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Nuclear engineering has undergone extensive progress over the years. In the past century, colossal developments have been made and with specific reference to the mathematical theory and computational science underlying this discipline, advances in areas such as high-order discretization methods, Krylov Methods and Iteration Acceleration have steadily grown.

Nuclear Computational Science: A Century in Review addresses these topics and many more; topics which hold special ties to the first half of the century, and topics focused around the unique combination of nuclear engineering, computational science and mathematical theory. Comprising eight chapters, Nuclear Computational Science: A Century in Review incorporates a number of carefully selected issues representing a variety of problems, providing the reader with a wealth of information in both a clear and concise manner. The comprehensive nature of the coverage and the stature of the contributing authors combine to make this a unique landmark publication.

Targeting the medium to advanced level academic, this book will appeal to researchers and students with an interest in the progression of mathematical theory and its application to nuclear computational science.



Chapter 1. Advances in Discrete-Ordinates Methodology

In 1968, Bengt Carlson and Kaye Lathrop published a comprehensive review on the state of the art in discrete-ordinates (SN) calculations [10]. At that time, SN methodology existed primarily for reactor physics simulations. By today’s standards, those capabilities were limited, due to the less-developed theoretical state of SN methods and the slower and smaller computers that were then available. In this chapter, we review some of the major advances in SN methodology that have occurred since 1968. These advances, combined with the faster speeds and larger memories of today’s computers, enable today’s SN codes to simulate problems of much greater complexity, realism, and physical variety. Since 1968, several books and reviews on general numerical methods for SN simulations have been published [32, 46, 71], but none of these covers the advanced work done during the past 20 years.
Edward W. Larsen, Jim E. Morel

Chapter 2. Second-Order Neutron Transport Methods

Among the approaches to obtaining numerical solutions for neutral particle transport problems, those classified as second-order or even-parity methods have found increased use in recent decades. First-order and second-order methods differ in a number of respects. Following discretization of the energy variable, invariably through some form of the multigroup approximation, the time-independent forms of both are differential in the spatial variable and integral in angle. They differ in that the more conventional first-order equation includes only first derivatives in the spatial variables, but requires solution over the entire angular domain. Conversely, the second-order form includes second derivatives but requires solution over one half of the angular domain. The two forms in turn lead to contrasting approaches to reducing the differential–integral equations to sets of linear equations and in the formulation of iterative methods suitable for the numerical solution of large engineering design problems. In what follows, we explore the state of methods used to solve the second-order transport equation, comparing them, where possible, to first-order methods.
E. E. Lewis

Chapter 3. Monte Carlo Methods

Monte Carlo methods comprise a large and still growing collection of methods of repetitive simulation designed to obtain approximate solutions of various problems by playing games of chance. Often these methods are motivated by randomness inherent in the problem being studied (as, e.g., when simulating the random walks of “particles” undergoing diffusive transport), but this is not an essential feature of Monte Carlo methods. As long ago as the eighteenth century, the distinguished French naturalist Compte de Buffon [1] described an experiment that is by now well known: a thin needle of length l is dropped repeatedly on a plane surface that has been ruled with parallel lines at a fixed distance d apart. Then, as Laplace suggested many years later [2], an empirical estimate of the probability P of an intersection obtained by dropping a needle at random a large number, N, of times and observing the number, n, of intersections provides a practical means for estimating π
Jerome Spanier

Chapter 4. Reactor Core Methods

This chapter addresses the simulation flow chart that is currently used for reactor-physics simulations. The methodologies presented are more appropriate to the context of power reactors, and the chapter focuses particularly on the three-dimensional (3D) aspect of core calculations. Software design that is currently used to achieve accurate numerical simulations of reactor cores is also studied from a practical nuclear engineering point of view. The focus here is on processes and the needs for reactor physicists or nuclear engineers to use modern-day software with confidence and reliability.
Robert Roy

Chapter 5. Resonance Theory in Reactor Applications

The most essential objective in reactor physics is to provide an accurate account of the intricate balance between the neutrons produced by the fission process and those lost due to the absorption process as well as those leaking out of the reactor. The presence of resonance structures in neutron cross sections obviously plays an important role in such processes. Therefore, the treatment of neutron resonance phenomena has constituted one of the most fundamental subjects in reactor physics since its conception. It is the area where the concepts of nuclear reaction and the treatment of the neutronic balance in reactor lattices over a wide span of energy become intertwined. The basic issue here is how to apply the microscopic neutron cross sections in the macroscopic reactor systems. Because of its importance to reactor physics, much of the existing nuclear data and a significant portion of all cross-section processing codes downstream are devoted to the treatment of resonance phenomena prior to any meaningful neutronic calculations via either the deterministic or Monte Carlo approaches.
R. N. Hwang†

Chapter 6. Sensitivity and Uncertainty Analysis of Models and Data

This chapter highlights the characteristic features of statistical and deterministic methods currently used for sensitivity and uncertainty analysis of measurements and computational models. 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 uncertainties with 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 the most efficient method for computing exactly the local sensitivities for large-scale nonlinear problems comprising many parameters. This efficiency 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.
Dan Gabriel Cacuci

Chapter 7. Criticality Safety Methods

The objective of this chapter is to examine the history of nuclear criticality safety calculations. To this end, we will review the history of criticality safety concerns and look at the various approaches in dealing with this particular type of calculation. The criticality safety methods that are the subject of this chapter are those that are used to determine the criticality safety of the handling, transportation, and storage of fissile materials outside nuclear reactors.
G. E. Whitesides, R. M. Westfall, C. M. Hopper

Chapter 8. Nuclear Reactor Kinetics: 1934–1999 and Beyond

When one of the editors of this book, Professor Yousry Azmy, asked me to give a lecture on the development of reactor kinetics during the twentieth century and the future direction of research in this area in the twenty-first century, my reaction was, “Wow! Review the developments of a century of reactor kinetics. That’s a lot to cover.” But then, upon reflection on the fact that the discipline of reactor kinetics, and reactor physics in general, did not even exist until the 1930s, I realized that I did not have to review a whole century of development, but rather, a mere two thirds of the century! Still a somewhat daunting task!
Jack Dorning


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