Advanced Monte Carlo Methods for Thermal Radiation Transport by Allan B. Wollaber A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Nuclear Engineering and Radiological Sciences and Scientific Computing) in the University of Michigan 2008 Doctoral Committee: Professor Edward W. Larsen, Chair Professor Tamas I. Gombosi Professor James P. Holloway Professor William R. Martin Allan B. Wollaber 2008 © All Rights Reserved To my family: my wife Carrie and my son Dobi ii Acknowledgements I wish to acknowledge the significant impact that my advisor, Prof. Edward Larsen has had upon my approach to technical problems. Through his continued demand for excellence, he has recognized and brought forth a potential that I did not know that I had. He has also been a wonderful person work with. I would like to thank my family for their encouragement and assistance. My wife Carrie deserves special recognition for her ability to manage almost everything required to keep me functional and grounded. My parents, Bruce and Debra, have also been supportive of me and my choice of career path. I would like to thank my fellow students in the NERS department who have pro- vided me with conceptual help, software advice, and stress relief. I would particularly like to thank Troy Becker and Greg Davidson, Troy for his indefatigable assistance in everything from integrating factors to cat-sitting, and Greg for his C++ tutelage and geometry library. Thanks is due to Todd Urbatsch and Jeff Densmore for providing distance support through Los Alamos National Laboratory during my first summer at the University, which motivated this work. It was also a pleasure working with them locally during my summer practicum. Bob Hill of Argonne National Laboratory deserves recognition for bringing me to Argonne and providing me with all the resources that I needed, despite the several setbacks that I encountered. Finally, I would like to thank my fellowship for their invaluable support, with- out which, none of this would have been possible. This work was performed under appointment by the Department of Energy Computational Science Graduate Fellow- ship, which is provided under grant number DE-FG0297ER25308. iii Table of Contents Dedication ..................................... ii Acknowledgements ............................... iii List of Figures .................................. viii List of Tables ................................... xiv Chapter I. Introduction .............................. 1 1.1 Linear Transport Solution Methodologies . 3 1.2 A Nonlinear Transport Problem . 7 1.3 Proposed Work and its Impact . 10 1.4 Thesis Synopsis . 16 II. Thermal Radiative Transfer: Basic Equations and Approxi- mations ................................. 19 2.1 The Thermal Radiative Transfer Equations . 19 2.1.1 Simple Material Models . 24 2.1.2 The Equilibrium Solution . 25 2.2 Common Approximations to the TRT Equations . 26 2.2.1 The Multigroup Approximation . 26 2.2.2 The Gray Approximation . 29 2.2.3 The Diffusion Approximation . 32 2.2.4 The Quasidiffusion Approximation . 34 2.2.5 A Simplifying Linearization . 35 iv III. The Implicit Monte Carlo Method and its Monte Carlo In- terpretation ............................... 37 3.1 Recasting the TRT Equations . 38 3.2 Time Discretization and Linearization . 40 3.2.1 Discussion . 46 3.3 MonteCarlo ........................... 49 3.4 Implicit Monte Carlo Implementation . 55 3.4.1 Summary of the IMC Procedure . 65 3.5 A Brief Introduction to Variance Reduction . 66 3.5.1 Absorption Weighting . 66 3.5.2 Russian Roulette . 67 3.5.3 Splitting ........................ 68 3.5.4 Weight Windows . 69 3.6 Summary............................. 69 IV. Stability Analysis of the IMC Method .............. 70 4.1 The Scaled, 1-D, Nonlinear TRT Equations . 72 4.2 The Scaled IMC Equations . 76 4.3 The Linearized, Near-Equilibrium IMC Equations . 79 4.4 Analysis.............................. 88 4.4.1 Unconditional Stability . 89 4.4.2 Damped Oscillations . 91 4.5 Numerical Results . 99 4.5.1 Single Fourier Modes . 100 4.5.2 Source Problems . 105 4.5.3 Marshak Waves . 111 4.6 A 0-D Testbed . 116 4.6.1 Analysis of a Linear Problem . 119 4.7 Summary............................. 127 V. Temperature Estimation and Evaluation ............. 131 5.1 A New Quasidiffusion Method . 133 v 5.1.1 Discussion . 136 5.1.2 Time Discretization . 137 5.1.3 An Average Interpolated Temperature . 141 5.1.4 Iterative Refinement . 143 5.1.5 Spatial Discretization . 146 5.1.6 Gray Procedure . 150 5.2 Stability Analysis . 151 5.3 Numerical Results . 154 5.3.1 Temporal Order of Accuracy . 155 5.3.2 Gray Marshak Waves . 157 5.3.3 Frequency-dependent Marshak Waves . 168 5.4 Summary............................. 175 VI. A Time-Dependent Fleck Factor .................. 180 6.1 IMC Equations with a Time-Dependent Fleck Factor . 183 6.1.1 Discussion . 187 6.1.2 Implementation Differences . 192 6.2 0-D Stability Analysis . 193 6.2.1 An Alternative Time-dependent Fleck Factor, a Cau- tionaryTale ...................... 198 6.3 Numerical Results . 200 6.3.1 Temporal Accuracy in 0-D Problems . 200 6.3.2 Gray Marshak Waves . 205 6.3.3 Frequency-Dependent Marshak Waves . 212 6.4 Summary............................. 216 VII. A Global Weight Window ...................... 221 7.1 A Global Weight Window . 226 7.1.1 The Global Weight Window Algorithm . 229 7.1.2 A Modification for Marshak Waves . 232 7.2 Numerical Results . 236 7.2.1 Gray Marshak Waves . 236 7.2.2 Frequency-Dependent Marshak Waves . 243 7.3 Summary............................. 249 vi VIII. Conclusions ............................... 253 8.1 Coda ............................... 258 Appendices .................................... 260 Bibliography ................................... 267 vii List of Figures Figure 4.1 The region in which λ is physically permissible. 85 4.2 Monotonicity conditions on ∆τ and ∆tσ0c for the 0-D, nonlinear, gray IMC equations. 93 4.3 A contour plot of the 0-D amplification factor ρ1,0........... 94 4.4 A contour plot of the amplification factor ρ1 for ξ = 0.001. 95 4.5 The values of q, ∆τ , and ξ for which ρ1 = 0. 96 4.6 The amplification factor ρ1 as a function of ∆τ for q = 1 and several fixed Fourier modes ξ.......................... 96 4.7 A contour plot of the amplification factor ρ2 for ξ = 0.1. 99 4.8 Experimentally-obtained damped oscillations in temperature profiles for ξ = 0.01, q = 0.14, ∆τ = 10, for which the theory predicts ρ1 = −0.2862............................... 101 4.9 Experimentally-obtained damped oscillations in temperature profiles for ξ = 0.01, q = 0.14, ∆τ = 1, for which the theory predicts ρ1 = −0.1861.................................. 102 4.10 Experimentally-obtained temperature profiles for ξ = 0.01, q = 0.14, ∆τ = 0.35 for which the theory predicts ρ1 = −0.0096. 103 4.11 Experimentally-obtained temperature profiles for ξ = 0.01, q = 0.14, ∆τ = 0.2 for which the theory predicts ρ1 = 0.1454. 104 4.12 Experimentally-obtained temperature profiles for ξ = 0.1, q = 0.1241, ∆τ = 4.32 for which the theory predicts ρ1 = −0.2538. 105 viii 4.13 Temperature profiles for ξ = 0.5, q = 0.1121, ∆τ = 0.805 for which ρ1 = −0.1809............................... 106 4.14 Experimentally-obtained temperature rise for an inhomogeneous source shut off at τ = 4 for a range of ∆τ ................... 106 4.15 The geometry for a two-region source problem. 108 4.16 Experimentally-obtained temperature profiles for a two-region source problem in which ∆τ,1 = 8 mean free times for emission (in region 1). 109 4.17 Experimentally-obtained temperature profiles for a two-region source problem in which ∆τ,1 = 4 mean free times for emission (in region 1). 110 4.18 Experimentally-obtained temperature profiles for a two-region source problem in which ∆τ,1 = 2 mean free times for emission (in region 1). 111 4.19 Time-dependent temperature profiles for a Marshak wave problem in which ∆τ = 0.1 and ∆x = 0.05.................... 113 4.20 Temperature profiles at τ = 8 for a Marshak wave problem in which the time step is varied using ∆τ = 0.1, 0.25, 0.5, 1, and 2 mean free times for emission. 114 4.21 Temperature profiles at τ = 8 for a Marshak wave problem in which the grid size is varied using ∆x = 0.025, 0.05, 0.1, and 0.2 cm. 114 4.22 A contour plot of the 0-D amplification factor ρ` of the IMC equa- tions applied to a linear problem. 125 4.23 Experimentally-obtained damped oscillations in energy density pro- files for ξ = 0.01, q = 0.14, ∆τ = 10 in a linear problem, for which the theory predicts ρ` ≈ −0.2525. 127 5.1 The average temperature T∗ as a function of the beginning and end of time step temperatures Tn and Tn+1................. 142 5.2 The zone-centered spatial discretization. 147 5.3 A contour plot of the amplification factor for the temperature esti- mationalgorithm............................. 154 ix 5.4 The time-dependent temperatures for IMC methods that that use data evaluated at Tn (blue) and at T∗ (red) for 10, 100, 500, and 1000timesteps.............................. 156 5.5 The numerically-calculated order of temporal error for the (1) IMC- T∗, (2) IMC-Tn+1, and (3) traditional IMC temperature solutions. 157 5.6 Temperature profiles at τ = 8 for a Marshak wave problem in which (a) the time step is varied and (b) the spatial grid size is varied using “traditional”IMC............................. 159 5.7 Time-dependent temperature profiles for a Marshak wave problem in which ∆τ = 1.0 and ∆x = 0.05. Solid lines refer to the IMC-T∗ solution, while dashed lines refer to the estimate produced by the Quasidiffusion method. 161 5.8 Temperature profiles at τ = 8 for a Marshak wave problem in which (a) the time step is varied and (b) the spatial grid size is varied for the IMC-T∗ method (without iteration).