Goal-based adaptive methods applied to the spatial and angular dimensions of the transport

212 Pages·2015·11.62 MB·English

by  Mark Goffin

Applied Modelling and Computation Group
Department of Earth Science and Engineering
Imperial College London
Goal-based adaptive methods applied to the
spatial and angular dimensions of the
transport equation
Mark Andrew Goffin
Sep 2014
Supervised by
Prof. C. Pain and Dr M. Eaton
Submitted in part fulfilment of the requirements for the degree of Doctor of
Engineering in Nuclear Engineering of Imperial College London and the
Diploma of Imperial College London
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Abstract
This thesis contains research into adaptive methods for the spatial and angular dimensions
of the neutral particle transport equation. Adaptive methods have been developed for two
angular discretisations: the spherical harmonics method and an octahedron-based wavelet
discretisation. The spatial discretisationused is a sub-grid scalefinite elementmethod. The
primary focus of the research is goal-based adaptive methods which optimise a particular
functional of the solution. The error measures that drive the adaptive methods are pre-
sented along with the novel and efficient techniques that are used to approximate them.
Adaptivealgorithmsarefirstdevelopedandpresentedforthespatialandangulardiscretisa-
tionsseparately. Theadaptivemethodsfortheangulardimensionsproducevariableangular
resolution across the space and energy dimensions of the equation. The adaptive methods
for the spatial dimensions use an anisotropic mesh optimisation algorithm which repositions
nodes and elements of the mesh. The adaptive wavelet discretisation allows anisotropic res-
olution of the angular domain at each point in space and energy which can be very efficient.
The ultimate outcome of the research is an algorithm that adapts the angular and spatial
resolution simultaneously. This is achieved using the wavelet discretisation by combining
the individual adaptive procedures. All adaptive methods developed are shown to produce
results with a given accuracy for a smaller number of degrees of freedom. The performance
of the methods heavily depends on the physical system that is being modelled. Typically
performing best in shielding type calculations. The benefits from the adaptive methods are
two-fold: (i) the reduction in degrees of freedom can lead to smaller computational times,
and (ii) the automated adaptive process can reduce the overall user time spent performing
convergence analysis.
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Declaration of Originality
I hereby declare that the content in this thesis is my own work unless stated otherwise in
which instance it will be appropriately referenced.
3
Copyright Declaration
The copyright of this thesis rests with the author and is made available under a Creative
CommonsAttributionNon-CommercialNoDerivativeslicence. Researchersarefreetocopy,
distributeortransmitthethesisontheconditionthattheyattributeit,thattheydonotuse
it for commercial purposes and that they do not alter, transform or build upon it. For any
reuse or redistribution, researchers must make clear to others the licence terms of this work.
4
Acknowledgements
IwouldfirstlyliketothankmyacademicsupervisorsProf.ChrisPainandDrMatthewEaton.
Chrishasprovidedvitalinputandadvisedonallmattersthroughouttheresearch. Hisadvice
hashelpedovercomeavarietyoftheproblemsthatwereencountered. Matthewensuredthat
I was fully prepared to undertake the research by providing an excellent training program
and support material. I have the sincerest appreciation for all the assistance provided by
Dr Andrew Buchan, a research fellow at Imperial College. He has provided a lot of his time
andeffortinhelpingmeunderstandalmosteverythingIknowaboutradiationtransportand
various other mathematical topics.
I want to thank my industrial supervisor Prof. Paul Smith who has always provided
positive encouragement and offered advice for real-world applications. This thanks extends
to all those at the ANSWERS team from AMEC who made me feel welcome whilst based
at the Winfrith office. Especially those that provided support and advice on using their
software.
IwishtothankProf.RichardSmedley-Stevenson,avisitingprofessoratImperialCollege
fromAWE.Hehasprovidedinsightandadvicewithregardstomathematicalconceptsrelated
to radiation transport. In addition, he provided various recommendations for optimisations
and improvements to the research code.
IwanttonotemyappreciationforallthehelpIreceivedfromDrChrisBaker,whoatthe
time was finishing his doctorate. He spent a lot of time teaching me about the concepts of
thisresearchproject. Healsoprovidedthefirstversionofthecodewhichwasusedtoproduce
one of the results chapters of this thesis. I would also like to thank Dr Brendan Tollit for
providing the framework for the latest version of the code and helping me understand some
oftheintricacieswithintheframework. AfurtherthanksisextendedtoDrAncaBelmewho
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althoughonlypresentatthecollegeforashorttimehelpeddeveloptheadjointcapabilityin
thecode. AmoregeneralthankyouisrequiredfortheAMCGgroupwhichhasexperiencein
many topics related to numerical modelling and provided some of the fundamental libraries
used in this research.
I want to thank two fellow doctorate students who began at the same time as I did, Dan
Ayres and Chris Cooling. They have provided both academic and personal support over
the years. This extends to the whole EngD cohort in which I found a group of like-minded
peers to share experiences with. More recently, I have also had valuable discussions with
Alexandros Adam and Dr Steven Dargaville for which I am very grateful.
From a more personal level, I would like to express my thanks to my parents and my
brotherwhohavealwayssupportedeverythingIhaveundertaken. Myparentshaveprovided
fantastic support over the years without ever seriously questioning my perpetual student
status.
The biggest thanks is expressed to my loving girlfriend Emily who has been very patient
with me for the last four years. She ensured that I always kept a good work-life balance and
provided much needed joy throughout the experience.
I would like to express my gratitude to both the EPSRC and AMEC for funding the
research.
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Contents
1 Introduction 22
1.1 Deterministic vs. Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Deterministic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 Forms of the Transport Equation . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 Spatial Discretisation Methods . . . . . . . . . . . . . . . . . . . . . . 28
1.2.3 Angular Discretisation Methods. . . . . . . . . . . . . . . . . . . . . . 29
1.3 Adaptive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.1 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.2 Adaptive Methods within Radiation Transport . . . . . . . . . . . . . 33
1.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Solving the Transport Equation 40
2.1 Multigroup Energy Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Sub-Grid Scale Finite Element Spatial Discretisation . . . . . . . . . . . . . . 45
2.3 General Angular Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Spherical Harmonics Basis . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2 Discrete Ordinates Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.3 Linear Octahedral Wavelet Basis . . . . . . . . . . . . . . . . . . . . . 53
2.4 Source Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 External Source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4.2 Scatter Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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CONTENTS
2.4.3 Fission Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Solving the Sub-Grid Scale Equations . . . . . . . . . . . . . . . . . . . . . . 62
2.7 Adjoint Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.7.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.7.2 The Adjoint Transport Equation . . . . . . . . . . . . . . . . . . . . . 64
2.8 Solving the Adjoint Sub-Grid Scale Equations . . . . . . . . . . . . . . . . . . 65
3 The Goal-Based Error Measure 67
3.1 Continuum Functional Error Measure . . . . . . . . . . . . . . . . . . . . . . 67
3.1.1 Functional Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Adjoint Error Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1.3 Forward Error Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Discrete Functional Error Measure . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Goal Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.1 Average Scalar Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.3 Surface Current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.4 Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Representing the Solution Error in the Trial Space . . . . . . . . . . . . . . . 76
4 Error Metric for Spatial Adaptivity 79
4.1 Approximating the Solution Error . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.2 Hessian Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Residual Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Metric Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Metric Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 Maximal Inner Ellipse Method . . . . . . . . . . . . . . . . . . . . . . 84
4.4.2 Minimal Outer Ellipse Method . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Metric Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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CONTENTS
4.6 Adapting the Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Goal-based Spatial Adaptivity Applied to the Eigenvalue 90
5.1 Adaptive Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.1 Single-mesh Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 Multi-mesh Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Example 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 Example 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.3 Example 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Angular Adaptivity using Spherical Harmonics 111
6.1 Spherical Harmonics Angular Error Indicator . . . . . . . . . . . . . . . . . . 111
6.2 Residual Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Spherical Harmonics Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 Adaptive Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4.1 Continuous Error Indicator . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4.2 Nodal Error for Regular Adaptivity . . . . . . . . . . . . . . . . . . . 118
6.4.3 Nodal Error for Goal-Based Adaptivity . . . . . . . . . . . . . . . . . 119
6.4.4 Adaptive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5.1 Example 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5.2 Example 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.3 Example 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5.4 Example 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.5.5 Example 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7 Angular Adaptivity using Octahedral Wavelets 149
7.1 Octahedral Wavelets Angular Error Indicator . . . . . . . . . . . . . . . . . . 150
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CONTENTS
7.2 Residual Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Adaptive Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3.1 Continuous Error Indicator . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3.2 Goal-Based Error Indicator . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3.3 Adaptive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4.1 Example 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.4.2 Example 7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.4.3 Example 7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.4.4 Example 7.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8 Simultaneous Spatial and Angular Adaptivity using Octahedral Wavelets180
8.1 Spatial Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.1.1 Hessian Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.1.2 Metric Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2 Adaptive Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3.1 Example 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3.2 Example 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.3.3 Example 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Conclusions and Further Work 197
9.1 Chapter Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.1.1 Goal-based Spatial Adaptivity Applied to the Eigenvalue . . . . . . . 197
9.1.2 Angular Adaptivity using Spherical Harmonics . . . . . . . . . . . . . 198
9.1.3 Angular Adaptivity using Octahedral Wavelets . . . . . . . . . . . . . 199
9.1.4 SimultaneousSpatialandAngularAdaptivityusingOctahedralWavelets199
9.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
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