Monte Carlo Methods for Particle Transport

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Acknowledgement
About the Author
Chapter 1: Introduction
	1.1 HISTORY OF MONTE CARLO SIMULATION
	1.2 STATUS OF MONTE CARLO CODES
	1.3 MOTIVATION FOR WRITING THIS BOOK
	1.4 AUTHOR’S MESSAGE TO INSTRUCTORS
Chapter 2: Random Variables and Sampling
	2.1 INTRODUCTION
	2.2 RANDOM VARIABLES
		2.2.1 Discrete random variable
		2.2.2 Continuous random variable
		2.2.3 Notes on pdf and cdf characteristics
	2.3 RANDOM NUMBERS
	2.4 DERIVATION OF THE FUNDAMENTAL FORMULATION OF MONTE CARLO (FFMC)
	2.5 SAMPLING ONE DIMENSIONAL DENSITY FUNCTIONS
		2.5.1 Analytical inversion
		2.5.2 Numerical inversion
		2.5.3 Probability mixing method
		2.5.4 Rejection technique
		2.5.5 Numerical evaluation
		2.5.6 Table lookup
	2.6 SAMPLING MULTIDIMENSIONAL DENSITY FUNCTIONS
	2.7 EXAMPLE PROCEDURES FOR SAMPLING A FEW COMMONLY USED DISTRIBUTIONS
		2.7.1 Normal distribution
		2.7.2 Watt spectrum
		2.7.3 Cosine and sine functions sampling
	2.8 REMARKS
Chapter 3: Random Number Generator (RNG)
	3.1 INTRODUCTION
	3.2 RANDOM NUMBER GENERATION APPROACHES
	3.3 PSEUDO RANDOM NUMBER GENERATORS (PRNGS)
		3.3.1 Congruential Generators
		3.3.2 Multiple Recursive Generator
	3.4 TESTING RANDOMNESS
		3.4.1 χ2 − Test
			3.4.1.1 χ2 − distribution
			3.4.1.2 Procedure for the use of χ2 − test
		3.4.2 Frequency test
		3.4.3 Serial test
		3.4.4 Gap test
		3.4.5 Poker test
		3.4.6 Moment test
		3.4.7 Serial correlation test
		3.4.8 Serial test via plotting
	3.5 EXAMPLE FOR TESTING A PRNG
		3.5.1 Evaluation of PRNG based on period and average
		3.5.2 Serial test via plotting
	3.6 REMARKS
Chapter 4: Fundamentals of Probability and Statistics
	4.1 INTRODUCTION
	4.2 EXPECTATION VALUE
		4.2.1 Single variable
		4.2.2 Useful formulation for the expectation operator
		4.2.3 Multivariable
	4.3 SAMPLE EXPECTATION VALUES IN STATISTICS
		4.3.1 Sample mean
		4.3.2 Sample variance
	4.4 PRECISION AND ACCURACY OF A SAMPLE AVERAGE
	4.5 COMMONLY USED DENSITY FUNCTIONS
		4.5.1 Uniform density function
		4.5.2 Binomial density function
			4.5.2.1 Bernoulli process
			4.5.2.2 Derivation of the Binomial density function
		4.5.3 Geometric density function
		4.5.4 Poisson density function
		4.5.5 Normal (
	4.6 LIMIT THEOREMS AND THEIR APPLICATIONS
		4.6.1 Corollary to the de Moivre-Laplace limit theorem
		4.6.2 Central limite theorem
			4.6.2.1 Demonstration of the Central Limit Theorem
	4.7 GENERAL FORMULATION OF THE RELATIVE UNCERTAINTY
		4.7.1 Special case of a Bernoulli random process
	4.8 CONFIDENCE LEVEL FOR FINITE SAMPLING
		4.8.1 Student’s t-distribution
		4.8.2 Determination of confidence level and application of the t-distribution
	4.9 TEST OF NORMALITY OF DISTRIBUTION
		4.9.1 Test of skewness coefficient
		4.9.2 Shapiro-Wilk test for normality
Chapter 5: Integrals and Associated Variance Reduction Techniques
	5.1 INTRODUCTION
	5.2 EVALUATION OF INTEGRALS
	5.3 VARIANCE REDUCTION TECHNIQUES FOR DETERMINATION OF INTEGRALS
		5.3.1 Importance sampling
		5.3.2 Control variates technique
		5.3.3 Stratified sampling technique
		5.3.4 Combined sampling
	5.4 REMARKS
Chapter 6: Fixed-Source Monte Carlo Particle Transport
	6.1 INTRODUCTION
	6.2 INTRODUCTION TO THE LINEAR BOLTZMANN EQUATION
	6.3 MONTE CARLO METHOD FOR SIMPLIFIED PARTICLE TRANSPORT
		6.3.1 Sampling path length
		6.3.2 Sampling interaction type
			6.3.2.1 Procedure for N (> 2) interaction type
			6.3.2.2 Procedure for a discrete random variable with N outcomes of equal probabilities
		6.3.3 Selection of scattering angle
	6.4 A 1-D MONTE CARLO ALGORITHM
	6.5 PERTURBATION VIA CORRELATED SAMPLING
	6.6 HOW TO EXAMINE STATISTICAL RELIABILITY OF MONTE CARLO RESULTS
	6.7 REMARKS
Chapter 7: Variance reduction techniques for fixed-source particle transport
	7.1 INTRODUCTION
	7.2 OVERVIEW OF VARIANCE REDUCTION FOR FIXED SOURCE PARTICLE TRANSPORT
	7.3 PDF BIASING WITH RUSSIAN ROULETTE
		7.3.1 Implicit capture or survival biasing with Russian roulette
			7.3.1.1 Russian roulette technique
		7.3.2 Path-length biasing
		7.3.3 Exponential transformation biasing
		7.3.4 Forced collision biasing
	7.4 PARTICLE SPLITTING WITH RUSSIAN ROULETTE
		7.4.1 Geometric splitting
		7.4.2 Energy splitting
		7.4.3 Angular splitting
	7.5 WEIGHT-WINDOW TECHNIQUE
	7.6 INTEGRAL BIASING
		7.6.1 Importance (adjoint) function methodology
		7.6.2 Source biasing based on the importance sampling
	7.7 HYBRID METHODOLOGIES
		7.7.1 CADIS methodology
			7.7.1.1 FW-CADIS technique
	7.8 REMARKS
Chapter 8: Scoring/Tallying
	8.1 INTRODUCTION
	8.2 MAJOR PHYSICAL QUANTITIES IN PARTICLE TRANSPORT
	8.3 TALLYING IN A STEADY STATE SYSTEM
		8.3.1 Collision estimator
		8.3.2 Path-length estimator
		8.3.3 Surface-crossing estimator
			8.3.3.1 Estimation of partial and net currents
			8.3.3.2 Estimation of flux on a surface
		8.3.4 Analytical estimator
	8.4 TIME DEPENDENT TALLYING
	8.5 FORMULATION OF TALLIES WHEN VARIANCE REDUCTION USED
	8.6 ESTIMATION OF RELATIVE UNCERTAINTY OF TALLIES
	8.7 UNCERTAINTY IN A RANDOM VARIABLE DEPENDENT ON OTHER RANDOM VARIABLES
	8.8 REMARKS
Chapter 9: Geometry and particle tracking
	9.1 INTRODUCTION
	9.2 COMBINATORIAL GEOMETRY APPROACH
		9.2.1 Definition of a surface
		9.2.2 Definition of cells
		9.2.3 Examples for irregular cells
	9.3 DESCRIPTION OF BOUNDARY CONDITIONS
	9.4 PARTICLE TRACKING
	9.5 REMARKS
Chapter 10: Eigenvalue (criticality) Monte Carlo method for particle transport
	10.1 INTRODUCTION
	10.2 THEORY OF POWER ITERATION FOR EIGENVALUE PROBLEMS
	10.3 MONTE CARLO EIGENVALUE CALCULATION
		10.3.1 Random variables for sampling fission neutrons
			10.3.1.1 Number of fission neutrons
			10.3.1.2 Energy of fission neutrons
			10.3.1.3 Direction of fission neutrons
		10.3.2 Procedure for Monte Carlo Eigenvalue simulation
			10.3.2.1 Estimators for sampling fission neutrons
		10.3.3 A method to combine the estimators
	10.4 ISSUES ASSOCIATED WITH THE STANDARD EIGENVALUE MONTE CARLO SIMULATION PROCEDURE
	10.5 DIAGNOSTIC TESTS FOR SOURCE CONVERGENCE
		10.5.1 Shannon entropy technique
			10.5.1.1 Concept of Shannon entropy
			10.5.1.2 Application of the Shannon entropy to the fission neutron source
		10.5.2 Center of Mass (COM) technique
	10.6 STANDARD EIGENVALUE MONTE CARLO CALCULATION  PERFORMANCE, ANALYSIS, SHORTCOMINGS
		10.6.1 A procedure for selection of appropriate eigenvalue parameters
		10.6.2 Demonstration of the shortcomings of the standard eigenvalue Monte Carlo calculation
			10.6.2.1 Example problem
			10.6.2.2 Results and analysis
	10.7 REMARKS
Chapter 11: Fission matrix methods for eigenvalue Monte Carlo simulation
	11.1 INTRODUCTION
	11.2 DERIVATION OF FORMULATION OF THE FISSION MATRIX METHODOLOGY
		11.2.1 Implementation of the FM method - Approach 1
		11.2.2 Implementation of the FM method - Approach 2
			11.2.2.1 Issues associated with the FMBMC approach
	11.3 APPLICATION OF THE FM METHOD - APPROACH 1
		11.3.1 Modeling spent fuel facilities
			11.3.1.1 Problem description
			11.3.1.2 FM coefficient pre-calculation
			11.3.1.3 Comparison of RAPID to Serpent - Accuracy and Performance
		11.3.2 Reactor cores
		11.3.3 A few innovative techniques for generation or correction of FM coeffiicients
			11.3.3.1 Geometric similarity
			11.3.3.2 Boundary correction
			11.3.3.3 Material discontinuity
		11.3.4 Simulation of the OECD/NEA benchmark
	11.4 DEVELOPMENT OF OTHER FM MATRIX BASED FORMULATIONS
	11.5 REMARKS
Chapter 12: Vector and parallel processing of Monte Carlo particle transport
	12.1 INTRODUCTION
	12.2 VECTOR PROCESSING
		12.2.0.1 Scalar computer
		12.2.0.2 Vector computer
		12.2.1 Vector performance
	12.3 PARALLEL PROCESSING
		12.3.1 Parallel performance
			12.3.1.1 Factors affecting the parallel performance
	12.4 VECTORIZATION OF THE MONTE CARLO PARTICLE TRANSPORT METHODS
	12.5 PARALLELIZATION OF THE MONTE CARLO PARTICLE TRANSPORT METHODS
		12.5.1 Other possible parallel Monte Carlo particle transport algorithms
	12.6 DEVELOPMENT OF A PARALLEL ALGORITHM USING MPI
	12.7 REMARKS
Appendix A: Appendix 1
	A.1 INTEGER OPERATIONS ON A BINARY COMPUTER
Appendix B: Appendix 2
	B.1 DERIVATION OF A FORMULATION FOR THE SCATTERING DIRECTION IN A 3 D DOMAIN
Appendix C: Appendix 3
	C.1 SOLID ANGLE FORMULATION
Appendix D: Appendix 4
	D.1 ENERGY DEPENDENT NEUTRON NUCLEAR INTERACTIONS IN MONTE CARLO SIMULATION
	D.2 INTRODUCTION
	D.3 ELASTIC SCATTERING
	D.4 INELASTIC SCATTERING
	D.5 SCATTERING AT THERMAL ENERGIES
Appendix E: Appendix 5
	E.1 SHANNON ENTROPY
		E.1.1 Derivation of the Shannon entropy - Approach 1
		E.1.2 Derivation of the Shannon entropy - Approach 2
Bibliography
Index