Monte Carlo Methods for Particle Transport

Alireza Haghighat

• Éditeur: CRC Press
• Année: 2020
• ISBN:
  9780429582202
• Édition: 2
• Sujet: Mathematics
• Sous-sujet: Probability & Statistics

Fully updated with the latest developments in the eigenvalue Monte Carlo calculations and automatic variance reduction techniques and containing an entirely new chapter on fission matrix and alternative hybrid techniques. This second edition explores the uses of the Monte Carlo method for real-world applications, explaining its concepts and limitations. Featuring illustrative examples, mathematical derivations, computer algorithms, and homework problems, it is an ideal textbook and practical guide for nuclear engineers and scientists looking into the applications of the Monte Carlo method, in addition to students in physics and engineering, and those engaged in the advancement of the Monte Carlo methods.

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• Describes general and particle-transport-specific automated variance reduction techniques

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• Presents Monte Carlo particle transport eigenvalue issues and methodologies to address these issues

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• Presents detailed derivation of existing and advanced formulations and algorithms with real-world examples from the author's research activities

Chapter 1

Introduction

CONTENTS

1.1History of Monte Carlo Simulation

1.2Status of Monte Carlo Codes

1.3Motivation for Writing this Book

1.4Author’s Message to Instructors

The Monte Carlo method is a statistical technique which is capable of simulating a mathematical or physical experiment on a computer. In mathematics, it can provide the expectation value of functions and evaluate integrals; in science and engineering, it is capable of simulating complex problems which are comprised of various random processes with known or assumed probability density functions. To be able to simulate the random process, i.e., sample from a probability function for an event, it uses random numbers or pseudo-random numbers. Just like any statistical process, the Monte Carlo method requires repetition to achieve a small relative uncertainty, and therefore, may necessitate impractically large simulation times. To overcome this difficulty, parallel algorithms and variance reduction techniques are needed. This book attempts to address major topics affecting development, utilization, and performance of a Monte Carlo algorithm.

1.1History of Monte Carlo Simulation

The birth of Monte Carlo simulation can be traced back to WWII. At that time, because of the Manhattan project, there was significant urgency in understanding nuclear fission and generating special nuclear materials. Great minds from all over the world were assembled in the United States to work on the Manhattan project. This coincided with another initiative: building of the first electronic computer. The first computer, ENIAC, had over 17,000 vacuum tubes in a system with 500,000 solder joints and was built at the University of Pennsylvania in Philadelphia under the leadership of physicist John Mauchly and engineer Presper Eckert [75]. The story is that, Mauchly was inspired to build an electronic computer to perform the work done in large rooms filled with mostly women calculating integrals for firing tables (ranges versus trajectories) for the U.S. Army. John von Neumann, who was a consultant for both the Army and Los Alamos National Lab (LANL) and was well aware of Edward Teller’s new initiative in the area of thermonuclear energy, became interested in using ENIAC for testing models for thermonuclear reactions. He convinced the U.S. Army that it was beneficial for Los Alamos scientists (Nicholas Metropolis and Stan Frankel) to simulate thermonuclear reactions using ENIAC. The work started in 1945, before the end of the war, and its initial phase ended after the war in 1946. In addition to Metropolis, Frankel, and von Neumann, another scientist named Stanislaw (Stan) Ulam participated in a national project review meeting at LANL. It was Ulam’s observation that the new electronic computer could be used for performing the tedious statistical sampling that was somewhat abandoned because of the inability to do large numbers of calculations. Von Neumann became interested in Ulam’s suggestion and prepared an outline of a statistical approach for solving the neutron diffusion problem. Several people, including Metropolis, became interested in exploring the new statistical simulation approach; Metropolis suggested the name “Monte Carlo,” which was inspired from the fact that Ulam’s uncle used to borrow money from his family members because he “just had to go to Monte Carlo,” a popular gambling destination in the Principality of Monaco. For the initial simulation, von Neumann suggested a spherical core of fissionable material surrounded by a shell of tamper material. The goal was to simulate neutron histories as they go through different interactions. To be able to sample the probability density functions associated with these interactions, he invented a pseudo-random number generator algorithm [104] referred to as middle-square digits, which was later replaced with more effective generators by H. Lehmer [64].It was quickly realized that the Monte Carlo method was more flexible for simulating complex problems as compared to differential equations. However, since the method is a statistical process and requires the achievement of small variances, it was plagued by long times because of the need for a significant amount of computation! It is important to note that Enrico Fermi had already used the method for studying the moderation of neutrons using a mechanical calculator in the early 1930s while living in Rome. Naturally, Fermi was delighted with the invention of ENIAC; however, he came up with the idea of building an analog device called FERMIAC (shown in Figure 1.1) for the study of neutron transport. This device was built by Percy King and was limited to two energy groups, fast and thermal, and two dimensions. Figure 1.2 shows a demonstration of its application. Meanwhile, Metropolis and von Neumann’s wife (Klari) designed a new control system for ENIAC to be able to process a set of instructions or a stored-program as opposed to the plugboard approach. With this new capability, Metropolis and von Neumann were able to solve several neutron transport problems. Soon other scientists in the thermonuclear group started studies with different geometries and for different particle energies. Later, mathematicians Herman Khan, C. J. Everett, and E. Cashwell became interested in the Monte Carlo method and published several articles on algorithms and use of the method for particle transport simulation [57, 56, 22, 33, 32].

Figure 1.1 Photo of FERMIAC (courtesy of Mark Pellegini, the Bradbury Museum, Los Alamos, NM)

Figure 1.2 Application of FERMIAC (from N. Metropolis. 1987. Los Alamos Science 15: 125)

1.2Status of Monte Carlo Codes

The above brief history indicates that early development of Monte Carlo particle transport techniques were mainly conducted by the scientists at LANL. As a result, LANL has been the main source of general-purpose Monte Carlo codes, starting with MCS (Monte Carlo Simulation) in 1963 and followed by MCN (Monte Carlo Neutron) in 1965, MCNG (Monte Carlo coupled Neutron and Gamma) in 1973, and MCNP (Monte Carlo Neutron Photon) in 1977. MCNP has been under continuous development and the latest version, MCNP6, was released in 2013 [80]. The progress made over the past 50 years demonstrates the sustained effort at LANL on development, improvement, and maintenance of Monte Carlo particle transport codes. There has also been simultaneous development of and improvement in nuclear physics parameters, i.e., cross sections, in the form of the cross-section library referred to as the Evaluated Nuclear Data File (ENDF). Currently, ENDF/B-VIII, the 8th version, is in use.

In addition to LANL, other groups, both in the United States and abroad, have developed Monte Carlo codes. The author’s group has developed the A³MCNP (Automated Adjoint Accelerated MCNP) code system [43] for automatic variance reduction using the CADIS (Consistent Adjoint Driven Importance Sampling) methodology [46]. The Oak Ridge National Lab (ORNL) has developed a number of codes, including MORSE [31] and KENO [82], and more recently the ADVANTG code system [78] that uses CADIS and its alternate, “forward” CADIS (FW-CADIS) [109], methodology. Internationally, there are two general codes, MCBEND [120] and TRIPOLI [13], and a few specialized codes, including EGS [50], GEANT [1], and PENELOPE [89]. GEANT, which is an open-source code, has been developed in support of nuclear physics experiments. The other two codes, EGS and PENELOPE, have special focus on electron-photon transport for medical applications.

Finally, it is important to mention that in the past two decades there have been efforts on the development of codes based on the hybrid deterministic-Monte Carlo techniques. Chapter 11 is devoted to this topic.

1.3Motivation for Writing this Book

Until the early 1990s, Monte Carlo methods were mainly used for benchmarking studies by scientists and engineers at national labs who had access to advanced computers. This situation, however, changed drastically with the advent of high performance computers (with fast clock cycles), parallel computers, and, more recently, PC clusters. To this effect, at the 8th International Conference on Radiation Shielding in 1994, over 70% of papers utilized deterministic methods for simulation of different problems or addressed new techniques and formulations. Since the late 1990s, this situation has been reversed, and Monte Carlo methods have become the first or, in some cases, only tool for performing particle transport simulations for a variety of applications. On the positive side, the method has enabled numerous people with different levels of knowledge and preparation to be able to perform particle transport simulations. However, on the negative side, it has created the potential of drawing erroneous results by novice users who do not appreciate the limitations o...

Table des matières

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Acknowledgement
• About the Author
• Chapter 1: Introduction
• Chapter 2: Random Variables and Sampling
• Chapter 3: Random Number Generator (RNG)
• Chapter 4: Fundamentals of Probability and Statistics
• Chapter 5: Integrals and Associated Variance Reduction Techniques
• Chapter 6: Fixed-Source Monte Carlo Particle Transport
• Chapter 7: Variance reduction techniques for fixed-source particle transport
• Chapter 8: Scoring/Tallying
• Chapter 9: Geometry and particle tracking
• Chapter 10: Eigenvalue (criticality) Monte Carlo method for particle transport
• Chapter 11: Fission matrix methods for eigenvalue Monte Carlo simulation
• Chapter 12: Vector and parallel processing of Monte Carlo particle transport
• Appendix A: Appendix 1
• Appendix B: Appendix 2
• Appendix C: Appendix 3
• Appendix D: Appendix 4
• Appendix E: Appendix 5
• Bibliography
• Index