FW/CADIS-Ω: An Angle-Informed Hybrid Method for Neutron Transport

197 Pages·2017·5.49 MB·English

UC Berkeley
UC Berkeley Electronic Theses and Dissertations
Title
FW/CADIS-Ω: An Angle-Informed Hybrid Method for Neutron Transport
Permalink
https://escholarship.org/uc/item/26c5k0tq
Author
Munk, Madicken
Publication Date
2017

Peer reviewed|Thesis/dissertation
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University of California
FW/CADIS-Ω: An Angle-Informed Hybrid Method for Neutron Transport
by
Madicken Munk
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Engineering - Nuclear Engineering
in the
Graduate Division
of the
University of California, Berkeley
Committee in charge:
Assistant Professor Rachel N. Slaybaugh, Chair
Assistant Professor Massimiliano Fratoni
Professor John Harte
Dr. Tara Pandya
Summer 2017
FW/CADIS-Ω: An Angle-Informed Hybrid Method for Neutron Transport
Copyright 2017
by
Madicken Munk
1
Abstract
FW/CADIS-Ω: An Angle-Informed Hybrid Method for Neutron Transport
by
Madicken Munk
Doctor of Philosophy in Engineering - Nuclear Engineering
University of California, Berkeley
Assistant Professor Rachel N. Slaybaugh, Chair
The development of methods for deep-penetration radiation transport is of continued
importance for radiation shielding, nonproliferation, nuclear threat reduction, and medical
applications. As these applications become more ubiquitous, the need for transport methods
that can accurately and reliably model the systems’ behavior will persist. For these types
of systems, hybrid methods are often the best choice to obtain a reliable answer in a short
amount of time. Hybrid methods leverage the speed and uniform uncertainty distribution
of a deterministic solution to bias Monte Carlo transport to reduce the variance in the
solution. At present, the Consistent Adjoint-Driven Importance Sampling (CADIS) and
Forward-Weighted CADIS (FW-CADIS) hybrid methods are the gold standard by which to
model systems that have deeply-penetrating radiation. They use an adjoint scalar flux to
generate variance reduction parameters for Monte Carlo. However, in problems where there
exists strong anisotropy in the flux, CADIS and FW-CADIS are not as effective at reducing
the problem variance as isotropic problems.
This dissertation covers the theoretical background, implementation of, and characteri-
zation of a set of angle-informed hybrid methods that can be applied to strongly anisotropic
deep-penetration radiation transport problems. These methods use a forward-weighted ad-
joint angular flux to generate variance reduction parameters for Monte Carlo. As a result,
they leverage both adjoint and contributon theory for variance reduction. They have been
named CADIS-Ω and FW-CADIS-Ω.
To characterize CADIS-Ω, several characterization problems with flux anisotropies were
devised. These problems contain different physical mechanisms by which flux anisotropy
is induced. Additionally, a series of novel anisotropy metrics by which to quantify flux
anisotropy are used to characterize the methods beyond standard Figure of Merit (FOM)
and relative error metrics. As a result, a more thorough investigation into the effects of
anisotropy and the degree of anisotropy on Monte Carlo convergence is possible.
The results from the characterization of CADIS-Ω show that it performs best in strongly
anisotropic problems that have preferential particle flowpaths, but only if the flowpaths are
not comprised of air. Further, the characterization of the method’s sensitivity to determin-
2
istic angular discretization showed that CADIS-Ω has less sensitivity to discretization than
CADIS for both quadrature order and P order. However, more variation in the results
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were observed in response to changing quadrature order than P order. Further, as a result
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of the forward-normalization in the Ω-methods, ray effect mitigation was observed in many
of the characterization problems.
The characterization of the CADIS-Ω-method in this dissertation serves to outline a path
forward for further hybrid methods development. In particular, the response that the Ω-
method has with changes in quadrature order, P order, and on ray effect mitigation are
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strongindicatorsthatthemethodismoreresilientthanitspredecessorstostronganisotropies
in the flux. With further method characterization, the full potential of the Ω-methods can
be realized. The method can then be applied to geometrically complex, materially diverse
problems and help to advance system modelling in deep-penetration radiation transport
problems with strong anisotropies in the flux.
i
This dissertation is dedicated to the internet of cats: a series of interconnected paws, tails,
fur, space, and autotune.
Thanks also to the human family, friends, and mentors for the support and assistance that,
sometimes, only opposable thumbs can provide.
ii
Contents
Contents ii
List of Figures v
List of Tables vii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review 4
2.1 Monte Carlo Variance Reduction . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Statistical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1.1 Population Statistics . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1.2 The Central Limit Theorem . . . . . . . . . . . . . . . . . . 7
2.1.1.3 The Figure of Merit . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Variance Reduction Methods for Monte Carlo Radiation Transport . 9
2.1.3 Automated Variance Reduction Methods for Monte Carlo Radiation
Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Importance Functions for Variance Reduction . . . . . . . . . . . . . . . . . 14
2.2.1 The Concept of Importance . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 The Adjoint Solution for Importance . . . . . . . . . . . . . . . . . . 16
2.2.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 The Contributon Solution for Importance . . . . . . . . . . . . . . . . 19
2.3 Automated Variance Reduction Methods for Local Solutions . . . . . . . . . 22
2.3.1 CADIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Becker’s Local Weight Windows . . . . . . . . . . . . . . . . . . . . . 24
2.4 Automated Variance Reduction Methods for Global Solutions . . . . . . . . 25
2.4.1 Cooper’s Isotropic Weight Windows . . . . . . . . . . . . . . . . . . . 26
2.4.2 Becker’s Global Weight Windows . . . . . . . . . . . . . . . . . . . . 26
iii
2.4.3 FW-CADIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 Other Notable Methods . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Automated Angle-Informed Variance Reduction Methods . . . . . . . . . . . 30
2.5.1 Angular Biasing with Population Control Methods . . . . . . . . . . 31
2.5.1.1 AVATAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1.2 Simple Angular CADIS . . . . . . . . . . . . . . . . . . . . 36
2.5.1.3 Cooper’s Weight Windows . . . . . . . . . . . . . . . . . . . 39
2.5.2 Angular Biasing Using the Exponential Transform . . . . . . . . . . . 40
2.5.2.1 Early Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.2.2 LIFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Variance Reduction in Large Application Problems . . . . . . . . . . . . . . 44
3 Methodology 47
3.1 Theory: Angle-Informed Importance Maps for CADIS and FW-CADIS . . . 47
3.1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.2 The Ω Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2.1 CADIS-Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2.2 FW-CADIS-Ω . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Computational Success Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Anisotropy Quantification . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1.1 The Scalar Contributon Ratio . . . . . . . . . . . . . . . . . 51
3.2.1.2 The Ratio of Adjoint Fluxes . . . . . . . . . . . . . . . . . . 52
3.2.1.3 The Maximum to Average Flux Ratio . . . . . . . . . . . . 53
3.2.1.4 The Maximum to Minimum Flux Ratio . . . . . . . . . . . 54
3.2.2 Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2.1 Relative Error . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2.2 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Denovo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 ADVANTG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Characterization Problems and Results 60
4.1 Description of the Characterization Problems . . . . . . . . . . . . . . . . . 60
4.1.1 Identification of Anisotropy-Inducing Physics . . . . . . . . . . . . . . 61
4.1.2 Problem Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Introduction to Data Visualization and Analysis . . . . . . . . . . . . 69
4.2 Characterization Problem Results . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Computational Specifications . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Single Turn Labyrinth . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.3 Multiple Turn Labyrinth . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.4 Steel Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.4.1 Air Channel Variant . . . . . . . . . . . . . . . . . . . . . . 101
iv
4.2.4.2 Concrete Channel Variant . . . . . . . . . . . . . . . . . . . 103
4.2.5 U-Shaped Corridor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.6 Shielding with Rebar . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2.7 Therapy Room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.3 Sensitivity to Deterministic Parameter Choice . . . . . . . . . . . . . . . . . 135
4.3.1 Parametric Study Description . . . . . . . . . . . . . . . . . . . . . . 136
4.3.2 Quadrature Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3.3 Scattering (P ) Order . . . . . . . . . . . . . . . . . . . . . . . . . . 145
N
4.3.4 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.4 Method Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.4.1 Problem Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.4.2 Deterministic Solver Choice . . . . . . . . . . . . . . . . . . . . . . . 154
4.4.3 Lessons Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5 Conclusions 157
5.1 Assessment of the Ω-methods . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2.1 Software Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2.2 Characterization Problem Extension . . . . . . . . . . . . . . . . . . 160
5.2.3 Application Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography 165
A Software for this Project 173
A.1 Omega Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
A.2 Anisotropy Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.3 Inputs and Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.3.1 Parametric Study Problems . . . . . . . . . . . . . . . . . . . . . . . 181
A.3.2 Postprocessing Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3.3 Supporting Repositories . . . . . . . . . . . . . . . . . . . . . . . . . 183
v
List of Figures
2.1 Weight window illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.1 Single turn labyrinth geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Multi-turn labyrinth geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Steel plate embedded in concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 U-shaped corridor in concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 Concrete shielding with rebar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Nuclear medicine therapy room. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Sample results for a characterization problem tally. . . . . . . . . . . . . . . . . 72
4.8 Example distribution of all anisotropy metrics for highest, intermediate, and low-
est energy groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.9 Different ways of visualizing M for a characterization problem. . . . . . . . . . 76
4
4.10 M violin plots using different selections of the metric data. . . . . . . . . . . . . 77
2
4.11 Sample scatterplots of M distribution against the relative error improvement
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factor, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
RE
4.12 Tally results comparison between methods for single turn labyrinth. . . . . . . . 84
4.13 Tally relative error comparison between methods for single turn labyrinth . . . . 85
4.14 Flux map slice of single turn labyrinth. . . . . . . . . . . . . . . . . . . . . . . . 88
4.15 Ω-flux flux map for lowest energy group, single turn labyrinth. . . . . . . . . . . 89
4.16 Tally results comparison between methods for multiple turn labyrinth. . . . . . 91
4.17 Tally relative error comparison between methods for multiple turn labyrinth . . 92
4.18 Flux map slice of multiple turn labyrinth. . . . . . . . . . . . . . . . . . . . . . 94
4.19 Violin plots of M distribution using values above the mean contributon flux for
3
labyrinth problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.20 M distributions at problem midplane for labyrinth problems . . . . . . . . . . . 96
4
4.21 Relative error improvement factor as a function of M distribution statistics. . . 97
3
4.22 Figure of Merit improvement factor as a function of M distribution statistics. . 98
3
4.23 Tally results comparison between methods for steel bar embedded in concrete. . 100
4.24 Tallyrelativeerrorcomparisonbetweenmethodsforsteelbarembeddedinconcrete.101
4.25 Flux maps for steel beam in concrete. . . . . . . . . . . . . . . . . . . . . . . . . 105
4.26 M distribution plots for material variants of steel beam in concrete. . . . . . . . 107
2
4.27 M distribution plots for material variants of steel beam in concrete. . . . . . . . 108
4