Nuclear Computational Science (eBook)

Nuclear Computational Science 
2 
Preface 6
Obituary Composed by Dr. Roger Blomquistfor Dr. Ely Meyer Gelbard 9
1 Advances in Discrete-Ordinates Methodology 13
1.1 Introduction 13
1.2 Basic Concepts 14
1.3 Three Challenging Physical Problems 26
1.3.1 Thermal Radiation Transport in the Stellar Regime 26
1.3.2 Charged-Particle Transport 30
1.3.3 Oil-Well Logging Tool Design 33
1.4 Advances in Spatial Discretizations 35
1.4.1 Characteristic Methods 35
1.4.2 Linear Discontinuous Method 37
1.4.3 Nodal Methods 39
1.4.4 Solution Accuracy in the Thick Diffusion Limit 42
1.5 Advances in Angular Discretizations 48
1.5.1 Angular Derivatives 48
1.5.2 Anisotropic Scattering 50
1.6 Advances in Fokker–Planck Discretizations 57
1.6.1 The Continuous-Scattering Operator 57
1.6.2 The Continuous-Slowing-Down Operator 61
1.7 Advances in Time Discretizations 64
1.8 Advances in Iteration Acceleration 66
1.8.1 Fourier Analysis 67
1.8.2 Diffusion-Synthetic Acceleration 69
1.8.3 DSA-Like Methods for Outer Iteration Acceleration 72
1.8.4 A Deficiency in Multidimensional DSA and DSA-Like Methods 74
1.9 Krylov Methods 75
1.9.1 The Central Theme of Krylov Methods 75
1.9.2 Convergence and Preconditioning of Krylov Methods 79
1.9.3 Applying Krylov Methods to the SN Equations 81
1.10 Future Challenges 86
References 88
2 Second-Order Neutron Transport Methods 97
2.1 Introduction 97
2.2 The Transport Equation 98
2.2.1 First- and Second-Order Forms 99
2.2.2 Weak Forms 100
2.2.3 Variational Formulation 103
2.3 The Discretized Diffusion Equation 104
2.3.1 The Diffusion Formulation 104
2.3.2 Finite Element Discretization 106
2.4 The Discretized Transport Equation 108
2.4.1 Anisotropic Scattering 108
2.4.2 Angular Approximations 112
2.4.2.1 Spherical Harmonics Expansions 112
2.4.2.2 Discrete Ordinates Approximations 113
2.4.2.3 Simplified Angular Approximations 115
2.4.3 Spatial Discretization 116
2.5 Hybrid and Integral Methods 118
2.5.1 A Variational Nodal Method 118
2.5.2 An Even-Parity Integral Method 122
2.5.3 Combined Methods 124
2.6 Discussion 124
References 125
3 Monte Carlo Methods 128
3.1 Introduction 128
3.2 Organizing Principles 130
3.2.1 Generating Sequences 131
3.2.2 Error Analysis 131
3.2.3 Error Reduction 131
3.2.4 Foundations/Theoretical Developments 132
3.3 Historical Perspectives 132
3.4 Generating Sequences 133
3.4.1 Pseudorandom Sequences 134
3.4.2 Quasirandom Sequences 140
3.4.3 Hybrid Sequences 142
3.4.4 State of the Art 144
3.5 Error Analysis 144
3.5.1 The Pseudorandom Case 144
3.5.2 The Quasi-random Case 145
3.5.3 The Hybrid Case 151
3.5.4 Current State of the Art 151
3.6 Error Reduction 152
3.6.1 Introduction 152
3.6.2 Control Variates 153
3.6.3 Importance Sampling 157
3.6.4 Stratified Sampling 160
3.6.5 Use of Expected Values 161
3.6.6 Other Error Reduction Strategies 162
3.6.7 State of the Art 165
3.7 Foundations/Theoretical Developments 166
3.7.1 State of the Art 168
3.8 Challenges 168
References 169
4 Reactor Core Methods 177
4.1 Introduction 177
4.2 Analytic Methods and Early Calculation Schemes 177
4.3 Lattice Cell and Assembly Codes 181
4.3.1 Lattice Physics Calculations 183
4.3.1.1 Producing Cross-section Libraries 186
4.3.1.2 Self-Shielding and Multigroup Approximation 186
4.3.1.3 Generic Multigroup Solver 188
4.3.1.4 Discrete Ordinates 190
4.3.1.5 Method of Characteristics 192
4.3.1.6 Collision Probability Method 194
4.3.1.7 Bn Solutions and Diffusion Coefficients 196
4.3.1.8 Lattice Solvers 197
4.3.1.9 Putting It All Together into Lattice Codes 199
4.3.2 Homogenization Process 200
4.3.2.1 Reaction Rates and Homogenized Cross Sections 200
4.3.2.2 Generalized Equivalence Theory and Discontinuity Factors 202
4.4 Reactor Core Solvers 202
4.4.1 Pn Approximations and Diffusion 203
4.4.1.1 Spherical Harmonics and the Even-Parity Transport Equation 203
4.4.1.2 The Diffusion Approximation 204
4.4.1.3 SPn and Improved Diffusion 206
4.4.2 Diffusion-Like Methods 207
4.4.2.1 Transverse Integrated Nodal Methods 208
4.4.2.2 Analytic Nodal Methods 209
4.4.2.3 Core Harmonics and Modal Synthesis 210
4.4.3 Variational Formulation and Finite Elements 211
4.4.3.1 Classical Spatial Finite Elements 211
4.4.3.2 Mixed and Hybrid Finite Elements 212
4.4.4 Putting It All Together into Reactor Codes 213
4.5 Core Applications 214
4.5.1 Pin Power Reconstruction in LWR Reactors 216
4.5.2 Estimates of Zonal Powers in CANDU Reactors 217
4.5.3 Teaching Modern Reactor Core Methods 219
4.6 Concluding Remarks 221
References 223
5 Resonance Theory in Reactor Applications 226
5.1 Introduction 226
5.1.1 Historical Perspective 227
5.1.2 Self-shielding Effects in Perspective 229
5.2 Representation of Microscopic Cross Sections 230
5.2.1 Brief Description of R-Matrix Theory 230
5.2.1.1 Wigner–Eisenbud Version 230
5.2.1.2 Kapur–Peierls Version 232
5.2.2 Practical Representations Currently in Use 233
5.2.2.1 Single Level Breit–Wigner Approximation (SLBW) 233
5.2.2.2 Multilevel Breit–Wigner Approximation (MLBW) 234
5.2.2.3 Adler–Adler Approximation (AA) 235
5.2.2.4 Reich–Moore Approximation 237
5.2.3 Other Alternative: Generalized Pole Representation 238
5.3 Doppler-Broadening of Cross Sections 239
5.3.1 Practical Doppler-Broadening Kernels in Use 240
5.3.1.1 Ideal Gas Model 240
5.3.1.2 Accommodation of Crystalline Binding Effects via Effective Temperature Model 241
5.3.2 Analytical Broadening via Doppler-Broadened Line-Shape Functions 242
5.3.2.1 Traditional Doppler-Broadened Line-Shape Functions 243
5.3.2.2 Generalization of Doppler-Broadened Line-Shape Functions 243
5.3.2.3 Evaluations of the Doppler-Broadened Functions 244
5.3.3 Direct Numerical Doppler-Broadening of Point-Wise Cross Sections 246
5.3.3.1 Kernel-Broadening Approach 247
5.3.3.2 Heat Equation Approach Based on the Finite Difference Method 249
5.4 Resonance Absorption in Homogeneous Media 249
5.4.1 Average Scattering Kernel for Practical Applications 250
5.4.2 Characteristics of the Slowing-Down Equation 251
5.4.2.1 Slowing-Down Density 252
5.4.2.2 Concept of Placzek Oscillations 253
5.4.3 Resonance Integrals and Their Applications 255
5.4.3.1 Traditional Resonance Integral Concept 255
5.4.3.2 Various Resonance Integral Approximations 256
5.4.4 Various Developments Motivated by the Emergence of the Fast Reactor Program 259
5.4.4.1 Generalization and Computation of the J-Integral 259
5.4.4.2 Connections Between the Resonance Integral and Traditional Multigroup Cross Section Processing 261
5.4.4.3 Rigorous Treatment of Resonance Absorption via Numerical Means 263
5.5 Resonance Absorption in Heterogeneous Media 266
5.5.1 Traditional Collision Probability Methods for a Two-Region Cell 266
5.5.1.1 General Features of Collision Probability of Practical Interest 267
5.5.1.2 Various Earlier Methods Based on Approximate Escape Probabilities for Isolated Fuel Lumps 269
5.5.2 Traditional Collision Probability Treatment in a Closely Packed Lattice 271
5.5.2.1 General Features of the Escape Probability 272
5.5.2.2 Fine-Tuning of the Rational Approximation for Routine Applications 274
5.5.3 Connections to Resonance Integral and Multigroup Cross-section Calculations 275
5.5.3.1 Rational Approximation and Approximate Flux Based Approaches 275
5.5.3.2 Nordheim's Method 276
5.5.4 Rigorous Treatment of Resonance Absorption in a Unit Cell with Multiple Regions and Many Resonance Isotopes 276
5.5.4.1 Kier's Method for Cylindrical Unit Cells 277
5.5.4.2 Olson's Method for Unit Cell with Many Plates 280
5.6 Treatment of Unresolved Resonances 281
5.6.1 Statistical Theory Basis 281
5.6.1.1 Some Statistical Theory Fundamentals 282
5.6.1.2 Statistical Distributions of Practical Interest 283
5.6.1.3 Conceptual Aspects of Computing Average Cross Sections 285
5.6.2 Average Unshielded Cross Sections and Fluctuation Integrals 287
5.6.3 Traditional Approaches for Computing Self-shielded Average Cross Sections 288
5.6.3.1 Methods Based on Direct Integrations 288
5.6.3.2 Methods Using the Statistically Generated Resonance Ladders 290
5.6.4 Probability Table Methods 291
5.6.4.1 Conceptual Basis 291
5.6.4.2 Methods for Computing the Tabulated Quantities 292
5.7 Future Challenges 294
References 296
6 Sensitivity and Uncertainty Analysis of Models and Data 300
6.1 Introduction 300
6.2 Sensitivities and Uncertainties in Measurementsand Computational Models: Basic Concepts 300
6.2.1 Measurement Uncertainties 302
6.2.2 Propagation of Errors (Moments) 306
6.3 Statistical Methods for Sensitivity and Uncertainty Analysis 311
6.3.1 Sampling-Based Methods 312
6.3.2 Reliability Algorithms: FORM and SORM 316
6.3.3 Variance-Based Methods 317
6.3.4 Design of Experiments and Screening Design Methods 319
6.4 Deterministic Methods for Sensitivity and Uncertainty Analysis 323
6.4.1 The Forward Sensitivity Analysis Procedure (FSAP) for Nonlinear Systems 327
6.4.2 The Adjoint Sensitivity Analysis Procedure (ASAP) for Nonlinear Systems 330
6.4.3 The Adjoint Sensitivity Analysis Procedure (ASAP) for Responses Defined at Critical Points 333
6.4.4 ASAP for Linear Systems 339
6.5 Paradigm Applications of the ASAP 340
6.5.1 Application of the ASAP to Compute the Variance of the Maximum Flux of Particles in a Particle Diffusion Problem 340
6.5.2 ASAP for a Ricatti Equation 348
6.5.3 ASAP for a System of Linear Ordinary Differential Equations 352
6.6 Computational Considerations and Open Problems 355
References 359
7 Criticality Safety Methods 363
7.1 Introduction 363
7.1.1 Overview 363
7.1.2 Historical Background 363
7.2 Nuclear Criticality Safety: The Early Years 364
7.2.1 The First Criticality Concerns 364
7.2.2 Early Attempts at Criticality Safety Computations 365
7.3 Criticality Safety Versus Reactor Design Calculations 365
7.3.1 Computational Requirements for Reactor Design 365
7.3.2 Special Requirements for Criticality Safety Calculations 366
7.3.3 Early Computational Tools for Criticality Safety Calculations 366
7.4 Role of the Sn, or Discrete Ordinates Method 367
7.4.1 Impetus for the Early Sn Method Development 368
7.4.2 Later Sn Method Development 368
7.5 Role of the Monte Carlo Method 368
7.5.1 Early Monte Carlo Calculational Methods 369
7.6 Critical Experiments, Benchmarks, and Validation 370
7.7 Evaluation of the Various Methods and Their Role in Current Criticality Safety Calculations 370
7.7.1 Role of the Sn Method 370
7.7.2 Evaluation of the Role of the Sn Method 371
7.7.3 Role of the Monte Carlo Method 371
7.7.4 Evaluation of the Monte Carlo Method 372
7.7.5 Summary of the Monte Carlo Criticality Safety Software 373
7.7.6 The N(BN)2 Method 373
7.8 The Role of Cross-Section Representation 374
7.8.1 Multigroup Cross Sections 374
7.8.2 Point-Wise Cross Sections 375
7.9 Elements of a Complete Nuclear Criticality Safety Computational Tool Set 375
7.9.1 Cross-section Selection 375
7.9.2 Using All Available Tools to Ensure Economical and Accurate Computations 376
7.10 The Future 376
7.11 Summary 377
References 377
8 Nuclear Reactor Kinetics: 1934–1999 and Beyond 382
8.1 Introduction 382
8.2 Prologue: The Historical Origins of the Equations of Reactor Kinetics 383
8.2.1 The Time-Dependent Neutron Diffusion Equation 383
8.2.2 The Point Reactor Kinetics Model 385
8.3 The Point Reactor Kinetics Equations 389
8.3.1 The Basics: From the One-Group Diffusion Equation with Delayed Neutrons for a Bare Homogeneous Reactor to the Point Reactor Kinetics Equations 389
8.3.2 More General Developments of the Point Reactor Kinetics Equations: ``Shape Functions,'' ``Time Functions,'' and ``Neutron Importance'' 395
8.3.3 Variational Formulations and Asymptotic Formulations of Point Reactor Kinetics and the Appearance of ``Additional Terms'' 401
8.4 A Digression on the Kinetics of a Pulse of Neutrons in Non-multiplying Systems and Subcritical Multiplying Systems: Pulsed Neutron Experiments and Their Analysis 406
8.4.1 Neutron Thermalization, Exponential Decay, and Diffusion Cooling 406
8.4.2 Non-Exponential Decay and the Theory of Pulsed Neutron Die-Away: The Continuous Spectrum of the Boltzmann Operator 409
8.4.3 Exponential and Non-exponential Decay in Subcritical Fast Multiplying Assemblies 418
8.5 Space–Time Reactor Kinetics 424
8.5.1 Finite-Difference Schemes for the Time-Dependent Multigroup Neutron Diffusion Equations 426
8.5.2 Variational, Modal, Synthesis, and Related Methods for the Time-Dependent Multigroup Diffusion Equations 430
8.5.3 Coarse-Mesh and Nodal Methods for Space–Time Reactor Kinetics 436
8.5.4 Homogenization Theories for Space–Time Kinetics Calculations and for Point Kinetics Calculations 441
8.5.5 Adaptive-Model Kinetics Calculations 447
8.6 Reactor Dynamics 448
8.7 Epilogue: 1934–1999 (and Prologue: 2000 – and Beyond) 454
8.7.1 Adaptive-Model Reactor Kinetics 455
8.7.2 Reactor Dynamics of Advanced Reactors 456
8.7.3 Reactor Dynamics in the Twenty-First Century 457
References 458
Index 465

"Chapter 6 Sensitivity and Uncertainty Analysis of Models and Data (p. 291-293)

Dan Gabriel Cacuci

6.1 Introduction

This chapter highlights the characteristic features of statistical and deterministic methods currently used for sensitivity and uncertainty analysis of measurements and computationalmodels. The symbiotic linchpin between the objectives of uncertainty analysis and those of sensitivity analysis is provided by the “propagation of errors” equations, which combine parameter uncertaintieswith the sensitivities of responses (i.e., results of measurements and/or computations) to these parameters.

It is noted that all statistical uncertainty and sensitivity analysis methods first commence with the “uncertainty analysis” stage, and only subsequently proceed to the “sensitivity analysis” stage. This procedural path is the reverse of the procedural (and conceptual) path underlying the deterministic methods of sensitivity and uncertainty analysis, where the sensitivities are determined prior to using them for uncertainty analysis.

In particular, it is emphasized that the Adjoint Sensitivity Analysis Procedure (ASAP) is themost e?cientmethod for computing exactly the local sensitivities for large-scale nonlinear problems comprising many parameters. This e?ciency is underscored with illustrative examples. The computational resources required by the most popular statistical and deterministic methods are discussed comparatively. A brief discussion of unsolved fundamental problems, open for future research, concludes this chapter.

6.2 Sensitivities and Uncertainties in Measurements and Computational Models: Basic Concepts

In practice, scientists and engineers often face questions such as: How well does the model under consideration represent the underlying physical phenomena? What confidence can one have that the numerical results produced by the model are correct? How far can the calculated results be extrapolated? How can the predictability and/or extrapolation limits be extended and/or improved?

Answers to such questions are provided by sensitivity and uncertainty analyses. As computerassisted modeling and analyses of physical processes have continued to grow and diversify, sensitivity and uncertainty analyses have become indispensable investigative scientific tools in their own right. Since computers operate on mathematical models of physical reality, computed results must be compared to experimental measurements whenever possible.

Such comparisons, though, invariably reveal discrepancies between computed and measured results. The sources of such discrepancies are the inevitable errors and uncertainties in the experimental measurements and in the mathematical models. In practice, the exact forms of mathematical models and/or exact values of data are not known, so their mathematical form must be estimated. The use of observations to estimate the underlying features of models forms the objective of statistics.

This branch of mathematical science embodies both inductive and deductive reasoning, encompassing procedures for estimating parameters from incomplete knowledge and for refining prior knowledge by consistently incorporating additional information. Thus, assessing and, subsequently, reducing uncertainties in models and data require the combined use of statistics together with the axiomatic, frequency, and Bayesian interpretations of probability."