Monte Carlo Methods for Particle Transport

287 Pages·2014·5.115 MB·English

by  Alireza Haghighat

Nuclear physics H
a
g
“This is an outstanding reference and textbook on applied stochastic methods. It h Monte Carlo Methods
i
is a must-have for scientists, students, and practitioners interested in Monte Carlo g
h
methods for solving particle transport problems. This book provides an excellent a
t
description of the fundamentals through numerous example problems and a rich
discussion of advantages and pitfalls of the Monte Carlo method. The chapter M for Particle Transport
on solving eigenvalue problems is long overdue, where diagnosing convergence
o
of the fission source in reactor physics problems with high dominance ratio is
n
challenging and as a result has been a subject of much research.”
t
—Farzad Rahnema, Georgia Institute of Technology, Atlanta, USA e

C
“This is a very solid book for graduate students in nuclear engineering to learn how
a
the Monte Carlo method can be used to solve reactor physics problems. It covers
r
l
the fundamentals of Monte Carlo simulations before discussing how the technique o
can be used to solve fixed and fission sources of neutron transport problems.
M
Excellent examples are provided in the main text, in addition to a complete set of
homework problems at the end of each chapter. This makes it an ideal textbook for e
those teaching a course on simulation methods in reactor physics.” t
h
—G. Marleau, École Polytechnique de Montréal, Québec, Canada
o
d
“Professor Haghighat has succeeded in writing a book that is concise, but also
s
includes all ingredients in the Monte Carlo method. … [This book is] an excellent
f
addition to the bookshelf of teachers, students, researchers, and practitioners in the o
field of nuclear reactor design and radiation shielding applications.” r

—Nam Zin Cho, Korea Advanced Institute of Science and Technology, P
Daejeon, South Korea a
r
t
i
c
l
e

T
r
a
n
Alireza Haghighat
s
p
o
K20567
6000 Broken Sound Parkway, NW r
Suite 300, Boca Raton, FL 33487 ISBN: 978-1-4665-9253-7 t
711 Third Avenue 90000
New York, NY 10017
an informa business
2 Park Square, Milton Park
www.crcpress.com Abingdon, Oxon OX14 4RN, UK
9 781466 592537
w w w. c r c p r e s s . c o m
K20567 mech rev.indd 1 10/15/14 8:20 AM
Monte Carlo Methods
for Particle Transport
K20567_Book.indb 1 10/24/14 12:15 PM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Monte Carlo Methods
for Particle Transport
Alireza Haghighat
K20567_Book.indb 3 10/24/14 12:15 PM
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2015 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20141014
International Standard Book Number-13: 978-1-4665-9254-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copy-
right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
vides licenses and registration for a variety of users. For organizations that have been granted a photo-
copy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
I would like dedicate this book to
my wife, son, and mother.
v
K20567_Book.indb 5 10/24/14 12:15 PM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Acknowledgments xv
About the author xvii
1 Introduction 1
1.1 History of Monte Carlo simulation 1
1.2 Status of Monte Carlo codes 4
1.3 Motivation for writing this book 4
1.4 Overview of the book 5
1.5 Recommendations to instructors 8
1.6 Author’s expectation 9
References 9
2 Random variables and sampling 11
2.1 Introduction 11
2.2 Random variables 12
2.2.1 Discrete random variable 12
2.2.2 Continuous random variable 13
2.2.3 Notes on pdf and cdf characteristics 15
2.3 Random numbers 15
2.4 Derivation of the fundamental formulation of
Monte Carlo (FFMC) 16
2.5 Sampling one-dimensional density functions 18
2.5.1 Analytical inversion 18
vii
K20567_Book.indb 7 10/24/14 12:15 PM
viii Contents
2.5.2 Numerical inversion 18
2.5.3 Probability mixing method 20
2.5.4 Rejection technique 21
2.5.5 Numerical evaluation 22
2.5.6 Table lookup 23
2.6 Sampling multidimensional density functions 24
2.7 Example procedures for sampling a few
commonly used distributions 26
2.7.1 Normal distribution 27
2.7.2 Watt spectrum 28
2.7.3 Cosine and sine function sampling 29
2.8 Remarks 29
References 30
Problems 30
3 Random number generation (RNG) 35
3.1 Introduction 35
3.2 Random number generation approaches 36
3.3 Pseudorandom number generators (PRNGs) 38
3.3.1 Congruential generators 38
3.3.2 Multiple recursive generator 45
3.4 Testing randomness 47
3.4.1 χ2-test 47
3.4.2 Frequency test 49
3.4.3 Serial test 49
3.4.4 Gap test 49
3.4.5 Poker test 49
3.4.6 Moment test 49
3.4.7 Serial correlation test 50
3.4.8 Serial test via plotting 50
3.5 Examples for PRNG tests 50
3.5.1 Evaluation of PRNG based on period
and average 50
3.5.2 Serial test via plotting 53
3.6 Remarks 55
References 55
Problems 56
4 Fundamentals of probability and statistics 59
4.1 Introduction 59
4.2 Expectation value 60
4.2.1 One-dimensional density function 60
K20567_Book.indb 8 10/24/14 12:15 PM
Contents Ix
4.2.2 Multidimensional density function 62
4.2.3 Useful theorems associated with the
“true variance” 63
4.3 Definition of sample expectation values
used in statistics 63
4.3.1 Sample mean 63
4.3.2 Expected value of the sample variance 64
4.4 Precision and accuracy of a statistical process 66
4.4.1 Uniform distribution 67
4.4.2 Bernoulli and binomial distributions 67
4.4.3 Geometric distribution 71
4.4.4 Poisson distribution 72
4.4.5 Normal (“Gaussian”) distribution 74
4.5 Limit theorems and their applications 81
4.5.1 Corollary to the de Moivre–Laplace
limit theorem 81
4.5.2 Central limit theorem 85
4.6 Formulations of uncertainty and relative error
for a random process 88
4.6.1 General random process 88
4.6.2 Special case of a Bernoulli process 89
4.7 Confidence interval for finite sampling 90
4.7.1 Introduction to Student’s t-distribution 90
4.7.2 Determination of confidence interval
and application of the t-distribution 92
4.8 Test of normality of distribution 93
4.8.1 Test of skewness coefficient 93
4.8.2 Shapiro–Wilk test for normality 93
References 94
Problems 94
5 Integrals and associated variance reduction
techniques 99
5.1 Introduction 99
5.2 Estimation of integrals 100
5.3 Variance reduction techniques associated with
integrals 101
5.3.1 Importance sampling 102
5.3.2 Correlation sampling technique 106
5.3.3 Stratified sampling technique 107
5.3.4 Combined sampling 118
K20567_Book.indb 9 10/24/14 12:15 PM