Download or read book A Rapidly Converging Alternative to Source Iteration for Solving the Discrete Ordinates Radiation Transport Equations in Slab Geometry written by Nicholas J. Wager and published by . This book was released on 2004-03 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: I present a numerical technique to solve the time independent Boltzmann Transport Equation for the transport of neutrons and photons. The technique efficiently solves the discrete ordinates equations with a new iteration scheme. I call this new scheme the angle space distribution iteration method because it combines a non-linear, high angular-resolution flux approximation within individual spatial cells with a coarse angular-resolution flux approximation that couples all cells in a spatial mesh. This shown to be an efficient alternative to source iteration. The new method is implemented using the step characteristic and exponential characteristic spatial quadrature schemes. The latter was introduced in 1993 and has been shown to accurate for both optically thin and optically thick spatial meshes and to produce strictly positive angular fluxes. The discrete ordinates equations can be solved using the conventional source iteration method. However, it is well known that this method converges prohibitively slowly for optically-thick problems with regions that are dominated by scattering rather than absorption. The new scheme converges rapidly even for such problems. Numerical results show that the new scheme is reliably accurate for the problems intended, and that it is fast and efficient in use of memory. The angle space distribution iteration method is demonstrated in slab geometry, for a single energy group, using isotropic cross sections, with exponential and step characteristic spatial quadrature.
Download or read book New Splitting Iterative Methods for Solving Multidimensional Neutron Transport Equations written by Jacques Tagoudjeu and published by Universal-Publishers. This book was released on 2011-04 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis focuses on iterative methods for the treatment of the steady state neutron transport equation in slab geometry, bounded convex domain of Rn (n = 2,3) and in 1-D spherical geometry. We introduce a generic Alternate Direction Implicit (ADI)-like iterative method based on positive definite and m-accretive splitting (PAS) for linear operator equations with operators admitting such splitting. This method converges unconditionally and its SOR acceleration yields convergence results similar to those obtained in presence of finite dimensional systems with matrices possessing the Young property A. The proposed methods are illustrated by a numerical example in which an integro-differential problem of transport theory is considered. In the particular case where the positive definite part of the linear equation operator is self-adjoint, an upper bound for the contraction factor of the iterative method, which depends solely on the spectrum of the self-adjoint part is derived. As such, this method has been successfully applied to the neutron transport equation in slab and 2-D cartesian geometry and in 1-D spherical geometry. The self-adjoint and m-accretive splitting leads to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of minimal residual and preconditioned minimal residual algorithms using Gauss-Seidel, symmetric Gauss-Seidel and polynomial preconditioning are then applied to solve the matrix operator equation. Theoretical analysis shows that the methods converge unconditionally and upper bounds of the rate of residual decreasing which depend solely on the spectrum of the self-adjoint part of the operator are derived. The convergence of theses solvers is illustrated numerically on a sample neutron transport problem in 2-D geometry. Various test cases, including pure scattering and optically thick domains are considered.
Download or read book Distribution Iteration written by Nicholas J. Prins and published by . This book was released on 2008 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: The discrete ordinates method is widely used to solve the Boltzmann transport equation for neutral particle transport for many engineering applications. Source iteration is used to solve the discrete ordinates system of equations, but can be slow to converge in highly scattering problems. Synthetic acceleration techniques have been developed to address this shortcoming; however, recent research has shown synthetic acceleration to lose effectiveness or diverge for certain problems. LTC Wager introduced an alternative to source iteration and demonstrated it in slab geometry. Here the method is further developed, enhancing efficiency in various ways, and demonstrated in XY-geometry as well as slab geometry. It is shown to be efficient even for those problems for which diffusion-synthetic and transport-synthetic accelerations fail or are ineffective. The method has significant advantages for massively-parallel implementations.
Download or read book Dissertation Abstracts International written by and published by . This book was released on 2009 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nuclear Science Abstracts written by and published by . This book was released on 1964 with total page 2300 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book An Analysis of the Finite differenced Even parity Discrete ordinates Equations in Slab Geometry written by and published by . This book was released on 1990 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considerable effort has been expended in recent years in finding improved spatial differencing schemes for the neutron and radiation transport equations. Standard criteria used to select a candidate scheme are its order of spatial convergence for small mesh size and its positivity in the sense that positive solutions emerge from positive input data. More recently, it has become clear that truly robust schemes must behave well in diffusing regions and must be compatible with an effective iteration acceleration method. Recently, Morel and Larsen reported their work on a promising new method called the multiple balance method that has virtually all the desirable characteristics. Here we study a different approach to the problem by considering discrete-ordinates approximations to the even-parity transport equations. We analyze three spatial difference approaches: diamond differencing, cell-edge differencing, and cell-center differencing. For the case of isotropic scattering and sources, the latter two approaches are shown to be strictly positive, to be second-order accurate, to be compatible with derived diffusion synthetic acceleration methods, and to possess the necessary diffusion limits. Unlike previous work with the even-parity equation, we do not use finite elements or variational principles. 5 refs., 1 tab.
Download or read book INIS Atomindex written by and published by . This book was released on 1984 with total page 1858 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Spatial Multigrid Iterative Method for Two dimensional Discrete ordinates Transport Problems written by Brian David Lansrud and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel's one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel's basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.
Download or read book ASME Technical Papers written by and published by . This book was released on 1966 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Isolate written by Glen E. Putnam and published by . This book was released on 1962 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: The digital computer program, ISOLATE, is designed to solve the Boltzmann transport equation for problems with the following properties: monodirectional sources incident at any angle to a boundary; plane parallel geometry; anisotropic scattering in a se ies of contiguous media; monoenergetic radiation or particle consideration; anisotropic diffuse sources at the slab boundaries; and isotropic or symmetrical reflection at the slab boundaries. Problems are solved by expressing the Boltzmann transport equation and the flexible boundary condition equations in difference equation form on the phase space. The difference equations are solved by a direct me hod, which takes advantage of special properties of the matrices. It is a novel approach to solutions of such problems; and it has proved to be a much faster and more efficient method than the usual iterative methods in many practical instances. The formulation and the solutions of the difference equations are described and detailed instructions are included to facilitate both the application of the code to physical problems and the actual operation of the code. The code is written in the language of FORTRAN II. Problem running times with the direct method of solution rarely exceed fifteen minutes on the IBM 7090. (Author).
Download or read book Paper written by and published by . This book was released on 1986 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Physics Briefs written by and published by . This book was released on 1988 with total page 1042 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Time Dependent Transport Equation Solver written by Leonard Woodrow Lee and published by . This book was released on 1991 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nuclear Computational Science written by Yousry Azmy and published by Springer Science & Business Media. This book was released on 2010-04-15 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Handbook of Nuclear Engineering written by Dan Gabriel Cacuci and published by Springer Science & Business Media. This book was released on 2010-09-14 with total page 3701 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an authoritative compilation of information regarding methods and data used in all phases of nuclear engineering. Addressing nuclear engineers and scientists at all levels, this book provides a condensed reference on nuclear engineering since 1958.
Download or read book Treatment Planning in Radiation Oncology written by Faiz M. Khan and published by Lippincott Williams & Wilkins. This book was released on 1998 with total page 645 pages. Available in PDF, EPUB and Kindle. Book excerpt: Radiation oncology is integrated with medical physics for a unique look at the state of the art in cancer patient care. World-renowned authors provide a complete discussion of treatment planning featuring the clinical, physical, and technical aspects involved. Coverage includes 3-D conformal treatment and other advancements in computer technology and medical imaging to bring the reader a modern perspective on treatment planning.
Download or read book Numerical Methods in the Theory of Neutron Transport written by Guriĭ Ivanovich Marchuk and published by Harwood Academic Publishers. This book was released on 1986 with total page 632 pages. Available in PDF, EPUB and Kindle. Book excerpt:
