Acceleration Methods for Monte Carlo Particle Transport
Simulations
by MASSACHUSETTS INSTITUTE
Lulu Li
OFTECHNOLOGYS.M., Nuclear Science and Engineering, 2013
JUL
19
2017
Massachusetts Institute of Technology
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B.S., Physics, 2011
University of Illinois Urbana-Champaign
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Submitted to the Department of Nuclear Science and Engineering
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in Nuclear Science and Engineering
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Massachusetts Institute of Technology
February 2017
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Kord S. Smith, Ph.D.
KEPCO Professor of the Practice of Nuclear Science and Engineering
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Benoit Forget, Ph.D.
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Acceleration Methods for Monte Carlo Particle Transport Simulations by
Lulu Li
Submitted to the Department of Nuclear Science and Engineering on December 20, 2016, in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy in Nuclear Science and Engineering
Abstract
Performing nuclear reactor core physics analysis is a crucial step in the process of both design-ing and understanddesign-ing nuclear power reactors. Advancements in the nuclear industry demand more accurate and detailed results from reactor analysis. Monte Carlo (MC) eigenvalue neutron transport methods are uniquely qualified to provide these results, due to their accurate treatment of space, angle, and energy dependencies of neutron distributions. Monte Carlo eigenvalue sim-ulations are, however, challenging, because they must resolve the fission source distribution and accumulate sufficient tally statistics, resulting in prohibitive run times. This thesis proposes the Low Order Operator (LOO) acceleration method to reduce the run time challenge, and provides analyses to support its use for full-scale reactor simulations. LOO is implemented in the contin-uous energy Monte Carlo code, OpenMC, and tested in 2D PWR benchmarks.
The Low Order Operator (LOO) acceleration method is a deterministic transport method based on the Method of Characteristics. Similar to Coarse Mesh Finite Difference (CMFD), the other acceleration method evaluated in this thesis, LOO parameters are constructed from Monte Carlo tallies. The solutions to the LOO equations are then used to update Monte Carlo fission sources. This thesis deploys independent simulations to rigorously assess LOO, CMFD, and unaccelerated Monte Carlo, simulating up to a quarter of a trillion neutron histories for each simulation.
Analysis and performance models are developed to address two aspects of the Monte Carlo run time challenge. First, this thesis demonstrates that acceleration methods can reduce the vast number of neutron histories required to converge the fission source distribution before tallies can be accumulated. Second, the slow convergence of tally statistics is improved with the ac-celeration methods for the earlier active cycles. A theoretical model is developed to explain the observed behaviors and predict convergence rates.
Finally, numerical results and theoretical models shed light on the selection of optimal sim-ulation parameters such that a desired statistical uncertainty can be achieved with minimum neutron histories. This thesis demonstrates that the conventional wisdom (e.g., maximizing the number of cycles rather than the number of neutrons per cycle) in performing unaccelerated MC simulations can be improved simply by using more optimal parameters. LOO acceleration provides reduction of a factor of at least 2.2 in neutron histories, compared to the unaccelerated Monte Carlo scheme, and the CPU time and memory overhead associated with LOO are small. Thesis Supervisor: Kord S. Smith
Title: Professor of the Practice of Nuclear Science and Engineering Thesis Supervisor: Benoit Forget
ACKNOWLEDGMENTS
I am extremely grateful to my thesis co-advisor, Prof. Kord Smith, for his seemingly endless support and encouragement. His door has always been open, and he is more than generous in sharing his wealth of knowledge and insights. The amount of energy he possesses, his passion and work ethics have also had a profound influence on me.
I also want to express my gratitude towards my co-advisor, Prof. Benoit Forget, who provided me one of the first exposures to reactor physics back in 2010. He has always been patient, and provided the vision and guidance that makes this thesis possible.
I would also like to thank Koroush Shirvan, whose generous guidance started when I was an undergraduate, and extended into the years that we share the same office. His knowledge on reactor technologies and the diligence he possesses have benefited me tremendously, and I am extremely grateful for them.
I also owe my gratitude to my mentors Rodolfo Ferrer and Tamer Bahadir from Studsvik Scan-dpower, Inc. The three summers I interned at Studsvik offered me such an unique opportunity to learn from the best industry practices, and I want to thank everyone I met from the company for being encouraging and helpful, especially Arthur DiGiovine, David Dean, and Joel Rhodes. A number of PhD graduates from my research group have been instrumental for my time at MIT, including Paul Romano, Bryan Herman, Nathan Gibson, and Jeremy Roberts. Paul authored OpenMC which serves as the code platform of this thesis. He has also been one of the people that encouraged me to join MIT and this research group, which I owe my gratitude to. Bryan and Nathan were teaching assistants for reactor physics courses, though for me their office hours extended way beyond the semesters, and they have always been my to-go persons for questions and discussions. I also benefited tremendously from the various fascinating theoretical discussions with Jeremy Roberts, even though many of them are not directly related to our own research projects.
I would like to thank my best friends, Lindsay O'Brien and Jessica Hunter, for being fun, supportive and intelligent. My time at MIT would have been significantly different without them. Other strong personalities that provided me with mental intelligence and support include Marina Dang, Heather Barry, Mareena Robinson, and Lisa Bleheen. They have each taught me, in their own ways, valuable lessons that I am extremely grateful for.
My fellow graduate students Will Boyd and Sam Shaner have been great support since our earlier years on OpenMOC, and their initiatives have always been inspiring.
I would like to acknowledge Emmy Dabbelt for volunteering to edit this thesis, as it would not be in its current form without her. I would also like to thank all my other friends at MIT, UIUC, and from home. In particular, I would like to acknowledge my boyfriend, Palmer Dabbelt, whose endless support and encouragement kept me going and made this thesis possible.
Finally, I would like to acknowledge funding support from Studsvik Fellowship, and from Department of Energy (DOE) Center for Exascale Simulation of Advanced Reactors (CESAR), and Consortium for Advanced Simulation of LWRs (CASL).
CONTENTS
I INTRODUCTION 16
1.1 B ackground . . . . 16
1. 1. 1 Future full-core reactor analyses demand new methods . . . . 16
1.1.2 Advantages and challenges of Monte Carlo . . . . 17
1.2 The Monte Carlo method and OpenMC . . . . 19
1.3 O bjective . . . 20
2 ACCELERATION METHODS 2.1 Literature review of acceleration methods . . . . 2.1.1 Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) method . . . ... 2.1.2 Uniform Fission Site (UFS) method . . . . 23
2.1.3 Wielandt's method . . . .23
2.1.4 Superhistory method, batching algorithm . . . .24
2.1.5 Fission matrix method . . . . 25
2.1.6 Limited-Collision Monte Carlo (LCMC) . . . .26
2.1.7 Coarse Mesh Finite Difference (CMFD) . . . . 26
2.2 Coarse Mesh Finite Difference (CMFD) . . . .29
2.2.1 CMFD-MC theory . . . .29
2.2.2 Choices of key parameters for CMFD-MC in OpenMC . . . . 32
2.2.3 Diffusion coefficients . . . . 35
2.2.4 Spatial acceleration mesh size . . . . 36
2.2.5 Tally reset scheme . . . . 36
2.3 Low-Order Operator (LOO) . . . . 43
2.3.1 Review of Methods of Characteristics (MOC) . . . . 43
2.3.2 LOO-accelerated MC (LOO-MC) concepts . . . . 46
2.3.3 LOO-MC steps . . . . 47
2.3.4 Implementation of LOO in OpenMC . . . . 52
2.4 Sum m ary . . . . 53
3 FRAMEWORK OF TEST PROBLEMS 54 3.1 1D homogeneous problem . . . . 55
3.2 2D homogeneous problem . . . . 55
3.3 2D repeating lattices problem . . . . 56
3.4 2D BEAVRS problem . . . . 57
3.5 Sum m ary . . . . 57
4 ACCELERATION'S IMPACT ON THE FISSION SOURCE STATIONARITY 59 4.1 Literature review of fission source characterization methods . . . . 59
4.2 Stationarity metrics and heuristics . . . . 61
4.2.1 The Shannon entropy metric . . . . 61
4.2.2 The first moment entropy metric . . . . 64
4.2.4 Stationarity heuristics . . . . 72
4.3 Impact of acceleration methods . . . . 72
4.3.1 Reduction in number of inactive cycles . . . . 72
4.3.2 Improvement in stationarity fission source . . . . 73
4.3.3 Effect of the cycle size . . . . 75
4.4 Error composition model and spectral analysis . . . . 79
4.4.1 Proposed model for Monte Carlo (MC) error convergence . . . . 79
4.4.2 Spectral analysis . . . . 83
4.5 Summary . . . . 89
5 ACCELERATION'S IMPACT ON THE ACTIVE CYCLE CONVERGENCE 90 5.1 Terminology . . . . . . . "... ... ... 90
5.2 Characterization of the solutions . . . . 92
5.2.1 Fission source distributions and their uncertainties . . . . 92
5.2.2 Normality test . . . . 93
5.2.3 Two-sample Welch's t-test . . . . 95
5.3 Numerical convergence analysis . . . . 103
5.3.1 Method for assessing convergence rate . . . . 103
5.3.2 Parametric study on number of independent simulations . . . . 104
5.3.3 Numerical convergence results . . . . 106
5.4 Theoretical convergence analysis . . . 117
5.4.1 Conventional ACC model for predicting MC convergence . . . . 117
5.4.2 Limitation of the conventional ACC model . . . .
120
5.4.3 Extended correlation coefficient model to predict accelerated methods . 120 5.5 Summary . . . . 132
6 OPTIMAL SCHEME FOR ACHIEVING TARGET UNCERTAINTIES 134 6.1 Method for numerically deriving optimal simulation parameters . . . . 135
6.2 Optimal strategy for achieving 0.5% uncertainty . . . . 137
6.2.1 Performing a single simulation . . . . 137
6.2.2 Performing multiple simulations . . . . 140
6.3 Optimal strategy for achieving 0.25% uncertainty . . . . 141
6.3.1 Performing a single simulation . . . . 141
6.3.2 Performing multiple simulations . . . . 141
6.4 CPU time and memory overhead . . . . 143
6.5 Summary and recommendations . . . 145
7 CONCLUSIONS 147 7.1 Development of low-order acceleration methods . . . 147
7.2 Impact on inactive cycles . . . 147
7.3 Impact on active cycles . . . . 148
7.4 Optimal simulation schemes . . . . 149
7.5 Recommendations for future work . . . . 149
1
Appendix
152
B DERIVATION OF THE HOMOGENEOUS MIXTURE 156
C NUMERICAL EXPERIMENTS FOR OPTIMAL SCHEME 159
LIST OF FIGURES
Figure 1.1 Current deterministic multi-level approach to reactor analysis. .... 17
Figure 2.1 Shannon entropies for ten independent runs each using quarter
assem-bly vs. assemassem-bly acceleration mesh in CMFD-accelerated MC (CMFD-MC), 2D the Benchmark for Evaluation and Validation of Reactor
Simula-tions (BEAVRS) problem for 500 cycles of 4 million neutrons/cycle 37
Figure 2.2 Illustration of the point reset scheme for the first 30 cycles, where reset happens at the 10th and 25th cycle . . . . 39
Figure 2.3 Illustration of the moving window scheme with a window size of 10
for the first 30 cycles . . . . 40
Figure 2.4 Illustration of the expanding window approach for the first 30 cycles 42
Figure 2.5 Illustration of net currents (left) vs. quadrant currents (right) in a 2D mesh m. Notice the arrows in the right plot are only representations of the phase-space of the quadrant currents. . . . . 46 Figure 2.6 Quadrant fluxes (left) and low-order characteristic tracks (right)
pro-jected onto a 2D plane . . . . 48
Figure 3.1 Geometry of the 2D repeating lattices problem . . . . 56
Figure 3.2 Radial geometry of the 2D BEAVRS problem . . . . 58
Figure 4.1 Plot of Shannon entropy vs. cycle for the 2D BEAVRS problem, 10
million neutrons/cycle . . . . 62
Figure 4.2 Plot of Shannon entropy vs. cycle for the 2D BEAVRS problem, 10
million neutrons/cycle, ten independent simulations for each method 63
Figure 4.3 Plot of first moment entropy vs. cycle for the 2D BEAVRS problem,
10 million neutrons/cycle, ten independent simulations for each method 65
Figure 4.4 Plot of first moment entropy vs. cycle for the 2D BEAVRS problem,
10 million neutrons/cycle . . . . 66
Figure 4.5 Root Mean Square (RMS) percent error of instantaneous MC Fission
Source (FS), 2D BEAVRS problem, 10 million neutrons/cycle . . . . 68
Figure 4.6 RMS percent error of instantaneous MC FS, unaccelerated MC method,
2D BEAVRS problem, 10 million neutrons/cycle, ten independent sim-ulations. See Table 4.1 for the mean and std averaged over ten simula-tion s. . . . . 69
Figure 4.7 RMS percent error of instantaneous MC FS, CMFD-MC method, 2D
BEAVRS problem, 10 million neutrons/cycle, ten independent simula-tions. See Table 4.1 for the mean and std averaged over ten simulasimula-tions. 70
Figure 4.8 RMS percent error of instantaneous MC FS, LOO-MC method, 2D
BEAVRS problem, 10 million neutrons/cycle, ten independent simula-tions. See Table 4.1 for the mean and std averaged over ten simulasimula-tions. 71
Figure 4.9 Instantaneous fission source RMS percent error, ID homogeneous
prob-lem, simulated using 1 million neutrons/cycles for 2000 cycles, 10 cm x 10 cm x 10 cm mesh . . . . 74
Figure 4.10 Entropy vs. cycle for 2D BEAVRS problem, simulated with 2.5 million
Figure 4.11 Percent error of fission source for 2D BEAVRS problem, simulated with 2.5 million neutrons/cycle vs. 10 million neutrons/cycle . . . . . 77
Figure 4.12 Effect of cycle size in instantaneous fission source RMS percent error, ID homogeneous problem, 10 cm x 10 cm x 10 cm mesh. The means and standard deviations are tabulated in Table 4.2. . . . . 78
Figure 4.13 Reference fission source distribution for the 2D BEAVRS problem
used in the spectral analysis . . . . 84
Figure 4.14 MC's error by frequencies, 2D BEAVRS problem simulated with 10
million neutrons/cycle . . . . 87
Figure 4.15 LOO-MC's error by frequencies, 2D BEAVRS problem simulated with
10 million neutrons/cycle . . . . 87
Figure 4.16 MC's error by frequencies, 2D BEAVRS problem simulated with 2.5
million neutrons/cycle . . . . 88 Figure 5.1 Plot of MC fission source distribution and its uncertainty for the 2D
BEAVRS problem simulated with 10 million neutrons/cycle. . . . . . 93
Figure 5.2 Plot of LOO-MC fission source distribution and its uncertainty for the
2D BEAVRS problem simulated with 10 million neutrons/cycle. . . . 94
Figure 5.3 Plot of CMFD-MC fission source distribution and its uncertainty for
the 2D BEAVRS problem simulated with 10 million neutrons/cycle. 94
Figure 5.4 Plots of the rejected regions using the D'Agostino-Pearson normality test with a significant level of 5%. 2D BEAVRS problem simulated with 10 million neutrons/cycle. . . . . 96
Figure 5.5 Histograms of fission sources generated by 50 independent
simula-tions, illustrating what is likely a normal distribution (left plots), and likely not a normal distribution (right plots). 2D BEAVRS problem simulated with 10 million neutrons/cycle. . . . . 97
Figure 5.6 Comparison of the MC and the octant-symmetric MC fission source
distributions. 2D BEAVRS problem. . . . . 100
Figure 5.7 Comparison of the LOO-MC and the octant-symmetric MC fission
source distributions. 2D BEAVRS problem. . . . . 101
Figure 5.8 Comparison of the CMFD-MC and the octant-symmetric MC fission
source distributions. 2D BEAVRS problem. . . . . 102 Figure 5.9 Effect of number of simulations on the convergence behavior. 2D
ho-mogeneous problem case #2 (DR = 0.99) simulated with 1 million neu-trons/cycle. . . . . 105
Figure 5.10 Effect of number of simulations on the convergence behavior. Case: 2D
homogeneous problem case #2 (DR = 0.99) simulated with 1 million neutrons/cycle. . . . . 106
Figure 5.11 Sample standard deviation of the accumulated fission source from 200
independent simulations at active cycles for ID homogeneous problem
(DR = 0.97), simulated with 1 million neutrons/cycle, quarter assembly
acceleration meshes . . . . 108 Figure 5.12 Sample standard deviation of the accumulated fission source from 200
independent simulations at active cycles for 2D homogeneous problem
# 1 (DR = 0.976), simulated with 1 million neutrons/cycle, quarter
Figure 5.13 Standard deviation of the accumulated fission source from 100 inde-pendent simulations at active cycles for 2D homogeneous problem # 2
(DR = 0.990), simulated with 1 million neutrons/cycle, quarter
assem-bly acceleration meshes . . . . 110
Figure 5.14 Sample standard deviation of the accumulated fission source of 100 independent runs at active cycles for 2D repeating lattices problem (DR
= 0.974), simulated with 4 million neutrons/cycle, quarter assembly acceleration meshes . . . . I11 Figure 5.15 Sample standard deviation of the accumulated fission source of 50
in-dependent simulations at active cycles for the 2D BEAVRS problem, simulated with 2.5 million neutrons/cycle, quarter assembly accelera-tion m eshes . . . . 113 Figure 5.16 Sample standard deviation of the accumulated fission source of 50
in-dependent simulations at active cycles for the 2D BEAVRS problem, simulated with 10 million neutrons/cycle, quarter assembly accelera-tion m eshes . . . . 114 Figure 5.17 Sample standard deviation of the accumulated fission source of 50
in-dependent simulations for the 2D BEAVRS problem, quarter assem-bly acceleration meshes. The x-axis shows the total neutron histories, and the statistics on tallies are only generated during the active cycles, which is why the starting points of the curves depend on the cycle size 115
Figure 5.18 Correlogram of 1D homogeneous problem simulated using
unacceler-ated MC, generunacceler-ated with the conventional ACC formulation . . . . 121
Figure 5.19 Correlogram of ID homogeneous problem simulated using
CMFD-MC, generated with the conventional ACC formulation . . . . 121
Figure 5.20 Correlogram of 1D homogeneous problem simulated using LOO-MC,
generated with the conventional ACC formulation . . . . 122
Figure 5.21 Numerical results and conventional ACC-predicted model for the 1D
homogeneous problem. Observations: conventional ACC-predicted model agrees well with numerical results for unaccelerated MC, and disagrees
for CMFD-MC and LOO-MC . . . . 123
Figure 5.22 Correlation coefficient's dependency on cycle number i and cycle lag
k for the 1D homogeneous problem simulated using unaccelerated MC 124
Figure 5.23 Correlation coefficient's dependency on cycle number i and cycle lag
k for the 1D homogeneous problem simulated using CMFD-MC . . . 125
Figure 5.24 Correlation coefficient's dependency on cycle number i and cycle lag
k for the 1D homogeneous problem simulated using LOO-MC . . . . 125
Figure 5.25 Effective ACC Pk generated using expanded ACC model for the 1D
homogeneous problem . . . . 126
Figure 5.26 Expanded correlation coefficient model applied to ID homogeneous
problem simulated with CMFD-MC and LOO-MC. Numerical results are from Fig. 5.11. . . . . 127
Figure 5.27 Expanded correlation coefficient model applied to 2D BEAVRS
prob-lem simulated with CMFD-MC . . . . 129
Figure 5.28 Expanded correlation coefficient model applied to 2D BEAVRS
Figure 5.29 Asymptotic convergence behaviors predicted by the expanded correla-tion coefficient model for the 2D BEAVRS problem . . . . 131
Figure 6.1 Possible schemes for achieving 0.5% uncertainty . . . . 137
Figure 6.2 Optimal schemes for achieving 0.5% uncertainty via a single simula-tion, and numerical results using a single LOO-MC simulation . . . . 139
Figure 6.3 Optimal schemes for achieving 0.25% uncertainty . . . . 142
Figure C. 1 Numerical experiments for achieving 0.5% uncertainty using unaccel-erated M C . . . . 160
Figure C.2 Numerical experiments for achieving 0.5% uncertainty using CMFD-MC 161
Figure C.3 Numerical experiments for achieving 0.5% uncertainty using LOO-MC 162
Figure C.4 Numerical experiments for achieving 0.25% uncertainty using
unac-celerated M C . . . . 163
Figure C.5 Numerical experiments for achieving 0.25% uncertainty using CMFD-MC164
LIST OF TABLES
Table 2.1 CMFD tallied quantities . . . .29
Table 2.2 Comparison of transport-corrected diffusion coefficients and CMM gen-erated diffusion coefficients for a 2D 3.1% enriched BEAVRS assembly 36 Table 2.3 LOO tallied quantities . . . . 48
Table 3.1 Composition of the homogeneous mixture . . . . 55
Table 3.2 Basic characteristics of the 2D homogeneous problem case #1 and #2 . 55 Table 4.1 RMS percent errors averaged over 10 independent simulations. The second column is the average of the ten "mean" values, and the third column is the average of the ten "std" values. . . . . 68
Table 4.2 Mean and standard deviation in stationary fission sources, ID homo-geneous problem . . . . 75
Table 4.3 Summary of leading MC errors and their characteristics . . . . 83
Table 5.1 Summary of test problems and simulation parameters . . . . 107
Table 6.1 Optimal simulation parameters to achieve 0.5% uncertainty via a single sim ulation . . . . 138
Table 6.2 Minimum number of neutron histories needed to achieve 0.5% uncer-tainty . . . .
138
Table 6.3 Optimal simulation parameters to achieve 0.5% uncertainty via multi-ple sim ulations . . . . 140
Table 6.4 Optimal simulation parameters to achieve 0.25% uncertainty via a sin-gle sim ulation . . . 141
Table 6.5 Minimum number of neutron histories needed to achieve 0.25% uncer-tainty . . . .
142
Table 6.6 Optimal simulation parameters to achieve 0.25% uncertainty via mul-tiple sim ulations . . . 143
Table 6.7 LOO-MC's wall time breakdowns for 10 independent simulations for the 2D BEAVRS problem, each simulates 2.5 billion neutron histories 144 Table 6.8 CMFD-MC's wall time breakdowns for 10 independent simulations for the 2D BEAVRS problem, each simulates 2.5 billion neutron histories 145 Table A.1 Notation for indexes used in this work . . . . 153
Table A.2 Notation for variables used in Ch.2 . . . . 154
Table A.3 Notation for variables used in Ch.5 . . . . 154
Table A.4 Notation for general statistical concepts . . . . 155
Table A.5 Notation for specific statistical concept used in this work . . . . 155
Table B.
1
Modified heterogeneous pin cell material compositions . . . 156Table B.2 Main modification made and their results to generate a homogeneous mixture representative of a BEAVRS 3.1% enriched fuel pin . . . 158
ACRONYMS
ACC Autocorrelation Coefficient
BEAVRS the Benchmark for Evaluation and Validation of Reactor Simulations CMFD Coarse Mesh Finite Difference
CMFD-MC CMFD-accelerated MC DFT Discrete Fourier Transform
DR dominance ratio
FW-CADIS Forward-Weighted Consistent Adjoint Driven Importance Sampling
FS Fission Source
HZP Hot Zero Power LWR Light Water Reactor LOO Low Order Operator
LOO-MC LOO-accelerated MC
MC Monte Carlo
MCMC Markov Chain Monte Carlo MOC Methods of Characteristics OpenMC Open Monte Carlo PWR Pressurized Water Reactor RMS Root Mean Square
1
INTRODUCTIONPerforming nuclear reactor core physics analysis is a crucial step in designing and understand-ing nuclear power reactors. Advancements in the nuclear industry demand more accurate and detailed results from reactor analysis. This thesis aims to enable more efficient high-fidelity Monte Carlo simulations so that such simulations can be more readily used in routine analysis.
1.1 BACKGROUND
1.1.1 Future fidl-core reactor analyses demand new methods
The challenges reactor physicists face include, but are not limited to: producing power dis-tributions and any other parameters on increasingly finer spatial meshes (e.g., on pellet and sub-pellet levels); tracking thousands of materials, containing hundreds of isotopes; supporting three-dimensional whole-core depletion analysis with uncertainty quantification (e.g., including uncertainties in cross sections and manufacturing tolerances); accommodating a wider range of
nuclear reactor designs with spectra deviating from those of conventional Light Water Reac-tors (LWRs) [I].
In light of these challenges, existing methods employed in production-level neutronics tools often prove insufficient to supply the high-fidelity simulations desired. While the ultimate goal is to address challenges of time dependence, nuclide depletion and multi-physics coupling, the first step is to solve the simpler steady-state neutron transport problem. Solving the steady-state integro-differential form of the Boltzmann neutron transport equation in its full spatial, angular and energy complexities is impossible:
0 .VT(-, 0, E) +Et (Y,E)T(-r, n,E)
= jdn' d E' E s( Y , 6' -+ 0, E' - E )( F
n
E )+ 4
JT
dE'vf Ef (F, E')p(-, E') (1.1)A deterministic method discretizes this transport equation in space, angle and energy. Given
the coupling between the dependencies, the discretizations cannot be separated completely. Therefore, the currently employed approaches use a multi-level scheme descending in energy
complexity and ascending in spatial complexity as illustrated in Fig. 1.1.1 For example, first a zero-dimensional (i.e., infinite medium) calculation is performed for each isotope using
Figure
1. 1:
Current deterministic multi-level approach to reactor analysis.wise cross sections to generate thousands of multi-group cross sections; next a one-dimensional calculation is performed for each pin cell to reduce the multi-group cross sections from
thou-sands to hundreds; then a two-dimensional angular-dependent lattice physics calculation gen-erates few-group spatially-homogenized parameters for each lattice type; finally, few-group coarse-mesh parameters are used to perform a three-dimensional full core nodal simulations us-ing diffusion theory. This process, designed and perfected over the last two decades for LWRs, has been demonstrated to be accurate and computationally efficient [3, 4, 5, 6].
Current production-level tools make a number of approximations at each level specific for LWRs in order to be computationally efficient. Unfortunately, these approximations hinder ap-plications to future reactor analysis. For example, attempting to apply the lattice and nodal methods to non-LWR cores would involve re-evaluating all of approximations made, including the choice of energy and spatial discretization at each level.
One extension of the whole-core deterministic method is to solve the full reactor problem at
once, as
opposed
to performing singlelattice
calculations followed by a nodal core calculation.The whole-core deterministic method eliminates approximations from the last step in the
multi-level scheme, but nevertheless is still constrained by the previous approximations such as the multi-group cross section generation.
1 .1.2 Advantages and challenges of Monte Carlo
As opposed to the deterministic approach discussed above that solves Eq. (1. 1) by numerical discretizations, MC tackles the same physical problem from an entirely different approach. A Monte Carlo simulation determines stochastically individual neutron's interactions (reaction types, location, energy, etc) by sampling from probability distribution functions built from
MC methods have been deployed to solve the neutron transport problem since the 1950s and 60s [7, 8, 9, 10, 11, 12]. MC methods continued to develop, and often served as benchmark tools, given their accurate treatment of space and energy. Unfortunately the computational burden associated with following millions and billions of neutron histories renders MC unusable for routine full-core realistic reactor core simulations.
Advantages of MC for reactor criticality problems
Compared to deterministic methods, continuous-energy MC methods boast a number of major advantages in solving current and future reactor problems. Deterministic formulations rely on a spatial discretization whose truncation error is reduced as the mesh size is reduced. For reactors with complex geometries, only user intervention can select the appropriate spatial mesh for a given problem. MC methods, on the other hand, involve little or no approximation in handling arbitrary geometry.
MC methods also contain few approximations in handling the energy distribution of neutrons. The directly-resolved resonance treatment provided by continuous-energy MC has a huge ad-vantage over deterministic counterparts.
Taken together, the two previous aspects illustrate that MC is free from the critical approxi-mations made by the multi-level scheme to generate few-group homogenized cross sections for deterministic methods. Accurate spatial, angular and energy treatments in MC enable reliable simulations of any reactor design that would otherwise be cumbersome, if at all possible, using deterministic methods.
Challenges of MC for reactor criticality problems
The major roadblock in the process of adapting MC for routine reactor criticality calculations is its significant run time requirement. For example, Kelly et al. [13] report 56,700 CPU-hours to perform a high-fidelity analysis of a realistic Pressurized Water Reactor (PWR) problem. The issue of prohibitive run time can be addressed from two directions. One is MC's slow rate in evolving an initial fission source before tallies can be accumulated. Secondly, MC requires a large number of histories to reach desirable statistics on tallies.
The implementation of MC methods still face various challenges. One such issue lies in the methods' large memory requirement in depletion calculations to store cross section data and tal-lies, which are estimated to be on the order of terabytes [14]. Martin [15] discusses the various challenges in coupling MC neutronics simulations to multiphysics feedback. For example, the coupled neutronics and thermal hydraulics scheme typically has a lower tolerance of statistical fluctuations than the neutronics simulation alone. Additionally, temperature-dependent neutron cross section generation proves challenging in continuous-energy MC simulations. These chal-lenges are being addressed by on-going development in the MC community, and are not the focus of this thesis.
1.2 THE MONTE CARLO METHOD AND OPENMC
An open-source general MC neutron transport code, Open Monte Carlo (OpenMC) [16], is the platform used for implementing and evaluating algorithms in this thesis. OpenMC has been under development at MIT beginning in 2009 as part of Paul Romano's dissertation, and it provides a computationally efficient platform with realistic geometry modeling via constructive solid geometry representations, and full-physics capability. Details of the implementation can be found in Romano and Forget [17], Romano [18] and Romano et al. [19]. Key features of MC eigenvalue calculations are summarized below. These concepts presented below are common to all MC codes.
A Monte Carlo method simulates many neutron histories. A simple description of a neutron
history in an analog MC eigenvalue simulation is the following. For each neutron, its position, direction, and energy are initialized. Its distance to collision is sampled, i.e. stochastically de-termined from the probability distribution functions, based on the property of the material that the neutron resides in, and the neutron's energy and direction of movement. After updating the neutron's location by the sampled distance, its type of interaction is sampled based on the cross sections of the material and the neutron's energy. If the reaction type is scattering, the neutron's existing direction and energy are sampled, and the simulation continues to the next collision. If the reaction type is absorption or fission, or the neutron's history is completed. In the case of a fission event, the emitted neutrons are stored into the fission bank. Each neutron also has a
weight in OpenMC that can be adjusted.'
To obtain information like reaction rates, tallies are accumulated during each neutron history. There are different estimators, which accumulate tallies. The analog estimator, the simplest type, is used as an example: To obtain the scattering reaction rates tally, a Monte Carlo simulation accumulates the pre-collision neutron weight every time a neutron undergoes a scattering event. The normalized accumulated results are the scattering reaction rates. Users can further specify tallies for specific spatial regions, meshes, and for specific energy ranges.
In a fixed-source calculation, the neutron source is known, and the simulation could
accumu-late tallies from all neutron histories. In a Monte Carlo eigenvalue - or criticality - calculation,
however, the fission source distribution is unknown. MC eigenvalue calculations employ the
method of successive generations [11, 20], analogous to the power iteration approach to solve
an eigenvalue problem. That is, a Monte Carlo cycle2 follows a certain number of neutron
his-tories as described above. The fission bank generated is then sampled to produce a source bank, which serves as the initial source distribution for the next cycle. The term fission source
distri-1
The rational of neutron weights is that when there is a reason to change the neutron distributions, e.g., be-cause of variance reduction and acceleration method (will be discussed later), it is hard to justify alteration of the physical locations of individual neutrons. Instead, we can alter the neutron weights, which represent their relative "importance."
2
The term "generation" is often used in literature to describe this basic unit of neutron histories. Another term
bution is used in this thesis to refer to the spatial distribution of neutrons formed by the source
bank.
The fission source distribution is likely to be incorrect for the earlier cycles, and accumulation during these cycles would contaminate tallies. Hence MC eigenvalue calculations first perform
inactive cycles' during which the fission source distribution is evolved, and no accumulation
is performed. Once a stationary fission source is achieved, i.e., when the instantaneous fission source's error is a statistical noise whose expected value does not reduce, active cycles2 start, and tallies are accumulated.
This thesis uses cycle size to describe the number of neutron histories simulated per cycle.
The instantaneous fission source refers to the nu-fission reaction rates at each mesh and energy
group, generated by each inactive or active MC cycle. For an active cycle, the term accumulated
fission source refers to the average of instantaneous fission sources from the first active cycle to
the current active cycle.
1.3 OBJECTIVE
The overarching objective of this thesis is to enable more efficient high-fidelity Monte Carlo simulations so that such simulations can be more readily used in routine analysis. The following aspects will be discussed:
" Incorporation of a new low-order acceleration method into a continuous-energy MC code to reduce the time to stationarity, improve the stability, and improve the overall conver-gence rate
" Assessment of the performance of accelerated and unaccelerated MC calculations via numerical analyses and theoretical models.
" Identification of procedure for selecting optimal schemes for performing MC simulations A survey of the existing literature comprise chapter 2. This thesis provides a new approach for understanding the convergence of fission source during a Monte Carlo simulation, aided by decomposing the error by spatial frequencies (chapter 4).
This thesis proposes a new low-order method, LOO, to accelerate the fission source con-vergence in continuous-energy MC simulations. Its performance is evaluated against the best alternative acceleration method, Coarse Mesh Finite Difference (CMFD), and the baseline unac-celerated MC method. Chapter 4 provides a comprehensive evaluation of the low-order methods' performance in accelerating the fission source to reach stationarity. An analysis of the methods' impacts on the accumulated tallies is provided in chapter 5. In order to provide a comprehensive IMore broadly, burn-in is a loosely defined term used in the Markov Chain Monte Carlo (MCMC) community to refer to the process of throwing away some earlier iterations before getting to an equilibrium condition.
2
view, this thesis presents and discusses optimal schemes to achieve sample target statistics in chapter 6.
The numerical results presented in this thesis model real LWRs in 2D, with the understanding that MC and the low-order acceleration methods are applicable to a wider range of reactor types in 3D. The MC method discussed in this thesis is the steady-state continuous-energy MC eigenvalue/criticality calculation. The low-order acceleration methods of interest in this thesis, CMFD and LOO, are formulated as multi-group in energy and Cartesian meshes in space.
2 ACCELERATION METHODS
This chapter provides the theories for the acceleration methods evaluated in this thesis. Sec-tion 2.1 provides a comprehensive overview of the previous literature regarding accelerating Monte Carlo (MC) eigenvalue problems. Section 2.2 describes the theories of the Coarse Mesh Finite Difference (CMFD) acceleration method, and Sec. 2.3 presents a new low-order transport-based acceleration method, Low Order Operator (LOO), applied to Monte Carlo calculations. Choices of key parameters are discussed, and the implementations described in this chapter are
used to generate results in chapter 4, chapter 5 and chapter 6. The notations used in this section
are listed in Tables A. I and A.2.
2.1 LITERATURE REVIEW OF ACCELERATION METHODS
This section provides a review of the methods that have been proposed to improve MC eigen-value calculations by reducing the number of inactive cycles needed to achieve stationarity, or improving the accuracy of the results during the active cycles, or flattening the spatial distribu-tion of the statistical uncertainties.
2.1.1 Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) method
The Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) method has been successfully applied in MC fixed-source problems, e.g., detector response problems. Wagner et al. [23] provide a recent summary, and numerous publications on this method date back to Wagner and Haghighat [24]. Wagner and Mosher [25] and Wagner et al. [26] extend
FW-CADIS to a multi-group MC eigenvalue problem by performing two deterministic
compu-tations prior to the MC simulation, and using the deterministic solutions to generate the source and weight window for the MC simulation. The cited studies demonstrate that FW-CADIS en-ables the uncertainties of the fission source distribution to be spatially flatter (1.0-6.6%) than the conventional MC simulation (0.6-16.2%), for an eight group 2D Pressurized Water Reac-tor (PWR) quarter core model.
Adapting FW-CADIS to continuous energy MC eigenvalue calculation is not straight forward, because the deterministic computations require macroscopic cross-sections and a fission source. Kelly et al. [13] propose a sandwich scheme, wherein a conventional MC calculation is first performed to generate a fission source and other parameters for the deterministic calculations. The deterministic calculations are then carried out to generate importance functions, before
finally performing a Monte Carlo calculation. Even so the cited study notes that it is difficult to achieve a very flat uncertainty distribution in both radial and axial directions.
2.1.2 Uniform Fission Site (UFS) method
The Uniform Fission Site (UFS) method is designed to spatially flatten the radial and axial rel-ative uncertainties in fission sources [13, 27, 28]. The rationale behind UFS is that the variance is inversely proportional to the number of neutrons that contribute to that tally. Therefore, for a power reactor problem, the regions with fewer neutrons tend to have a larger variance or un-certainty. UFS tackles the larger uncertainty in lower neutron population regions by sampling
more source neutrons, thereby decreasing the uncertainty, with smaller neutron weights, which conserve the expected number of fission sites in each mesh cell.
Specifically, the expected value of the number of fission sites generated at a collision event in a conventional MC simulation without UFS is k E, where w is the weight of the neutron
k Et
that collides, k is the neutron multiplication factor generated by the previous cycle, and
vEf
andEt
are, respectively, the neutron production and total cross sections. Then the expected value ofthe number of fission sites generated at a collision event in a Monte Carlo simulation with UFS
Wl'v 1K
is , where i is the index of the mesh in which the neutron resides, Vi is the fraction
k Et Si
of the volume of mesh i over the total volume, and
Si
is the fraction of fission source of meshi over total fission source. To avoid biasing the problem, the weights of the neutrons sampled
are multiplied by .That is, the regions with lower numbers of neutrons in conventional MC
simulations are simulated with more neutrons, each with a smaller weight in the UFS scheme. As demonstrated in the cited studies, UFS effectively flattens the distribution of the relative uncertainties on the power density tallies. But it does not reduce the number of inactive cycles.
2.1.3 Wielandt's method
Wielandt's method was originally developed for use in the deterministic method, and was ap-plied to MC criticality calculations by Yamamoto and Miyoshi [29]. Wielandt's method is de-signed for problems with dominance ratios (DRs) close to unity. The method modifies the in-verse of the eigenvalue by the inin-verse of a user-specified value.
The implementation of Wielandt's method in MC consists of modifying the expected number
SVEf f to
o-,
isiVEf where o is the weight of the neutron thatk Et k ke Et
collides, k is the eigenvalue generated by the previous cycle, ke is user-specified eigenvalue, and
vEf and
Et
are the neutron production and total cross sections respectively. The method also Vchanged the expected number of daughter neutrons generated at a fission event from v to ,
of daughter neutrons are tracked within the same MC cycle, effectively multiplying the number of neutron histories in a single cycle.
Wielandt's method is subsequently reviewed in Dufek [30] and Shim and Kim [31]. The cited studies appear to agree that Wielandt's method reduces the number of MC cycles necessary to achieve stationarity, and also helps achieve a reliable determination of stationarity. Unfortu-nately, because the number of neutron histories simulated at each cycle is effectively increased, the total number of neutron histories to obtain stationarity is not improved. In fact She et al. [32] demonstrate theoretically that the conventional Wielandt method cannot save computational time in obtaining fission source stationarity.
2.1.4 Superhistory method, batching algorithm
Brissenden and Garlick [33] implement the superhistory powering method in the UK code MONK to overcome the underestimation of variance due to cycle-to-cycle correlation. More specifically, the method follows each neutron in the source bank for multiple fission genera-tions, hence the name superhistory, as opposed to a single generation in the conventional MC approach. Blomquist and Gelbard [34] further the discussion and demonstrate, with simplified models, that the superhistory method is as efficient as conventional MC, if not more so in spe-cial cases. One must, however, consider that in a superhistory method the number of histories followed in each cycle is greater than that of the number of neutrons the source bank includes. She et al. [32] show theoretically that similar to the Wielandt method, the superhistory method cannot reduce the computational time it takes to reach fission source stationarity.
Although different in exact implementation, the non-overlapping batch mean method by Ueki
[35], the history-based batch method by Shim et al. [36], and the batching algorithm by Kelly
et al. [13] all take root in exhibit the similar rationale: group multiple consecutive fission gener-ations into a single group referred to as batch, thus reducing the batch-to-batch correlation and improving the computed confidence interval. A more recent study by Park et al. [37] compares Gelbard's batch method and Shim's history-based batch method,' and concludes that Shim's history-based batch method provides the best estimation of the real variance.
In summary, various methods based on combining multiple successive generations of neutron histories into a single batch have been proven to reduce batch-to-batch correlation and improve the computed confidence interval. These methods do not, however, theoretically change the underlying fission source distribution so that the fission source stationarity can be achieved with fewer neutron histories.
1
2.1.5 Fission matrix method
One of the earliest mentions of the fission matrix method is the matrix method from Kaplan [7].
In his book, Kaplan divides the neutrons by space and velocity into multiple regions, forming a matrix M whose entry mij accounts for the neutrons born in region i and terminated in region
j.
Then the matrix problem Api = pinij [7, eq.(lc), p.12] is solved, where pi, pi refers tothe proportion of neutrons in region i and j, respectively. The cited book subsequently discusses
a number of factors that affect the selection of the discretized phase space and other parameter values given the concerns of computational efficiency.
In various studies from the following decade, the fission matrix method is discussed as an approach for accelerating fission source convergence, including Hammersley and Handscomb
[9], Morrison et al. [10], Mendelson [12], Carter and McCormick [38]. The method lost
trac-tion for about two to three decades, until Kadotani et al. [39], Urbatsch [40] and Kitada and Takeda [41] implemented the method in ORNL's KENO, LANL's MCNP and JAERI's GMVP, respectively. Carney et al. [42] provide a comprehensive overview of the method, and consider
its requirement on storage - which scales as n2, where n is the number of regions in space and
energy -to be the main reason that hinders widespread use of the fission matrix method after its
inception.
As computational capabilities developed, the past decade bore witness to renewed interest in the fission matrix method. Dufek and Gudowski [43] investigate the stability and convergence
rate of the fission matrix method
1D
homogeneous slab problem, and conclude that the methodrequires sufficient neutrons per cycle to avoid corruption of active cycles' fission source. Also, the mesh size must be reasonably small because the larger the mesh, the slower the conver-gence.' Subsequently, Dufek and Gudowski [44] propose a related method, the fission matrix based MC, which states that, given a sufficient number of meshes, the fission matrix becomes independent of the error in the fission source, although the study does not specify the number of meshes needed. Thttelberg and Dufek [45] propose using the fission matrix eigenvector to estimate the error in the cumulative fission source, under the assumption that the statistical er-ror in the fundamental mode eigenvector of the fission matrix is not correlated to the statistical error in the accumulated fission source. Dufek and Holst [46] show numerically that there is a strong error correlation between the fission matrix eigenvector and the cumulative fission source when the mesh is coarse. This conclusion is inline with the current understanding of the cycle-to-cycle correlation present in the unaccelerated MC simulation, namely, that the cycle-cycle-to-cycle correlation is stronger as the tally mesh becomes coarser. Therefore the eigenvector of the fission matrix nevertheless still suffers from the serially correlated nature of the method of successive generation or power iteration in MC.
1
The cited study discusses the propagation of systematic errors in the fission matrix due to the feedback between the fission matrix and the fission source.Carney et al. [42, 47, 48, 49] provide a comprehensive discussion of the fission matrix im-plementation and capability in MCNP6. In particular, two options are offered to reduce the storage requirement associated with the fission matrix method. One option is the sparse fission matrix storage method, which allocates storage space for matrix bands ad hoc by introducing approximations. The other method is the compressed-row storage scheme, which stores only the non-zero entries of the fission matrix, making no approximation in its presentation of that fission matrix.
In addition, She et al. [50] use solutions of the fission matrix method for undersampling diagnostics. Nielsen et al. [51] apply the acceleration method to simulating advanced reactors. Pan et al. [52] discuss the instability in the fission matrix method, and consider it to be a result of the statistical errors. The cited study then improves the stability of the method by limiting the number of inner iterations of the fission matrix method.
In summary, the main advantage of the fission matrix method is that, given sufficiently re-fined mesh, the neutron transition rates can be accurately determined without knowledge of the fission source. The method has been demonstrated to accelerate fundamental mode fission source convergence. The fission matrix equation also provides the forward and adjoint fission sources, as well as higher mode sources. But, the fission matrix method's formulation of tracking neutrons' region-to-region dependency results in two drawbacks. The method requires storage scales roughly proportional to the number of meshes squared. This issue could be partially miti-gated with a storage compression scheme. Secondly, the mesh-to-mesh tallies are likely to con-tain more statistical fluctuations relative to the mesh-averaged tallies, which can require more neutrons to remain stable.
2.1.6 Limited-Collision Monte Carlo (LCMC)
Keady [53] develops the Limited-Collision Monte Carlo (LCMC) which modifies the MC scheme that limits each neutron's history to no more than a certain number of collisions. LCMC has the advantage of reducing the CPU time per cycle at the cost of not being able to converge to the correct source distribution. Thus, the cited study applies the method during the active cycles only, and demonstrates that it is advantageous for 1D multi-group problems, though the advan-tages diminish in the 2D two-group pin cell problem, which is far removed from the full core continuous energy problem.
2.1.7 Coarse Mesh Finite Difference (CMFD)
The background and application of CMFD-accelerated MC (CMFD-MC) is reviewed in this section. The theory's detailed description is reserved for Sec. 2.2.
Origin and early development
CMFD has been successfully applied to deterministic methods for the last three decades. A
detailed description of the past studies and variations of CMFD's application in deterministic methods can be found in Li [54, Sec.2.2]. The first published study applying a diffusion-based low-order acceleration method to MC is Cho et al. [55], which employs p-CMFD rebalance externally to MCNP calculations as a feasibility investigation. Additionally, Lee et al. [56, 57] successfully accelerate simple 1 D and 2D multi-group MC calculations with CMFD.
Lee et al. [58], Lee et al. [59], Lee [60], and Lee et al. [61] add the multiset approach, which accumulates low-order tallies over a set of multiple MC cycles, with acceleration parameters generated using the average of multiple cycles. The low-order tallies are reset at the end of each set in order to remove bias from poor fission sources at earlier results. Also, the feedback to
MC using the low-order solutions is disabled during the first cycle of each set. The cited studies
successfully demonstrate that CMFD acceleration can rapidly converge 3D multi-group PWR
MC calculations in 20 cycles.
A very similar approach, referred to as the hybrid deterministic/MC using the nonlinear
dif-fusion acceleration (NDA), is reported by Willert [62], Willert et al. [63, 64, 65]. This method's low-order formulation appears identical to that of the CMFD, and the high-order MC simulation
is constrained to scattering-free and fixed-source calculations.
1
D one-group and 2D two-grouptest problems are reported.
Application to 3D continuous energy MC
Young et al. [66] apply CMFD to MCNP5, which is a 3D continuous MC code. The investigation concludes that the non-linear diffusion coefficient term is highly sensitive to statistical noise, and
that CMFD-MC generates inconsistent results compared with unaccelerated MC.1
Kelly et al. [67] and Herman [2] apply CMFD to 3D continuous energy MC codes MC21 and Open Monte Carlo (OpenMC), respectively. Both studies perform benchmark using a realistic
3D PWR, the the Benchmark for Evaluation and Validation of Reactor Simulations (BEAVRS)
problem, and observe significant reduction in the number of inactive cycles required. Kelly et al. [67] report a reduction from 200 inactive cycles without CMFD to 40 inactive cycles with CMFD, using 4 million neutrons per cycle; Herman [2] reports a reduction from 200 inactive cycles without CMFD to 20 inactive cycles with CMFD, using 4, 10, 20, and 50 million neutrons per cycle (1 million neutrons/cycle generated unstable results with CMFD acceleration).
'Upon closer inspection of the tally accumulation scheme for the acceleration parameters, this thesis suggests that the lack of acceleration tally reset might contribute to the inconsistencies between the conclusion drawn in the cited study and the ones discussed next. The tally reset scheme determines how the tallies that generate the acceleration parameters are accumulated. More details are provided in Sec. 2.2.5.
Wolters' modification term to diffusion coefficients
Wolters et al. [68], Wolters [69] and Wolters et al. [70] propose a modification term to the non-linear diffusion coefficient. The method takes the ID mono-energetic and isotropic transport
equation in Eq. (2.1), and computes its zeroth angular moment form using
f'I(.)
dy and firstangular moment form using
f!
()p
dpi.y
P(x, p) + Et (x)l(x, ) = (Ex) + k 1x(x,
p')
du' (2.1)The first angular moment formulation is then multiplied by a spatial weighting function, and integrated over each acceleration mesh. The resulting relation is manipulated into an identity for each acceleration mesh such that, as the scalar fluxes approach the right solutions, the identities approach zero. Hence, the Wolters' modification subtracts these terms in the formulation of the non-linear diffusion coefficients. The rationale behind this subtraction is that the diffusion coefficients suffer from statistical fluctuations when generated using tallies from a single cycle. Taking advantage of the fact that the statistical noises in the non-linear diffusion coefficients and in the correction terms stem from the same random events, the subtraction should reduce the statistical noise in the resulting diffusion coefficients. The cited studies demonstrate that
for
1D
one group MC simulation, the Wolters' modification improves the CMFD-MC approachusing low-order parameters generated from a single MC cycle.
Keady [53, Section 3.3, 3.5] extends the modification to 2D and shows that, for the seven group 2D C5G7 benchmark problem [71], the Wolters' correction's "effectiveness is signifi-cantly diminished." Based on the numerical results presented in Keady [53, Sec. 3.3], including Shannon entropy, apparent relative standard deviation, and relative error (using a "high-fidelity reference MC pin power solution"), this thesis considers that in the scope of multi-group MC, there is no clear performance difference between the Wolters' formulation and accumulating
low-order parameters over five cycles.1
Stability of CMFD-MC
Keady [53] concludes that CMFD-MC is not unconditionally stable in the infinite-particle limit. The cited study reports from numerical results that CMFD-MC is unstable when using
assem-bly size acceleration mesh in the 2D seven energy group C5G7 benchmark problem [71]. The
same calculation, without changing the number of neutrons per cycle, becomes stable once the acceleration mesh is reduced to quarter-assembly size.
1
One complication is that, as stated in Keady [53, p.91], Wolters' variant is performed using quarter-assembly acceleration meshes, while the LJLS variant (i.e., accumulating low-order parameters over five cycles) is performed using assembly meshes. From the cited study and previous literature, smaller acceleration meshes improve the con-vergence rate. Hence it is unclear how the two variants compare in terms of performance when using the same mesh size.
2.2 COARSE MESH FINITE DIFFERENCE (CMFD)
Theory of CMFD-MC is reviewed in Sec. 2.2.1, followed by choices of key parameters in OpenMC (Sec. 2.2.2). Notations used in this section can be found in Table A.I and Table A.2.
2.2.1 CMFD-MC theory
While there are several variations of the CMFD formulation applied to deterministic methods, its application in MC method remains fairly consistent in the current literature, with the exception of how tallies for generating low-order parameters are accumulated.1 To start, an acceleration spatial mesh and energy structure is selected. This thesis uses uniform Cartesian spatial meshes.2 Then for an arbitrary cycle - for example, the n-th cycle - of a Monte Carlo simulation, CMFD acceleration consists of the following four steps:
1. Acceleration tallies are accumulated. During the stochastic MC process, volume-integrated reaction rates and surface-integrated currents are accumulated in their corresponding bins. The tallies are listed in Table 2.1, where g, g' are energy group indexes, and "I is the mesh index. A" is the area of the surface of the m-th mesh that is perpendicular to the u-th direction. "I is the net current of the u surface at the m-th mesh, where u
E-
(x, y, z)designates the direction, and designates whether the surface is on the positive (+) or negative (-) side along the u-th direction.3
Notation Meaning
KPg,mVm)
volume-integrated flux(Et,g,mPg,mVm) volume-integrated total reaction rates
Vs Es,g' -g,m Pg',m Vm) volume-integrated scattering production rates
KVsEsi,g'
-g,m Pg',m Vm) volume-integrated P1 scattering production rates((XV )fg'g,m Pgm Vm) volume-integrated fission production rates
Kj
"A") surface-integrated currentTable 2.1: CMFD tallied quantities
2. The restriction step is performed to obtain acceleration parameters from acceleration tal-lies. The following volume-averaged parameters are computed from the tallies:
1The options are detailed in Sec. 2.2.5.
2
Non-Cartesian spatial meshes have been used in CMFD applied to deterministic methods [72].
3The CMFD implementation in OpenMC actually accumulates partial currents for the albedo boundary
condi-tions at the core/reflector interface as defined in Eq. (2.17). For the rest of the formulation, only net currents are
needed. Hence to simplify the notation in discussing the standard CMFD theory, the notion of net currents are used
