97394 76.04 EP-SS8 0.96026 89.64 Higher order EP-SSN methods yield relatively accura te results compared to Monte Carlo; however, note that keff relative differences are higher for case 2. This can be attributed to the strong transport effects in troduced by the control rod. The CRWs (Eq. 6.4) obtained with the EP-SSN and Monte Carlo methods are given in Table 6-17. Table 6-17. CRWs estimated with the EP-SSN and Monte Carlo methods. Method CRW Monte Carlo 1.47e-02 EP-SS2 1.54e-02 EP-SS4 1.47e-02 EP-SS6 1.47e-02 EP-SS8 1.46e-02 Except for the SS2 order, all the other SSN orders yield a very accurate CRW compared to the Monte Carlo solution. Figures 6-28 and 6-29 show the 3-D scalar flux distribution obtained with the EP-SS8 method for both cases in energy groups 1 and 4, respectively. In Figure 6-28A and 6-29A the e ffect of the sodium channel is visible, especially in group 4, where neutron modera tion occurs. In Figures 6-28B and 6-29B, the flux distortion due to the half-inserte d control rod is cl early noticeable. 5The relative difference with Monte Carlo is calculated in pcm as keff=1.0e5*[ keff(EP-SSN)keff(MC)]/ keff(MC).

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123 Figure 6-28. Scalar flux dist ribution in energy group 1: A) Case 1; B) Case 2. Figure 6-29. Scalar flux dist ribution in energy group 4: A) Case 1; B) Case 2.

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124 6.4.8 The MOX 2-D Fuel Assembly Benchmark Problem The MOX 2-D Fuel Assembly benchmark problem44 has been proposed by NEA/OECD to test the current capabilities of radiation transport codes to perform wholecore calculations without sp atial homogenization. For this benchmark both 2-D and 3-D versions of the problem were developed and accurate Monte Carlo solutions were obtained. The benchmark problem is the sixt een assembly (quarter core symmetry) C5 MOX fuel assembly problem proposed by Cavarec.46 The 2-D mesh distribution is shown in Figure 6-30. Figure 6-30. Mesh distribution of the M OX 2-D Fuel Assembly Benchmark problem. The model consists of 81 coarse meshes, discretized with a to tal of 112,425 fine meshes. I have calculated the criticality eigenvalue and power distribution for this problem with the EP-SSN method. Table 6-18 shows th e criticality eigenvalues ( keff)

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125 calculated with different EP-SSN method and their relative error compared to the Refrence Monte Carlo solution.44 Table 6-18. Criticality eigenvalues and re lative errors for the MOX 2-D benchmark problem. Method keff Relative error (pcm) Monte Carlo 1.86550EP-SS2 1.19335573.0901 EP-SS4 1.19017305.0862 EP-SS6 1.1907 349.7535 The EP-SS4 method yields the most accurate solution in terms of the criticality eigenvalue. The increased accura cy obtained with the EP-SS4 method compared to the diffusion method is due to the better representation of th e transport effects due to heterogeneous regions with fuel-moderator interfaces. The accuracy obtained with the EP-SS6 method slightly degrades due to the fact that the spatial mesh is not refined for increasing SSN orders. The power distribution, normalized over the number of fuel pins,44 estimated for the inner UO2 fuel assembly (see Figure 6-30) is 485.3, which differs by -1.5% compared to the MCNP reference solution (492.8 0.1%). For the MOX and the outer UO2 fuel assemblies, I estimated a normalized power e qual to 212.2 and 144.4, respectively. These results differ by ~0.3% and ~3.3% as compared to the Monte Carlo results (MOX: 211.70.18%, Outer UO2: 139.80.20%), respectively. Note that the EP-SS2 solution was obtained in 30 minutes running on 27 processors with spatial decomposition; the EP-SS4 solution required 52.5 minutes on 18 processo rs with a hybrid domain decomposition (2angle, 9-space), while the EP-SS6 method took 86.3 minutes on 81 processors (3-angle, 27-space) The EP-SS2 and EP-SS4 solutions were obtained on the PCPENII Cluster owned by the Nuclear & Radiologi cal Department at the University of Florida. The EP-

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126 SS6 solution was obtained on the Zeta-Cluster (64 processors) a nd Kappa-Cluster (40 processors), part of the CARRI ER Computational Lab Grid at the University of Florida. Figure 6-31 shows the scalar flux distribut ion for each energy group obtained with the EP-SS4 method. Figure 6-31. Scalar flux di stribution for the 2-D MOX Fuel Assembly benchmark problem (EP-SS4): A) Energy group 1; B) En ergy group 2; C) Energy group 3; D) Energy group 4; E) Energy gr oup 5; F) Energy group 6; G) Energy group 7. Figure 6-32 shows the normalized pin power distribution obtained with the EP-SS4 method.

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127 A B Figure 6-32. Normalized pin power distri bution for the 2-D MOX Fuel Assembly benchmark problem (EP-SS4): A) 2-D view; B) 3-D view.

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128 CHAPTER 7 PARALLEL ALGORITHMS FO R SOLVING THE EP-SSN EQUATIONS ON DISTRIBUTED MEMORY ARCHITECTURES This chapter describes the parallel algorith ms developed for the PENSSn code in distributed-memory architectures. I will desc ribe the domain decomposition strategies developed, including spatial, angular a nd hybrid (spatial/angular) decompositions. The parallel performance of PENSSn for a test problem, based on the speed-up, parallel efficiency and parall el fraction of the code is m easured. Further, the parallel efficiency of the Krylov subspace based iter ative solvers, and a methodology to improve their performance are discussed. Finally, I will present the parallel perfor mance obtained with PENSSn for the solution of the MOX 2-D Fuel Assembly Be nchmark problem discussed in Chapter 6. 7.1 Parallel Algorithms for the PENSSn Code PENSSn is designed to run on distribut ed memory architectures, where each processor is an independent uni t with its own memory bank. This type of architecture is composed usually of PC-workstations linke d together via a ne twork backbone. The interconnection scheme among the processors is fundamental for distributed memory architectures because it affects, in part, the performance of the system. For cluster-type architectures, the processors are connected using a switch, which allows data transfer among the units. For this type of system, the limited bandwidth available for processor intercommunication can be a limiting factor. Current network switches are capable of

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129 1/10 GBit/sec bandwidth. Therefore, the parallel algorithm must minimize the communication time in order to yield an acceptable parallel performance. PENSSn is written in Fortran-90 and it is parallelized with the MPI (Message Passing Interface) libraries.27 This approach guarantees full portability of the code on a large number of platforms. The code solves the multigroup EP-SSN equations with anisotropic scattering of arbitrary order for fixed source and criticality problems. Three decomposition strategies have been implemented: spatial, angular and hybrid (spatial/angular) domain decompositions. The basic philosophy of this approach is to decompose part of the phase space on the processors, through a mapping function which defines the parallel virtual topology. The mapping function or parallel vector, assigns portions of the domain to the processors; hence the calcula tion is performed locally by ea ch processor on the allocated sub-domain. Note that on each processor only part of the domain is allocated in memory; this type of approach is defi ned as parallel memory, and it allows solving large problems which would be impossible to solve on a single workstation. The main advantages of a parallel algorithm can be summarized in parallel tasking and memory partitioning. The first aspect relates to the computation time reduction achievable with a parallel computer; in an ideal situation, where no communication time is considered, p processors would solve the problem p -times faster than a single unit. In the remainder of this chapter, I will show that in practice, this leve l of performance is not achieved. Memory partitioning allows the subdiv ision of the problem in RAM memory, hence, allowing the treatment of large simulatio n models. This aspect also eliminates the

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130 need for scratch files on hard drives; the ove rall performance benefits from this aspect due to faster access of memory banks compared to hard drives. 7.2 Domain Decomposition Strategies In order to parallelize the EP-SSN equations, we partition th e spatial domain into a number of coarse meshes and allocate them to different processors. Similarly, the angular domain is partitioned by allocating indivi dual angles or groups of angles to each processor. The hybrid spatial and angular domain decomposition allows for simultaneous processing of spatial and angular sub-domains. Once the system is partitioned and the parallel vector is specified, the PENSSn c ode proceeds to sequentially allocate different sub-domains onto different processors, ge nerating the so-called virtual topology. 7.2.1 Angular Domain Decomposition The angular domain is partitioned base d on a decomposition vector, which assigns the angles or group of angles to independent processors. Each pro cessor locally solves angular fluxes for a subset of the total angular domain. After an inner iteration is completed, the moments of the even-parity angular flux are calcul ated using collective operations of the MPI library to minimize the communication overhead and to maintain data parallelism. In the PENSSn code, a subroutine is dedicated for the angular integration of the even-parity angular fluxe s on the parallel envir onment, yielding total quantities such as scalar flux, curre nts, etc. The collective operation MPI_ALLREDUCE is used for this purpose;27 note that when angular integratio n is performed, the values of the total quantities are also updated on each pr ocessor. Hence, this subroutine represents also a synchronization point.

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131 7.2.2 Spatial Domain Decomposition The spatial domain is partitioned into co arse meshes, as discussed in Chapter 5; each coarse mesh is then sequentially alloca ted to the processors through a decomposition vector. Every processor solves for the even-parity angular fluxes only on its assigned spatial sub-domain. The synchronization algor ithm consists of a master/slave algorithm and a scheduling array, which contains inform ation related to the al location of the phasespace on every processor. The master/slave algorithm consists of a paired MPI_SEND/MPI_RECEIVE between two pro cessors which share a coarse mesh interface. The scheduling array toggles each processor between send and receive modes, and it provides information on which portion of the phase-space has to be transferred. Note that before the sending processor ini tiates the communication phase, the projection algorithm, described in Chapter 5, is i nvoked. When every processor has updated the interface values on each coarse mesh, the calculation is continued. As for the angular decomposition algorithm, this point represents a sync hronization phase. 7.2.3 Hybrid Domain Decomposition The hybrid domain decomposition is a combination of spatial and angular decompositions. The hybrid decomposition takes advantage of both speed-up and memory partitioning offered by the angular a nd spatial decomposition, respectively. This decomposition strategy is based on the same algorithms described in the previous sections. Figure 7-1 shows an example of hybrid domain decomposition.

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132 Figure 7-1. Hybrid decomposition for an EP-SS6 calculation (3 directions) for a system partitioned with 4 coarse meshes on 6 processors. 7.3 Parallel Performance of the PENSSn Code The parallel performance of PENSSn is a ssessed using a test problem composed of 64 coarse meshes; each coarse mesh is discre tized with 4,000 fine meshes for a total of 256,000 fine meshes. The problem is characte rized by a homogeneous material with onegroup P0 cross-sections; the total cross-section is equal to 1.0 cm-1, while the scattering cross-section is equal to 0.5 cm-1. A uniform distributed source is present in the system, emitting 1.0 particles/cm3/sec. An SS8 order is used for the calculations, which yields a total of 4 directions. Reflec tive boundary conditions are pres cribed on boundary surfaces at x =0.0 cm, y =0.0 cm, z =0.0 cm, and vacuum boundary conditions are prescribed at x =24.0 cm, y =24.0 cm, z =24.0 cm. The convergence criterion for the angular flux is set to 1.0e-4, while it is set to 1.0e-6 for the Krylov solver. Calculations have been performed on tw o different PC-Clusters: PCPENII at the Nuclear & Radiological Engin eering Department and the Kapp a Cluster at the Electrical

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133 and Computer Engineering De partment, part of the CARRI ER Computational Lab Grid. The specifications for the PCPE NII Cluster are the following: 8 nodes (16 processors) Dual Intel Xeon pro cessors with 2.4 GHz clock frequency, with hyper-threading 4 GB per node of DDR RAM me mory on a 533 MHz system bus. 1 Gb/s full duplex Ethernet network architecture. 40 GB hard drives per each node. 512 KB L2 type cache memory for each processor. The Kappa Cluster has the following technical specifications: 20 nodes (40 processors) Dual 2.4GHz In tel Xeon processors with 533MHz frontside bus with hyper-threading. Intel server motherboard with E7501 Chipset. On-board 1 Gb/s Ethernet. 1GB of Kingston Registered ECC DDR PC2100 (DDR266) RAM. 40GB IDE drive @ 7200 RPM. The analysis of the parallel performance of PENSSn is based on the definition of speed-up, parallel efficiency and parallel fract ion. The speed-up is the direct measure of the time reduction obtained due to parallel ta sking; the mathematical definition of speedup is given by p s pT T S (7.1) where p is the number of processors, Ts is the wall-clock time for the serial run and Tp is the wall-clock time for the parallel run on p processors. The parallel efficiency measures the performance of the domain decomposition algorithm. The definition of pa rallel efficiency is given by p Sp p. (7.2) The speed-up and parallel efficiency are a ffected by communication time and idle time for each processor, by load-imbalance, and by th e parallel fraction in the Amdahl’s law.

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134 Finally, using the Amdahl’s law for expr essing the theoretical speed-up, we can estimate the parallel fraction. s c p p pT T p f f S ) 1 ( 1 (7.3) where fp is the parallelizable frac tion of the code running on p processors and Tc is the parallel communication time. Eventually a ll these quantities are affected by the loadimbalance which may be caused by the different amount of workload. Figure 7-2 shows the speed-up obtained for different decompos ition strategies on the two PC Clusters. 0.00 1.00 2.00 3.00 4.00 5.00 6.00Ser i al 2 S 2 A 4 S 4A 2A/ 2S 8 S 2A /4S 4A/ 2 S 16 S 2A/8S 4A /4S 32SDecomposition strategySpeed-up Kappa Cluster HCS-UF PCPEN2 NRE-UF Figure 7-2. Speed-up obtained by running PE NSSn on the Kappa and PCPENII Clusters. In Figure 7-2, the “decompos ition strategy” refers to the number of processors and the type of decomposition used; “S” refers to spatial decomposition and “A” refers to angular decomposition, and “/” identifies hybrid decompositi ons. Except for the 8-spatial domain decomposition, the speed-up is comp arable for the two clusters up to 4 processors. The maximum speed-up achieved is 5.27 and 4.62, for the PCPENII and Kappa Cluster, respectively, for a spatialdecomposition strategy on 16 processors. Note that as the number of processors increas es, the speed-up obtained does not increase as

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135 well. This behavior is directly related to the concept of granularity and to the communication time. The granularity represen ts the amount of work-load available to each processor; a large grain size leads to a more efficient usage of the machines. In contrast, a small grain size leads to a large communication overhead and, hence, to lower parallel efficiencies. By increasing the number of processors for a fixed problem size, we effectively reduce the granularity with s ubsequent degradation of the speed-up and parallel efficiency as shown in Figure 7.3. 0.0% 20.0% 40.0% 60.0% 80.0% 100.0%S eria l 2 S 2 A 4 S 4 A 2A/ 2 S 8 S 2A /4S 4A/2S 16S 2A /8S 4A /4S 32SDecomposition strategyParallel efficiency (%) Kappa Cluster HCS-UF PCPEN2 NRE-UF Figure 7-3. Parallel effici ency obtained by running PEN SSn on the Kappa and PCPENII Clusters. The spatial discretization of the test pr oblem does not introduce any load imbalance per se; however, Figure 7-3 shows a difference in terms of parallel efficiency between the angularand spatial-decomposition strategies on the same number of processors. This difference is due to load imbalance introduced by the Krylov iterative solver. Table 7-1 presents the data supporting the load im balance generated by the Krylov solver.

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136 Table 7-1. Data relative to the load imbalance generated by the Krylov solver. Decomposition ProcessorDirection Krylov iterations Communication + Idle Time (sec) Grain size 2A/1S 1 1 9749 83.769 28.5% 2 32796 2 3 48877 3.967 71.5% 4 57804 The current strategy for partitioning the phase-space on each processor is based on a sequential allocation of sub-domains. Howe ver, processor 1 is characterized by a smaller grain size compared to processor 2; therefore, the idle time of processor 1 is much larger than processor 2. This is a clea r example of load imbalance and its effect is observed in the relatively low speed-up of 1. 41 and parallel efficiency of 70.3%. This behavior is not observed for the spatial decomposition where the entire angular domain is locally stored on each processor. Table 7-1 shows also a measure of the gr ain size as the rati o between the total number of Krylov iterations required by each processor and the total number of Krylov iteration required by all the processors. Note that the load imbalance is clearly shown by the grain size calculated on each processor th at is 28.5% and 71.5% on processors 1 and 2, respectively. Theoretically, the load imbalance due to the Krylov solvers could be overcome by adopting an automatic load balancing algorithm. As discussed in Chapter 6, the spectral condition number for the matrix operators incr eases as the direction cosine approaches unity. Based on these results, the sequential al location of the angular sub-domains is not optimal. The best angular decomposition algor ithm is based on progr essively pairing the directions which yield the highest an d lowest spectral condition numbers.

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137 Figure 7-4. Angular domain decomposition based on the automatic load balancing algorithm. Figure 7-4 shows the angular domain partitioning based on the automatic load balancing algorithm. If this algorithm is a pplied, the grain size would change to 45.3% and 54.7% for processor 1 and 2, respectively. To complete the analysis of the parallel performance of PENSSn, I have calculated the parallel fraction of the code, by using th e Amdahl’s law in Eq. 7.3. Figure 7-5 shows the speed-up obtained on the PCPENII cluste r compared to the theoretical speed-up predicted by the Amdhal’s Law when the communication time is neglected. 0.00 1.00 2.00 3.00 4.00 5.00 6.00Serial 2 S 2A 4S 4A 2 A /2 S 8S 2 A /4 S 4A/2 S 1 6SDecomposition strategySpeed-up PCPEN2 NRE-UF Theoretical Speed-up (fp=0.87 Tc=0) Figure 7-5. Parallel fraction obt ained with the PENSSn code. 1 2 3 4 Processor1 Processor2

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138 I obtained a 87% parallel fraction ( fp) with a maximum relative difference of 19% between the theoretical speed -up prediceted by the Amdahl’s law and the speed-up observed on PCPENII. Tables 7-2 and 7-3 summ arize the supporting data associated with the PCPENII and Kappa Clusters respectively. Table 7-2. Parallel performance data obtained on PCPENII Cluster. DD # of processors Parallel Vector Wall-clock time (sec) Speed-up Efficiency Memory MB/proc Parallel Fraction Serial 1 1A/1S 207.4 158.8 S 2 1A/2S 116.24 1.78 89.2% 73.2 88.0% A 2 2A/1S 147.58 1.41 70.3% 79.9 61.6% S 4 1A/4S 67.95 3.05 76.3% 43.1 89.8% A 4 4A/1S 84.28 2.46 61.5% 53.2 84.1% H 4 2A/2S 86.04 2.41 60.3% 46.4 78.4% S 8 1A/8S 42.68 4.86 60.7% 28 90.8% H 8 2A/4S 50.09 4.14 51.8% 29.7 86.9% H 8 4A/2S 49.78 4.17 52.1% 33.1 87.1% S 16 1A/16S 39.32 5.27 33.0% 20.5 86.5% DD stands for Domain Decomposition (A – Angular, H – Hybrid, S – Spatial). Table 7-3. Parallel performance data obtained on Kappa Cluster. DD # of processors Parallel Vector Wall-clock time (sec) Speed-up Efficiency Memory MB/proc Parallel Fraction S 2 1A/2S 116.14 1.76 88.1% 73.2 86.9% A 2 2A/1S 145.58 1.41 70.3% 79.9 61.7% S 4 1A/4S 63.55 3.22 80.5% 43.1 92.1% A 4 4A/1S 82.11 2.49 62.3% 53.2 83.6% H 4 2A/2S 82.2 2.49 62.3% 46.4 80.2% S 8 1A/8S 71.9 2.85 35.6% 28 74.3% H 8 2A/4S 47.45 4.31 53.9% 29.7 88.0% H 8 4A/2S 51.08 4.01 50.1% 33.1 86.0% S 16 1A/16S 44.32 4.62 28.9% 20.5 83.6% H 16 2A/8S 55.42 3.69 23.1% 21.4 77.9% H 16 4A/4S 47.25 4.33 27.1% 23 82.3% S 32 1A/32S 49.29 4.15 13.0% 16.8 78.4% S 64 1A/64S 54.19 3.78 5.9% 14.9 74.8% These results indicate that the PENSSn code is characterized by a relatively high parallel performance; also the PCPENII a nd Kappa Clusters yield almost the same performance. The main advantage of the PCPE NII Cluster is the la rge amount of memory

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139 available, which allows the simulation of large 3-D models without compromising the parallel performance due to communication overhead. 7.4 Parallel Performance of PENSSn App lied to the MOX 2-D Fuel Assembly Benchmark Problem The MOX 2-D benchmark problem discussed in Chapter 6 is used to assess the parallel performance of PEN SSn for a large criticality eigenvalue problem. The problem specifications are discussed in Chapter 6, and an SS4 order is used for the performance testing. Table 7-4 presents the parallel performance data obtained on the PCPENII Cluster. Table 7-4. Parallel performance data for the 2-D MOX Fuel Assembly Benchmark problem (PCPENII Cluster). Decomposition Strategy Number of processors Parallel Vector Wall-clock time (sec) Speed-up Efficiency Memory MB/proc Serial 1 1A/1S 511.0 1.0 341.6 Angular 2 2A/1S 263.7 1.9 95.0% 202.0 Spatial 3 1A/3S 198.7 2.6 86.6% 105.3 Hybrid 6 2A/3S 132.6 3.9 65.0% 71.7 Spatial 9 1A/9S 107.3 4.8 53.3% 39.5 Note that for this problem the angular decomposition yields the best speed-up, because for the low order SSN methods the load-imbalance due to the Krylov solver is not so significant. For this problem, high order EP-SSN methods provided a relatively accurate solution, both in terms of critical ity eigenvalue and power distribution. As indicated, the hybrid decomposition is used mainly to increase the speed-up. Moreover, the PENSSn code yielded good parallel pe rformance both in terms of speed-up and memory utilization.

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140 CHAPTER 8 DEVELOPMENT OF A NEW SYNTHETI C ACCELERATION METHOD BASED ON THE EP-SSN EQUATIONS The inversion of the trans port operator is obtained usi ng the Source Iteration or Richardson iteration method. It is known that this iterative technique is very ineffective for problems with optically th ick regions and scattering ratio close to unity. In these conditions, the spectral radius of the tran sport operator tends to unity and the convergence process becomes very slow.3 Hence, it is necessary to develop acceleration schemes which can increase the rate of convergence.17 In principle, synthetic acceleration schemes consist of two distinct opera tors: a higher order operator (e.g., SN) and a lower order operator, usually a diffusion-like equation. The idea is to correct the solution of the diffusion-like equation using the transport solution, with subs equent acceleration of the convergence process. In the late 1960s, Gelbard and Hageman developed a synthetic acceleration method based on the diffusion equation and the S4 equations.28 Later, Reed independently derived a similar synthetic acceleration scheme40 and pointed out some limitations of the method derived by Gelbard and Hageman. The synt hetic method developed by Reed has the advantage of being very effective for small mesh sizes, but it is unstable for mesh size greater than ~1 mfp. Later, Alcouffe i ndependently derived the Diffusion Synthetic Acceleration (DSA) method. He addressed th e issue of stability of the method and derived an unconditionall y stable DSA algorithm.29 Alcouffe pointed out that in order to obtain an unconditionally stable method, the diffusion equation must be derived from the

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141 discretized version of the transport equation. In this way, the consistency between the two operators is preserved. However, Warsa, Wa reing and Morel, recently observed a loss in the effectiveness of DSA schemes, especially for multi-dimensional heterogeneous problems.31-32 This chapter addresses the development of a new synthetic acceleration method based on the EP-SSN method. Since the discretization of the EP-SSN equations is not consistent with the discretization of the SN method, the acceleration method is limited to mesh sizes up to ~1 mean free path. The first part of this chapter discusses the theory involved in developing the EP-SSN synthetic acceleration method, and the second part presents the numerical results obtained for a test problem. The performance of the EPSSN synthetic acceleration algor ithm will be compared with the Simplified Angular Multigrid (SAM) acceleration method.39 In conclusion, I will point out strengths and weaknesses of the new method, and I will build the foundations for the FAST (Flux Acceleration Simplified Transport) preconditio ning algorithm, discussed in Chapter 9. 8.1 The EP-SSN Synthetic Acceleration Method This section describes the general theory of a general synthe tic acceleration method and, hence, its applicatio n to the solution of the SN equations. Solutions to many engineeri ng problems of practical inte rest can be obtained via a balance equation written in operator form as follows q Kf Tf (8.1) In the case of the SN equations, the operators T and K are defined as tT ˆ, (8.2) 0 4) ˆ ˆ (E E r d dE Ks (8.3)

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142 Commonly, Eq. 8.1 is solved using the Ri chardson iterative method formulated as q Kf Tfl l 1. (8.4) In Eq. 8.4, l is the iteration index; hen ce, by inverting the operator T in Eq. 8.4, we obtain the following q T Mf fl l 1 1 (8.5) where K T M 1 (8.6) In Eq. 8.5 the operator M is usually referred to as iteration operator In this derivation the spectral ra dius of the iteration operator is assumed to be less than unity; note that this is a realistic assumption fo r the problems encountered in engineering applications. To discuss the synthetic opera tor, I introduce the re sidual term given by l l lf f r 1 1. (8.7) Using the residual formulation, Eq. 8.5 reduces to i i Mr r 1. (8.8) The sum over the residuals following the lth iteration is given by l k l k lr M I M r M M r 1 2...) (. (8.9) Where, I is the identity matrix. Hence, using the above formulation, the exact solution to Eq. 8.5 is obtained by 1 k k l lr f f (8.10) or l lr M I M f f (8.11)

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143 Note that using the definition of the iteration operator M (Eq. 8.6), Eq. 8.11 reduces to l lKr K T f f1) ( (8.12) Eq. 8.12 is just another formulation for Eq. 8.1 and its solution is as difficult. However, Eq. 8.12 separates the solution to Eq. 8.1 into two terms. The synthetic acceleration method is based on approximating the high order operator (e.g., transport) ) ( K T with a lower order operator WL. The lower order operator must possess two fundamental properties in order for the synthe tic method to be effective: 1) the lower order operator has to be a good approximation to the high order operator; and 2) it has to be easy to invert. The synthe tic acceleration method formulated based on Eq. 8.12 can be written as l L l lKr W f f1 2 / 1 1 ~ (8.13) where q T Mf fl l 1 2 / 1 ~ (8.13) and l l lf f r 2 / 1~ (8.14) The philosophy of the synthetic accelerati on method described above consists in utilizing a lower order operator to project th e residual term on the sub-space generated by the operator K The projection operation with the low order operator is performed in a fraction of the time required for the soluti on of the higher order operator, thereby producing a significant speed-up of the iterati on process with a cons equent reduction of the numerical spectral radius. The synthetic acceleration method discussed above can be readily applied to the SN equations. The method is designed to accelerate the convergence of the inner iteration in

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144 each energy group. In the following derivation we assume isotropic scattering and source for simplicity. In this particular case, the high order operator is the transport equation. q K Hl S l n 0 2 / 1 ~ (8.15) where t n nH ˆ, (8.16) and 0 0 s SK (8.17) In the SI method, the scalar flux (0) is initially guessed and substituted into Eq. 8.15; hence, Eq. 8.15 is solved and the value of the scalar flux at the new iteration is simply evaluated by ) 2 ( 1 2 / 1 1~N N n l n n lw (8.18) and the iteration process is continued un til convergence. The synthetic acceleration method described above substitutes the update of the scalar flux in Eq. 8.18, with the following expression l l lp 2 / 1 1~ (8.19) Note that the term indicated with the tild a symbol in Eq. 8.19 is obtained from the SN method in Eq. 8.15. The projection of the residual term ( pl) on the scattering kernel is performed using the EP-SSN equations as follows l S m L l mr K W p0 1 ~ (8.20) The residual in Eq. 8.20 is calculated as the difference between the transport solution and the accelerated solution for the scalar flux

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145 l l lr 2 / 1 ~ (8.21) where t t m m LW 2 ,, (8.22) and 2 / ) 1 ( 1 ,~L m l m e m lp w p. (8.23) The SN equations given in Eq. 8.15 are used to compute an uncollided scalar flux, which is then used to evaluate the residual wi th Eq. 8.21. The residual is then projected onto the scattering kernel via the EP-SSN operator. Hence, the inversion of the EP-SSN operator generates a projection of the residua l onto the scattering kernel, which is then used to complete the calculation of the scal ar flux at the new iteration using Eq. 8.19. 8.2 Spectral Analysis of the EP-SSN Synthetic Acceleration Method In this section, I will study the theoretical performance of the synthetic acceleration method based on the EP-SSN equations. For this purpose I will analyze the spectrum of eigenvalues of the synthetic operator in th e Fourier transformed space. The performance of iterative methods can be a ssessed by studying the spectral radius. For an infinite homogeneous medium, the spec tral radius is equal to MAX, (8.24) where represents the spectrum of eigenvalues in the transformed space. In general, the spectral radius is an i ndication of the error reduction for the iterative method at each iteration. Hence, if the spectral radius appr oaches unity the method presents very slow convergence.

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146 The spectrum of eigenvalues for the EP-SSN synthetic acceleration algorithm can be written as N N NA A A 1, (8.25) where the transport operator in the transformed space is formulated as 1 0 2 2) arctan( 1 c d c A (8.26) and the lower order EP-SSN operator is defined by N Nw c A1 2 21 (8.27) It is well known that the spectral radius of the unaccelerated tr ansport operator in an infinite homogeneous medium is equal to c c A MAXA ) arctan( lim ) (0 (8.28) Therefore, as discussed in the beginning of this chapter, the unaccelerated source iteration method presents very slow convergen ce for scattering-dominated problems, and where the leakage probability is relatively small. Figure 8-1 s hows the spectrum of eigenvalues for the unaccelerated source it eration and for the synthetic acceleration methods obtained with different SSN orders.

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147 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.01.53.04.56.07.59.010.512.013.515.016.518.019.521.022.524.025.527.028.5Fourier modesEigenvalues Source Iteration SS-2 Synthetic SS-4 Synthetic SS-6 Synthetic SS-8 Synthetic Figure 8-1. Spectrum of eigenvalues for th e Source Iteration and Synthetic Methods based on different SSN orders. As indicated by Figure 8-1, the spec tral radius of different EP-SSN synthetic methods decreases with increasing order; this be havior is due to the fact that higher order EP-SSn methods resolve higher frequenc y modes of the spectrum, which are characterized by higher order a ngular dependencies. Also, the spectral radius obtained for the different methods are listed in Table 8-1. Table 8-1. Spectral radius for the different iterative methods. Method Spectral Radius Source Iteration 0.99 SS2 Synthetic 0.221391 SS4 Synthetic 0.109545 SS6 Synthetic 0.072785 SS8 Synthetic 0.054489 Based on these results, higher order EP-SSN equations present significantly smaller spectral radii than diffusion based synthetic accel eration algorithms and, therefore, better

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148 acceleration performance. However, the numeri cal results will show that in practice theoretical performance is not achieved. 8.3 Analysis of the Algorithm Stabili ty Based on Spatial Mesh Size In this section, I will analy ze the stability of the EP-SSN synthetic acceleration method with respect to the spatial mesh size. In this phase of th e investigation, the discretization of the EP-SSN formulation is not consistent with the transport operator; hence, the stability of the method depends on the size of the spatial mesh. The EP-SSN acceleration method has been implemente d into the PENTRAN Code System.15 For this analysis, I have considered a simple 3-D cube with a homogeneous material. The size of the cube is 10x10x10 cm3, discretized with a 1.0 cm uniform mesh along the three axes. The total cross-section is varied in order to change the dimension of the system in terms of mean free paths ( mfp ), and the c-ratio is set equal to 0.99. The boundary conditions prescribed are re flective on the planes at x =0.0 cm, y =0.0 cm, z =0.0 cm and vacuum at x =10.0 cm, y =10.0 cm and z =10.0 cm. An isotropic source, with magnitude 1.0 [n/cm3/sec] is uniformly distributed inside the system. The point-wise convergence tolerance for the scalar flux is set to 1. 0e-5. Figure 8-2 shows the number of inner iteration required by the EP-SSN synthetic methods as a function of the mesh size and different order of the lower-order EP-SSN operator.

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149 Figure 8-2. Number of inner iterations re quired by each acceleration method as a function of the mesh size. Due to the inconsistent discreti zation of the transport and EP-SSN operators, the synthetic acceleration method degrades in term s of performance as the size of the mesh increases, and for mesh sizes greater than 1.0 mfp the acceleration technique becomes unstable. Table 8-2 compares the EP-SSN synthetic and the unaccelerated transport methods based on the number of inner iterations for a 1.0 mfp mesh size. Table 8-2. Comparison of the number of inner iterati on between EP-SSN synthetic methods and unaccelerated transport. Method Inner iterations EP-SS2 12 EP-SS4 28 EP-SS6 38 EP-SS8 40 Unaccelerated transport 262 The EP-SSN synthetic methods reduce the number of inner iteration from ~6 to ~21 times with respect to the unaccelerated tr ansport calculation. Note that as the SSN order is increased, the acceleration performance is degrad ed; this behavior is due to the increasing number of inner iterations requi red to solve higher order EP-SSN equations.

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150 8.3.1 Comparison of the EP-SSN Synthetic Acceleration with the Simplified Angular Multigrid Method The EP-SSN synthetic acceleration method is compared with the Simplified Angular Multigrid (SAM).39 I have tested the effects of the scattering ratio and differencing schemes on the convergence rate. The test problem is a 10x10x10 cm3 box with homogeneous medium. A vacuum boundary condition is prescribed on all surfaces. A fixed source of magnitude 1.0 particles/cm3/s is placed in a region ranging from 4 to 6 cm along the x -, y -, and z -axes. The differencing schemes tested with the PENTRAN code are DZ, DTW and EDW. Th e problem is discretized with a 1.0 cm uniform mesh and an S8 level-symmetric quadrature set is used in the calculations. The point-wise flux convergence tolerance is set to 1.0e-6. Figures 8-3, 8-4, and 8-5 show the number of inner iterations required to converge as a function of the scattering ratio for the Source Iteration (SI), SAM and EP-SS2 synthetic acceleration method, using the DZ, DTW, and EDW differencing schemes, respectively. 0 20 40 60 80 100 120 140 160 180 200 0.10.20.30.40.50.60.70.80.91 Scattering ratioNumber of inner iterations SAM SI Synthetic SS-2 Figure 8-3. Number of inne r iterations as a function of the scattering ratio (DZ differencing scheme).

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151 The synthetic method improves the converg ence rate by a factor of ~6.5 for a scattering ratio of 1.0. Note also that th e performance of the synthetic method is not significantly affected by the scattering ratio. 0 20 40 60 80 100 120 140 160 180 200 0.10.20.30.40.50.60.70.80.91 Scattering ratioNumber of inner iterations SAM SI Synthetic SS-2 Figure 8-4. Number of inner iterations as a function of the scattering ratio (DTW differencing scheme). 0 20 40 60 80 100 120 140 160 180 200 0.10.20.30.40.50.60.70.80.91 Scattering ratioNumber of inner iterations SAM SI Synthetic SS-2 Figure 8-5. Number of inner iterations as a function of the scattering ratio (EDW differencing scheme). As shown in Figures 8-4 and 8-5, the synthetic acceleration method improves the convergence rate compared to the SI and SAM methods. For the DTW and EDW

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152 schemes the synthetic method reduces the numbe r of inner iterations by a factor of ~10 and ~8, respectively. The inc onsistent discretization of the operators does not yield significant instabilities in these cases; this is due to the fact that the fine-mesh size is adequate to yield a stable acceleration scheme. Figure 8-6 shows a comparison of the numbe r of inner iterations for the synthetic method with DZ, DTW, and EDW differencing schemes. 0 5 10 15 20 25 30 35 0.10.20.30.40.50.60.70.80.91Scattering ratioNumber of inner iterations DZ differencing scheme DTW differencing scheme EDW differencing scheme Figure 8-6. Number of inne r iterations for the EP-SS2 synthetic method obtained with DZ, DTW, and EDW differencing schemes. All the differencing schemes perform sim ilarly for scattering ratios up to 0.7; however, for scattering ratios greater than 0. 8, the DTW differencing scheme yields the best convergence performance. The degraded performance of the DZ differencing scheme is due to the flux fix-up performed on the solution. The EDW differencing scheme degrades the performance of the s ynthetic method, because for scattering ratios close to unity, the physics of the problem is dominated by scattering processes, while the EDW differencing scheme predicts an expone ntial behavior of the particle flux.

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153 8.4 Limitations of the EP-SSN Synthetic Acceleration Method Based on the analysis of the EP-SSN synthetic acceleration method, I have identified the following limitations: Stability of the method is dependent on a mesh size smaller than 1.0 mfp The method is affected by numerical os cillations for multidimensional problems with heterogeneous materials. Domain decomposition algorithms in parall el computing environments may worsen the performance of the synthetic method. As previously discussed, these limitations are mainly due to the inconsistent discretization of the transport and EP-SSN operators. However, if this condition is met, it does not necessarily imply unconditional stab ility of the method. Hence, for large heterogeneous multi-dimensional problem, this method is of limited applicability with current formulations. To address this problem, I have decoupled the SN and EP-SSN methods by using the last one as a preconditioner. The philosophy be hind this approach is to use the PENSSn code to obtain an initial solution in a fr action of the time required by the transport calculation; then the solution is introduced as an initial guess into the transport code. This approach has led to the development of the Flux Acceleration Simplified Transport (FAST) system, which is a fully automated preconditioning system for the discrete ordinates method.

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154 CHAPTER 9 FAST: FLUX ACCELERATION SIMPLIFI ED TRANSPORT PRECONDITIONER BASED ON THE EP-SSN METHOD In this chapter, I will discuss the development and implementation of FAST in the PENTRAN Code System. FAST is based on the PENSSn code and it is a fully automated system, which is integrated into PENTRAN-SSn. The FAST algorithm is used to precondition and speed-up both cr iticality and fixed source calculations. I will present the performance of the new system for both criticality and shielding calculations. Three problems with significant transport effects will be used to asses the capability of the algorithm. The first problem was introduced in Section 6.4.3, and it is characterized by regions of air and a considerable leakage probability. The second problem, introduced in Section 6.4.5, will demonstrate the effectiveness of the FAST algorithm in dealing with fixed source problem s characterized by anis otropic scattering. The third problem is a 3-D whole-core calc ulation based on the MOX 3-D Fuel Assembly Benchmark extension47 proposed by OECD/NEA. For these test problems, I will discuss the speed-up obtained with FAST compared to a conventio nal transport calculation without preconditioning. 9.1 Development and Implementation of FAST The FAST code is derived from the PENSSn co de described in Chapters 6 and 7; the code retains every feature of PENSSn except for balance table generation and advanced input features which are part of the stand-alone co de only. These modifications have been necessary in order to completely integrate the code into PENTRAN-SSn. The

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155 system is fully automated and in order to invoke it, only an additional input card is required in a standard PENTRAN deck, as shown in Figure 9-1. Figure 9-1. Card required in PENT RAN-SSn input deck to initiate SSN preconditioning. More details about the usage of the FAST preconditioner can be found in the PENTRAN manual.48 The SSN card is used to instruct PENTRAN to generate the files needed by the FAST algorithm such as problem defin ition, cross-sections and fixed source distribution. The FAST algorithm prepares the initial solution for the SN code and creates output files for s calar flux, currents, and cri ticality eigenvalue. The output files are dumped for each energy group and they are split into file_a for the first half of the spatial domain and file_b for the second half. This format is used to co ntain the file sizes under the limits handled by current opera ting systems. The files containing the initial solution are read successively by PENT RAN and used as initial guess for starting the calculation. Figure 9.2 shows th e flowchart for PENTRAN-SSn system.

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156 Figure 9-2. Flow-chart of th e PENTRAN-SSn Code System. As shown in Figure 9-2, the transport calculation is composed of three main phases: Phase 1 consists of processing the input deck and generating input files for the FAST algorithm; in phase 2, the FAST preconditioning algorithm generates the initial solution and dumps it to the output files; phas e 3 completes the transport calculation by running the preconditioned PENTRAN-SSn code. PENTRAN-SSn ( Run with SSNcard ) Input files generated for FAST: ssnvars (problem definition) ssnxs (cross-sections) FAST algorithm generates the initial solution EXPRESSN generates fixed source distribution files (.fst) from PENTRAN input deck Fixed source calculation Criticality eigenvalue calculation Preconditioned PENTRAN-SSn generates the final solution Phase 1 Phase 2 Phase 3

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157 As I will show in the next sections, the FAST preconditioner produces an accurate solution within a fraction of the computat ion time required by a standard transport calculation; hence, by starting the transport calculation with this initial solution, convergence is rapidly achieved. The acceleratio n performance can also be explained in terms of the Fourier transform of the transp ort operator. The Maclau rin expansion of the transport operator in the frequency domain for0 is given by 21 1 1 o i i A (9.1) Note that 0 modes correspond to low-fre quency Fourier eigenmodes and, hence, to long wavelengths; th ese are error modes that span large optical distances and have weak spatial gradients. The low-fre quency Fourier modes formulated in Eq. 9.1 have also weak angular dependencies. These modes of the spectrum are rapidly resolved by diffusive-like algorithms such as EP-SSN, while the standard SI method efficiently suppresses the error modes with strong spatial and a ngular variations, where 0 Here, the preconditioning phase quickly resolv es the low-frequency error modes, while the successive transport calculation resolv es the remaining high-frequency modes. 9.2 Testing the Performance of the FAST Preconditioning Algorithm In this section, I will present the results obtained with the FAST preconditioner and PENTRAN-SSn for a critic ality eigenvalue, a fixed s ource, and the MOX 3-D Fuel Assembly benchmark problems. 9.2.1 Criticality Eigenvalue Problem The objective of this test is to verify the performance of FAST and PENTRANSSn for a criticality eigenvalue problem char acterized by significant transport effects. The problem considered is described in Secti on 6.4.3. The system is characterized by air

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158 regions where the EP-SSN method is also affected by nume rical difficulties. In order to remedy this issue, I have utilized the Dens ity Factor Multiplier (DFM) described in Section 6.4.3. I have calculated the ratio of the total nu mber of inner iterations required to solve the problem with the preconditioned PENT RAN-SSn and with the non-preconditioned PENTRAN. Figure 9-3 presents th is ratio for different EP-SSN methods with DFM disabled and enabled. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 EP-SS2EP-SS4EP-SS6EP-SS8 MethodInner iterations ratio Density Factor Disabled Density Factor Enabled Figure 9-3. Ratio of total num ber of inner iterations require d to solve the problem with preconditioned PENTRAN-SSn and non-preconditioned PENTRAN. Two aspects are clearly apparent in Figure 9-3; in first instance, the acceleration performance obtained with DFM enabled is obv iously superior to the case where DFM is disabled. Also, by enabling DFM, the convergence of the EP-SSN method is significantly improved, as well as the accuracy of the solutio n, as shown in Table 9-1. Note that by disabling DFM, all the EP-SSN methods did not converge within the maximum number of outer iterations set to 50.

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159 Table 9-1. Criticality eigenvalues obtained with the preconditioned PENTRAN-SSn code for different EP-SSN orders. DFM Disabled DFM Enabled Method FAST PENTRAN-SSn FAST PENTRAN-SSn EP-SS2 0.694370.95212 0.776210.95213 EP-SS4 0.830970.95212 0.905440.95212 EP-SS6 0.8464 0.95213 0.921380.95212 EP-SS8 0.848940.95213 0.924360.95215 Figure 9-3 shows also that higher order EP-SSN methods improve the acceleration performance of PENTRAN-SSn. This behavior is expected since the air regions and the significant boundary leakage introduce substa ntial angular dependency; hence, as predicted high order EP-SSN methods yield a more accu rate solution and better acceleration performance. 9.2.2 Fixed Source Problem The purpose of this test problem is to evaluate the performance of the FAST algorithm in accelerating fixed source calcu lations. The problem configuration is described in section 6.4.5. I have preconditioned PENTRAN-SSn using the EP-SS4 and EP-SS6 methods. Figures 9-4 and 9-5 show the rela tive change in flux value as a function of the inner iteration number for energy groups 1 and 2. In group 1, the non-preconditioned calculation presents a slight error reduction in the first 8 iterations, while PENTRAN-SSn logarithmically reduces the relative error from the first iteration. This behavior is due to the capability of the EP-SSN method of resolving the low-frequency e rror modes and, hence, to bypass the plateau region where the SI method alone is experiencing difficulties in reducing the error.

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160 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1234567891011121314151617 Inner iteration numberRelative change in flux value PENTRAN (No preconditioning) PENTRAN-SS4 PENTRAN-SS6 Figure 9-4. Relative change in flux value in group 1. Figure 9-5 shows a similar behavior for energy group 2. Due to the small size of this problem the speed-up obtai ned with PENTRAN-SSn is roughl y equal to a factor of 2. It is also worth noticing that the EP-SS4 method is sufficient to provide an accurate solution for an efficient preconditioning; the EP-SS6 method does not yield significant benefits and, actually, it requires more comput ation time. This behavior can be explained by considering that the EP-SS6 method does not introduce a far more accurate solution compared to EP-SS4; however, the computational time required by FAST for EP-SS6 increases by ~60%, therefore explai ning the behavior discussed above. 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 12345678910111213141516 Inner iteration numberRelative change in flux value PENTRAN (No preconditioning) PENTRAN-SS4 PENTRAN-SS6 Figure 9-5. Relative change in flux value in group 2.

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161 9.3 The MOX 3-D Fuel Assembly Benchmark Problem The MOX 3-D Fuel Assembly Benchmark problem extension47 has been proposed by OECD/NEA to test the capab ility of current transport me thods and codes in dealing with whole-core simulations without spatia l homogenization. The benchmark geometry is the sixteen assembly (quarter core symme try) C5 MOX fuel assembly proposed by Cavarec.46 Each fuel assembly consists of a 17x17 lattice of square pin cells. The side length of each pin cell is 1.26 cm and all the fuel pins a nd guide tubes have a 0.54 cm radius. The benchmark extension has introduced three different configurations of the reactor core: Unrodded configuration. Rodded-A configuration. Rodded-B configuration. The unrodded configuration does not include any control rod in the model; in the second configuration Rodded-A, a control rod cl uster is inserted 1/3 of the way into the inner UO2 assembly (refer to Figure 6-29 for a model view on the x y plane). In the Rodded-B configuration, the control rod clusters are inserted 2/3 of the way into the inner UO2 assembly and 1/3 of the way into both MOX assemblies. The seven-group, transport corrected, isot ropic scattering cross-sections for each material were obtained using the collision probability code DRAGON,49 which uses the WIMS-AECL 69-group library; these cross-sect ions include up-scatte ring processes also. These seven-group cross-sections proved the most difficult to solve and thus, were chosen to enhance transport difficu lties of heterogeneous problems. This problem presents significant trans port effects due to the heterogeneous configuration, and highly angular dependent flux on the fuel-moderator interface. The

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162 model is discretized with 946,080 fine meshes, and an S6 level symmetric quadrature set is used. The PENTRAN Code System re quired a memory allocation of 1381.2 MB/Processor with an 8-space and 2-angle domain decomposition strategy. An EP-SS4 method is used for the FAST preconditioner, requiring 548.6 MB/Processor with a 16space domain decomposition strategy. In the PENTRAN code, the point-wise flux convergence tolerance was chosen equal to 1.0e-3; while the conve rgence tolerance on the criticality eigenvalue was set equal to 1.0e-5. The FAST preconditioner used the same convergence criteria specified fo r PENTRAN. Both preconditioned and nonpreconditioned transport calculations ran on the 16-processors PCPENII Cluster. In the following sections, I will present th e results and performance obtained with PENTRAN-SSn using the FAST preconditioner for the aforementioned three configurations. 9.3.1 MOX 3-D Unrodded Configuration In this case, the model of the reactor core does not contai n any control rod; therefore, the FAST preconditioner yields the most accura te solution. Table 9-2 presents the results obtained with PENTRAN and PE NTRAN-SSn as compared to a Monte Carlo reference solution. Table 9-2. Results obtained for the M OX 3-D in the Unrodded configuration. Method Criticality eigenvalue Relative error (pcm) Inner iterations Total Time Time ratio Inner it. ratio Monte Carlo 1.143080.0026PENTRAN 1.1466 307.94 2755 2.97d 1 1 PENTRAN-SSn 1.14477 147.85 413 15.2h 4.7 6.67 FAST 1.15891 1384.85 6.9h The inner iterations and time ratios presen ted in Table 9-2 are calculated using the values obtained with non-preconditi oned PENTRAN and PENTRAN-SSn. Since no

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163 synthetic acceleration is employed within PENTRAN-SSn, the values of time ratio and inner iterations ratio are similar, as expected. For this case, I observed an improvement in the accuracy of the criticality eigenvalue due to the preconditioned solution. It is worth mentioni ng that improvements in accuracy have been observed al so for the pin-power distributions. Figure 9-6 shows that for this case th e accurate critical flux distribution and eigenvalue provided by FAST yield a non-oscillatory beha vior of the criticality eigenvalue for PENTRAN-SSn, leading to a very steep error reduction as shown in Figure 9-7. Figure 9-7 includes also a separate chart th at shows the behavior of the criticality eigenvalue relative error for PENTRAN-SSn. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 11223344556677889100111122133144155166177188 Outer iteration numberCriticality eigenvalue PENTRAN (No preconditioning) PENTRAN-SSn PENTRAN-SSn k-eff=1.14477 (MCNP Rel. Diff.=147.85 pcm) PENTRAN k-eff=1.14660 (MCNP Rel. Diff.= 307.94 pcm) Figure 9-6. Behavior of the cr iticality eigenvalue as a func tion of the outer iterations.

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164 Figure 9-7. Convergence behavior of the criticality eigenvalue. Figure 9-8 shows that the total comput ation time could be further reduced by terminating the preconditioning phase earlier in th e calculation process, since the variation on the soluti on obtained with FAST is not significant. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1591317212529333741454953576165697377818589 Outer iteration numberCriticality eigenvalue FAST (EP-SS4 time=6.9h on 16 processors) PENTRAN-SSn (time=8.3h on 16 processors) Figure 9-8. Preconditioning and transport calculation phases with relative computation time required.

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165 9.3.2 MOX 3-D Rodded-A Configuration The results of PENTRAN and PENTRANSSn, as compared to the reference Monte Carlo solution, are shown in Table 9-3. Table 9-3. Results obtained for th e MOX 3-D Rodded-A configuration. Method Criticality eigenvalue Relative error (pcm) Inner iterations Total Time Time ratio Inner it. ratio Monte Carlo 1.128060.0027 PENTRAN 1.12753 -46.98 2714 3.4d 1.0 1.0 PENTRAN-SSn 1.12890 74.46 468 15.9h 5.1 5.8 FAST 1.14582 1574.38 7h For this case PENTRAN-SSn yield a speed-up of 5.1 compared to the nonpreconditioned transport calculation. Note al so that PENTRAN-SSn overestimates the criticality eigenvalue, which is a conservative solution from an engineering point of view. Figure 9-9 shows the behavior of the cri ticality eigenvalue as a function of the outer iterations; note that PENTRAN-SSn reduces the necessary number of outer iterations by a factor of ~17 as co mpared to non-preconditioned PENTRAN. 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 11223344556677889100111122133144155166177188199 Outer iteration numberCriticality eigenvalue PENTRAN (No preconditioning) PENTRAN-SSn PENTRAN-SSn k-eff=1.12890 (MCNP Rel. Diff.=74.46 pcm) PENTRAN k-eff=1.12753 (MCNP Rel. Diff.= -46.98 pcm) Figure 9-9. Behavior of the cr iticality eigenvalue as a func tion of the outer iterations.

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166 Figure 9-10 shows the behavior of the re lative error as a function of the outer iterations; this figure incl udes a chart which shows the convergence behavior for PENTRAN-SSn only. Figure 9-10. Convergence behavior of the criticality eigenvalue. As clearly shown in Figure 9-10, the relativ ely accurate critical flux and criticality eigenvalue provided by FAST, produce a steep erro r reduction of ~3 out er iterations for each order of magnitude. As previously discussed, the EP-SSN method is very efficient in resolving the low-frequency error modes while the transport calculation corrects the preconditioned solution by resolvin g high-frequency error modes. Figure 9-11 presents the beha vior of the criticality eigenvalue calculation during the preconditioning and transport calculation phases. Note that the total computation time of 15.9 hours could be further reduced by earlier termination of the preconditioning phase, because from about the 40th iteration the relative change in the criticality eigenvalue is small.

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167 0 0.2 0.4 0.6 0.8 1 1.2 1.4 159131721252933374145495357616569737781858993 Outer iteration numberCriticality eigenvalue FAST (EP-SS4 time=7h on 16 processors) PENTRAN-SSn (time=8.9h on 16 processors) Figure 9-11. Preconditioning and transport calc ulation phases with relative computation time required. 9.3.3 MOX 3-D Rodded-B Configuration For this configuration the control rods are inserted at different positions inside the reactor core. Table 9-4 presents PENTRAN and PENTRAN-SSn results as compared to the reference Monte Carlo solution. Table 9-4. Results obtained for th e MOX 3-D Rodded-B configuration. Method Criticality eigenvalue Relative error (pcm) Inner iterations Total Time Time ratio Inner it. ratio Monte Carlo 1.077770.0027PENTRAN 1.06772 -932.48 1352 1.9d 1 1 PENTRAN-SSn 1.07356 -390.62 526 17.7h 2.6 2.57 FAST 1.09553 1647.85 8.2h The control rods introduce strong angular dependencies in the particle flux distribution; hence, as expected the FAST preconditioner yields a less accurate solution compared to the Rodded-A or Unrodded cases. However, the accuracy is sufficient to

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168 accelerate the transport calculation by a factor of ~2.6. Figure 9-12 shows the behavior of the criticality eigenvalue as a function of the outer iteration number. 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 13579111315171921232527 Outer iteration numberCriticality eigenvalue PENTRAN (No preconditioning) PENTRAN-SSn PENTRAN-SSn k-eff=1.07356 (MCNP Rel. Diff.=-390.62 pcm) PENTRAN k-eff=1.06772 (MCNP Rel. Diff.= -932.48 pcm) Figure 9-12. Behavior of the cr iticality eigenvalue as a func tion of the outer iterations. PENTRAN-SSn achieve the converged solu tion in half of the outer iterations required by the non-preconditioned PENTRAN; moreover, for this case the FAST algorithm increases the accuracy of the soluti on compared to Monte Carlo. The improved accuracy is due to the fact th at the solution provided by FAST reduces the numerical diffusion phenomenon caused by the relativ ely coarse discretization along the z -axis. Figure 9-13 presents the convergence beha vior of the crit icality eigenvalue obtained with PENTRAN-SSn and PENTRAN.

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169 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 13579111315171921232527 Outer iteration numberCriticality eigenvalue relative erro r PENTRAN (No preconditioning) PENTRAN-SSn Figure 9-13. Convergence behavior of the criticality eigenvalue. The relative error drops sharply in the fi rst iterations due to the preconditioned solution. Figure 9-14 presents the behavi or of the criticality eigenvalue in the preconditioning and transport calculation phase s; again, the tota l computational time could be further reduced by stopping the FAST preconditioner at an early stage in the calculation process. 0 0.2 0.4 0.6 0.8 1 1.2 159131721252933374145495357616569737781 Outer iteration numberCriticality eigenvalue FAST (EP-SS4 time=8.2h on 16 processors) PENTRAN-SSn (time=9.5h on 16 processors) Figure 9-14. Preconditioning and transport calc ulation phases with relative computation time required.

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170 In conclusion, the FAST preconditioning algorithm integrated into the PENTRAN-SSn code has been proven very effective in acceler ating the transport calculations for a large whole-core 3-D mode l. The computational time is reduced by a factor of 3 to 5, depending on the probl em, as compared to non-preconditioned calculations. Moreover, I have also observed a slight improvement in the accuracy of the criticality eigenvalue and power distribution.

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171 CHAPTER 10 SUMMARY, CONCLUSION, AND FUTURE WORK In this research work, I have developed advanced quadrature sets including the PNEW and the PN-TN, along with a new biasing tec hnique named Regional Angular Refinement (RAR). These quadrature sets do not present negative weights for any SN order and they are suitable for problems charact erized by highly angular dependent fluxes and/or sources. Based on the results obtained, the PN-TN quadrature set yields very accurate results and it is currently implem ented in the PENTRAN Code System. The RAR technique has been proven very effectiv e in dealing with highly angular dependent sources; the quadrature sets biased with RAR lessen the ray-effects, therefore yielding accurate results in a fraction of the tim e required by a standard quadrature set. These new quadrature set generation techniqu es are very suitable for the simulation of medical physics applications and devices, wh ere large regions of air require advanced quadrature sets in order to remove ray-effects from the solution. I have investigated the Even-Parity Simplified SN (EP-SSN) formulation and it has been proven to be very accurate for a wide range of problems, incl uding fixed source and criticality calculations. Therefore, I devel oped a new 3-D code, named PENSSn (Parallel Environment Neutral-particle Simplifi ed Sn) which is based on the EP-SSN equations. The code is designed for pa rallel computing environments and it solves the EP-SSN equations with anisotropic scattering of arb itrary order, for fixed source and criticality problems. The code has been benchmarke d using realistic 2-D and 3-D problems, including a small LWR and an FBR model, and the MOX 2-D/3-D Fuel Assembly

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172 Benchmark problem. The code yiel ds very accurate results within the limitations of the method. In summary, the main limiting factors of the EP-SSN methodology are the following: Optically thin media. Low density or void-like regions. Strong spatial/angular flux variations. Systems characterized by highly peaked anisotropic scattering. The parallel capabilities of PENSSn ha ve been tested on the PCPENII cluster (Nuclear and Radiological Engineering) and the Zeta cluster (Ele ctrical and Computer Engineering High Performance Computing Lab) at the University of Florida; the code present a parallel fraction of ~87% and the parallel performance achieved follows the predictions of Amdahl’s Law. To speed-up the convergence of the SN method, I have developed a new synthetic acceleration method based on the EP-SSN equations; however, I found limited applicability for this method, due to instabili ties which appear for mesh sizes greater than 1 mfp and for highly heterogeneous problems. Further, I have developed a new preconditioning algorithm based on the EP-SSN equations, for the acceleration of the SN method. The Flux Acceleration Simplified Transport (FAST) preconditioner is a fully automated system based on the kernel of the PENSSn code. FAST is currently implemented in the PENTRAN-SSn transport code. This approach is very effective for acceler ating large radiation transport problems in parallel computing environments, such as the MOX 3-D Fuel Assembly Benchmark problem. For this problem, the FAST preconditioner has reduced the total computation time by a factor ranging between ~2.6 and ~5.1 compared to a standard non-accelerated transport calculation.

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173 In conclusion this research work has culminated in the development of new methodologies that enhance the accuracy and feas ibility of transport calculations for large realistic models. The new quadrature sets developed will improve the accuracy of dose calculations for medical physics applications. Due to its accuracy and limited executi on times, the PENSSn code is an ideal candidate for core physics, core design a nd certain shielding applications. A high performance has been obtained using Krylov subspace iterative solvers, which in the future may become a standard method fo r solving radiation transport problems. Moreover, a new formulation of the EP-SSN equation has been developed which proved to be a key aspect for the preconditioning/ acceleration algorithm designed for the SN method. Finally, the new FAST preconditioner, integrated into the PENTRAN-SSn Code System, represents a leap forward in comput ational physics; large 3D radiation transport calculations for core or shielding design can now be performed within a fraction of the computation time required in the past. The methods described in this dissertati on can be further enhanced and developed by studying the following issues: The calculation of the point-weights for the PN-TN quadrature set could be improved by solving the linear system of equations obtainable from the evenand odd-moment conditions of the direction cosines. A selection method for the biasing re gion in the RAR technique should be developed based on the physics of the problem. An automatic load-balancing algorithm for the Krylov solv ers should be developed, following the ideas described in Chapter 8. This new algorit hm may significantly improve the parallel performance of the angular domain decomposition strategy. Memory usage optimization and fine -tuning of the domain decomposition algorithms in the PENSSn code. Extension of the PENSSn code w ith time-dependent capabilities. The PENSSn code will be reviewed for QA.

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174 The new synthetic acceleration method based on the EP-SSN equations could be investigated further; a consistent discretization between the EP-SSN and SN discretized operators may yield a stable algorithm for a wider range of problems. An optimization study on the FAST preconditioner should be undertaken, in order to identify the necessary level of accuracy which yields the best performance in terms of speed-up.

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175 APPENDIX A EXPANSION OF THE SCATTERING TERM IN SPHERICAL HARMONICS The angular dependency of the scatteri ng cross section can be approximated by expanding the function using a complete ba sis of polynomial functions: spherical harmonics. L l l g g sl g g sP r l r0 0 0 ,) ( ) 1 2 ( , (A.1) where 1 1 0 0 0 ,) ( 2 P r d rg g s g g sl (A.2) The scattering cross section is assumed to be dependent only on the cosine of the scattering angle, i.e. '0, where and are the directions of the particle before and after the scattering process. Note that this assumption implies the fact that the probability of scattering into the direction and energy group g’ does not depend on the initial direction of the particle. For very low energy ranges, i.e. “cold neutrons”, this assumption is only approximate. In Eqs. A.1 and A.2 the angular variable in normalized on the unit sphere as follows 1 2 21 1 2 0 d d d. (A.3)

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176 The direction vector is defined by the polar ( 0) and azimuthal ( 2 0 ) angles. Hence the direction co sine between the directions and is given by ) cos( ) 1 ( ) 1 ( '2 / 1 2 2 / 1 2 0 (A.4) The Legendre addition theorem states the following in terms of orthogonal spherical surface harmonics l k k l k l lY Y l P1 0) ( ) ' ( ) 1 2 ( 1 (A.5) The spherical harmonics functions ( Y ) are defined in terms of the Associated Legendre polynomials ) exp( ) ( )! ( )! ( ) 1 2 ( ) (, ik P k l k l l Yk l k l (A.6) ) ( ) 1 ( ) (* , k l k k lY Y (A.7) By using Eqs. A.5, A.6 and A.7, the Lege ndre polynomials are rewritten as follows l k k l k l l l lk P P k l k l P P P1 0) ( cos ) ( ) ( )! ( )! ( 2 (A.8) So that the complete scattering kernel expa nded in terms of spherical harmonics becomes )]}. sin( ) , ( ) cos( ) , ( [ ) ( )! ( )! ( 2 ) , ( ) ( ){ , ( ) 1 2 (' 1 1 '0 ' k z y x k z y x P k l k l z y x P z y x lk g Sl k g Cl l k k l G g L l g l l g g sl (A.9)

PAGE 194

177 APPENDIX B PERFORMANCE OF THE NEW EP-SSN FORMULATION The new formulation of the EP-SSN equations discussed in Chapter 4 is tested for the small FBR problem described in Chapter 6. The new formulation de rived is useful to accelerate the solution of the EP-SSN equations via the source iteration method. The small FBR model is simulated with the new formulation EP-SSN method and its performance is compared to the standard EP-SSN formulation. Table B-1 shows the number of iterations required to converge and the relative computation time for different EP-SSN methods derived with th e standard formulation. Table B-1. Performance data for the standard EP-SSN formulation. Method Krylov iterations Inner iterations Outer iterations Computation time (sec) EP-SS2 375882 1237 91 48.3 EP-SS4 635326 1038 74 79.6 EP-SS6 884580 953 67 110.1 EP-SS8 1137518 907 63 144.7 Table B-2 shows the number of ite rations required by the new EP-SSN formulation for solving the small FBR benchmark problem. Table B-2. Performance data for the new EP-SSN formulation. Method Krylov iterations Inner iterations Outer iterations Computation time (sec) EP-SS2 194071 343 17 23.2 EP-SS4 385411 517 31 47.8 EP-SS6 614960 589 37 75.9 EP-SS8 865288 636 41 107.6 The new EP-SS4 formulation reduces by more th an 50% the number of inner iterations and computation time, compared to th e standard formulation. This behavior is due to the reduction in terms of spect ral radius achieved by the new EP-SSN formulation.

PAGE 195

178 However, the new formulation increases the spectral condition number of the matrix operators; this behavior is detected in the ratio between Kryl ov iterations and inner iterations, shown in Table B-3. The new EP-SSN formulation presents higher values for this ratio, meaning that the matrix operators are characterized by larger spectral condition numbers. Table B-3. Ratio between Krylov iterations and inner iterations. Method Standard EP-SSN Modified EP-SSN EP-SS2 304 566 EP-SS4 612 745 EP-SS6 928 1044 EP-SS8 1254 1361 I have observed also a degradation of the performance of the new EP-SSN formulation for higher SSN orders. Table B-4 compares the inner iterations ratio and time ratio between the standard and new EP-SSN formulations; note that the speed-up decreases for increasing SSN orders. Table B-4. Inner iterations and time ratios for different SSN orders. Method Inner iterations ratio Time ratio EP-SS2 3.6 2.1 EP-SS4 2.0 1.7 EP-SS6 1.6 1.5 EP-SS8 1.4 1.3 The speed-up degradation can be expl ained by observing that the direction dependent removal cross section in Eq. B.1 de pends on the weights of the quadrature set. For high order quadrature sets, th e value of the weight is de creased accordingly. Due to this aspect, the removal cross section is le ss affected by the scattering term as the quadrature set order increases, therefore lead ing to a degradation of the method. This

PAGE 196

179 argument explains also the be havior observed for the trans port equation, verified using the SN formulation, where no signif icant benefits are observed. 1 .. 2 0 2 ,) ( ) ( 1 2 ) ( ) (L even l g g sl m m l g t R g mr w P l r r (B.1) Moreover, note that in Table B-4 the re duction in terms of inner iterations does not match necessarily the reduction in computati on time; clearly, this is due to the larger number of Krylov iterations required by the new EP-SSN formulation compared to the standard formulation.

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180 LIST OF REFERENCES 1. Bell G.I. and Glasstone S., Nuclear Reactor Theory Robert E. Krieger Publishing CO. Inc., Malabar, FL, USA, 1985. 2. Carlson B.G., Transport Theory: Discrete Ordi nates Quadrature over the Unit Sphere Los Alamos Scientific Laboratory Report, LA-4554, 1970. 3. Lewis E.E. and Miller W.F. Jr., Neutron Transport American Nuclear Society, La Grange Park, IL, 1993. 4. Carlson B.G. and Lathrop K.D., Discrete Ordinates Angul ar Quadrature of the Neutron Transport Equation Los Alamos Scientific Laboratory Report, LA-3186, 1965. 5. Lathrop K.D., “Remedies for Ray Effect s,” Nuclear Science and Engineering, Vol. 45, pp. 255-268, 1971. 6. Fletcher J. K., “The Solution of the Multigroup Neutron Transport Equation Using Spherical Harmonics,” Nuclear Science and Engineering, Vol. 84, pp. 33-46, 1983. 7. Carlson B.G., Tables of Equal Weight Quadrature EQN Over the Unit Sphere Los Alamos Scientific Laboratory Report, LA-4734, 1971. 8. Carew J.F. and Zamonsky G., “Uniform Positive-Weight Quadratures for Discrete Ordinate Transport Calculations,” Nuclear Science and Engineering, Vol. 131, pp.199-207, 1999. 9. Brown J.F. and Haghighat A., “A PEN TRAN Model for a Medical Computed Tomography (CT) Device,” Proceedings of Radiation Protection for our National Priorities (RPSD 2000), Spokane, Washington, September 17-21, 2000, on CDROM, American Nuclear Society, Inc., Lagrange Park, IL, 2000. 10. Sjoden G. E. and Haghighat A., “PENTRAN – Parallel Environment Neutralparticle TRANsport in 3-D Cartesian Geometry,” Proceedings of the Joint International Conference on Mathem atical Methods and Supercomputing for Nuclear Applications Vol. 1, pp. 232-234, Saratoga Springs, NY, October 6-10, 1997. 11. Longoni G. et al., “Investigation of New Qu adrature Sets for Discrete Ordinates Method with Application to Non-conventional Problems,” Trans. Am. Nucl. Soc. Vol. 84, pp. 224-226, 2001.

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181 12. Longoni G. and Haghighat A., “Development of New Quadrature Sets with the Ordinate Splitting Technique,” Proceedings of the ANS International Meeting on Mathematical Methods for Nu clear Applications (M&C 2001) Salt Lake City, UT, September 9-13, 2001, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2001. 13. Longoni G. and Haghighat A., “Simulation of a CT-Scan Device with PENTRAN Using the New Regional Angular Refinement Technique,” Proceedings of the 12th Biennial RPSD Topical Meeting of the Radiation Protection and Shielding Division of the American Nuclear Society Santa Fe, NM, April 14-18, 2002, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2002. 14. Longoni G. and Haghighat A., “Development of the Regional Angular Refinement and Its Application to the CT-Scan Device,” Trans. Am. Nucl. Soc. Vol. 86, pp. 246-248, 2002. 15. Longoni G. and Haghighat A., “Developmen t and Application of the Regional Angular Refinement Technique and it s Application to Non-conventional Problems,” Proceedings of PHYSOR 2002 ANS T opical Meeting International Conference on the New Frontiers of Nuclea r Technology: Reactor Physics, Safety and High-Performance Computing, Seoul, Korea, October 7-10, 2002, on CDROM, American Nuclear Society, Inc., Lagrange Park, IL, 2002. 16. Kucukboyaci V. et al., “PENTRAN Modeli ng for Design and Optimization of the Spherical-Shell Transmission Experiments,” Trans. Am. Nucl. Soc. Vol. 84, pp. 156-159, 2001. 17. Adams M. L. and Larsen E. W., “Fast It erative Methods for Discrete-Ordinates Particle Transport Calculations ,” Progress in Nuclear Energy, Vol. 40, n. 1, 2002. 18. Gelbard E., Davis J., and Pearson J. “Iterative Solutions to the Pl and Double-Pl Equations,” Nuclear Scie nce and Engineering, Vol. 5, pp. 36-44, 1959. 19. Ferziger J. H. and Milovan P., Computational Methods for Fluid Dynamics Second Edition Springer-Verlag, Berlin Heidelberg, Germany, 1999. 20. Golub G. and Ortega J.M., Scientific Computing An Introduction with Parallel Computing Academic Press, San Diego, CA, 1993. 21. Lewis E. E. and Palmiotti G., “Simplified Spherical Harmonics in the Variational Nodal Method,” Nuclear Science and Engineering, Vol. 126, pp. 48-58, 1997. 22. Brantley P.S. and Larsen E.W., “The Simplified P3 Approximation,” Nuclear Science and Engineering, Vol. 134, pp. 1-21, 2000.

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182 23. Longoni G. and Haghighat A., “The Even-Parity Simplified SN Equations Applied to a MOX Fuel Assembly Benchmark Problem on Distributed Memory Environments,” PHYSOR 2004 – The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments Chicago, IL, April 25-29, 2004, on CDROM, American Nuclear Society, Inc., Lagrange Park, IL, 2004. 24. Gamino R.G., “Three-Dimensional Noda l Transport Using the Simplified PL Method,” Proceedings of the International Topical Meeting Advances in Mathematics, Computati ons, and Reactor Physics, Pittsburgh, PA, April 28-May 2, 1991, on CD-ROM, American Nuclear Society, Inc., Lagrange Park, IL, 1991. 25. Longoni G., Haghighat A., and Sjoden G., “Development and Application of the Multigroup Simplified P3 (SP3) Equations in a Distribu ted Memory Environment,” Proceedings of PHYSOR 2002 ANS Topical Meeting International Conference on the New Frontiers of Nuclear Technol ogy: Reactor Physics, Safety and HighPerformance Computing Seoul, Korea, October 7-10, 2002, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2002. 26. Longoni G. and Haghighat A., “Developm ent and Applications of the SPL Methodology for a Criticality Eige nvalue Benchmark Problem,” Proceedings of the ANS Topical Meeting on Nuclear Mathema tical and Computational Sciences: A Century In Review – A Ce ntury Anew (M&C 2003) Gatlinburg, TN, April 6-11, 2003, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2003. 27. Gropp W., Lusk E., and Skjellum A., Using MPI Portable Parallel Programming with the Message Passing Interface The MIT Press, Cambridge, Massachussetts, 1999. 28. Gelbard E. M. and Hageman L. A., “The Synthetic Method as Applied to the SN Equations,” Nuclear Scie nce and Engineering, Vol. 37, pp. 288-298, 1969. 29. Alcouffe R. E., “Diffusion Syntheti c Acceleration Methods for the DiamondDifferenced Discrete-Ordinates Equations ,” Nuclear Science and Engineering, Vol. 64, pp. 344-355, 1977. 30. Chang J. and Adams M., “Analysis of Tr ansport Synthetic Acceleration for Highly Heterogeneous Problems,” Proceedings of the ANS Topical Meeting on Nuclear Mathematical and Computational Sciences : A Century In Review – A Century Anew (M&C 2003) Gatlinburg, TN, April 6-11, 2003, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2003. 31. Warsa J. S., Wareing T. A. and Morel J. E., “On the Degraded Effectiveness of Diffusion Synthetic Acceleration for Multidimensional SN Calculations in the Presence of Material Discontinuities,” Proceedings of the ANS Topical Meeting on Nuclear Mathematical and Computational Sciences: A Century In Review – A Century Anew (M&C 2003) Gatlinburg, TN, April 6-11, 2003, on CD-ROM, American Nuclear Society, Inc., La Grange Park, IL, 2003.

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183 32. Warsa J. S., Wareing T. A. and Morel J. E., “Krylov Iterat ive Methods and the Degraded Effectiveness of Diffusion S ynthetic Acceleration for Multidimensional SN Calculations in Problems with Materi al Discontinuities,” Nuclear Science and Engineering, Vol. 147, pp. 218-248, 2004. 33. Longoni G. and Haghighat A., “A New Synt hetic Acceleration Technique based on the Simplified Even-Parity SN Equations,” accepted for publication on Transport Theory and Statistical Physics, 2004. 34. Sjoden G. and Haghighat A., “The Expone ntial Directional Weighted (EDW) SN Differencing Scheme in 3D Cartesian Geometry,” Proceedings of the Joint International Conference on Mathem atical Methods and Supercomputing in Nuclear Applications Vol. II, pp. 1267-1276, Saratoga Springs, NY, October 6-10, 1997. 35. Lathrop K., “Spatial Differencing of the Transport Equation: Positivity vs. Accuracy,” Journal of Computational Physics, Vol. 4, pp. 475-498, 1969. 36. Petrovic B. and Haghighat A., “Analysi s of Inherent Oscillations in Multidimensional SN Solutions of the Neutron Transport Equation,” Nuclear Science and Engineering, Vol. 124, pp. 31-62, 1996. 37. Sjoden G. E., “PENTRAN: A Parallel 3-D SN Transport Code With Complete Phase Space Decomposition, Adaptive Differencing, and Iterative Solution Methods,” Ph.D. Thesis in Nuclear E ngineering, Penn State University, 1997. 38. Nakamura S., Computational Methods in Engineering and Science Wiley, New York, 1977. 39. Kucukboyaci V.N. and Haghighat A., “Angular Multigrid Acceleration for Parallel Sn Method with Application to Shielding Problems,” Proceedings of PHYSOR 2000 ANS International T opical Meeting on Advances in Reactor Physics and Mathematics and Computation into the Next Millennium Pittsburgh, PA, May 712, 2000, on CD-ROM, American Nuclear Soci ety, Inc., La Grange Park, IL, 2000. 40. Reed W. H., “The Effectiv eness of Acceleration Techni ques for Iterative Methods in Transport Theory,” Nuclear Science and Engineering, Vol. 45, pp. 245-254, 1971. 41. Larsen E. W., “Unconditionally Stable Diffusion-Synthetic A cceleration Methods for the Slab Geometry Discrete Ordinate s Equations. Part I: Theory,” Nuclear Science and Engineering, Vol. 82, pp. 47-63 1982. 42. Kobayashi K., Sugimura N. and Nagaya Y., 3-D Radiation Transport Benchmark Problems and Results for Simple Ge ometries with Void Regions OECD/NEA report, ISBN 92-64-18274-8, Issy-les-M olineaux, France, November 2000.

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184 43. Takeda T. and Ikeda H., 3-D Neutron Transport Benchmarks NEACRP-L-330 OECD/NEA, Osaka University, Japan, March 1991. 44. Benchmark on Deterministic Transpo rt Calculations Without Spatial Homogenization OECD/NEA report, ISBN 9264-02139-6, Issy-les-Molineaux, France, 2003. 45. Haghighat A., Manual of PENMSH Version 5 – A Cartesian-based 3-D Mesh Generator University of Florida, Florida, June, 2004. 46. Cavarec C., The OECD/NEA Benchmark Calcula tions of Power Distributions within Assemblies Electricit de France, France, September 1994. 47. Benchmark Specification for Deterministic MOX Fuel Assembly Transport Calculations Without Sp atial Homogenisation (3-D Extension C5G7 MOX) OECD/NEA report, Issy-les-Mo lineaux, France, April, 2003. 48. Sjoden G. and Haghighat A., PENTRAN: Parallel Environment Neutral-particle TRANsport SN in 3-D Cartesian Geometry – Users Guide to Version 9.30c University of Florida, Florida, May 2004. 49. Marleau G., A. Hbert, and R. Roy, A User’s Guide for DRAGON Ecole Polytechnique de Montral, Canada, December 1997.

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185 BIOGRAPHICAL SKETCH Gianluca Longoni was born in Torino, Italy, on 31st of October, 1975; he is the son of Giancarlo and Annalisa, and has a brother, Daniele, who is an excellent student and prospective aerospace engineer In 1994, Gianluca enrolled in the nuclear engineering program at Politecnico di Torino, located in Torino. He obtained the degree Laurea in Ingegneria Nucleare in March 2000; he pe rformed his research work under the supervision of Piero Ravetto, and he developed a new 2-D ra diation transport code based on the characteristics method in hexagonal geometry, for Accelerator Driven Systems (ADS). He moved to the USA in August 2000 to pur sue his Ph.D. with Professor Alireza Haghighat at Penn State University, Pennsylva nia. In fall 2001 he moved to University of Florida in Gainesville, as Prof. Haghi ghat joined the Nuclear and Radiological Engineering Department; Gianluca continued hi s research work in Florida. Gianluca has presented his research work in a number of international conferences in the US, South Korea and Europe. Gianluca is fond of basketball, having been a player in the pro-league in Italy. He is now a senior student in different martia l art styles, including Iwama-Ryu aikido, Mudokwan taekwondo, and Iaido Japanese swordsmanship.


Permanent Link: http://ufdc.ufl.edu/UFE0007560/00001

Material Information

Title: Advanced quadrature sets, acceleration and preconditioning techniques for the discrete ordinates method in parallel computing environments
Physical Description: Mixed Material
Language: English
Creator: Longoni, Gianluca ( Dissertant )
Haghighat, Alireza ( Thesis advisor )
Gamino, Ray G. ( Reviewer )
Glover, Joseph ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2004
Copyright Date: 2004

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Nuclear and Radiological Engineering

Notes

Abstract: In the nuclear science and engineering field, radiation transport calculations play a key-role in the design and optimization of nuclear devices. The linear Boltzmann equation describes the angular, energy and spatial variations of the particle or radiation distribution. The discrete ordinates method (Sn) is the most widely used technique for solving the linear Boltzmann equation. However, for realistic problems, the memory and computing time require the use of supercomputers. This research is devoted to the development of new formulations for the Sn method, especially for highly angular dependent problems, in parallel environments. The present research work addresses two main issues affecting the accuracy and performance of Sn transport theory methods: quadrature sets and acceleration techniques. New advanced quadrature techniques which allow for large numbers of angles with a capability for local angular refinement have been developed. These techniques have been integrated into the 3-D Sn PENTRAN (Parallel Environment Neutral-particle TRANsport) code and applied to highly angular dependent problems, such as CT-Scan devices, that are widely used to obtain detailed 3-D images for industrial/medical applications. In addition, the accurate simulation of core physics and shielding problems with strong heterogeneities and transport effects requires the numerical solution of the transport equation. In general, the convergence rate of the solution methods for the transport equation is reduced for large problems with optically thick regions and scattering ratios approaching unity. To remedy this situation, new acceleration algorithms based on the Even-Parity Simplified Sn (EP-SSn) method have been developed. A new stand-alone code system, PENSSn (Parallel Environment Neutral-particle Simplified Sn), has been developed based on the EP-SSn method. The code is designed for parallel computing environments with spatial, angular and hybrid (spatial/angular) domain decomposition strategies. The accuracy and performance of PENSSn has been tested for both criticality eigenvalue and fixed source problems. PENSSn has been coupled as a preconditioner and accelerator for the Sn method using the PENTRAN code. This work has culminated in the development of the Flux Acceleration Simplified Transport (FASTcopyright) preconditioning algorithm, which constitutes a completely automated system for preconditioning radiation transport calculations in parallel computing environments.
Abstract: acceleration, conjugate, discrete, neutrons, nuclear, parallel, radiation, simplified, Sn, transport
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 202 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2004.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003165717
System ID: UFE0007560:00001

Permanent Link: http://ufdc.ufl.edu/UFE0007560/00001

Material Information

Title: Advanced quadrature sets, acceleration and preconditioning techniques for the discrete ordinates method in parallel computing environments
Physical Description: Mixed Material
Language: English
Creator: Longoni, Gianluca ( Dissertant )
Haghighat, Alireza ( Thesis advisor )
Gamino, Ray G. ( Reviewer )
Glover, Joseph ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2004
Copyright Date: 2004

Subjects

Subjects / Keywords: Nuclear and Radiological Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Nuclear and Radiological Engineering

Notes

Abstract: In the nuclear science and engineering field, radiation transport calculations play a key-role in the design and optimization of nuclear devices. The linear Boltzmann equation describes the angular, energy and spatial variations of the particle or radiation distribution. The discrete ordinates method (Sn) is the most widely used technique for solving the linear Boltzmann equation. However, for realistic problems, the memory and computing time require the use of supercomputers. This research is devoted to the development of new formulations for the Sn method, especially for highly angular dependent problems, in parallel environments. The present research work addresses two main issues affecting the accuracy and performance of Sn transport theory methods: quadrature sets and acceleration techniques. New advanced quadrature techniques which allow for large numbers of angles with a capability for local angular refinement have been developed. These techniques have been integrated into the 3-D Sn PENTRAN (Parallel Environment Neutral-particle TRANsport) code and applied to highly angular dependent problems, such as CT-Scan devices, that are widely used to obtain detailed 3-D images for industrial/medical applications. In addition, the accurate simulation of core physics and shielding problems with strong heterogeneities and transport effects requires the numerical solution of the transport equation. In general, the convergence rate of the solution methods for the transport equation is reduced for large problems with optically thick regions and scattering ratios approaching unity. To remedy this situation, new acceleration algorithms based on the Even-Parity Simplified Sn (EP-SSn) method have been developed. A new stand-alone code system, PENSSn (Parallel Environment Neutral-particle Simplified Sn), has been developed based on the EP-SSn method. The code is designed for parallel computing environments with spatial, angular and hybrid (spatial/angular) domain decomposition strategies. The accuracy and performance of PENSSn has been tested for both criticality eigenvalue and fixed source problems. PENSSn has been coupled as a preconditioner and accelerator for the Sn method using the PENTRAN code. This work has culminated in the development of the Flux Acceleration Simplified Transport (FASTcopyright) preconditioning algorithm, which constitutes a completely automated system for preconditioning radiation transport calculations in parallel computing environments.
Abstract: acceleration, conjugate, discrete, neutrons, nuclear, parallel, radiation, simplified, Sn, transport
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 202 pages.
General Note: Includes vita.
Thesis: Thesis (Ph.D.)--University of Florida, 2004.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003165717
System ID: UFE0007560:00001


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ADVANCED QUADRATURE SETS, ACCELERATION
AND PRECONDITIONING TECHNIQUES
FOR THE DISCRETE ORDINATES METHOD IN
PARALLEL COMPUTING ENVIRONMENTS













By

GIANLUCA LONGONI


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


2004

































Copyright 2004

by

GIANLUCA LONGONI




























I dedicate this research work to Rossana and I thank her for the support and affection
demonstrated to me during these years in college. This work is dedicated also to my
family, and especially to my father Giancarlo, who always shared my dreams and
encouraged me in pursuing them.





"Only he who can see the invisible, can do the impossible."
By Frank Gaines















ACKNOWLEDGMENTS

The accomplishments achieved in this research work would have not been possible

without the guidance of a mentor such as Prof. Alireza Haghighat; he has been my

inspiration for achieving what nobody else has done before in the radiation transport area.

I wish to thank Prof Glenn E. Sjoden for his endless help and moral support in my

formation as a scientist. I also express my gratitude to Dr. Alan D. George, for providing

me with the excellent computational facility at the High-Performance Computing and

Simulation Research Lab. I am thankful to Prof. Edward T. Dugan for his useful

suggestions and comments, as well as to Dr. Ray G. Gamino from Lockheed Martin -

KAPL and Dr. Joseph Glover, for being part of my Ph.D. committee. I also thank

UFTTG for the interesting conversations regarding radiation transport physics and

computer science.

I am also grateful to the U.S. DOE Nuclear Education Engineering Research

(NEER) program, the College of Engineering, and the Nuclear and Radiological

Engineering Department at the University of Florida, for supporting the development of

this research work.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ....................................................... ............ ....... ....... ix

LIST OF FIGURES ......... ........................................... ............ xi

ABSTRACT ........ .............. ............. ...... ...................... xvi

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 O v erv iew ............... ..................................................... ................ 1
1.2 The Linear B oltzm ann Equation................................ ....................................1
1.3 Advanced Angular Quadrature Sets for the Discrete Ordinates Method...........2
1.4 Advanced Acceleration Algorithms for the SN Method on Parallel Computing
E n v iro n m en ts .................................................................... .. 5
1.5 The Even-Parity Sim plified SN M ethod .............................................................6
1.6 A New Synthetic Acceleration Algorithm Based on the EP-SSN Method ......10
1.7 An Automatic Preconditioning Algorithm for the SN Method: FAST (Flux
Acceleration Simplified Transport) ...................................... ............... 12
1.8 Outline ............... ...... ... .......... ...................................13

2 THE DISCRETE ORDINATES METHOD..................................................14

2.1 D iscrete Ordinates M ethod (SN) ........................................... ............... 14
2.1.1 Discretization of the Energy Variable............... .... ...............14
2.1.2 Discretization of the Angular Variable............................................ 17
2.1.3 Discretization of the Spatial Variable............. ...................20
2.1.4 D ifferencing Schem es ...................................... ....................... ......... 21
2.1.4.1 Linear-Diamond Scheme (LD).......................... ..............22
2.1.4.2 Directional Theta-Weighted Scheme (DTW) ............................23
2.1.4.3 Exponential Directional-Weighted Scheme (EDW) ...................24
2.1.5 The Flux M om ents ................ ........... .....................................25
2.1.6 B oundary C conditions ........................................ ......................... 25
2.2 Source Iteration M ethod ............................................................................ 25
2.3 Pow er Iteration M ethod .................................... .... ........................ 26
2.4 Acceleration Algorithms for the SN M ethod.............................................. 27









3 ADVANCED QUADRATURE SETS FOR THE SN METHOD ...........................29

3.1 Legendre Equal-Weight (PN-EW) Quadrature Set .......................................30
3.2 Legendre-Chebyshev (PN-TN) Quadrature Set..............................................31
3.3 The Regional Angular Refinement (RAR) Technique .................................33
3.4 Analysis of the Accuracy of the PN-EW and PN-TN Quadrature Sets..............34
3.5 Testing the Effectiveness of the New Quadrature Sets.................................38
3.5.1 K obayashi Benchm ark Problem 3 ..................................... ................... 38
3.5.2 CT-Scan Device for Medical/Industrial Imaging Applications.............43

4 DERIVATION OF THE EVEN-PARITY SIMPLIFIED SN EQUATIONS ............47

4.1 Derivation of the Simplified Spherical Harmonics (SPN) Equations..............48
4.2 Derivation of the Even-Parity Simplified SN (EP-SSN) Equations..................51
4.2.1 Boundary Conditions for the EP-SSN Equations ..................................55
4.2.2 Fourier Analysis of the EP-SSN Equations .......................................... 56
4.2.3 A New Formulation of the EP-SSN Equations for Improving the
Convergence Rate of the Source Iteration Method..............................59
4.3 Comparison of the Pi Spherical Harmonics and SP1 Equations...................60

5 NUMERICAL METHODS FOR SOLVING THE EP-SSN EQUATIONS ...............65

5.1 Discretization of the EP-SSN Equations Using the Finite-Volume Method ....65
5.2 Numerical Treatment of the Boundary Conditions........................................72
5.3 The Compressed Diagonal Storage Method ......................................... 74
5.4 Coarse M esh Interface Projection Algorithm ............. .................................. 75
5.5 Krylov Subspace Iterative Solvers..................... ............ ...... ............. 80
5.5.1 The Conjugate Gradient (CG) M ethod ..................................................82
5.5.2 The Bi-Conjugate Gradient M ethod ............... ................... ............83
5.5.3 Preconditioners for Krylov Subspace M ethods ......................................84

6 DEVELOPMENT AND BENCHMARKING OF THE PENSSn CODE..................86

6.1 Development of the PENSSn (Parallel Environment Neutral-particle
Sim plified Sn) Code..................... ........ ..................................... 87
6.2 Numerical Analysis of Krylov Subspace Methods........................................92
6.2.1 Coarse M esh Partitioning of the M odel................................................92
6.2.2 B oundary C conditions ........................................ ........................ 95
6.2.3 M material H eterogeneities....................... .................. .....................96
6.2.4 Convergence Behavior of Higher EP-SSN Order Methods...................97
6.3 Testing the Incomplete Cholesky Conjugate Gradient (ICCG) Algorithm .....99
6.4 Testing the Accuracy of the EP-SSN M ethod .............. ....... ....................100
6 .4 .1 S catterin g R atio .......................................................... ..................... 10 0
6.4.2 Spatial Truncation Error ............................................ ............... 103
6.4.3 Low D ensity M aterials...................................... ........................ 104
6.4.4 M material Discontinuities................... ..... ........................ 108
6.4.5 Anisotropic Scattering .............. ... ............. ............ ............... 111









6.4.6 Small Light Water Reactor (LWR) Criticality Benchmark Problem.... 115
6.4.7 Small Fast Breeder Reactor (FBR) Criticality Benchmark Problem.... 120
6.4.8 The MOX 2-D Fuel Assembly Benchmark Problem.........................124

7 PARALLEL ALGORITHMS FOR SOLVING THE EP-SSN EQUATIONS ON
DISTRIBUTED MEMORY ARCHITECTURES ............................................. 128

7.1 Parallel Algorithm s for the PEN SSn Code...................................................128
7.2 D om ain D ecom position Strategies ..................................... ............... ..130
7.2.1 Angular D om ain D ecom position........................................................ 130
7.2.2 Spatial D om ain D ecom position............. ...................... ... ...............131
7.2.3 Hybrid Domain Decomposition ............. .................. ....................131
7.3 Parallel Performance of the PENSSn Code .................................. .............. 132
7.4 Parallel Performance of PENSSn Applied to the MOX 2-D Fuel Assembly
B enchm ark Problem ........................................................... ............... 139

8 DEVELOPMENT OF A NEW SYNTHETIC ACCELERATION METHOD BASED
ON THE EP-SSN EQUATION S ........................................ ......................... 140

8.1 The EP-SSN Synthetic Acceleration M ethod ................................................141
8.2 Spectral Analysis of the EP-SSN Synthetic Acceleration Method...............145
8.3 Analysis of the Algorithm Stability Based on Spatial Mesh Size ...............148
8.3.1 Comparison of the EP-SSN Synthetic Acceleration with the Simplified
A ngular M ultigrid M ethod................................................ ............... 150
8.4 Limitations of the EP-SSN Synthetic Acceleration Method ..........................153

9 FAST: FLUX ACCELERATION SIMPLIFIED TRANSPORT
PRECONDITIONER BASED ON THE EP-SSN METHOD .............................154

9.1 Development and Implementation of FAST .............................................154
9.2 Testing the Performance of the FASTc Preconditioning Algorithm........... 157
9.2.1 Criticality Eigenvalue Problem ................................. ............... 157
9.2.2 Fixed Source Problem ....................................................................... 159
9.3 The MOX 3-D Fuel Assembly Benchmark Problem................................161
9.3.1 MOX 3-D Unrodded Configuration............................... ............... 162
9.3.2 MOX 3-D Rodded-A Configuration................... .................165
9.3.3 M OX 3-D Rodded-B Configuration....... ........................................ 167

10 SUMMARY, CONCLUSION, AND FUTURE WORK ........................................171

APPENDIX

A EXPANSION OF THE SCATTERING TERM IN SPHERICAL HARMONICS..175

B PERFORMANCE OF THE NEW EP-SSN FORMULATION ...............................177

L IST O F R E F E R E N C E S ......... .. ............... ................. .............................................. 180









BIOGRAPH ICAL SKETCH .............................................................. ............... 185
















LIST OF TABLES


Table pge

3-1. Even-moments obtained with a PN-EW S30 quadrature set. .....................................35

3-2. Even-moments obtained with a PN-TN S30 quadrature set .......................................36

3-3. CPU time and total number of directions required for the CT-Scan simulation........45

6-1. Comparison of number of iterations required to converge for the CG and Bi-CG
algorithm s ......... ... ......... .. ............................... ........ ..... .. ............ 93

6-2. Number of Krylov iterations required to converge for the CG and Bi-CG algorithms
with different boundary conditions. .............................................. ............... 95

6-3. Number of Krylov iterations required to converge for the CG and Bi-CG algorithms
for heterogeneous the box in a box problem. ........................... ................................ 96

6-4. Number of Krylov iterations required to converge for the CG and Bi-CG algorithms
for the EP-SSs equations. ............................................... .............................. 97

6-5. Number of iterations for the ICCG and CG algorithms. ........................ ..........99

6-6. Two groups cross-sections and fission spectrum. ..................................................106

6-7. Comparison of keff obtained with the EP-SSN method using DFM versus PENTRAN*
S6 (Note that DFM=1.0 implies no cross-sections scaling). ................................106

6-8. Balance tables for the EP-SSN and S16 methods and relative differences versus the
S 16 so lu tio n ........................................... ................... ....... ........ 1 1 1

6-9. Integral boundary leakage for the EP-SSN and S16 methods and relative differences
versus the S16 solution. .................. .......... .. ........ .. ............ .............. 111

6-10. Fixed source energy spectrum and energy range .................................................112

6-11. Maximum and minimum relative differences versus the S8 method for energy
group 1 and 2....................................................................... ...... ....... 113

6-12. Two-group cross-sections for the small LWR problem .............. ... .................116

6-12. Two-group cross-sections for the small LWR problem (Continued).....................116









6-13. Fission spectrum and energy range for the small LWR problem...........................117

6-14. Criticality eigenvalues calculated with different EP-SSN orders and relative error
compared to M onte Carlo predictions ....................................... ...... ...........117

6-15. CRWs estimated with the EP-SSN method ......................................................118

6-16. Criticality eigenvalues for the small FBR model. .............................................122

6-17. CRWs estimated with the EP-SSN and Monte Carlo methods............................122

6-18. Criticality eigenvalues and relative errors for the MOX 2-D benchmark problem.125

7-1. Data relative to the load imbalance generated by the Krylov solver........................136

7-2. Parallel performance data obtained on PCPENII Cluster............... ... .................138

7-3. Parallel performance data obtained on Kappa Cluster. ...........................................138

7-4. Parallel performance data for the 2-D MOX Fuel Assembly Benchmark problem
(P C PE N II C luster)............ .......................................................... ................... 139

8-1. Spectral radius for the different iterative methods. .............................................147

8-2. Comparison of the number of inner iteration between EP-SSN synthetic methods and
unaccelerated transport ........................................................................... ..... .... 149

9-1. Criticality eigenvalues obtained with the preconditioned PENTRAN-SSn code for
different E P -SSN orders................................................. ............................. 159

9-2. Results obtained for the MOX 3-D in the Unrodded configuration.........................162

9-3. Results obtained for the MOX 3-D Rodded-A configuration. ...............................165

9-4. Results obtained for the MOX 3-D Rodded-B configuration................................167

B-1. Performance data for the standard EP-SSN formulation.......................................177

B-2. Performance data for the new EP-SSN formulation..........................................177

B-3. Ratio between Krylov iterations and inner iterations. ...........................................178

B-4. Inner iterations and time ratios for different SSN orders............... .... ..............178
















LIST OF FIGURES


Figure pge

2-1. Cartesian space-angle coordinates system in 3-D geometry. ....................................16

2-2. Point weight arrangement for a S8 level-symmetric quadrature set .........................19

2-3. S20 LQN quadrature set. ......................................................................20

3-1. S28 PN-EW quadrature set. ............................................... .. .. ................................ 31

3-2. S28 PN-TN quadrature set ................................................... .............................. 33

3-3. PN-TN quadrature set (S16) refined with the RAR technique. ....................................34

3-4. Configuration of the test problem for the validation of the quadrature sets ..............37

3-5. Relative difference between the currents Jx and Jz for the test problem.....................37

3.6. Mesh distribution for the Kobayashi benchmark problem 3: A) Variable mesh
distribution; B) Uniform mesh distribution.............. ............................................39

3-7. Comparison of S20 quadrature sets in zone 1 at x=5.0 cm and z=5.0 cm. .................40

3-8. Comparison of S20 quadrature sets in zone 2 aty=55.0 cm and z=5.0 cm. ................40

3-9. Comparison of PN-EW quadrature sets for different SN orders in zone 1 at x=5.0 cm
an d z= 5 .0 cm ...................................................... ................. 4 1

3-10. Comparison of PN-EW quadrature sets for different SN orders in zone 2 aty=55.0
cm and z=5.0 cm ............. .... .......................... ...... .. ........... 41

3-11. Comparison of PN-TN quadrature sets for different SN orders in zone 1 aty=5.0 cm
and z= 5.0 cm ...................................................... ................. 42

3-12. Comparison of PN-TN quadrature sets for different SN orders in zone 2 aty=55.0 cm
and z= 5.0 cm ...................................................... ................. 42

3-13. Cross-sectional view of the CT-Scan model on the x-y plane.............................43

3-14. Scalar flux distribution on the x-y plane obtained with an S20 level-symmetric
quadrature set. ........................................................................44









3-15. Scalar flux distribution on the x-y plane obtained with an S50 PN-TN quadrature set.44

3-16. Scalar flux distribution on the x-y plane obtained with an S30 PN-TN quadrature set
biased with RAR. ............................. ..... ... ....... ....... .............. .. 45

3.17. Comparison of the scalar flux at detector position (x=72.0 cm).............................46

5.1. Fine mesh representation on a 3-D Cartesian grid ................ ............. ...............67

5.2. View of a fine mesh along the x-axis ........ .. .... ............................. .... ........... 68

5.3. Representation of a coarse mesh interface................................. ......... ............. 76

5.4. Representation of the interface projection algorithm between two coarse meshes. ...79

6-1. D description of PEN SSn input file .............. ................................... ............... 88

6-2. Flow-chart of the PEN SSn code ...................................................... ..................90

6-2. Flow-chart of the PENSSn code (Continued)..........................................................91

6-3. Configuration of the 3-D test problem ............................................. ............... 92

6-4. Convergence behavior of the CG algorithm for the non-partitioned model ..............94

6-5. Heterogeneous configuration for the 3-D test problem...................... ...............96

6-6. Distribution of eigenvalues for the EP-SSs equations.........................................98

6-7. Configuration of the 2-D criticality eigenvalue problem. ........................................101

6-8. Criticality eigenvalues as a function of the scattering ratio (c) for different methods. 101

6-9. Relative difference for criticality eigenvalues obtained with different EP-SSN
methods compared to the S16 solution (PENTRAN code). ...................................102

6-10. Plot of criticality eigenvalues for different mesh sizes........................................103

6-11. Plot of the relative difference of the EP-SSN solutions versus transport S16 for
different m esh sizes .................. ............................. .. .... ..... ............ 104

6-12. Uranium assembly test problem view on the x-y plane ........................................105

6-13. Relative difference of physical quantities of interest calculated with EP-SSN method
compared to the S6 PENTRAN solution. .................................... .................107

6-14. Convergence behavior of the PENSSn with DFM=100.0 and PENTRAN S6 ...... 108

6-15. Geometric and material configuration for the 2-D test problem ...........................109









6-16. Scalar flux distribution at material interface (y=4.84 cm)...................................109

6-17. Relative difference versus S16 calculations at material interface (y=4.84 cm).......110

6-18. Fraction of scalar flux values within different ranges of relative difference (R.D.) in
en erg y g rou p 1 ................................................................... 1 12

6-19. Fraction of scalar flux values within different ranges of relative difference (R.D.) in
en ergy g rou p 2 ....................................................................................... 1 13

6-20. Front view of the relative difference between the scalar fluxes obtained with the
EP-SSs and S8 methods in energy group 1................... ........................... 114

6-21. Rear view of the relative difference between the scalar fluxes obtained with the EP-
SS8 and S8 m ethods in energy group 1....................................... ...............115

6-22. Model view on the x-y plane. A) view of the model from z=0.0 cm to 15.0 cm, B)
view of the model from z=15.0 cm to z=25.0 cm .................................................. 115

6-23. M odel view on the x-z plane .................................................... .................. 116

6-24. Normalized scalar flux for case 1, in group 1 along the x-axis at y=2.5 cm and z=7.5
c m ................... ........................................................ ................ 1 1 8

6-25. Scalar flux distributions. A) Case 1 energy group 1, B) Case 2 energy group 1, C)
Case 1 energy group 2, D) Case 2 energy group 2 ..............................................119

6-26. View on the x-y plane of the small FBR model................... ...................................120

6-27. View on the x-z plane of the small FBR model................... ...................................121

6-28. Scalar flux distribution in energy group 1: A) Case 1; B) Case 2..........................123

6-29. Scalar flux distribution in energy group 4: A) Case 1; B) Case 2..........................123

6-30. Mesh distribution of the MOX 2-D Fuel Assembly Benchmark problem. ............124

6-31. Scalar flux distribution for the 2-D MOX Fuel Assembly benchmark problem (EP-
SS4): A) Energy group 1; B) Energy group 2; C) Energy group 3; D) Energy group
4; E) Energy group 5; F) Energy group 6; G) Energy group 7. ..........................126

6-32. Normalized pin power distribution for the 2-D MOX Fuel Assembly benchmark
problem (EP-SS4): A) 2-D view; B) 3-D view..................................................127

7-1. Hybrid decomposition for an EP-SS6 calculation (3 directions) for a system
partitioned with 4 coarse meshes on 6 processors............ ................................. 132

7-2. Speed-up obtained by running PENSSn on the Kappa and PCPENII Clusters........134









7-3. Parallel efficiency obtained by running PENSSn on the Kappa and PCPENII
C lu ste rs ...................................... ................................................. 1 3 5

7-4. Angular domain decomposition based on the automatic load balancing algorithm. 137

7-5. Parallel fraction obtained with the PENSSn code. ................................................137

8-1. Spectrum of eigenvalues for the Source Iteration and Synthetic Methods based on
different SSN orders............. .... ........................................................ .... .... .... 147

8-2. Number of inner iterations required by each acceleration method as a function of the
m esh size. ..........................................................................149

8-3. Number of inner iterations as a function of the scattering ratio (DZ differencing
scheme e)....................................................... ................... .. .... ... ... . 150

8-4. Number of inner iterations as a function of the scattering ratio (DTW differencing
scheme e)....................................................... ................... .. .... .. ... . 15 1

8-5. Number of inner iterations as a function of the scattering ratio (EDW differencing
scheme e)....................................................... ................... .. .... .. ... . 15 1

8-6. Number of inner iterations for the EP-SS2 synthetic method obtained with DZ, DTW,
and EDW differencing schemes. .................................................. ......... ...... 152

9-1. Card required in PENTRAN-SSn input deck to initiate SSN preconditioning.........155

9-2. Flow-chart of the PENTRAN-SSn Code System............... ...... .................156

9-3. Ratio of total number of inner iterations required to solve the problem with
preconditioned PENTRAN-SSn and non-preconditioned PENTRAN .................158

9-4. Relative change in flux value in group 1 ...................................... ............... 160

9-5. Relative change in flux value in group 2.............. ......................... ...... ............ 160

9-6. Behavior of the criticality eigenvalue as a function of the outer iterations............163

9-7. Convergence behavior of the criticality eigenvalue. ..............................................164

9-8. Preconditioning and transport calculation phases with relative computation time
requ ired ........................................................ ............................. 164

9-9. Behavior of the criticality eigenvalue as a function of the outer iterations..............165

9-10. Convergence behavior of the criticality eigenvalue. ............................................166

9-11. Preconditioning and transport calculation phases with relative computation time
requ ired ........................................................ ............................. 167









9-12. Behavior of the criticality eigenvalue as a function of the outer iterations ..........168

9-13. Convergence behavior of the criticality eigenvalue. ............................................169

9-14. Preconditioning and transport calculation phases with relative computation time
re q u ire d ......... .............. ...................... .................................................16 9















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

ADVANCED QUADRATURE SETS, ACCELERATION
AND PRECONDITIONING TECHNIQUES
FOR THE DISCRETE ORDINATES METHOD IN
PARALLEL COMPUTING ENVIRONMENTS


By

Gianluca Longoni

December 2004

Chair: Alireza Haghighat
Major Department: Nuclear and Radiological Engineering

In the nuclear science and engineering field, radiation transport calculations play a

key-role in the design and optimization of nuclear devices. The linear Boltzmann

equation describes the angular, energy and spatial variations of the particle or radiation

distribution. The discrete ordinates method (SN) is the most widely used technique for

solving the linear Boltzmann equation. However, for realistic problems, the memory and

computing time require the use of supercomputers. This research is devoted to the

development of new formulations for the SN method, especially for highly angular

dependent problems, in parallel environments. The present research work addresses two

main issues affecting the accuracy and performance of SN transport theory methods:

quadrature sets and acceleration techniques.

New advanced quadrature techniques which allow for large numbers of angles with

a capability for local angular refinement have been developed. These techniques have









been integrated into the 3-D SN PENTRAN (Parallel Environment Neutral-particle

TRANsport) code and applied to highly angular dependent problems, such as CT-Scan

devices, that are widely used to obtain detailed 3-D images for industrial/medical

applications.

In addition, the accurate simulation of core physics and shielding problems with

strong heterogeneities and transport effects requires the numerical solution of the

transport equation. In general, the convergence rate of the solution methods for the

transport equation is reduced for large problems with optically thick regions and

scattering ratios approaching unity. To remedy this situation, new acceleration algorithms

based on the Even-Parity Simplified SN (EP-SSN) method have been developed. A new

stand-alone code system, PENSSn (Parallel Environment Neutral-particle Simplified Sn),

has been developed based on the EP-SSN method. The code is designed for parallel

computing environments with spatial, angular and hybrid (spatial/angular) domain

decomposition strategies. The accuracy and performance of PENSSn has been tested for

both criticality eigenvalue and fixed source problems.

PENSSn has been coupled as a preconditioner and accelerator for the SN method

using the PENTRAN code. This work has culminated in the development of the Flux

Acceleration Simplified Transport (FASTc) preconditioning algorithm, which constitutes

a completely automated system for preconditioning radiation transport calculations in

parallel computing environments.














CHAPTER 1
INTRODUCTION

1.1 Overview

In the nuclear engineering field, particle transport calculations play a key-role in

the design and optimization of nuclear devices. The Linear Boltzmann Equation (LBE) is

used to describe the angular, energy and spatial variations of the particle distribution, i.e.,

the particle angular flux.1 Due to the integro-differential nature of this equation, an

analytical solution cannot be obtained, except for very simple problems. For real

applications, the LBE must be solved numerically via an iterative process. To solve large,

real-world problems, significant memory and computational requirements can be handled

using parallel computing environments, enabling memory partitioning and multitasking.

The objective of this dissertation is to investigate new techniques for improving the

efficiency of the of the SN method for solving problems with highly angular dependent

sources and fluxes in parallel environments. In order to achieve this goal, I have

investigated two major areas:

1. Advanced quadrature sets for problems characterized with highly angular
dependent fluxes and/or sources.
2. Advanced acceleration/preconditioning algorithms.

1.2 The Linear Boltzmann Equation

The LBE is an integro-partial differential equation, which describes the behavior of

neutral particle transport. The Boltzmann equation, together with the appropriate

boundary conditions, constitutes a mathematically well-posed problem having a unique

solution. The solution is the distribution of particles throughout the phase space, i.e.,









space, energy, and angle. The distribution of particles is, in general, a function of seven

independent variables: three spatial, two angular, one energy, and one time variable. The

time-independent LBE in its general integro-differential form1 is given by Eq. 1.1.

Q. V 7 (F, E, Q) + o, (F, E)VC(F, E, Q) = qe (r, E, Q)

+ JdE' JQ'o-, (F,'- E' ,Q'Q)(FE',Q') (1.1)
0 4;T

+ XE dE' JdQ r(F, E')(F, E', Q').
47r 0 4;T

In Eq. 1.1, I have defined the following quantities:

V.: Angular flux [particles/cm2/MeV/sterad/sec] .
F: Particle position in a 3-D space [cm].
E: Particle energy [MeV].
0: Particle direction unit vector.
a,: Macroscopic total cross-section [1/cm/MeV].
qe,,: External independent source [particles/cm3/MeV/sterad/sec] .
ao: Macroscopic double-differential scattering cross-section [1/cm/sterad/MeV].
S: Fission spectrum [1/MeV].
v : Average number of neutrons generated per fission.
7f : Macroscopic fission cross-section [1/cm/MeV].

1.3 Advanced Angular Quadrature Sets for the Discrete Ordinates Method

The discrete ordinates method (SN) is widely used in nuclear engineering to obtain

a numerical solution of the integro-differential form of the Boltzmann transport equation.

The method solves the LBE for a discrete number of ordinates (directions) with

associated weights.2 The combination of discrete directions and weights is referred to as

quadrature set.3 The major drawback of the SN method is the limited number of directions

involved, which, in certain situations, may lead to the so called ray-effects, which appear

as unphysical flux oscillations. In general, this behavior occurs for problems with highly









angular dependent sources and/or fluxes, or when the source is localized in a small region

of space, in low density or highly absorbent media.

In the past, several remedies for ray-effects have been proposed; the most obvious

one is to increase the number of directions of the quadrature set, or equivalently the SN

order. However, this approach can lead to significant memory requirements and longer

computational times. Carlson and Lathrop proposed a number of remedies45 for ray-

effects based on specialized quadrature sets, which satisfy higher order moments of the

direction cosines. Remedies based on first-collision approximations have also been

investigated.3 The source of particles generated from first-collision processes is often less

localized than the true source; hence, the flux due to this source is usually less prone to

ray-effects than the flux from the original source. If the true source is simple enough,

analytic expressions can be obtained for the uncollided flux and used to produce a first

collision source; however, for general sources and deep-penetration problems, this

method can be very time-consuming. An alternative approach6 is to expand the angular

flux in terms of spherical harmonics (PN). The PN method does not suffer from ray-

effects, because the angular dependency in the angular flux is treated using continuous

polynomial functions. However, the PN method has found limited applicability due to its

computational intensive requirements.

One of the most widely used techniques for generating quadrature sets is the level-

symmetric3 (LQN) method; however, the LQN method yields negative weights beyond

order S20. In problems with large regions of air or highly absorbent materials, higher

order (>20) quadrature sets are needed in order to mitigate ray-effects; therefore, it is

necessary to develop techniques which allow for higher order or biased quadrature sets.









In the past, different techniques have been investigated to remedy this issue. The

equal weight quadrature set (EQN), developed by Carlson,7 is characterized by positive

weights for any SN order. Other quadrature sets have been derived, by relaxing the

constraints imposed by the LQN method;2 for this purpose, the Gauss quadrature formula

and Chebyshev polynomials have been used for one-dimensional cylindrical or two-

dimensional rectangular geometries.4 In a recent study, the uniform positive-weight

quadrature sets8 (UEN and UGN) have been derived. The UEN quadrature set is derived by

uniformly partitioning the unit sphere into the number of directions defined by the SN

order while the UGN quadrature set selects the ordinates along the z-axis as roots of

Legendre polynomials.

A new biasing technique, named Ordinate Splitting (OS), has been developed9 for

the Equal Weight (EW) quadrature set; the OS technique is a method which is suitable to

solve problems in which the particle angular flux and/or source are peaked along certain

directions of the unit sphere. The idea is to select a direction of flight of the neutron and

split it into a certain number of directions of equal weights, while conserving the original

weight. This new biasing technique has been implemented in the PENTRAN code10 and

it has been proven very effective for medical physics applications such as CT-Scan

devices. 1115

In this research work, I have developed new quadrature sets11-12 based on Legendre

(PN) and Chebyshev (TN) polynomials. The Legendre-Chebyshev (PN-TN) quadrature set

is derived by setting the polar angles equal to the roots of the Legendre polynomial of

order N, and the azimuthal angles are calculated by finding the roots of the Chebyshev

polynomials.









The Legendre Equal-Weight (PN-EW) quadrature set is derived by choosing the

polar angles as the roots of the Legendre polynomial of order N, while the azimuthal

angles are calculated by equally partitioning a 90 degree angle. The set of directions is

then arranged on the octant of the unit sphere similar to the level-symmetric triangular

pattern. The main advantage of these new quadrature sets is the absence of negative

weights for any SN order, and their superior accuracy compared to other positive schemes

such as equal weight quadrature sets.

Moreover, I have developed a new refinement technique, referred to as Regional

Angular Refinement (RAR), which leads to a biased angular quadrature set.13-15 The RAR

technique consists of fitting an auxiliary quadrature set in a region of the unit sphere,

where refinement is needed. These quadrature sets have been applied successfully to

large problems, such as a CT-Scan device used for medical/industrial imaging9 and a

Time-of-Flight (TOF) experiment simulation.16 The benefit of using biased quadrature

sets is to achieve accurate solutions for highly angular dependent problems with reduced

computational time.

1.4 Advanced Acceleration Algorithms for the SN Method on Parallel Computing
Environments

Radiation transport calculations for realistic systems involve the solution of the

discretized SN equations. A typical 3-D transport model requires the discretization of the

SN equations in -300,000 spatial meshes, 47 energy groups, and considering an Ss

calculation, a total of 80 directions on the unit sphere. These figures yield approximately

1.13 billion unknowns. In terms of computer memory, this number translates into 9

GBytes of RAM for storage (in single precision) of only the angular fluxes. It is clear that

an efficient solution for such a problem is out of the scope of a regular workstation









available with current technology. Therefore, it is necessary to develop new algorithms

capable of harnessing the computational capabilities of supercomputers. Based on this

philosophy, in the late 1990s G. Sjoden and A. Haghighat have developed a new 3-D

parallel radiation transport code: PENTRAN (Parallel Environment Neutral-particle

Transport).10

However, besides the size and complexity of the problem being solved, other

aspects come into play, especially when dealing with criticality eigenvalue problems.

Because of the physics of these problems, the convergence rate of the currently used

iterative methods is quite poor. For realistic problems, such as whole-core reactor

calculations performed in a 3-D geometry, the solution of the SN equations may become

impractical if proper acceleration methods17 are not employed.

The main philosophy behind the novel acceleration algorithms developed in this

work is to employ a simplified mathematical model which closely approximates the SN

equations, yet can be solved efficiently.

The new acceleration/preconditioning algorithms have been developed during the

course of this research in three major phases:

1. Development of the PENSSn code based on the Even-Parity Simplified SN (EP-
SSN) method.
2. Investigation of a new synthetic acceleration algorithm based on the EP-SSN
method.
3. Development of an automated acceleration system for the SN method on parallel
environments: FAST (Flux Acceleration Simplified Transport).

1.5 The Even-Parity Simplified SN Method

The Even-Parity Simplified SN (EP-SSN) method is developed based on the

philosophy considered in the PN and Simplified Spherical Harmonics (SPN) methods.18









The spherical harmonics (PN) approximation to the transport equation is obtained

by expanding the angular flux using spherical harmonics functions truncated to order N,

where Nis an odd number; these functions form a complete basis in the limit of the

truncation error. In the limit of N oc, the exact transport solution is recovered.1 In 3-D

geometries, the number of PN equations grows as (N + 1)2. The PN equations can be

reformulated in a second-order form, cast as (N + 1)2 / 2 diffusion-like equations,

characterized by an elliptic operator. However, the number of these equations is very

large, and the coupling involves not only angular moments, but also mixed spatial

derivatives of these moments.6

Because of these issues, to reduce the computing time in the early 1960s, Gelbard

et al. proposed the Simplified Spherical Harmonics18 or SPN method. The Gelbard

procedure consists of extending the spatial variable to 3-D by substituting the second

order derivatives in the 1-D PN equations with the 3-D Laplacian operator. As a result,

coupling of spatial derivatives is eliminated, yielding only (N+ 1) equations as compared

to (N + 1)2. Further, since the SPN equations can be reformulated as second-order elliptic

equations, effective iterative techniques such as Krylov subspace19-20 methods, can be

employed.

The main disadvantage of the SPN equations is that the exact solution to the

transport equation is not recovered as N oc, due to terms that are inherently omitted in

replacing a 1-D leakage term with a simplified 3-D formulation. However, for idealized

systems characterized by homogeneous materials and isotropic scattering, the SPN and

the SN equations yield the same solution, given the same quadrature set and spatial

discretization is used for both methods.









Despite this fact, the SPN equations yield improved solutions2122 compared to the

currently used diffusion equation. The theoretical basis for the SPN equations has been

provided by many authors using variational methods and asymptotic analysis.22 It has

been shown that these equations are higher-order asymptotic solutions of the transport

equation. Moreover, Pomraning has demonstrated that the SPN equations, for odd N, are a

variational approximation to the one-group even-parity transport equation with isotropic

scattering in an infinite homogeneous medium.

Recently, the SPN formulation has received renewed interest, especially in reactor

physics applications. For applications such as the MOX fuel assemblies22-23 or for reactor

problems with strong transport effects,24 diffusion theory does not provide accurate

results, while the SPN equations improve the accuracy of the solution within a reasonable

amount of computation time.

Initially, I derived the SP3 equations starting from the 1-D P3 equations;25 however,

for developing a general order algorithm, I derived a general formulation using the even-

parity form of the SN transport equation.26 The Even-Parity Simplified SN (EP-SSN)

formulation has some interesting properties that make it suitable to develop algorithms of

any arbitrary order. Chapter 4 is dedicated to this issue. To make the method more

effective, the convergence properties of the EP-SSN equations were improved by

modifying the scattering term; it will be shown that this improved derivation is problem

dependent but can reduce the number of iterations significantly.

I have developed a general 3-D parallel code,23 PENSSn (Parallel Environment

Neutral-particle Simplified SN), based on the EP-SSN equations. The EP-SSN equations

are discretized with a finite-volume approach, and the spatial domain is partitioned into









coarse meshes with variable fine grid density in each coarse mesh emulating the

PENTRAN grid structure. A projection algorithm is used to couple the coarse meshes

based on the values of the even-parity angular fluxes on the interfaces. PENSSn uses

iterative solvers based on Krylov subspace19 methods: the Conjugate Gradient (CG) and

the Bi-Conjugate Gradient (Bi-CG) solvers. In addition, I have developed a

preconditioner based on the Incomplete Cholesky factorization20 for the CG method. The

finite-volume discretization of the EP-SSN equations in a 3-D Cartesian geometry yields

sparse matrices with a 7-diagonal sparse structure. Therefore, I optimized the memory

management of PENSSn by using a Compressed Diagonal Storage (CDS) approach,

where only the non-zero elements on the diagonals are stored in memory.

The PENSSn code is designed for parallel computing environments with angular,

spatial and hybrid (angular/spatial) domain decomposition algorithms.23 The space

decomposition algorithm partitions the 3-D Cartesian space into coarse meshes which are

then distributed among the processors while the angular decomposition algorithm

partitions the directions among the processors. The hybrid decomposition algorithm is a

combination of the two algorithms discussed above. Note that the hybrid decomposition

combines the benefits of memory partitioning offered by the spatial decomposition

algorithm, and the speed-up offered by the angular decomposition algorithm. The code is

written in Fortran 90, and for seamless parallelization, the MPI (Message Passing

Interface) library27 is used.

I have tested the PENSSn code for problems characterized by strong transport

effects, and have shown that the improvements over the diffusion equation can be

significant.26 The solutions obtained with the EP-SSN method are in good agreement with









SN and Monte Carlo methods; however, the computation time is significantly lower.

Hence, these results indicate that the EP-SSN method is an ideal candidate for the

development of an effective acceleration or preconditioning algorithm for radiation

transport calculations.

1.6 A New Synthetic Acceleration Algorithm Based on the EP-SSN Method

As mentioned earlier in this chapter, the solution of the linear Boltzmann equation

is obtained numerically via an iterative process. The most widely used technique to

iteratively solve the transport equation is the Source Iteration (SI) method3 or Richardson

iteration. The convergence properties of this method are related to the spectral radius of

the transport operator. It can be shown that for an infinite, homogeneous medium, the

spectral radius of the transport operator is dominated by the scattering ratio c, given by


c =- (1.2)
U t

where c, is the macroscopic scattering cross-section and is the macroscopic total cross-

section. Note that Eq. 1.2 can be obtained by Fourier analysis in an infinite homogeneous

medium. The asymptotic convergence rate v. is defined as

v, log(c). (1.3)

Eq. 1.3 indicates that for problems with scattering ratios close to unity, the

unaccelerated SI method is ineffective, because the asymptotic convergence rate tends

toward zero. Hence, the use of an acceleration scheme is necessary.

In the past, many acceleration techniques have been proposed25 for solving the

steady-state transport equation. The synthetic methods have emerged as effective

techniques to speed-up the convergence of the SI iterative process.28 In the synthetic









acceleration process a lower-order approximation of the transport equation (e.g.,

diffusion theory) is corrected using the transport equation at each iteration. In this way

the spectral radius of the accelerated transport operator is reduced with consequent speed-

up of the iteration process.

Two categories of synthetic methods have been investigated so far, the Diffusion

Synthetic Acceleration (DSA) and the Transport Synthetic Acceleration (TSA).29-30 The

first method has been proven to be very effective for 1-D problems and for certain classes

of multi-dimensional problems. However, recently it has been shown that for multi-

dimensional problems with significant material heterogeneities, the DSA method fails to

converge efficiently.3132 The same behavior, along with possible divergence, has been

reported also for TSA.30

I have developed and tested a new synthetic acceleration algorithm33 based on the

simplified form of the Even-Parity transport equations (EP-SSN). I tested the EP-SSN

algorithm for simple 3-D problems and I concluded that it is affected by instability

problems. These instabilities are similar in nature to what has been reported for DSA by

Warsa, Wareing, and Morel.3132 The main problems affecting the stability of the

synthetic methods are material heterogeneities and the inconsistent discretization of the

lower-order operator with the transport operator, which leads to divergence if the mesh

size is greater than -1 mean free path.

Moreover, because the synthetic acceleration method has been implemented into

the PENTRAN code, another consideration comes into play. The spatial differencing in

the PENTRAN code system is based on an Adaptive Differencing Strategy1o (ADS); with

this method the code automatically selects the most appropriate differencing scheme









based on the physics of the problem. Hence, the discretization of the lower-order operator

should be consistent with every differencing scheme present in the code. This task is

feasible if we consider only the linear-diamond (LD) differencing scheme; however, the

complexity increases if we consider the Directional Theta Weighted (DTW) or the

Exponential Directional Weighted (EDW) differencing schemes.34 Moreover, it has been

shown that even a fully consistent discretization of the lower-order operator does not

guarantee the convergence of the synthetic method.

Due to the issues discussed above, I have developed a system that utilizes the EP-

SSN method as preconditioner for the SN method.

1.7 An Automatic Preconditioning Algorithm for the SN Method: FAST, (Flux
Acceleration Simplified Transport)

The last phase of the development of an effective acceleration algorithm for the SN

method has culminated in the development of the FASTc system (Flux Acceleration

Simplified Transport). The FAST algorithm is based on the kernel of the PENSSn code.

The main philosophy followed in the development of the system is completely antithetic

to the synthetic acceleration approach. The idea is to quickly obtain a relatively accurate

solution with the EP-SSN method and to use it a preconditioned initial guess for the SN

transport calculation. This approach has the main advantage of decoupling the two

methods, hence avoiding all the stability issues discussed above.

The FASTc system is currently implemented into the PENTRAN-SSn Code

System for distributed memory environments, and it is completely automated from a user

point of view. Currently, the system has successfully accelerated large 3-D criticality

eigenvalue problems, speeding-up the calculations by a factor of 3 to 5 times, and hence

reducing significantly the spectral radius.









1.8 Outline

The remainder of this dissertation is organized as follows. Chapter 2 provides the

theory for the discrete ordinates method. The discretization of the phase space variables

in the transport equation will be discussed, along with the proper boundary conditions.

Chapter 3 discusses the theoretical development of the advanced and biased quadrature

sets for the discrete ordinates method. It also presents the application of the new

quadrature sets for the simulation of a CT-Scan device and for the Kobayashi benchmark

problem 3. Chapter 4 discusses the derivation of the EP-SSN equations. Chapter 5

discusses the numerical methods for the solution of the EP-SSN equations, along with the

iterative solvers based on Krylov subspace methods. Chapter 6 addresses the numerics

and accuracy of the EP-SSN equations. Chapter 7 presents the parallel algorithms

implemented in the PENSSn code for distributed memory architectures. Chapter 8

describes the development of a new synthetic acceleration algorithm based on the EP-SSN

method and its limitations. Chapter 9 focuses on the development of the FASTC

preconditioner; the performance of the algorithm is measured with two test problems and

a large 3-D whole-core criticality eigenvalue calculation. Chapter 10 will draw the

conclusions on the objectives accomplished and it will point out some aspects for future

development.














CHAPTER 2
THE DISCRETE ORDINATES METHOD

In this chapter, the Discrete Ordinates Method (SN) will be discussed in detail. The

discretized form of the transport equation is formulated in a 3-D Cartesian geometry. I

also address the iterative techniques and acceleration methods used to solve the SN

transport equations.

2.1 Discrete Ordinates Method (SN)

The Discrete Ordinates Method (SN) is widely used to obtain numerical solutions of

the linear Boltzmann equation. In the SN method, all of the independent variables (space,

energy and angle) are discretized as discussed below.

2.1.1 Discretization of the Energy Variable

The energy variable of the transport equation is discretized using the multigroup

approach.3 The energy domain is partitioned into a number of discrete intervals

(g=1... G), starting with the highest energy particles (g=l), and ending with the lowest

(g=G). The particles in energy group g are those with energies between Eg-1 and Eg. The

multigroup cross-sections for a generic reaction process x are defined as

E1 dE dOQa- (, E)yf(F, E, Q)
g(F) = H ---4;T-- (2.1)
f'g l dE4 dQ^(F,E,,Q)


Based on the definition given in Eq. 2.1, the group constants are defined in Eqs.

2.2, 2.3, and 2.4, for the "total," "fission" and "scattering" processes, respectively.











t() g dE ) (2.2)
gdEf dQy (F,E,Q)

dEg1 dE 4OQf (F, E)V(F, E, Q)
Of(, () ,= (2.3)
g 1dE d (F, E,Q)

JE 1 dE'S dn'z (F, E' E, E'.)y(F, E, Q')
( sgg(7)= 4 -' (2.4)
E' l dE'I dV (F, E, )
fE, 4;T

With the group constants defined above, the multigroup formulation of the transport

equation is written as

G
Q V /i (F, Q) + Ug, (F)Vg (F, Q) = JI'Jgg (F, YQ'Q)g, (F, Q')
g V'= 14 (2.5)
1 G
+ x _Cv f,g,(rF)Og,(rF) + q9( ,),
g'=l

for g 1, G,

where the angular flux in group g is defined as

g,(rF,) = J ldEj (F,E,Q ). (2.6)
fg

In Eq. 2.5, q (F, Q) is the angular dependent fixed source; in general, for

criticality eigenvalue problems, this term is set to zero. The scalar flux in Eq. 2.5 is

defined as

g (F) = JdfQ (Fg,). (2.7)
4;T

In a 3-D Cartesian geometry, the "streaming" term can be expressed as

a a a
.V=,- + +7 (2.8)
Ox Qy Oz









where the direction cosines are defined as

u=a Qi, r = j, 5 =Qk.

Figure 2-1 shows the Cartesian space-angle system of coordinates in three

dimensions.


(2.9)


Figure 2-1. Cartesian space-angle coordinates system in 3-D geometry.

The multigroup transport equation, with the scattering kernel expanded in terms of

Legendre polynomials and the angular flux in terms of spherical harmonics is given by

Eq. 2.10. The complete derivation of the scattering kernel expansion in spherical

harmonics shown in Eq. 2.10 is given in Appendix A.










P +7 + ) +JCg (x,Y,z)JVif (x,y,z,,u(P)=

G L
1 (2/ + 1)o g (x, yz){ ()q0' (, y, z) +
g'= ll= (2.10)

21 kP" ( g)[ (X, y, Z) cos(kp) + q~,g (X, y, z) sin(ko)] }
k=1 ( + k)!

+ g- f v g,g (x, y, yz)O,g (x, y, z) + q (F, ),
k g1

where

u : direction cosine along the x-axis
r : direction cosine along the y-axis
S: direction cosine along the z-axis
ag: total macroscopic cross-section

(: azimuthal angle, i.e. arctan -

Vyg (x, y,, z, ), (): angular flux in energy group g
,sl,g' g: 1 t moment of the macroscopic transfer cross-section
P, (u) : /th Legendre Polynomial
Sg (//): /th flux moment
Pk (/) : associated th, kth Legendre Polynomial
kc,g (/)): cosine associated th, kth flux moment
u,g (u): sine associated th, kth flux moment
,g: group fission spectrum
k: criticality eigenvalue
va ,g : fission neutron generation cross-section

2.1.2 Discretization of the Angular Variable

The angular variable, 0, in the transport equation is discretized by considering a

finite number of directions, and the angular flux is evaluated only along these directions.

Each discrete direction can be visualized as a point on the surface of a unit sphere with an

associated surface area which mathematically corresponds to the weight of the integration

scheme. The combination of the discrete directions and the corresponding weights is






18


referred to as quadrature set. In general, quadrature sets should satisfy the following

properties:3

* The associated weights must be positive and normalized to a constant, usually
chosen to be one
M
.m =1.0. (2.11)
m=l

* The quadrature set is usually chosen to be symmetric over the unit sphere, so the
solution will be invariant with respect to a 90-degree axis rotation and reflection.
This condition results in the odd-moments of the direction cosines having the
following property

,n = Wmr7m = C',m =0.0, for n odd. (2.12)
m=l m=l m=l

* The quadrature set must lead to accurate values for moments of the angular flux
(i.e., scalar flux, currents); this requirement is satisfied by the following conditions
on the even-moments of the direction cosines as follows
M M M 1
IWm/ = Wmr,' = Wm ,for n even. (2.13)
m=l m=l m=l + 1

A widely used method for generating a quadrature set is the level-symmetric

technique (LQN). In this technique, the directions are ordered on each octant of the unit

sphere along the z-axis () on N/2 distinct levels. The number of directions on each level

N
is equal to +1, for i=1, N/2. It is worth noting that in 3-D geometries, the total
2

number of directions is M=N(N+2), where N is the order of the SN method.

22 N
Considering + + 2 + = 1 and i + j +k = + 2, where N refers to the number
2

of levels and i, j, k are the indices of the direction cosines, we derive a formulation for

determining the directions as follows

A2 = ,2 +(i-1)A, (2.14)


where










2(1- 3 2)2 N 1
= 2(1- 2) for 2< i< ,and 0< 1,2 < (2.15)
(N 2) 2 3

In Eq. 2.14 the choice of ul determines the distribution of directions on the octant.

If the value of/ul is small, the ordinates will be clustered near the poles of the sphere;

alternatively, if the value of kl is large, the ordinates will be placed far from the poles.

The weight associated to each direction, called a point weight, is then evaluated

with another set of equations. For example, in the case of an Ss level-symmetric

quadrature set, this condition can be formulated as follows

2p, + 2p2 =i, (2.16a)

2p2 +3 w2, (2.16b)

2p2 w3, (2.16c)


lp1 = w4, (2.16d)

wherepi, p2 andp3 are point weights associated with each direction, and w1, w2, w3, W4

are the weights associated with the levels, as shown in Figure 2-2.


point weights (p)

Figure 2-2. Point weight arrangement for a Ss level-symmetric quadrature set.


Level weights
w4
- w3
__ w2

__ wi









As an example, Figure 2-3 shows the S20 LQN quadrature set for one octant of the

unit sphere.
















Figure 2-3. S20 LQN quadrature set.

Note that, this quadrature set is limited by unphysical negative weights beyond

order S20. Therefore, if a higher order quadrature set is needed beyond S20, another

formulation has to be developed, which satisfies the even- and odd-moments conditions.

To address this issue, I have developed new quadrature techniques based on the Gauss-

Legendre quadrature formula and on the Chebyshev polynomials.

2.1.3 Discretization of the Spatial Variable

The linear Boltzmann equation, given in Eq. 2.10, can be rewritten in an

abbreviated form as


Pm + 7my + + m C (F) ,f,(F) =Qmg(F(), (2.17)


form = 1,M and g=1,G.

The angle and energy dependence are denoted by the indices m and g, respectively.

The right hand side of Eq. 2.17 represents the sum of the scattering, fission and external

sources.









The spatial domain is partitioned into computational cells, bounded by x1/2, X3/2,...,

xI+1i2; yi/2, y3/2, ..., yJ+~/2; z1/2, z3/2, ..., ZK+12. The cross-sections are assumed to be constant

within each cell and they are denoted by cr,,,k. Eq. 2.17 is then integrated over the cell

volume V,k A Ayj Azk, and then divided by the cell volume to obtain the volume- and

surface-averaged fluxes, therefore reducing to


P" (w +/1 2,j,k,m,g T 1 -2,j,k,m,g ( 1,j+1 2,k,m,g T 1 j- 2,k,m,g)
J(2.18)

+ jM 1,k+1 2,m,g ,j,k-11 2,m,g ) 7 i,j,k i,j,k,m,g i,j ,k,m,g
Azk

In Eq. 2.18, the indices i, j, k represent the cell-center values, while i+1/2,j+1/2, k+1/2

refer to the surface values.

2.1.4 Differencing Schemes

For the SN method, different classes of differencing schemes are available. Low-

order differencing schemes require only the angular fluxes, and the average values at the

cell boundaries to be related at the cell average value. Various forms of Weighted

Difference (WD) schemes belong to this class. High-order differencing schemes require

higher order moments, and may be linear or non-linear. Discontinuous, characteristic, and

exponential schemes are examples of high-order differencing schemes.35

The solution of the SN equations is obtained by marching along the discrete

directions generated in each octant of the unit sphere; this process is usually referred to as

a transport sweep.3 For each computational cell, the angular fluxes on the three incoming

surfaces are already known, from a previous calculation or boundary conditions. The cell-

center fluxes and the fluxes on the three outgoing surfaces must be calculated, hence









additional relationships are needed. The additional relationships are referred to as the

"differencing schemes ". The general form of WD schemes can be expressed as

T ,j,k,mg al,J,k,m,g T +1 /2,j,k,,g ,,,k,m,g)R 1/2,j,k,m,g, (2.19a)


~ ,j,k,m,g ,,k,g b,k,, +1/2,k,m,g +(1- b,,j,k,m,g ) ,J- 1/2,k,m,g, (2.19b)

Y,,j,k,m,g = C,j,k,m,g Tj,,k+1/2,m,g + (- C,,J,k,m,g )Tj,k-1/2,m,g (2.19c)

The values a,,j,km,g, bi,j,k,m,g, and c,,k,m,g are determined based on the type of weighted

scheme employed.

2.1.4.1 Linear-Diamond Scheme (LD)

In the LD scheme, the cell-average flux is an arithmetic average of any two

opposite boundary fluxes; hence the weights are set to constant values a=b=c=1/2.


,,j,k,m,g ( z +1/2,j,k,m,g z- 1/2,j,k,m,g)

2 i ,J+l/2,k,m,g ,J-1/2,k,m,g2.20)

I (q ,j,k+1/2,m,g + j7,k-1/2m,g)


For example, in the direction /, > 0, 77 > 0, > 0 the outgoing fluxes are obtained as

follows

S1+1/2,j,k,m,g = 2T,,,k,m,g ~- 1/2,],k,m,g, (2.21a)

S,j+1/2,k,m,g = 2 ,km ,j-1/2,k,m,g (2.21b)

T,],k+1/2,m,g = 2i,],k,m,g ,],k-1/2,m,g 2 (2.21c)

We can then eliminate the fluxes on the outgoing surfaces in Eq. 2.18 and obtain

the center-cell angular flux









r"k Y + 12m g +"iQJ + Qek
2u 1-/2,j,k,m,g + 2r Tl,j-1/2,k,m,g + 2 T ,j,,k-1 /2,m,g -[ i,j,k

,,m,g (2.22)
-j, +2m 2u 27 2(
Ax, Ayj Azk

The outgoing fluxes are then evaluated using Eqs. 2.21a, b, and c.

The LD differencing scheme may yield negative angular fluxes in regions where

the flux gradient is large, even if the incoming fluxes and scattering source are positive.

In this case, one approach is to set negative fluxes equal to zero, and then the cell-average

flux is recalculated to preserve the balance of particles. This approach is referred to as

negative flux fix-up (DZ). The DZ scheme performs better than the LD in practical

applications, but the linearity and accuracy of the LD equations is not preserved.

2.1.4.2 Directional Theta-Weighted Scheme (DTW)

This scheme uses a direction-based parameter to obtain the weighting factors a, b,

and c which are restricted to the range 0.5 and 1.0. The DTW scheme uses a direction-

based parameter to obtain an angular flux weighting factor, which ensures positivity of

the angular flux and removes the oscillations due to the spatial and angular

discretization.36 The DTW average cell angular flux is given by


ql,j,k + aI/A + Vlny + V
a,, ,mgx b y c Az
V ,j,,mg a,],k,m,g i,],k,m,g Cj,k,m,gAz (2.23)
U M n17 m n n
a,,km,gAx b,,k,m,g C,,,k,m,gA

In the DTW scheme, the weights (a,,km,g, by,km,g, c.,,km,,g) are restricted to the range

between 0.5 and 1.0, approaching second order accuracy when all weights are equal to

0.5, which is in this case is equivalent to the LD scheme.









2.1.4.3 Exponential Directional-Weighted Scheme (EDW)

The Exponential Directional Weighted (EDW) differencing scheme34, implemented

in PENTRAN, is a predictor-corrector scheme, which utilizes the DTW to predict a

solution that is then corrected using an exponential fit. The EDW is an inherently positive

scheme, and the auxiliary equations derived for this method are given in Eq. 2.24.

Vm (x,y, z) = aexp(AP, (x)/ /pm )exp(A P,(y)/ r7, )exp(AkPl (z) /n m). (2.24)

The DTW scheme is used to calculate the angular fluxes (f) needed for the

estimation of the coefficients A,, A and Ak given in Eqs. 2.25a, b, and c, respectively.




A, t,x Vmx )Pm, (2.25a)
2 4


A ( outjy V-ny)1m (2.25b)
2 A


Ak out,z fin,z )m (2.25c)
2R,

where the subscripts in and out refers to the incoming and outgoing surface averaged

angular fluxes. The cell-average angular flux formulated with the EDW scheme is given

in Eq. 2.26.

2A 2A 2e A
V4 = exp( )-1 exp( )- lexp(Ak) -1
S+ m Vx + i + m V (2.26)

SAhx y Az

where fl is calculated using the coefficients given in Eqs. 2.25a, b, and c.37










2.1.5 The Flux Moments

The flux moments are obtained from the angular fluxes using the following

formulation


,,Jk,l = Wm ,,kP (m). (2.27)
m=l

Note that Eq. 2.27 for /=0 yields the scalar flux. The Associated Legendre moments

are calculated using Eqs. 2.28 and 2.29.

M
ijk,C, = Wm T,,j,kPm() COS(I. .), (2.28)
m=l

M
",,k,Sl = m ,,,k P" ( m ) i n( .) (2.29)
m=l

2.1.6 Boundary Conditions

Three major boundary conditions can be expressed with a general formula as

YT( ,m) = aT(, i'), (2.30)

where 'm -n = -'m n.

Depending on the value of coefficient a, the three boundary conditions are:

a=l, reflective boundary condition.

a=0, vacuum or non-reentrant boundary condition.

a=P, albedo boundary condition.

2.2 Source Iteration Method

Due to the integro-differential nature of the transport equation, the solution of the

multigroup discrete ordinates equations is obtained by means of an iterative process,

named the source iteration.3 The source iteration method consists of guessing a source

(i.e., in-group scattering source), then sweeping through the angular, spatial and energy









domains of the discretized system with the appropriate boundary conditions. When the

sweep is completed, integral quantities such as scalar flux and flux moments are obtained

from the angular fluxes, and then a new in-group scattering source is calculated, and the

iteration process continues until a convergence criterion, shown in Eq. 2.31, is satisfied.

Typical tolerances for fixed source calculations range in the order of 1.0e-3 to 1.0e-4.


< e. (2.31)


If fission and/or up-scattering processes are present, outer iterations are performed

on the fission and transfer scattering sources. Acceleration techniques may be applied

between source iterations to speed-up the convergence rate by determining a better guess

for the flux moments and the source.

2.3 Power Iteration Method

Criticality eigenvalue problems are solved using the method of power iteration.38

For this method, it is assumed that the eigenvalue problem has a largest positive

eigenvalue, k>O, with an associated fission distribution F(F) that is nonnegative. Hence

by considering ko>O and F (F) > 0 as initial guesses, we calculate the eigenvalue at

iteration i as follows

JdE dfF' 1(F)
k1[ = k' fdEf- (2.32)
fdEfdrF' (F)

where F' (F) = voa (F, E)q~ (F, E) is the fission source distribution at iteration i. The

iterative process is continued until the desired convergence is reached, as shown in Eq.

2.33.









k' k'1
1 < E. (2.33)
k"'

Generally, the tolerance required for criticality calculation is 1. Oe-4 to 1.0e-6.

2.4 Acceleration Algorithms for the SN Method

Many acceleration methods have been proposed to speed-up the convergence of the

iterative methods used to solve the steady-state transport equation.17 There are a number

of problems where standard non-accelerated iterative methods converge too slowly to be

practical. Most of these problems are characterized by optically thick regions with

scattering ratio near unity.

The three major acceleration approaches are the Coarse Mesh Rebalance (CMR),

Multigrid, and Synthetic methods. The CMR approach is based on the fact that the

converged solution must satisfy the particle balance equation.3 By imposing this balance

condition on the unconverged solution over coarse regions of the problem domain, it is

possible to obtain an iteration procedure that usually converges more rapidly to the

correct solution. However, this method is highly susceptible to the choice of the coarse

mesh structure and can be unstable.

The multigrid approach has been used to accelerate the SN calculations; the basic

principle of the method is to solve the equations on a coarse grid and to project the

solution onto a finer grid. Different types of multigrid approaches exist, such as "/" Slash-

cycle, V-Cycle and W-Cycle, and the Simplified Angular Multigrid39 (SAM). In Chapter

8, I will compare the results obtained with the EP-SSN synthetic acceleration and the

SAM method.

The synthetic acceleration approach is based on using a lower-order operator,

generally diffusion theory, as a means to accelerate the numerical solution of the









transport equation.17 In the late 1960s, Gelbard and Hageman developed a synthetic

acceleration method based on the diffusion and the S4 equations.28 Later, Reed

independently derived a similar synthetic acceleration scheme40 and pointed out some

limitations of the method derived by Gelbard and Hageman. The synthetic method

developed by Reed has the advantage of being very effective for small mesh sizes, but it

is unstable for mesh sizes greater than -1 mfp (mean free path). Later, Alcouffe

independently derived the Diffusion Synthetic Acceleration (DSA) method.29 He

addressed the issue of stability of the method and derived an unconditionally stable DSA

algorithm. Alcouffe pointed out that in order to obtain an unconditionally stable method,

the diffusion equation must be derived consistently from the discretized version of the

transport equation. In this way, the consistency between the two operators is preserved.

Recently, the DSA method has been found to be ineffective30-32 for

multidimensional problems with strong heterogeneities, even when a consistent

discretization of the lower order and transport operators is performed.














CHAPTER 3
ADVANCED QUADRATURE SETS FOR THE SN METHOD

This chapter covers the development of advanced quadrature sets for solving the

neutron transport equation via the Discrete Ordinates (SN) method. The level-symmetric

(LQN) quadrature set is the standard quadrature set for SN calculations; although, as

discussed in the previous chapter, this quadrature set is limited to order 20. The Equal

Weight (EW) quadrature set has been proposed to resolve the issue of negative weights;

this quadrature set is generated by partitioning the unit sphere into M directions, where


M = N(N + 2) and by assigning an equal weight to each direction w, = The EW
M

quadrature set yields positive weights for any SN order; however, it does not completely

satisfy the even-moment conditions given in Eqs. 2-13.

I have developed and tested new quadrature sets based on the Legendre (PN) and

Chebyshev (TN) polynomials. In this chapter, I discuss the Legendre Equal-Weight (PN-

EW), the Legendre-Chebyshev (PN-TN) quadrature sets, and the Regional Angular

Refinement (RAR) technique for local angular refinements. The PN-EW and PN-TN have

no limitations on the number of directions, and the RAR technique is an alternative to the

Ordinate Splitting (OS) technique.1l The OS technique has been developed to refine the

directions of a standard quadrature set using equal-spaced and equal-weight directions,

while the RAR utilizes the PN-TN quadrature set over a subset of ordinates. The main

difference between the OS and RAR techniques is in the refining methodology; the OS

technique focuses on the refinement of each single direction, while the RAR considers a









sector of the unit sphere. Also the biased quadrature set generated with RAR satisfies the

conditions on the odd- and even-moments of the direction cosines (Eq. 2-12 and 2-13).

To examine the effectiveness of the new techniques for angular quadrature

generation, each technique has been implemented into the PENTRAN code10 (Parallel

Environment Neutral-Particle TRANsport), and utilized for a number of problems of

practical interest.

3.1 Legendre Equal-Weight (PN-EW) Quadrature Set

In order to develop a quadrature set which is not limited to order S20, I have

investigated the Gauss-Legendre quadrature technique.4 This quadrature set is

characterized by the same arrangement of directions as the LQN, but the directions and

weights are evaluated differently. Given the SN order for the discrete set of directions, we

apply the Gauss-Legendre quadrature formula using the following recursive formulation

(j + 1)P,+1 = (2j + 1)4P7 jPl, forj =0, N, (3.1)

where

-1< <1, P_() = 0, and P() = 1. (3.2)

The -levels or polar angles, along the z-axis are set equal to the roots of Eq. 3.1.

The values represent the levels of the quadrature set. Once we have evaluated the

ordinates along the z-axis, we obtain the weights associated with each level using the

following recursive formulation

2 N
w2 ,for i = 1, (3.3)

[l d J










In order to complete the definition of each discrete direction, the azimuthal angle is


evaluated on each level by equally partitioning a 90 degree angle into


N
-- i + 2 angular
2


intervals, where i=1, N/2. Hence, the weight associated with each direction is given by


w
, for i = ...
I


(3.4)


In Eq. 3.4, j =1... i +1 is the number of directions with equal weights on the ith
2

level. Figure 3-1 shows the directions and the associated weights for an S28 PN-EW

quadrature set on one octant of the unit sphere.


wdej@t
00106670
0D01 066?3
0010AE68
o o 1CO
0010068
O OC9W33
oOON6648M
0009A6437
00092M388
00o c634
O0tJ3M292
000366243
DOCWAMi
0006i 47
GO DEM88


Figure 3-1. S28 PN-EW quadrature set.

Note that in Figure 3-1, all directions on the same -level have the same weight, as


indicated by the color.

3.2 Legendre-Chebyshev (PN-TN) Quadrature Set

In the PN-TN quadrature set, similar to PN-EW, we choose the -levels on the z-axis

equal to the roots of the Gauss-Legendre quadrature formula given in Eqs. 3.1 and 3.2;

however, the azimuthal angles on each level are set equal to the roots of the Chebyshev









(TN) polynomials of the first kind. Chebyshev polynomials of the first kind are

formulated as follows

T, [cos(o)] cos(/o). (3.5)

The Chebyshev polynomials are orthogonal and satisfy the following conditions

0, I# k
dyT()(y y)( y2) 1/2 =z,l= k = 0
Izr/2,1 =k 0
y =cos(m) (3.6)

Again, using the ordering of the LQN quadrature set, we define the azimuthal

angles on each level using the following formulation


0J = 2i-2j+l1) (3.7)


where o), e 0, 2) and i= 1, N/2.


In Eq. 3.7, i is the level number and j = 1... -i +1. The level and point weights are
2

generated in the same way as for the PN-EW. Note that both PN-EW and PN-TN

quadrature sets do not present negative weights for SN orders higher than 20.

Figure 3-2 shows the directions and the associated weights for an S28 PN-TN

quadrature set on one octant of the unit sphere.


























Figure 3-2. S40 PN-TN quadrature set over the unit sphere.

3.3 The Regional Angular Refinement (RAR) Technique

The RAR method is developed for solving problems with highly peaked angular

fluxes and/or sources. The approach consists of two steps. In the first step, we derive a

PN-TN quadrature set of arbitrary order on one octant of the unit sphere, as described in

Section 3.2.

In the second step, we define the area (angular segment) of the octant to be refined

along with the order (N') of the PN'-TN' quadrature set to be used in this region. The area

to be refined is characterized by the polar range (-min, max) and the azimuthal range (pmin,

pmax). Generally, the biased region is selected based on the physical properties of the

model. For example, if a directional source is forward peaked along the x-axis, the

quadrature set will be refined on the pole along the x-axis.

The -levels of the PN'-TN' quadrature set are calculated using Eqs. 3.1 and 3.2;

therefore, they are mapped onto the (-min, max) sub-domain using the following formula:

max 2 + max + mm for i=1, N'2. (3.8)
2 2








Hence, the azimuthal angles are evaluated using the Chebyshev polynomials as

follows


(3.9)


where co, i 0o, i=1, N'/2 andj=l, '/2.
S2


(N,')2
The number of directions in the refined region is equal to The weights in the
S2

refined region are renormalized to preserve the overall normalization on the unit sphere.

Figure 3-3 shows the RAR technique applied to an S16 PN-TN quadrature set. In the

biased region, which extends from =0.0 to =0.2 along the z-axis, and from qp=0 to

(p=10 on the azimuthal plane, an Slo PN-TN quadrature set is fitted.


Figure 3-3. PN-TN quadrature set (S16) refined with the RAR technique.

3.4 Analysis of the Accuracy of the PN-EW and PN-TN Quadrature Sets

The main advantage of the LQN technique is the fact that it preserves moments of

the direction cosines, thereby leading to an accurate solution. The PN-EW and PN-TN


0)i ) N'-2j + 1 (Pmx (Pm +( max + (3,mm
2 N' 2 2










quadrature sets attempt to preserve these quantities independently along the a-, fl-, and 5-

axes.

Therefore, I verified the capability of the new quadrature sets in preserving the

even moments of the direction cosines, which are directly related to the accuracy of the

quadrature set. Table 3-1 compares the even-moments of the direction cosines calculated

with an S30 PN-EW quadrature set with the exact value.

Table 3-1. Even-moments obtained with a PN-EW S30 quadrature set.


M

1=1

0.333333333
0.194318587
0.135907964
0.103874123
0.083691837
0.06983611
0.059749561
0.052087133
0.046074419
0.041234348
0.037257143
0.033932989
0.031114781
0.028696358
0.026599209


0.333333333
0.194318587
0.135907964
0.103874123
0.083691837
0.06983611
0.059749561
0.052087133
0.046074419
0.041234348
0.037257143
0.033932989
0.031114781
0.028696358
0.026599209


0.333333333
0.2
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065


Exact value

1+n

0.333333333
0.2
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065


As expected, the PN-EW preserves exactly the even-moments conditions along the

-axis, while on the a-, rl-axes these conditions are only partially preserved; the

maximum relative difference between the even-moments calculated with PN-EW and the

exact solution is 17.0%.

Table 3-2 shows the comparison of the even-moments evaluated with an S30 PN-TN


quadrature set and the exact value.


Moment
Order
(n even)
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30









Table 3-2. Even-moments obtained with a PN-TN S30 quadrature set.


M


Moment
Order
(n even)
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30


0.333333333
0.199999962
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065


0.333333333
0.2
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065


Exact value
1+n

0.333333333
0.2
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065


It is clear from Table 3-2 that the PN-TN quadrature set completely satisfies the

even-moment conditions. This is possible because both roots of Legendre and Chebyshev

polynomials satisfy the even-moment conditions given by Eqs. 2.13.

In order to further verify the accuracy of the PN-EW and PN-TN quadrature sets, and

to check their accuracy, I used a simple test problem, consisting of a homogeneous

parallelepiped, where an isotropic source is placed in its lower left corner as shown in

Figure 3-4.

Due to the symmetry of the problem, it is expected that the particle currents


flowing out of regions A and B (see Figure 3-4) have the same value.


0.333333333
0.199999962
0.142857143
0.111111111
0.090909091
0.076923077
0.066666667
0.058823529
0.052631579
0.047619048
0.043478261
0.043478261
0.037037037
0.034482759
0.032258065












HUMh
a 9Z
25JU im


UuIEN
ar~flEBD LIC.


Y k.9- QK EL
Y 9161 (Wago ROeIKkW B.C.


I rnl
V-3CLPJM


B


Figure 3-4. Configuration of the test problem for the validation of the quadrature sets.

Figure 3-5 shows the relative difference between the particle current in regions A


and B for the EW, PN-EW, and PN-TN as compared to the LQN technique.



12

S10
8- 2 --------------------------------------------------------------------





5 4

So --- ---------------------------------------------


8 10 12 14 16 18 20
Sn Order

--EW --Pn-Tn -- Pn-EW

Figure 3-5. Relative difference between the currents Jx and Jz for the test problem.

Level-symmetric is considered as the reference because it preserves moments of


both azimuthal and polar direction cosines. The PN-TN yields almost perfect symmetry,


while the PN-EW and Equal Weight show maximum relative differences of 4% and 10%,


respectively. It is worth noting that the loss in accuracy of the EW quadrature set is


attributed to the fact that the even-moment conditions are not satisfied. The PN-EW yields









higher accuracy compared to EW, because the even-moment conditions are satisfied

along the z-axis.

3.5 Testing the Effectiveness of the New Quadrature Sets

In this section, the effectiveness of the new quadrature sets is examined by

simulating two test problems: Kobayashi benchmark problem 3 and a CT-Scan device for

industrial/medical imaging applications.

3.5.1 Kobayashi Benchmark Problem 3

To examine the effectiveness of the new quadrature sets, I have used the first axial

slice of the Kobayashi43 3-D benchmark problem 3 with pure absorber. Figures 3-6 show

two different mesh distributions: Figure 3-6A is obtained from a previous study42, where

an appropriate variable mesh was developed; Figure 3-6B shows a uniform mesh

distribution that I have developed for the current study. The uniform mesh is used in

order to separate the effects of the angular discretization from the spatial discretization.

The reference semi-analytical solutions are evaluated in two spatial zones shown in

Figures 3-6A and 3-6B (zone 1 along y-axis, at every 10.0 cm intervals between 5.0

and 95.0 cm; zone 2 along x-axis, y = 55.0 cm, every 10.0 cm, between 5.0 cm and 55.0

cm).









A B













5 -- -t 5 5 55------ 5 25 4 5 55 x

Figur sh distribution for the obayashi benhark probe ar e




distribution shown in Figure 3-6a. In the previous study, by taking advantage of the
l -- ""- --











Figuriable 3.6. Mesh distribution for the obayashi benchmark problevel-symmetric quadraturiable mesh







presented a maximum relative error of -6% in zone 1. In the current study, the solution
disobtained with the level-symmetric quadrature set and uniform spatial mesh yields attribution.

maximum relative 3-7 shows the ratio of10% in zone calculated to the exact solution (C/E) for the level-

symmetric, P-EW and PN-EW and P-TN quadrature sets undeof order 20 for zonestimate 1. Also, in this figure,x by

51.9%I present and 8.5%, solution34 obtained in a previous st point of zone 1; the variables due to the fact that the

distribution shown in Figure 3-6a. In the previous study, byaround tweaking advantage ofmpared to the

-TNvariable mesh distribution, the solution obtained with the level-symmetric quadrature sets
presented a maximum relative error of -6% in zone 1. In the current study, the solution

obtained with the level-symmetric quadrature set and uniform spatial mesh yields a

maximum relative error of-10% in zone 1.

In zone 1, the PN-EW and PN-TN quadrature sets underestimate the scalar flux by

-51.9% and -8.5%, respectively, on the last point of zone 1; this is due to the fact that the

PN-EW quadrature set has fewer directions clustered around the y-axis as compared to the

PN-TN and level-symmetric quadrature sets.












1.20E+00
1.00E+00 -
.2 8.00E-01
S 6.00E-01
O 4.00E-01
2.00E-01
0.00E+00
5 15 25 35 45 55 65 75 85 95
Y (cm)

LQn S20 (Uniform mesh) LQn S20 (Variable mesh)
-- Pn-Tn S20 -X Pn-EW S20

Figure 3-7. Comparison of S20 quadrature sets in zone 1 at x=5.0 cm and z=5.0 cm.

Figure 3-8 compares the scalar flux obtained in zone 2 of the benchmark problem.


While using a uniform spatial mesh, the PN-TN quadrature set yields slightly better results


compared to level-symmetric.



1.40E+00
1.20E+00
0 1.OOE+00

utj 6.00E-01
0 4.00E-01
2.00E-01
2.00E-01 --------------------------------------
0.OOE+00
5 15 25 35 45 55
X (cm)

--LQn S20 (Uniform mesh) LQn S20 (Variable mesh)
-- Pn-Tn S20 Pn-EW S20

Figure 3-8. Comparison of S20 quadrature sets in zone 2 aty=55.0 cm and z=5.0 cm.

In zone 2, the maximum relative error obtained with PN-TN is -18.6%, while for


level-symmetric it is -21.9% using the uniform mesh distribution, and -6% using


variable meshing. However, an error of -28.2% is observed for the PN-EW quadrature


set.


Figures 3-9 and 3-10, show the solutions obtained with the PN-EW quadrature set


for different SN orders compared to level-symmetric S20, in zone 1 and 2 respectively.












1.10E+00
1.00E+00 -
9.00E-01
8.00E-01
,u 7.00E-01
6.00E-01
5.00E-01
4.00E-01
5 15 25 35 45 55 65 75 85 95
Y (cm)

-- LQn S20 -I- Pn-EW S20 -- Pn-EW S22
-- Pn-EW S24 -- Pn-EW S26

Figure 3-9. Comparison of PN-EW quadrature sets for different SN orders in zone 1 at
x=5.0 cm and z=5.0 cm.


1.40E+00
1.30E+00
1.20E+00
1.10E+00 --- - -- - - - -
% 1.10E+00
,u 1.00E+00
9.00E-01
8.00E-01
7.00E-01
5 15 25 35 45 55
X (cm)

-4- LQn S20 -- Pn-EW S20 -- Pn-EW S22
Pn-EW S24 -- Pn-EW S26

Figure 3-10. Comparison of PN-EW quadrature sets for different SN orders in zone 2 at
y=55.0 cm and z=5.0 cm.

In zone 1 (Figure 3-9), the PN-EW is not as accurate as level-symmetric, because


fewer directions are clustered near the y-axis; however in zone 1, the solution improves


somewhat by increasing the SN order. In zone 2 (Figure 3-10) the PN-EW set yields


inaccurate results, with a maximum relative error of -36% for the S20 case.


Figure 3-11 compares the ratios of different computed solutions to the exact


solution; the computed solutions wered obtained with the PN-TN quadrature set for orders


S20, S22, S24, S26 and with the S20 level-symmetric quadrature set. It appears that the

increase in the quadrature order does not have a noticeable effect in improving the











accuracy. However, this behavior can be attributed to the fact we have retained the same


spatial mesh discretization.



1.15E+00
1.10E+00 -
S1.05E+00
S1.00E+00
,u 9.50E-01
9.00E-01
8.50E-01
8.00E-01
5 15 25 35 45 55 65 75 85 95
Y (cm)

4- LQn S20 -- Pn-Tn S20 -- Pn-Tn S22
-X Pn-Tn S24 -- Pn-Tn S26

Figure 3-11. Comparison of PN-TN quadrature sets for different SN orders in zone 1 at
y=5.0 cm and z=5.0 cm.

In zone 2 (Figure 3-12), the solution obtained with an S22 PN-TN quadrature set is


more accurate than what obtained with level-symmetric. The S22 PN-TN yields a


maximum relative error of -9% compared to -22% from level-symmetric. Again, in zone


2, the accuracy somewhat decreases as the SN order increases, because the spatial mesh is


not consistently refined.



1.30E+00
1.20E+00 ---------------
2 1.10E+00
W 1.00E+00
3 9.00E-01
8.00E-01
7.00E-01
5 15 25 35 45 55
X (cm)

--LQn S20 --Pn-Tn S20 ---Pn-Tn S22
-- Pn-Tn S24 -- Pn-Tn S26

Figure 3-12. Comparison of PN-TN quadrature sets for different SN orders in zone 2 at
y=55.0 cm and z=5.0 cm.









3.5.2 CT-Scan Device for Medical/Industrial Imaging Applications

The model of a CT-Scan device used for medical/industrial applications is used in

this section to verify the accuracy and performance of the RAR technique. A CT-Scan

device utilizes a collimated x-ray source (fan-beam) to scan an object or a patient. The

main components of a CT-Scan device are an x-ray source mounted on a rotating gantry

and an array of sensors. The patient is positioned on a sliding bed that is moved inside the

CT-Scan. The mesh distribution for this model, is shown in Figure 3-13.




20.0 cM





740 0m
Aimay ofsewtrs

Figure 3-13. Cross-sectional view of the CT-Scan model on the x-y plane.

Figure 3-13 shows the simplified PENTRAN model which represents the x-ray

directional source ("fan" beam), a large region of air and an array of sensors. The size of

this model is 74 cm along the y-axis and 20 cm along the x-axis. The array of detectors is

located at 72 cm from the source along the x-axis.

The materials are described using one-group cross-sections from the 20-group

gamma of the BUGLE-96 cross-sections library. The group corresponds to an x-ray

source emitting photons in an energy range of 100 KeV to 200 KeV. The cross-sections

were prepared using a P3 expansion for the scattering kernel.

Because of the presence of large void regions and a directional source, the solution

of the transport equation is significantly affected by the ray-effects.3 One remedy is to use











high order quadrature sets with biasing, such as RAR. We compared the solutions


obtained with an Ss0 PN-TN quadrature set. The RAR technique has been applied to an S30


PN-TN quadrature set; the biased region on the positive octant extends from z=0.0 cm to


z=0.3 cm and the azimuthal angle extends from 0.0 to 5.0 degrees. In the biased region an


Slo PN-TN quadrature set is used. The PN-TN quadrature set biased with RAR resulted in


142 directions per octant. The unbiased Ss0 PN-TN quadrature set yielded 325 directions


per octant. The S20 level-symmetric quadrature set yielded 55 directions per octant.


Figures 3-14, 3-15, and 3-16 show the flux distributions in the x-y plane, obtained


with the level-symmetric S20, Ss0 PN-TN and S30 PN-TN with RAR, respectively.

Fluxtpkm;/E)
1.54E-18
1.22E-f4
g69E-07
7,6 E407
20 26E4107
4 82E-07
3 2E407
234E-IO7

S10 1.93E407
1.91E-H7






11 E407
1.19E~07
9.47EO6
7.93E411B
5 94E10
10 20 30 40 50 60 70 71 EI
x (Ic) 3 73E4~
2 .6EiO














1 51E.O5
Figure 3-14. Scalar flux distribution on the x-y plane obtained with an S20 level-
symmetric quadrature set.


Flux (p.ra1lE)

8.19E4B6



6.34' E"



.D2E46O
2 3.23E20 30 40 50 60 70



X(Cm) 2.9E45G
2. 1956E-10

1.24E4DG
5.D2E*05
10 20 30 40 50 60 70 16C-4e0
x (cm) 2.SDEID

t.24E4DS

Figure 3-15. Scalar flux distribution on the x-y plane obtained with an S50 PN-TN
quadrature set.



















I 10
>,,il


10 20 30 40 50 60 70
x (cm)


FIIjh i..Ir,T, F I
S6 EtCS
4.41 E+0
3A9E 06
2.77 E-'
2 ,19 E-W
1.74E+46
138E+06
1JOSE4c6
5 j5 E+05
5.43 E.+5
I4.31E105
3.41E-B5
3.70E+OM
2.14E+D5

8.146E+05
.70 E+04


Figure 3-16. Scalar flux distribution on the x-y plane obtained with an S30 PN-TN
quadrature set biased with RAR.

The above results indicate that the level-symmetric quadrature set exhibits

significant ray-effects, while S50 PN-TN and S30 PN-TN with RAR quadrature sets, yield

similar solutions without any ray-effects. The main advantage of using a biased

quadrature set is the significant reduction in computational cost and memory requirement.

Table 3-3 compares the CPU time and memory requirements for the three calculations

presented above.

Table 3-3. CPU time and total number of directions required for the CT-Scan simulation.
Quadrature Set Directions CPU Memory ratio Time ratio
Time(sec)
S50 PN-TN 2600 166.4 1.0 1.0
S30 PN-TN RAR (S0o) 1136 79.4 0.51 0.47
S20 LQN 440 33.3 0.2 0.2
a memory and time ratio are referred to the S50 PN-TN quadrature set.

The RAR technique lessens the ray effect in the flux distribution and greatly

reduces the computational time by more than a factor of 2 compared to S50.

The new quadrature sets biased with the OS rather than the RAR technique have

also been examined based on the CT-Scan model.9 Figure 3-17 compares the results of

PENTRAN with a reference Monte Carlo solution. For all cases, the first direction of the











lowest level in quadrature set is split in 9 or 25 directions; for example, PN-TN 22-2-55

corresponds to PN-TN S22 with direction 55 split in 9 directions. All the quadrature sets

biased with the OS technique yield accurate results within the statistical uncertainty of

the Monte Carlo predictions.

Due to the significant ray-effects, the level-symmetric S20 quadrature set without

ordinate splitting yields poor accuracy. Note that, even by using high order quadrature

sets, such as PN-TN S28 (840 directions), the solution at detector position is under

predicted by -21%.


9.00E+08
8.50E+08 -
8.00E+08
7.50E+08
x 7.00E+08 -

> 6.50E+08
>c 6.00E+08 \
5.50E+08 -

5.00E+08 S20-LS (No OS)
4.50E+08
4.00E+08
0 2 4 6 8 10 12 14 16 18 20
Y-Axis Position (cm)

Pn-Tn22-2-55 Pn-Tn-20-3-46 MCNP Tally Flux S20-LS-3-46 S20-EW-5-46

Figure 3.17. Comparison of the scalar flux at detector position (x=72.0 cm).














CHAPTER 4
DERIVATION OF THE EVEN-PARITY SIMPLIFIED SN EQUATIONS

This chapter presents the initial derivation of the Simplified Spherical Harmonics

(SPN) equations starting from the PN equations in 1-D geometry, and it discusses the

issues related to the coupling of the SPN moments on the vacuum boundary conditions.

Because of this peculiarity, the implementation of the general SPN equations into a

computer code proved to be cumbersome. However, I will present the initial derivation of

the SP3 equations, successively implemented into a new computer code named PENSP3

(Parallel Environment Neutral-particles SP3).

To overcome the difficulties related to the coupling of the SPN moments in the

vacuum boundary conditions, I adopted a different formulation based on the Even-Parity

Simplified SN (EP-SSN) equations. These equations are derived starting from the 1-D SN

equations, and using the same assumptions made for the derivation of the SPN equations;

however, the main advantage of this formulation is the natural decoupling of the even-

parity angular fluxes for the vacuum boundary conditions.

Therefore, a Fourier analysis of the EP-SSN equations will follow, along with the

derivation of a new formulation to accelerate the convergence of the source iteration

method applied to the EP-SSN equations. This chapter is concluded with the derivation of

the 3-D P1 equations. I will compare the P1 equations with the SP1 equations, and I will

describe the assumptions made in the derivation of the SP1 equations and the relation

with the spherical harmonics P1 formulation.









4.1 Derivation of the Simplified Spherical Harmonics (SPN) Equations

The SPN equations were initially proposed by Gelbard18 in the early 1960s.

However, they did not receive much attention due to the weak theoretical support.

Recently, the theoretical foundations of the SPN equations have been significantly

strengthened using a variational analysis approach.2122

I derive the multigroup SPN equations starting from the 1-D multigroup PN

equations and by applying the procedure originally outlined by Gelbard. The multigroup

1-D PN equations are given by

n+1 a,^ (X) n QO i (X) r 1
+, + [1, g(X) Cs, (x)(., (x) = q,,,g(), (4.1)
2n +1 x 2n +1 x

for n=0, N and g=1, G,

where

G L 2+1 s(+1
qg (x) P 4 (/)og, (x)Og, (X) + qf,g (x) + (x) (4.2)
g'=1 1=0 4i 7
g' g

In Eqs. 4.1 and 4.2, I defined the following quantities:

o,g (x), total macroscopic cross-section in group g.
os,,g (x), Legendre moment of the in-group macroscopic scattering cross-section of
order n.
,sg',,g (x), Legendre moment of the group transfer macroscopic scattering cross-section
of order 1.
ng (x), Legendre moment of the angular flux of order n.
q,g(x), fission source.
S, (x), Legendre moment of the inhomogeneous source of order n.
G, total number of energy groups and
L, order of the Legendre expansion of the macroscopic scattering cross-section (L








In Eq. 4.1, the angular flux is expanded in terms of Legendre polynomials as

L21+1
,g (x, P) = 21 (p)1,g (x) (4.3)
-0o 4;r


In Eq. 4.1, the term g is defined to be identically zero when n=N. This
dx

assumption closes the PN equations, yielding N+ equations with N+ unknowns. The

procedure prescribed by Gelbard to obtain the 3-D SPN equations from the 1-D PN,

consists of the following steps:

4. Replace the partial derivative operator in Eq. 4.1 for even n with the divergence
operator (V-).
5. Replace the partial derivative operator in Eq. 4.1 for odd n with the gradient
operator (V).

By applying this procedure to Eq. 4.1, it reduces to

n+l n 4.4a)
V + (F)+ V (r) + tg () s g(), (F ,g (F) = q, (F) (4.4a)
2n +1 2n +1

forn=0,2,...,N-1, g=1, G, and re V,

n+l n
Vn+ I, () + g, (V ) + tg () agg(F)ng () = q (r), (4.4b)
2n +1 2n +1

forn=1,3,...,N, g=1, G, and F V.

The SPN equations can be reformulated in terms of a second-order elliptic operator,

if one solves for the odd-parity moments using Eqs. 4.4b i.e.,

1 n+1 1 n qg (F)
(+) 1= n V(1 lg() nI 1g()) + i (4.5)
ran,g 2n+ an,g 2n+1 Can,g

where

an,g = tt,g (F) sng-,gg () (4.6)

and then substitute Eqs. 4.5 into Eqs. 4.4a to obtain









n+l n+2 1 n+l n+l 1
2n + 1 2n + 3 an+1 g 2n + 1 2n + 3 an+lg
n n 1 n n-Il 1(
V Vng n2,g (4.7)
2n+1 2n-1 oan,,g 2n+12n -1 oan,,g

+n+1 q ng n qy ,
S g2 + 1 Can+,g 2n +1 Can,g


for n=0,2,...,N-1 andg=1, G.

Note that for simplicity, the spatial dependency of q has been eliminated. The SPN

equations yield a system of (N+1)/2 coupled partial differential equations that can be

solved using standard iterative methods, such as Gauss-Seidel or Krylov subspace

methods. The main disadvantage of this formulation is that it yields Marshak-like

vacuum boundary conditions, coupled through the moments. Because of this issue the

implementation of this formulation into a computer code becomes cumbersome.

However, to study the effectiveness of the SPN method, I developed a 3-D parallel SP3

code,25 PENSP3 (Parallel Environment Neutral-particles SP3). The PENSP3 code is based

on Eqs. 4.8a and 4.8b, which are derived assuming isotropic scattering and isotropic

inhomogeneous source.

DI, (F)VF(F)+ t,,g ()F(F)= 2,,t (F02g (r)+ ,,g g ()00,o (r)
+ 1 GCoG C-( (4.8a)
+ o^ (F' g' (F) + k g(F)' (F) + S (4.8a)
g1=1 k 9 1
g' g

D2,g 0'42,g 0 + -g (F)02,g 0


5 g'

gfor g 1, G,
forg=1, G,


where









1 9
(F) 202,g )+00g(F), Dg (F) = and D2,g(F) 3 g (4.9)


The SP3 Marshak-like vacuum boundary conditions for Eqs. 4.8a and 4.8b are given by

3 (f )+ D,) 0( V)F (f)= i 2/+/b ( 2 f, p, (p)d/d (4.10a)
2 8


2,;g ()+ D ( V) () = F ()+2P (, )d .(4.10 Ob)
40 40 g

Eqs. 4.10a and 4.10b are termed Marshak-like boundary conditions, because in 1-D

geometry they reduce to standard Marshak boundary conditions. Implementation of these

formulations into a computer code is difficult because of the coupling of the SPN

moments.

The reflective boundary condition is represented by setting the odd-moments equal

to zero on the boundary, i.e.,

i O (Fb)= 0, for n odd, (4.11)

where Fb e OV and h is the normal to the surface considered.

4.2 Derivation of the Even-Parity Simplified SN (EP-SSN) Equations

I have derived the Even-Parity Simplified SN (EP-SSN) equations starting from the

1-D SN equations given by


um a (X1 ) + (x)(x, )VX,l)= Q(X, um), form 1, N, (4.12)

where

L L
Q(x, P/m)= (2/ +1)P ( ) (m (x),j (x) + (2/ + i)P1 (,m)SI (x) + qf (x), (4.13)
1l0 1/0









1 1 N
qf(x) = v, (x)O (x); (x) = M WP (/Pm)(x,/ m). (4.14)
k 2 m-=

A Gauss-Legendre symmetric quadrature set (PN) is considered, where /m e (-1,1),

M
wm = 2.0, and M N(N+2). In Eq. 4.13, L is the order of the Legendre expansion for
m=1

both the macroscopic scattering cross-section and the inhomogeneous source (L
Therefore, the even- and odd-parity angular fluxes are defined by


1M = 1 [(x, m) + i(x,-m)] (even), (4.15a)
2


w = [y(x, m)- (x,-/m )] (odd). (4.15b)
2

To reformulate the 1-D SN equations in terms of the even- and odd-parity angular

fluxes, I rewrite Eq. 4.12 for- /m as


m ) + r, (x)y(x,-'m) = Q(x,-/m), for m 1, N. (4.16)
ox

Then, I add Eqs. 4.12 and Eq. 4.16 to obtain

am [(xm)- ( m + ot (X)[),(X, Pm)+ m(X,-m)]= Q(x, m) + Q(x,--m), (4.17)


and use the definitions of even- and odd-parity angular fluxes (given by Eqs. 4.15a and

4.15b), to obtain


2/1m O (x)+ 2c, (x), '(x) = (21+ l)[P~(/pm)+ P,(-p/m ) ,,(x)O,(x)+
1 0 (4.18)

-(21 + 1)[PI (pm) + P, (-/m)]S, (x) + 2qf(x).
1-0

Consider the following identities for the Legendre polynomials.3

P, (-/) = P, (p), for I even, (4.19a)






53


P( (-P) = -P (p/), for I odd. (4.19b)

Eq. 4.18 can be rewritten as


m m (x)+ C, (x)yi (x)= 1 (2/1+)P (m )[ (x)A (x) + +S (x)] + qf (x). (4.20)
1=0,2
even

Similarly, by subtracting Eq. 4.16 from Eq. 4.12, I obtain


m [(x, () -X)] + )[y(x, )- (x,-, )] Q(x, ,m) -Q(x,-/i), (4.21)


and by using the definitions of even- and odd-parity angular fluxes, I obtain

aE= L
2/,m l'(x) + 2 (x),y (x) = (2/+1)[Pl (,m)-P (-,m )] ,(x) l(x)+
L10 (4.22)

Z(2 + 1)[PI(/m) P(-/ ) ().
1-0

Following the use of the Legendre polynomial identities (Eqs. 4.19a and 4.19b), Eq. 4.22

reduces to


P A (x) +o a() (x)= f (2 +1) (1 (pm)[a (x)o (x) + S (x)]. (4.23)
x 1=1,3
odd

Now, the odd-parity angular fluxes are then obtained from Eq. 4.23 as


Vm (x)= V y (x)+ 1 (2/+l)P, (p,,) )(x)cl (x)()+S,(x)]. (4.24)
C (X) xt () 1-t1,3
odd

Then, using Eq. 4.24 in Eq. 4.20, I obtain

Ia k /a (X) + ct (-x)/ (X) = (2/ + 1)P (pm, )[a (x)x (x) + S (x)]
x o (x) x 1=0,2
evn (4.25)
a 'm (2/+ 1)PI (p,)[, (x)q (x) + S,(x)]+ qf (x).
S( dd) 1=1,3
odd










Finally, the EP-SSN equations in 3-D Cartesian geometry with anisotropic

scattering kernel and anisotropic inhomogeneous source of arbitrary order L, are obtained

by applying the procedure outlined by Gelbard (i.e., substitution of first order partial

differential operators with the gradient operator) to Eq. 4.25.

2 L-1
V Vl m(r)2 + Et(r))m (r) / (2/ + I)P, (km )[, (F)q1 (F) + S, (F)]
( ) 1=0,2
even
S\ (4.26)


() 0-1,3
odd

for m 1, N/2,

where


(F) = m -v (r)+ 1 (2/+l)P, (m) J (F) (F)+ (F) (4.27)
,t (r) m (F) 1,3
odd

Due to the symmetry of the Gauss-Legendre quadrature set, the EP-SSN equations

only need to be solved on half of the angular domain, e.g. / e (0,1). The moments of the

even- and odd-parity angular fluxes are evaluated by

N/2
S(r)= ,WmP (,m )Vf (), for / even, (4.28a)
m=l

and

N/2
((r) = ZwmP (m )V (0), for odd. (4.28b)
m=l

The multigroup form of Eqs. 4.26 and 4.27 with anisotropic scattering and source

are written as










S- VgE g) + (r)g, (F) = f (21 + 1)P, (m )c,,,g,'g (F)q,,g (r)

even

G L L-1
-V. g- 1 (21+1)P,(,m) ,(),g,(F) + (2/ + )P,(,m,)S,,(F)
tg (F )g'=1=1,3 1=0,2
odd even

G L
- -(2/+l)P, (),u),, (F) +qf ,g()
ut,g ( )g' =1 =1 3


(4.29)

and



t ) t,g ( ) g'=1=1,3
odd

(4.30)

form 11, N2 and g 1, G.

4.2.1 Boundary Conditions for the EP-SSN Equations

The boundary conditions for the EP-SSN equations are based on the assumption that

the angular flux on the boundary surface is azimuthally symmetric about the surface

normal vector. The EP-SSN boundary conditions follow directly from the 1-D even-parity

SN boundary conditions. Hence, by considering the positive half of the angular domain

for the EP-SSN equations, the 1-D source boundary condition at the right boundary face is

given by

w (x,, P/ )-- n (x,,/ )=V(x,--/m), (4.31)

and the corresponding condition in 3-D is given by

(F,,P) n ( 0 /(F Pr) = Vy( ,-p) (4.32)









Note that in 3-D geometry, the EP-SSN formulation requires the incoming boundary

flux to be azimuthally symmetric about the surface normal vector. The 3-D albedo

boundary condition is given by


n.(F,/m.) = (/m) (4.33)
1+a

Note that in Eq. 4.33, the vacuum boundary condition is obtained by setting a = 0

while the reflective boundary condition is obtained by setting a = 1.

Note that the main advantage of the EP-SSN formulation compared to SPN is the

decoupling of the even-parity angular fluxes for the vacuum boundary conditions.

4.2.2 Fourier Analysis of the EP-SSN Equations

The EP-SSN equations are solved iteratively using the source iteration method. This

method is based on performing iterative cycles on the scattering source; moreover, the

method has a clear physical interpretation that allows one to predict classes of problems

where it should yield fast convergence. The source iteration method for the EP-SSN

equations is defined by

HL,m,g (F)m (r) = qm,g-g +q', for m 1, N/2 and g 1, G, (4.34)

where, HL,mg is the EP-SSN leakage plus collision operator, q ,m,g-g is the in-group

scattering source, and q' is a fixed source term that includes scattering transfers from

energy groups other than g, the external source and fission sources. The iterative method

begins by assuming a flux guess; then, Eq. 4.34 is solved for / '+1 and the in-group

scattering source is updated. This process continues until a certain convergence criterion

is satisfied.









The convergence rate of any iterative method is characterized by the spectral

radius. For the source iteration method, in an infinite homogeneous medium, it is well

known that the spectral radius is equal to the scattering ratio (c), given by


c, = gC g (4.35)
'tg

The scattering ratio is bounded between 0 and 1; hence, a c-ratio approaching one

means that the problem will converge slowly, while, oppositely, a c-ratio close to zero,

indicates a fast converging problem.

Fourier analysis is the tool of choice to analyze the convergence behavior of

iterative methods. For simplicity, I will consider the 1-D EP-SSN equations with isotropic

scattering and source, given by

a fU2 a
S-(X) /2 (x, ) + U (x)Ql'1 2(x, ) = co (x)ol(x) + Sex(x), (4.36)
ox o- (x) ox

which following division by ua (x), reduces to

a f2+a (x, S) + 1 ) = c S(x)+ x(x-) (4.37)
x C 2(x) x / + V+ o 2- (x)

where c is the scattering ratio defined by Eq. 4.35.

The EP-SSN equations can be rewritten in terms of the error between two

consecutive iterations as

a /12 aU E 1 /2 (.,O) (4.38)
t- (x (x) 2(xx 2

where

E/2 ( P) = IY /z(X, P)- 1/ 2(x, P), (4.39)









o,1 (x) = 0 (x)- 0"_,1 (x). (4.40)

The error terms are then expanded in terms of the Fourier modes, considering an

infinite homogeneous medium, the Fourier ansatz is defined as follows

11/2 (, p) = f(p) exp(iAx), and 5o,1 (x) = exp(iix), (4.41)

where

i = and A (- o, o).

By substituting the above relations into Eq. 4.38, I obtain the EP-SSN equations

mapped onto the frequency domain, resulting in function f(/) given by


f(u) = .(4.42)
1+ -p


Therefore, the spectrum of eigenvalues is obtained by observing that the error in the

scalar flux at iteration 1+1 can be written as


so, (x) = d 1 i/2(x, ) =exp() exdP ) dJ (/A) =exp(iAx)Jd (4.43)
1+ -P


By performing the integration over the angular variable in Eq. 4.43, the spectrum of

eigenvalues is found to be equal to


arctan -
W(A) = arctan (4.44)

Ut

The result obtained in Eq. 4.44 is similar to what is obtained for the SN equations.

The spectral radius is found to be equal top = max[)(A)] = c However, the convergence

behavior of the EP-SSN equations is also affected by the value of the total scattering









cross-section. For optically thin media, where the total cross-section assumes small

values, Eq. 4.44 suggests that the convergence should be very fast, and in the limit as

ot -> 0, the spectral radius will tend to zero.

4.2.3 A New Formulation of the EP-SSN Equations for Improving the Convergence
Rate of the Source Iteration Method

As discussed in the previous paragraph, the performance of the source iteration

method applied to the EP-SSN equations is similar to the SN equations. However, I have

derived a new formulation of the EP- SSN equations which reduces the spectral radius for

the source iteration method. Appendix B addresses the performance of the new

formulation for a criticality eigenvalue benchmark problem; note that the new

formulation is a key aspect for the successful implementation of an acceleration method

for the SN equations. The main idea behind the new formulation is to remove the in-group

component of the scattering kernel for each direction. In order to reformulate the EP- SSN

equations, we note that the even-moments in the in-group portion of the scattering kernel

can be expanded as follows

L-1 L-1 N/2
1 (2/+ 1)P, (Im )a ,g (F)qg (F) = (2/ + I)P1 (um )Ci,^ (F) ] WmP, (/m ) ,g (F).
1=0,2 1=0,2 m=l
even even

(4.45)

The term V 'g (F) is consistently removed from the in-group portion of the

scattering kernel and from the collision term on the left-hand side of the EP- SSN

equations.










2 L-1 N/2
-V. / VV/,() + ,(F)R (F)= 1 (2/+l) I(/(m)cl,gg (F) 1 Wm,P, (m )Vf,g (F)
tg ) ) 1=-0,2 m'=1
even m' m

G L-1 G L
+ 0 (21+ I)P, (km,) g'-g ()1,g ( m 1 (2/ + )P, (m ),g'->g 1,g
g'=l 1=0,2 ut,g g-'=11=1,3
g' even odd

L-1 m G L
+ 1(21+1)P (,m)Sg (F)-. 1 (2/+l)P(),g() +f,g()
1=0,2tg (F '-1-1,3
even odd
for m 1, N/2 and g 1, G. (4.46)

Note that in Eq. 4.46, the total cross-section is replaced, in the collision term, with

a direction-dependent removal cross-section as follows

L-1
,g (F)= ctg (F)- (2/ + 1)P (/km)2 mQ ,gg(F). (4.47)
1=0,2
even

In this new formulation, the main idea is to remove a "degree of freedom" from the

iteration process in order to reduce the iterations on the component ,g (rF) This


modification leads to a drastic reduction of the spectral radius.

4.3 Comparison of the P1 Spherical Harmonics and SP1 Equations

In order to understand the assumptions on which the SPN and the EP-SSN equations

are based, it is useful to examine the 3-D P1 spherical harmonics equations.6 The

expansion in spherical harmonics of the angular flux can be written as follows


V(~, )= 1 ( (2/ +1)Pm (cos )[ 1m (F) cos(m)o) + /y, (F) sin(mp)l, (4.48)
1=0 m=0

where

0 < 9 < zn andO < ( < 2i .

In the following discussion, for simplicity I will assume a Pi expansion of the

angular flux in spherical harmonics, given by









, ) =PO (cos 9)o (F) + 3P0 (cos 9)/0oF (F) + 3P' (cos 9)[i/f, () cos ( + 72,(F) sin P].

(4.49)

By substituting the definitions of the Associated Legendre polynomials6 in Eq.

4.49, I obtain the Pi expansion for the angular flux:

,(, ) = Voo (F) + 3p/o (F)- 3 sin 9[,, (F) cos o + Yii (F) sin ], (4.50)

where

u = cos 9.

The derivation of the SPN equations outlined by Gelbard, assumes implicitly that

the angular flux be azimuthally independent, and hence symmetric with respect to the

azimuthal variable. By introducing this assumption on the P1 expansion of the angular

flux in Eq. 4.50, I obtain

(r, )= ) /, fdp = t oo (F) + 3V'o (F) -3 sin .9 [y9 (F)cos p + 7, (F)sin (pip.

(4.51)

Therefore, by performing the integration on Eq. 4.51, I obtain

(F,/ ) = 00 (F) + 3/ /0 (F). (4.52)

It is evident that the angular flux obtained in Eq. 4.52 is equivalent to the SPi

angular flux where, y,'o is the scalar flux and y/10 is the total current.

The general formulation of the multigroup PN equations6, with anisotropic

scattering and source, is obtained by substituting Eq. 4.48 into the linear Boltzmann

equation and deriving a set of coupled partial differential equations for the moments

, (F) and 7 (F).
/I I










2(1 + m + 1) + 2(1 m) +
Oz Oz Ox


+(l + m +2)(l + m + 1V) '- m
ax


+ 2(21+ 1)lo,gV I


-(1I


1 + Yg+l\m-
9y


1y


V'-l + 0) 1g+l


m- )( l-m) l 1m +
c 'x


2Slm,g ,


2(1+ m + 1) +l +2(1
Oz

+ (1 + m + 2)(1 + m + 1)

+2(21+ 1)ol,g = 2S'


nm) IlIm
Oz


'Ylg+ l+1 +
ay


+ Ov ay


0-/g+ lm
a IIm


+ aym M1
ax


ay


(1 m -1)( -m)


g
g


/ O lm 1
ay


(4.53b)


forg= 1, G,


where


,g = tg sl,g-g *


Therefore, the P1 equations are obtained by evaluating Eqs. 4.53a and 4.53b for

/=0, 1 and m=0, 1, as follows

(1=0, m=0)


2



8z
2 c "
z


+2K



+ 2


Ox



ay
ar1"
dr


+6a +2
x


+ 2 Vf
&


dy2110


2Soo g,


(4.54a)



(4.54b)


+ + 2-o,g /0
cYx


-6"21 910
S+ 67 VI 0


2S1o g
9<;g


(4.54c)


(4.53a)


+ a-iy } 1
a


(1 1, m=0)


k


4 2
Oz










4 20 +2 020 + 6
8z 8z


ay 21 + 60-1,Y = 2S'o g.


I y O
a y ax)J


20g



V20
ay


+0 +12K
9y


+-+ 60-,g vl11
x a, )


/20 12 22 a 22 + 6 71,gY


2SIg, (4.54e)


2S',g. (4.54f)


The terms with /1> and m> are dropped from Eqs. 4.54c through f, yielding the

following relationships


1 1
30-1,g


Og0 S10,gO
az 30-g
1, g


1 70 S',g
710 +
3017 az 307-g

01 0g+ 700 S g
/g 30 lg a Oy .3.g
30- 30-1g


(4.55a)



(4.55b)



(4.55c)


1 1 17 o
- 30,I Sy


Y0go + S 0'g
ax J 30-1


Then, by substituting Eqs. 4.55a, c and d in Eq. 4.54a, I obtain


,1 a 1
30-g OX30-0 +
30,7, gx 3(7,g


c 1
y 30-1,g


Oy~
+ O,gy00 = S00, + Sl,
8x


where


a S, S~S al (S'
S 3 0 ,g x3- g aJ g31, J
1,g Z 30,~, 8x (30,~, B


Analogously, by using Eqs. 4.55b, c and d in Eq. 4.54b, I obtain


(=1, m=1)


6 a~ f21 2 a
az y


(4.54d)


6 21g
8z


(4.55d)


(4.56)


(4.56a)









1 1 8' 8 1 8-gf
.- -V0 00 + 0+ og + g00 SOg + Sg (4.57)
3cg cx 3c g y 3cl,g ax

where

SSI I'og SI'g + Slg (4.57a)
Ozg az 30,,g ) 33,g) y 3,g"


Eqs. 4.56 and 4.57 constitute a coupled system of partial differential equations for

Vog and yog, which must be solved iteratively. Recall that the assumption made in the

SPN methodology is that the angular flux is azimuthally symmetric; therefore, to obtain

the SPI equations (Eq. 4.58 or 4.59), terms such as yg are dropped from Eqs. 4.56 and

4.57, as follows

V. g v +C00 =SO-V. g (4.58)
3cl, g 0 3c lg)


or


-V. + ,gq0 = S, V. g (4.59)
3cg Ig 3c g )

Here, I can also conclude that in the case of a homogeneous medium, with isotropic

scattering, the P1 and the SPi equations yield the same solution, because the azimuthal

dependency on the angular flux is removed. Note that this result can also be generalized

to the SPN equations.














CHAPTER 5
NUMERICAL METHODS FOR SOLVING THE EP-SSN EQUATIONS

This chapter addresses the numerical techniques utilized to solve the EP-SSN

equations; I will describe the discretization of the EP-SSN equations in a 3-D Cartesian

geometry using the finite-volume method, along with the matrix operator formulation

utilized and the boundary conditions. I will also introduce the Compressed Diagonal

Storage (CDS) method, which is fundamental for reducing the memory requirements and

the computational complexity of the iterative solvers. Further, a new coarse mesh based

projection algorithm for elliptic-type partial differential equations will be presented.

Finally, I will describe a class of iterative solvers based on the Krylov subspace

methods, such as the Conjugate Gradient (CG) and the Bi-Conjugate Gradient methods

(Bi-CG). The CG and Bi-CG methods have been implemented to solve the linear systems

of equations arising from the finite-volume discretization of the EP-SSN equations.

Furthermore, the issue of preconditioning of the CG methodology will be discussed.

5.1 Discretization of the EP-SSN Equations Using the Finite-Volume Method

The EP-SSN equations derived in Chapter 4 are discretized using the finite-volume

approach. For this purpose, I consider a general volume Vin a 3-D Cartesian geometry.

The volume Vis then partitioned into non-overlapping sub-domains Vj, called coarse

meshes. Note that, the coarse mesh sub-domains are generally defined along the

boundaries of material regions. As I will discuss in Chapter 7, the main purpose of this

approach is to partition the problem for parallel processing.









The discretization of the spatial domain is completed by defining a fine-mesh grid

onto each coarse mesh. I have derived a formulation of the discretized EP-SSN equations

which allows for variable fine mesh density on different regions of the problem; this

approach is very effective to generate an effective mesh distribution, because it allows a

finer refinement only in those regions where higher accuracy is needed.

The finite-volume discretization of the multigroup EP-SSN equations (Eqs. 4.29) is

obtained by performing a triple integration on a finite volume, dr dxdydz, as follows


-I -, ()dr = [Q ,m()+Qextg, m(F)Q Pr, (5.1)
v vtgQ

where

G L-1
Q,,,m (F)= (2 1+ )P, (/m ,),,,g',, (F) ,,g' (F)
g'= 1 0,2
even (5.1a)

m Z (21+1)P, (m)csl.g' >g (f),g' (g)
tg ( ) g'=1l=1,3
odd

L-1 L
Qextgm(F)= -(21+ )P (/,m)Sg() 'm (21+l)(P, um),g, (F), (5.1b)
1=0,2 ut,g (r)=1,3
even odd

and


Qfg () = VCf, ()0 (F) (5.1c)


For this derivation, I consider a central finite-difference scheme for generic mesh

element with coordinates x,, y, and Zk; an example of a fine mesh element and its neighbor

points is shown in Figure 5.1.











(i, j, k+1)
S (i,j+1, k)

(i-1,j,k) (i, j, k (i+1,j,k)
z

(i,j-1, k)

(i,j, k-1) x

Figure 5.1. Fine mesh representation on a 3-D Cartesian grid.

The generic fine mesh element is defined by the discretization step sizes, Axc, Ayc,


and Az, along the x-, y- and z-axis, respectively. Note that the discretization steps are


constant within each coarse mesh; hence, a non-uniform mesh distribution is not allowed.

The discretization steps are defined as follows


Lc Lc Lc
Ax = Ay = Az =-- and Avc = AxcAyAzc, (5.2)
Nc' N y Nc


force 1, Ncm

where, Ncm is the total number of coarse meshes; L L and Lc are the dimensions of


the coarse mesh (c), along the x-, y- and z-axis, respectively; and N Ny, and NC refer


to the number of fine meshes along the x-, y- and z-axis, respectively. Note that, Eq. 5.1 is

numerically integrated on a generic finite volume Ave.


I will first consider the integration of the elliptic or leakage operator (first term in

Eq. 5.1) as follows






68



J,2 r1: dx/ l/ dy k+1:d z: J ) \ (r)
S2
neihbo p Yins lon t fh k Vxa x


IL km 2 8x / 2
L- Kd1/ ^ d12 1 a iV wm' (r), =




L[A Vtg(rg) ) gg(F) Vg ()JJ

--11 2 1 +
2 7
A X CL (rF) Oyz V1M' g 2F(kY 1) Jm ,_12


For simplicity, I will derive the discretized operator along the x-axis; the treatment

is analogous along they- and z-axis. Figure 5.2 represents the view of a fine mesh and its

neighbor points along the x-axis.


Ax,
Ax,/2 Ax,/2










X;-1 X;-1/2 X X + 1/2 X,+1

x


Figure 5.2. View of a fine mesh along the x-axis.

In Figure 5.2, au represents a generic macroscopic cross-section (e.g., total,

fission, etc.) which is constant within the fine mesh. In Eq. 5.3, I evaluate the right-side

and left-side partial derivatives along the x-axis at x,+112.










fE()2 E( )(Xl/ g 2,y, Zk)- mg( ,yj,,zk)
Jtg(XI,Y,,Zk) )j/2
fE2( ) (X m mgl
fJm,g )mg/22Z ;,Zk)m= ,g Z(5.4)


'(E+) +2 ZVk I m,g( +1 ,YJ,Zk)- ~m', g)( +1/2,Yj,Zk)
1'g (,+l/2,yj,Zk)= mtg(Xl YJIZk) (5.5)
t,g (X+1 ,, Zk) x, c/2

In order for the elliptic operator to be defined, the function ,/ g (x, y, z) must be

continuous along with its first derivative f, (x, y, z) and second derivative, which

translates into the fact that the even-parity angular flux belongs to a C2 functional space,

or rg (x, y, z) e C2. Therefore, the following relationships hold true


Vg (X,+l/2 ,YjZk ) mg )(X/29 1,Yj,Zk)= m ,g+(X 1/2,YJ,Zk), (5.6)

and

fg(X+/2 1 YZk) fmg )(X1/2,Y,Zk)= (1/2 ,Zk). (5.7)


Therefore, I eliminate the value ofV/lg ( +1/2, Y, Z) in Eqs. 5.4, obtaining the

second order, central-finite differencing formula for the even-parity angular flux:

dx E dx E
E d+l1,],kmg/+1,jkmg + ,djkmg/,Jkmg
+l/2,,j,m,g (5.8)
d I+l,j,k,m,g d ,j,k,m,g

and the even-parity current density


z+l/2,j,k,m,g dx x +l,j,k,m,g V ,jk,m,g 5.9)
dj,k,m,g +d+l,j,k,m,g



In Eqs. 5.8 and 5.9, I have defined the pseudo-diffusion coefficients along the x-

axis, as

2 2 2
dx m dx km dx m (5.10)
i,],k,m,g z ,j,k,m,g i l-,]j,k,m,g 1
7t,c,j,k,g ct, +l,j,k,g c (t,,-1,,k,g c










Analogously, the expression for fE (x 1/2, k) is obtained as follows


E 2dX ,k,m,gd ,kg ( E E
-1/2,j,k,mg d + x (f ,],k,m,g i-1,j,k,m,g) (5.11)
Ij,k,m,g d l-,j,k,m,g

The partial derivatives along the y- and z-axis are discretized in a similar fashion,

yielding the finite-volume discretized elliptic operator given by


i d xJy Jl/Zdy J;k+lPdz K Q. 12m V 1,2( 1
Xmg 2 k2dj j Z 12m lk E g
S1/2 i 2 Y k1/2 d g ()

-, ,E H H, H.
A Az, + [ ,m,g E 1,,k,n,g l ,k,nm,g -,,n,g ( ,k,,g k g (512)








2dJk m,gdx-1J,k,m g 2dJ,k,m gd- 1,],k,m g
Y = d1 +d ,ak 1mg dx d, (5.13a)
,- Ax A [r j,k m,g +1,jkm,g kj-,k,m,g ( 1,j l,,k,gg
where



2d dx 2d d2xk
d,j,k,m,g +l,J ,k,m,g ,j,k ,m,g ,k,m,g 1 )
]'i,+l,m,g d +d l 'd'lmg +d m (5.13)
,jikmg +l,j,k,m,g i,j,k,m,g -Il,j ,k,m,g

2dy dY 2cY dY

S i, ,k,m,g i,j+l,k,m,g i, ,k,m,g d ,j-I,k,m,g
i,],k,m,g j+l,,,g i,],k,m,g Ij ij-,k,m,g


2Ykdk,m,g z ,k, z ,Yk,k,m,g ,k, ,gd (5.13
i,,j,k,m,g + d,-j,k+l,m,g i d j,k,m,g d, ,,k-1,m,g

and

2 2 2
S/m /m jf /m54
dy dy = dd = (5.14)
,,t,,j,k,g lc ,t,,j+l1,k,g (7c t,i,J-1,k,g y c

2 2 2
dz = mI z = (5.15)
I,j,k,m,g n- di,j,k+l,m,g d ,j,k-,m,g n (5.15)
tl,],k,g z c t,l,],k+l,g c t,l,],k-1,g Z c

Finally, by integrating the remaining terms of the EP-SSN equations, I obtain the

complete multigroup EP-SSN formulation with anisotropic scattering and source as

follows











- Ay Az,[a,+,m, l,,k,m,g V,k,m,g- ,z-i,m,g ( i,k,m,g

- Ax cAz [,l+l,m,g Vm(V,J +l,k,m,g V- ijk,m,g ) ,j-l,m,g ,(Vi,k,m,g

- AX Ay 1,/k ,g jk+l,m,g I,],k,m,g )Yk,k,m,g ( Ijk,m,g
G L-i
+ t,gj kV ,,k,m,g c = Av (2/+1)P ( )g,g'->g,,j,k,g',,j
g'= 1=0,2
even
AG L
-Ay Az, m Z(2/+l)P(m)csl,g'->g,,,k (1,g',+l/i2,,k
't,g,i,j,k g'= 1=1,3
odd
G L
-Ax Az, m Z(21+ l)P, (Um)gsl,g'->g,,,,k (1,g',,j+l/2,k
't,g,z,j,k g'= 1=1,3
odd
G L
SAx c (21+y )PI (Am)Crsl,g'-> g',,.klg,,k+l/ 2
t,g,z,j,k g'= 1=1,3
odd
L-1
+ 1(21+1)P (/m)S,g,,,j,kAv +
1 0,2
even
G L
Ay c m i:(21+1)1P(,m)sl,g'g,,J,k (Sl,g',+l/2,J,k
t,g,z,j,k g'= 1=1,3
odd
G L
-AxcAz, Z :(2/+ l)P(m)'sl,g'->g,,,k (,gj+ /2,k
0 t,g,z,],k g'= 1=1,3
odd

Axc, m Z (2+l)P (P sl,g'-g,,,,k (S,g' +/2
0t,g,z,],k g'= 1=1,3
odd

Qf,g,,j,k Av ,


V1,j ,k,m,g




,Av -
k c



l,g',-1/2,],k)



l,g',z,j-1/2,k



,g',,,k-1/2 )







-S I,g',-1/2,jk



-Sl,g',,j -1/2,k )-



-Sl,g',,j,k- 1/2)+


for c=1, Ncm, m=1, N/2, L=0, N-1, g=1, G.

The EP-SSN equations discretized with the finite-volume method can be expressed

in a matrix operator form characterized by a 7-diagonal banded structure.


(5.16)







72


-Dm g U Ug Um g
LX D Ux UY U
m,g m,g mg mg mg
Sm,g m,g mg mg
m,g g m,g g m,g
L LX D Ux
m g mg nm,g mg
Lz L Lx D Ux
Lg L LX D
mg mg m,g m,g

force 1, Nc,; m 1, N/2; g 1, G,

where

Dx = AycAzc(azn, +a1 ,,n, )+AxcZ c j,nim,g + ~,-~1,g
+AcAyc c(k,k+1mg + k ,k-im,g

Ug c =-AycAzca,+,, LX = -AycAzca,
mg mg mg C mg


UY, =-AxcAzc ,,,,g LY, = -Ax AzyP ,


g = c ycYk,k+l,m,g, g g = -AcycYk,k-1,m,g


for i=2, N -1, j=2, Ny -, k=2, N- 1, c=1, Ncm, m=1, N/2, g=1, G.


5.2 Numerical Treatment of the Boundary Conditions

The boundary conditions for the EP-SSN equations are discretized as well using the

finite-volume method. In general, the BCs can be prescribed at back (-Xb), front (+xb), left

(-yb), right (+yb), bottom (-Zb), and top (+Zb). The reflective boundary conditions are

simply derived as follows:

-Xb) f,g,2,,k = 0 +Xb) ',g,N 1 2,,k =0, (5.17a)


-Yb) ,g,, /2,k = 0 +yb) m,g,,N +1 = 0, (5.17b)


+Zb) )i,,2,,, N+1+2 = 0 .


(5.17c)


-Zb) V,g,,,12 -= 0,










The vacuum boundary conditions are obtained from Eq. 4.32, by setting a = 0.

Hence, the vacuum boundary conditions along the x-, y- and z-axis are given below:

Front side vacuum boundary condition x = +Xb


o aN,m,g E
VN,+1/2,],k,m,g Nx ,j,k,m,g
S+aN,,m,g

aNmg 1 (2/+1)Pl (m ))[s,g'>g,Njk g,N,+1/2,j,k +Sl,g',NJ+1/2,j,k
+aN,,m,g t,g,Nx,j,k g'=1=1,3
odd
(5.18a)

Back side vacuum boundary condition x = -Xb


o a,m,g E
V1/2,j,k,m,g Vl ,],k,m,g +
1 l,m,g
aMg G L (5.18b)
+ al,m,g 1 (21+1) P( Um)[ sl,g'->g,l,J,k I ,g',3/2,j,k +Sl,g',3/2,j,k
1+al,m,g (t,gl,j,k g'=11=1,3
odd

Right side vacuum boundary condition y = +Yb


0 bN,,m,g E
1,Ny+l/2,k,m,g I+b ,m,g Ny,k,m,g
bNymm,,g


1b G
+_:---- (21+1)PI(um) sl,g'->g,iNyk 1,g',iNy+1/2,k +SI,g',iN +1/2,k
+ Ny,m,g 't,g,l,Ny,k g'=l1=1,3
odd
(5.19a)

Left side vacuum boundary condition y = -yb


o bmg E
Vz,l1/2,k,m,g 1- +b 1,l,k,m,g
l,m,g

+-,m,g 1 ,(2/1+l)P(/m)[sg'->g,,l,kO,g',i,3/2,k +SI,g',,3/2,kl
+b 1,m,g t,g,i,l,k g'=11=1,3
odd







74


Bottom side vacuum boundary condition z = -Zb


o CN,m,g E
Vl,J,N,+1 2,m,g = ,],N ,m,g
S NN,mn,g
Nmg 1 (2 +1 )P,) ( o12 +S, 121
+ N,m,g gt,g,i,j,N, g'= =13
odd
(5.20a)

Top side vacuum boundary condition z = +Z


O Cl,m,g E
i,],j1 2 ,m,g + g V ',],,,g

Simg G L (5.20b)
+ ,m,g 1 Z Z (21+ 1)P,(, m)[-r,g g,z,j,o,g',',,3/2 +S,,,,,3/2
1+ ',mg ut,g,c,j,1 g'=1=1,3
odd

where

2 ,m 2 ,m


1,m,g A N,,m,g A ,

c c t,g,l,1,k A c t,gz,Ny,k


2,tm 2,tm
Cimg CN,,m,g
,g c g, 1 zc t,g,jN


and

Nc Ncm Ncm
N, = N ,, N, Y X: = Y Nc
c=l c=l c=1l

5.3 The Compressed Diagonal Storage Method

Due to the sparse structure of the matrices involved, I have adopted the

Compressed Diagonal Storage (CDS) method in order to efficiently store the matrix

operators. The CDS method stores only the non-zero elements of the coefficient matrix

and it uses an auxiliary vector to identify the column position of each element. Due to the









banded structure of the coefficients matrix, a mapping algorithm is easily defined for a

generic square matrix as follows:

a(i, j) e A,J a(i, d) E Ad
i= 1,I i= 1, jcol(i,d). (5.21)
j = 1, J d = -3,3

The algorithm defined in Eq. 5.21, maps the full structure of the matrix A into a

compressed diagonal structure, where for each element on row i, there is an associated

diagonal index ranging from -3 to 3, with index 0 being the main diagonal, and an

auxiliary vectorjcol, which stores the column position of each element. If we consider a

360x360 full matrix in single precision, with a total of 129600 elements, the memory

required for allocating the matrix is roughly 2.1 MB. However, if the CDS method is

used, the total number of non-zero elements to be stored is only 2520, for a total memory

requirement of 42 KB, which is a reduction of a factor of 50 compared to the full matrix

storage. Moreover, since the CDS method stores only non-zero elements, I have also

obtained a reduction in the number of operations involved in the matrix-vector

multiplication algorithms.

5.4 Coarse Mesh Interface Projection Algorithm

The partitioning of the spatial domain into non-overlapping coarse meshes leads to

a situation in which the EP-SSN equations have to be discretized independently for each

coarse mesh. Therefore, each coarse mesh is considered as an independent transport

problem; however, to obtain the solution on the whole domain, an interface projection

algorithm has to be used in conjunction with an iterative method. The matrix operators

have to be modified on the interfaces in order to couple the equations on each coarse










mesh. For explanatory purposes, consider Figure 5.3, which shows the interface region

between two coarse meshes.



Coarse mesh 1 Coarse mesh 2

I I
AxN A,














XN- XN-1/2 XN X1' X3/2' X2'

XN+1/2 X1/2'

Figure 5.3. Representation of a coarse mesh interface

The coordinates xN+1/2 and xl/2, represent the interface on coarse mesh 1 and 2,

respectively. As shown in Figure 5.3, the discretization of the elliptic operator for coarse

mesh 1, using the central finite difference method, would require the values of the even-

parity angular flux at points XN-1, xN, and xlr. Similarly, in coarse mesh 2, the

discretization would involve the value of the EP angular flux at points xN, xr,, and x2'.

However, the point xr, is located on coarse mesh 2 and point xN is located on coarse mesh

1; hence this term does not appear explicitly in the matrix operator for both coarse

meshes.

In order to couple the equations on the interface, I have reformulated the discretized

equations by bringing the unknown points on the right side of the equations. The

numerical discretization of the EP-SSN equations in coarse mesh 1 would yield










[ aNIg(m^g ,) -NN ia(-1,m,g 1mg im,g )]+
Et g, H, Ax( = AxQ Ax
+ ,g ENn,g N N= gN Next,g,mN NQf,g,N

(5.22)

where

2dx dlmg 2dx mgdXlmg (.3
aN,l',m,g 2d 'g and aNN lmg dXNg N-,g (5.23)
N,mg l',m,g N,mg N 1,m,g

The coefficient d,,,,g depends on the material properties and fine mesh

discretization of coarse mesh 2, and it is calculated a priori; however, in Eq. 5.22, the

term /,,,g is unknown, and hence has to be evaluated iteratively by placing it in the

source term, as shown in Eq. 5.24.

E E E E A,
aN,l,m,g N,m,g + aN,N-,m,g N,m,g N,N-,m,g N 1,m,g t,g,N Nmg

NQs,g,,N +N XNQext,g,mN + Nf,g,N+ N,1', mg g *

(5.24)

A similar equation can be formulated for coarse mesh 2, as follows

a,2' ( g i- 1,N,m,g (,,g lm,g)]
(5.25)
+ ag ,vj Rg1x = AQs,g,m,1 + extg,mi, + ZhlQf g,1,

or

E-aig^/mg +ai g g + a,N,m,g Vmg + g ,',m,g =
-- l',2',m,g f2',m,g ',2',m,g l',m,g V l',Hmg(l',m)
(5.26)
lQs,g,m,l' + AX'Qext,gml' + a 'Qf,g, + al',N,m,g N,m,g,

where

2dX dx 2dX dx
aI2 g 2'"mg lm and a Nmg (5.27)
1',2,mg d +dx d d
1',m,g 2',m,g l',m,g Nmg









Therefore, Eq. 5.24 and 5.26 are coupled through the value of the EP angular fluxes

',m,g and N,m,g The EP-SSN equations are solved iteratively starting in coarse mesh 1,

and assuming an initial guess for Vim,g. Once the calculation is completed the value of


N,,g in Eq. 5.26, is set equal to /N,g Hence, once the calculation is completed on

coarse mesh 2, the value obtained for /1 mg is used in Eq. 5.24, to update the value


of mg; this procedure continues until a convergence criterion is satisfied.

In a 3-D Cartesian geometry the coupling on the coarse mesh interfaces is achieved

exactly as described above; however, in this case the coarse meshes can be discretized

with different fine mesh grid densities. The variable grid density requires a projection

algorithm in order to map the EP angular fluxes and the pseudo-diffusion coefficients

among different grids. As stated earlier in this chapter, the variable density grid approach

is very effective to refine only those regions of the model where a higher accuracy is

needed; note that the main constraint on the fine mesh grid is the mesh size being smaller

than the mean free path for that particular material region. The main philosophy behind

the projection algorithm is derived from the multigrid method, where a

prolongation/injection operator is used to map a vector onto grids with different

discretizations.

Figure 5-4 shows the application of the projection algorithm along they-axis on the

interface between two coarse meshes.












Coarse mesh 1
(Finer grid)


Coarse mesh 2
(Coarser grid)


Grid 1 Grid 2


F3 F4


G5F G6F G7F G8F
F iF,t F.
G1F G2F G3F G4F
x x

Figure 5.4. Representation of the interface projection algorithm between two coarse
meshes.

For simplicity, I will consider the projection of a vector between two coarse

meshes, along the y-axis, as shown in Figure 5.4. The fine-to-coarse projection of a

vector is obtained by collapsing the values as follows


4
FIc Z W'FGQF,
1=1





wF=A for i=l, 4
Ac


(5.28)


(5.29)


In Eq. 5.29, AF and Ac, are the areas associated with the fine-mesh and coarse-


mesh grid, respectively. Conversely, the coarse-to-fine projection is obtained as follows

G1F = wIFF, (5.30a)


G2F = W2FF (5.30b)


where










G3F 3FFc, (5.30c)

G4F = 4FFC (5.30d)

In general, the fine-to-coarse mesh projection is obtained with the following formulation

NF
Fc =Z WFGJF, (5.31)
J=1

where

AJF
WF A (5.32)
AC

The weights in Eq. 5.32 are the ratios of the areas of the fine meshes intercepted by

the coarse meshes on which the values are being mapped. Similarly, the coarse-to-fine

mesh projection algorithm is defined as follows

N,
GF JC FJc, (5.33)
j=1

where

AF
wc (5.34)
A,,

By using the above formulations, the even-parity angular fluxes and the pseudo-

diffusion coefficients are projected among coarse meshes with different grid densities.

Note that the projected pseudo-diffusion coefficients need to be calculated only one time

at the beginning of the calculation, while, the projections for the EP angular fluxes have

to be updated at every iteration.

5.5 Krylov Subspace Iterative Solvers

Due to the size and sparse structure of the matrix operators obtained from the

discretization of the EP-SSN equations, direct solution methods such as LU









decomposition and Gaussian elimination do not perform effectively both in terms of

computation time and memory requirements. In contrast, the Krylov subspace iterative

methods, such as Conjugate Gradient (CG), are specifically designed to efficiently solve

large linear systems of equations characterized by sparse matrix operators.

Note that in many engineering applications, the matrix operators resulting from a

finite-difference discretization is usually positive-definite and diagonally dominant.

These conditions are fundamental in ensuring the existence of a unique solution. A matrix

is positive-definite if it satisfies the following condition

X' Ax > 0, for every vector # 0. (5.35)

Moreover, a matrix is defined to be diagonally dominant if the following condition holds

true.


a,, > a, for i 1, n. (5.36)
J-1


The CG algorithm is based on the fact that the solution of the linear system Ax = b

is equivalent to finding the minimum of a quadratic form given by


f(2)= -1 T A b T+c. (5.37)
2

The minimum of the quadratic form of Eq. 5.37 is evaluated by calculating its

gradient as follows


8x,
f '(x) = (5.38)









The gradient of a function is a vector field, and for a given point x, points in the

direction of the greatest increase of f(i)). Because the matrix A is positive-definite, the

surface defined by the function f(x) presents a paraboloid shape, which ensures the

existence of a global minimum. Moreover, the diagonal dominance of the matrix A

ensures the existence of a unique solution. By applying Eqs. 5.37 and Eq. 5.38, we derive

the formulation for the gradient of the function f(2), given by


f'(2) = AI + A A- b. (5.39)
2 2

If the matrix A is symmetric, Eq. 5.39 reduces to

f'(x) = Ax -b. (5.40)

Therefore, by setting f' (2) in Eq. 5.40 equal to zero, we find the initial problem

that we wish to solve.

5.5.1 The Conjugate Gradient (CG) Method

The CG method is based on finding the minimum of the function f(x) using a line

search method. The calculation begins by guessing a first set of search directions do

using the residual as follows:

d, = :0 = b Axo. (5.41)

The multiplier a for the search directions is calculated as follows

FTF
a, (5.42)
d, Ad

where i is the iteration index.

The multiplier a is chosen such that the function f(x) is minimized along the search

direction. Therefore, the solution and the residuals are updated using Eqs. 5.43 and 5.44.









x+1 = X, +ad, (5.43)

ft+ = a,Ad,. (5.44)

The Gram-Schmidt orthogonalization method is used to update the search

directions by requiring the residuals to be orthogonal at two consecutive iterations. The

orthogonalization method consists of calculating the search directions

S= +, (5.45)

where the coefficients / are given by
T +
1,+1 (5.46)
r,r,

Note that Eq. 5.44 indicates that the new residuals are a linear combination of the

residual at the previous iteration and Ad It follows that the new search directions are

produced by a successive application of the matrix operator A on the directions at a

previous iteration d,. The successive application of the matrix operator A on the search

directions d, generates a vector space called Krylov subspace, represented by

K, = span ..4 .. .4 d 0,..., A''do }. (5.47)

This iterative procedure is terminated when the residuals satisfy the following

convergence criterion

MAXr )< E, (5.48)

where E is the value of the tolerance, which is usually set to 1.0e-6.

5.5.2 The Bi-Conjugate Gradient Method

The Bi-Conjugate Gradient (Bi-CG) has been developed for solving non-symmetric

linear systems. The update relations for the residuals are similar to the CG method;