Nuclear Computational Science - A Century in Review

Nuclear Computational Science

Preface

Obituary Composed by Dr. Roger Blomquistfor Dr. Ely Meyer Gelbard

1 Advances in Discrete-Ordinates Methodology

1.1 Introduction

1.2 Basic Concepts

1.3 Three Challenging Physical Problems

1.3.1 Thermal Radiation Transport in the Stellar Regime

1.3.2 Charged-Particle Transport

1.3.3 Oil-Well Logging Tool Design

1.4 Advances in Spatial Discretizations

1.4.1 Characteristic Methods

1.4.2 Linear Discontinuous Method

1.4.3 Nodal Methods

1.4.4 Solution Accuracy in the Thick Diffusion Limit

1.5 Advances in Angular Discretizations

1.5.1 Angular Derivatives

1.5.2 Anisotropic Scattering

1.6 Advances in Fokker–Planck Discretizations

1.6.1 The Continuous-Scattering Operator

1.6.2 The Continuous-Slowing-Down Operator

1.7 Advances in Time Discretizations

1.8 Advances in Iteration Acceleration

1.8.1 Fourier Analysis

1.8.2 Diffusion-Synthetic Acceleration

1.8.3 DSA-Like Methods for Outer Iteration Acceleration

1.8.4 A Deficiency in Multidimensional DSA and DSA-Like Methods

1.9 Krylov Methods

1.9.1 The Central Theme of Krylov Methods

1.9.2 Convergence and Preconditioning of Krylov Methods

1.9.3 Applying Krylov Methods to the SN Equations

1.10 Future Challenges

References

2 Second-Order Neutron Transport Methods

2.1 Introduction

2.2 The Transport Equation

2.2.1 First- and Second-Order Forms

2.2.2 Weak Forms

2.2.3 Variational Formulation

2.3 The Discretized Diffusion Equation

2.3.1 The Diffusion Formulation

2.3.2 Finite Element Discretization

2.4 The Discretized Transport Equation

2.4.1 Anisotropic Scattering

2.4.2 Angular Approximations

2.4.2.1 Spherical Harmonics Expansions

2.4.2.2 Discrete Ordinates Approximations

2.4.2.3 Simplified Angular Approximations

2.4.3 Spatial Discretization

2.5 Hybrid and Integral Methods

2.5.1 A Variational Nodal Method

2.5.2 An Even-Parity Integral Method

2.5.3 Combined Methods

2.6 Discussion

References

3 Monte Carlo Methods

3.1 Introduction

3.2 Organizing Principles

3.2.1 Generating Sequences

3.2.2 Error Analysis

3.2.3 Error Reduction

3.2.4 Foundations/Theoretical Developments

3.3 Historical Perspectives

3.4 Generating Sequences

3.4.1 Pseudorandom Sequences

3.4.2 Quasirandom Sequences

3.4.3 Hybrid Sequences

3.4.4 State of the Art

3.5 Error Analysis

3.5.1 The Pseudorandom Case

3.5.2 The Quasi-random Case

3.5.3 The Hybrid Case

3.5.4 Current State of the Art

3.6 Error Reduction

3.6.1 Introduction

3.6.2 Control Variates

3.6.3 Importance Sampling

3.6.4 Stratified Sampling

3.6.5 Use of Expected Values

3.6.6 Other Error Reduction Strategies

3.6.7 State of the Art

3.7 Foundations/Theoretical Developments

3.7.1 State of the Art

3.8 Challenges

References

4 Reactor Core Methods

4.1 Introduction

4.2 Analytic Methods and Early Calculation Schemes

4.3 Lattice Cell and Assembly Codes

4.3.1 Lattice Physics Calculations

4.3.1.1 Producing Cross-section Libraries

4.3.1.2 Self-Shielding and Multigroup Approximation

4.3.1.3 Generic Multigroup Solver

4.3.1.4 Discrete Ordinates

4.3.1.5 Method of Characteristics

4.3.1.6 Collision Probability Method

4.3.1.7 Bn Solutions and Diffusion Coefficients

4.3.1.8 Lattice Solvers

4.3.1.9 Putting It All Together into Lattice Codes

4.3.2 Homogenization Process

4.3.2.1 Reaction Rates and Homogenized Cross Sections

4.3.2.2 Generalized Equivalence Theory and Discontinuity Factors

4.4 Reactor Core Solvers

4.4.1 Pn Approximations and Diffusion

4.4.1.1 Spherical Harmonics and the Even-Parity Transport Equation

4.4.1.2 The Diffusion Approximation

4.4.1.3 SPn and Improved Diffusion

4.4.2 Diffusion-Like Methods

4.4.2.1 Transverse Integrated Nodal Methods

4.4.2.2 Analytic Nodal Methods

4.4.2.3 Core Harmonics and Modal Synthesis

4.4.3 Variational Formulation and Finite Elements

4.4.3.1 Classical Spatial Finite Elements

4.4.3.2 Mixed and Hybrid Finite Elements

4.4.4 Putting It All Together into Reactor Codes

4.5 Core Applications

4.5.1 Pin Power Reconstruction in LWR Reactors

4.5.2 Estimates of Zonal Powers in CANDU Reactors

4.5.3 Teaching Modern Reactor Core Methods

4.6 Concluding Remarks

References

5 Resonance Theory in Reactor Applications

5.1 Introduction

5.1.1 Historical Perspective

5.1.2 Self-shielding Effects in Perspective

5.2 Representation of Microscopic Cross Sections

5.2.1 Brief Description of R-Matrix Theory

5.2.1.1 Wigner–Eisenbud Version

5.2.1.2 Kapur–Peierls Version

5.2.2 Practical Representations Currently in Use

5.2.2.1 Single Level Breit–Wigner Approximation (SLBW)

5.2.2.2 Multilevel Breit–Wigner Approximation (MLBW)

5.2.2.3 Adler–Adler Approximation (AA)

5.2.2.4 Reich–Moore Approximation

5.2.3 Other Alternative: Generalized Pole Representation

5.3 Doppler-Broadening of Cross Sections

5.3.1 Practical Doppler-Broadening Kernels in Use

5.3.1.1 Ideal Gas Model

5.3.1.2 Accommodation of Crystalline Binding Effects via Effective Temperature Model

5.3.2 Analytical Broadening via Doppler-Broadened Line-Shape Functions

5.3.2.1 Traditional Doppler-Broadened Line-Shape Functions

5.3.2.2 Generalization of Doppler-Broadened Line-Shape Functions

5.3.2.3 Evaluations of the Doppler-Broadened Functions

5.3.3 Direct Numerical Doppler-Broadening of Point-Wise Cross Sections

5.3.3.1 Kernel-Broadening Approach

5.3.3.2 Heat Equation Approach Based on the Finite Difference Method

5.4 Resonance Absorption in Homogeneous Media

5.4.1 Average Scattering Kernel for Practical Applications

5.4.2 Characteristics of the Slowing-Down Equation

5.4.2.1 Slowing-Down Density

5.4.2.2 Concept of Placzek Oscillations

5.4.3 Resonance Integrals and Their Applications

5.4.3.1 Traditional Resonance Integral Concept

5.4.3.2 Various Resonance Integral Approximations

5.4.4 Various Developments Motivated by the Emergence of the Fast Reactor Program

5.4.4.1 Generalization and Computation of the J-Integral

5.4.4.2 Connections Between the Resonance Integral and Traditional Multigroup Cross Section Processing

5.4.4.3 Rigorous Treatment of Resonance Absorption via Numerical Means

5.5 Resonance Absorption in Heterogeneous Media

5.5.1 Traditional Collision Probability Methods for a Two-Region Cell

5.5.1.1 General Features of Collision Probability of Practical Interest

5.5.1.2 Various Earlier Methods Based on Approximate Escape Probabilities for Isolated Fuel Lumps

5.5.2 Traditional Collision Probability Treatment in a Closely Packed Lattice

5.5.2.1 General Features of the Escape Probability

5.5.2.2 Fine-Tuning of the Rational Approximation for Routine Applications

5.5.3 Connections to Resonance Integral and Multigroup Cross-section Calculations

5.5.3.1 Rational Approximation and Approximate Flux Based Approaches

5.5.3.2 Nordheim's Method

5.5.4 Rigorous Treatment of Resonance Absorption in a Unit Cell with Multiple Regions and Many Resonance Isotopes

5.5.4.1 Kier's Method for Cylindrical Unit Cells

5.5.4.2 Olson's Method for Unit Cell with Many Plates

5.6 Treatment of Unresolved Resonances

5.6.1 Statistical Theory Basis

5.6.1.1 Some Statistical Theory Fundamentals

5.6.1.2 Statistical Distributions of Practical Interest

5.6.1.3 Conceptual Aspects of Computing Average Cross Sections

5.6.2 Average Unshielded Cross Sections and Fluctuation Integrals

5.6.3 Traditional Approaches for Computing Self-shielded Average Cross Sections

5.6.3.1 Methods Based on Direct Integrations

5.6.3.2 Methods Using the Statistically Generated Resonance Ladders

5.6.4 Probability Table Methods

5.6.4.1 Conceptual Basis

5.6.4.2 Methods for Computing the Tabulated Quantities

5.7 Future Challenges

References

6 Sensitivity and Uncertainty Analysis of Models and Data

6.1 Introduction

6.2 Sensitivities and Uncertainties in Measurementsand Computational Models: Basic Concepts

6.2.1 Measurement Uncertainties

6.2.2 Propagation of Errors (Moments)

6.3 Statistical Methods for Sensitivity and Uncertainty Analysis

6.3.1 Sampling-Based Methods

6.3.2 Reliability Algorithms: FORM and SORM

6.3.3 Variance-Based Methods

6.3.4 Design of Experiments and Screening Design Methods

6.4 Deterministic Methods for Sensitivity and Uncertainty Analysis

6.4.1 The Forward Sensitivity Analysis Procedure (FSAP) for Nonlinear Systems

6.4.2 The Adjoint Sensitivity Analysis Procedure (ASAP) for Nonlinear Systems

6.4.3 The Adjoint Sensitivity Analysis Procedure (ASAP) for Responses Defined at Critical Points

6.4.4 ASAP for Linear Systems

6.5 Paradigm Applications of the ASAP

6.5.1 Application of the ASAP to Compute the Variance of the Maximum Flux of Particles in a Particle Diffusion Problem

6.5.2 ASAP for a Ricatti Equation

6.5.3 ASAP for a System of Linear Ordinary Differential Equations

6.6 Computational Considerations and Open Problems

References

7 Criticality Safety Methods

7.1 Introduction

7.1.1 Overview

7.1.2 Historical Background

7.2 Nuclear Criticality Safety: The Early Years

7.2.1 The First Criticality Concerns

7.2.2 Early Attempts at Criticality Safety Computations

7.3 Criticality Safety Versus Reactor Design Calculations

7.3.1 Computational Requirements for Reactor Design

7.3.2 Special Requirements for Criticality Safety Calculations

7.3.3 Early Computational Tools for Criticality Safety Calculations

7.4 Role of the Sn, or Discrete Ordinates Method

7.4.1 Impetus for the Early Sn Method Development

7.4.2 Later Sn Method Development

7.5 Role of the Monte Carlo Method

7.5.1 Early Monte Carlo Calculational Methods

7.6 Critical Experiments, Benchmarks, and Validation

7.7 Evaluation of the Various Methods and Their Role in Current Criticality Safety Calculations

7.7.1 Role of the Sn Method

7.7.2 Evaluation of the Role of the Sn Method

7.7.3 Role of the Monte Carlo Method

7.7.4 Evaluation of the Monte Carlo Method

7.7.5 Summary of the Monte Carlo Criticality Safety Software

7.7.6 The N(BN)2 Method

7.8 The Role of Cross-Section Representation

7.8.1 Multigroup Cross Sections

7.8.2 Point-Wise Cross Sections

7.9 Elements of a Complete Nuclear Criticality Safety Computational Tool Set

7.9.1 Cross-section Selection

7.9.2 Using All Available Tools to Ensure Economical and Accurate Computations

7.10 The Future

7.11 Summary

References

8 Nuclear Reactor Kinetics: 1934–1999 and Beyond

8.1 Introduction

8.2 Prologue: The Historical Origins of the Equations of Reactor Kinetics

8.2.1 The Time-Dependent Neutron Diffusion Equation

8.2.2 The Point Reactor Kinetics Model

8.3 The Point Reactor Kinetics Equations

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

8.3.2 More General Developments of the Point Reactor Kinetics Equations: ``Shape Functions,'' ``Time Functions,'' and ``Neutron Importance''

8.3.3 Variational Formulations and Asymptotic Formulations of Point Reactor Kinetics and the Appearance of ``Additional Terms''

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

8.4.1 Neutron Thermalization, Exponential Decay, and Diffusion Cooling

8.4.2 Non-Exponential Decay and the Theory of Pulsed Neutron Die-Away: The Continuous Spectrum of the Boltzmann Operator

8.4.3 Exponential and Non-exponential Decay in Subcritical Fast Multiplying Assemblies

8.5 Space–Time Reactor Kinetics

8.5.1 Finite-Difference Schemes for the Time-Dependent Multigroup Neutron Diffusion Equations

8.5.2 Variational, Modal, Synthesis, and Related Methods for the Time-Dependent Multigroup Diffusion Equations

8.5.3 Coarse-Mesh and Nodal Methods for Space–Time Reactor Kinetics

8.5.4 Homogenization Theories for Space–Time Kinetics Calculations and for Point Kinetics Calculations

8.5.5 Adaptive-Model Kinetics Calculations

8.6 Reactor Dynamics

8.7 Epilogue: 1934–1999 (and Prologue: 2000 – and Beyond)

8.7.1 Adaptive-Model Reactor Kinetics

8.7.2 Reactor Dynamics of Advanced Reactors

8.7.3 Reactor Dynamics in the Twenty-First Century

References

Index