Numerical Solution of the k-Eigenvalue Problem Öffentlichkeit

Abstract

Obtaining solutions to the k-eigenvalue form of the radiation transport equation is an
important topic in the design and analysis of nuclear reactors. Although this has been an
area of active interest in the nuclear engineering community for several decades, to date no
truly satisfactory solution strategies exist. In general, existing techniques are either slow
to converge for difficult problems or suffer from stability and robustness issues that can
cause solvers to diverge for some problems. This work provides a comparison between a
variety of methods and introduces a new strategy based on the Davidson method that has
been used in other fields for many years but never for this problem. The Davidson method
offers an alternative to the nested iteration structure inherent to standard approaches and
allows expensive linear solvers to be replaced by a potentially cheap preconditioner. To
fill the role of this preconditioner, a strategy based on a multigrid treatment of the energy
variable is developed. Numerical experiments using the 2-D NEWT transport package are
presented, demonstrating the effectiveness of the proposed strategy.

Table of Contents

1 Introduction 1
2 Neutron Transport Equation 5
2.1 Continuous ............................ 5

2.1.1 Boundary Conditions................... 6

2.1.2 Scattering Integral .................... 7
2.2 Discretizations........................... 9

2.2.1 Energy........................... 10

2.2.2 Angle ........................... 11

2.2.3 Space ........................... 13

2.2.4 Alternate Discretizations................. 14
2.3 Discrete Formulation ....................... 15

2.3.1 Linear System....................... 15

2.3.2 k-Eigenvalue Problem .................. 16
2.4 Structure ............................. 18

2.4.1 Solution Vector ...................... 18

2.4.2 Transport Matrix..................... 19

2.4.3 Scattering and Fission Matrices . . . . . . . . . . . . . 21

2.4.4 Transfer Matrices..................... 22

2.4.5 Implementation Considerations . . . . . . . . . . . . . 24
2.5 Spectral Properties ........................ 26
2.5.1 Transport Operators ................... 26

2.5.2 k-Eigenvalue Problem................... 28
3 Eigensolvers 33
3.1 Fixed Point Methods....................... 34

3.1.1 Power Iteration ...................... 34

3.1.2 Shifted Power Iteration.................. 35

3.1.3 Rayleigh Quotient Iteration ............... 36

3.1.4 Newton's Method..................... 37
3.2 Subspace Methods ........................ 38

3.2.1 Arnoldi's Method..................... 40

3.2.2 Generalized Davidson Method.............. 41

3.2.3 Davidson Preconditioners ................ 42

3.2.4 Reduced Memory Subspace Methods . . . . . . . . . . 46

3.2.5 Schur Forms........................ 48

3.2.6 Restarting Subspace Methods .............. 49
4 Transport Solvers 52
4.1 Monoenergetic Solvers ...................... 52

4.1.1 Richardson Iteration ................... 53

4.1.2 Diffusion Synthetic Acceleration . . . . . . . . . . . . . 53

4.1.3 Other Linear Preconditioners .............. 56

4.1.4 Nonlinear Acceleration.................. 56

4.1.5 Krylov Methods...................... 58
4.2 Multigroup Solvers ........................ 59

4.2.1 Block Gauss-Seidel .................... 60

4.2.2 Upscatter Acceleration.................. 60

4.2.3 Krylov Methods...................... 62
4.3 Eigensolvers............................ 64

4.3.1 Power Iteration ...................... 64
4.3.2 Shifted Iterations ..................... 65

4.3.3 Nonlinear Acceleration.................. 65

4.3.4 Krylov Methods...................... 66

4.3.5 Newton's Method..................... 67
4.4 Parallelization........................... 68

4.4.1 Domain Decomposition.................. 68

4.4.2 KBA............................ 69

4.4.3 Unstructured Sweeps................... 70
5 Multigrid-in-Energy 72
5.1 Basic Structure .......................... 73

5.2 Grid Transfer ........................... 74

5.3 Smoothing Iterations ....................... 75

5.4 Coarse Angle Approximation................... 77

5.5 Parameter Selection........................ 78
6 Numerical Results 88
6.1 Test Problems........................... 89

6.2 Parametric Studies ........................ 95
6.2.1 Spatial Refinement .................... 96

6.2.2 Angular Refinement ................... 99

6.2.3 Energy Refinement ....................100

6.2.4 Scattering Order Refinement...............101

6.2.5 Subspace Dimensions...................103
6.3 Linear Solvers...........................107
7 Conclusion 112

A Linear Solver Parametric Studies 116

Bibliography 120

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics Engineering, Nuclear
Stichwort	eigenvalue solvers radiation transport
Committee Chair / Thesis Advisor	Benzi, Michele, Emory University
Committee Members	Clarno, Kevin, Oak Ridge National Laboratory Veneziani, Alessandro, Emory University Nagy, James, Emory University

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