A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

Abstract

A method for applying the discrete ordinates method for solution of the neutron transport equation in arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the traditional shape of a mesh element within a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. Using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons.The method of characteristics, originally developed to eliminate negative fluxes encountered in many finite difference approximations, had been previously applied to regular cell structures in multiple dimensions and various coordinate systems. In such geometries, the geometrical relationships between sides were determined analytically and incorporated directly into the numerical model. However, the present work makes no assumptions about the orientations or the number of sides in a given cell, and computes all geometric relationships between each set of sides in each cell for each discrete direction. A set of nonreentrant polygons can therefore be used to represent any given two-dimensional space.iv Results for a number of test problems have been compared to solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN computer programs. Numerical results were obtained for problems ranging from simple onedimensional geometry to complicated multidimensional configurations. These results have demonstrated the ability of the developed method to closely approximate complex geometrical configurations and to obtain accurate results for problems that are extremely difficult to model using traditional methods. v DEDICATION This work is dedicated to my wife, Leigh. Her patience, understanding and love over the long road leading to this point made the path seem much less rocky; her encouragement and willingness to listen to my problems helped to sustain my efforts; and the daughter she has given us has taught me that miracles are possible. vi ACKNOWLEDGEMENTS I wish to express my sincere and profound gratitude to Ron Pevey and Ted Parish, both of whom provided invaluable support and insight in my endeavors. Their guidance, encouragement, technical assistance and senses of humor were invaluable assets. I also wish to acknowledge the support given to me by the United States Department of Energy, the Westinghouse Savannah River Company, and most importantly, by several members of my direct management and by my colleagues within the Savannah Riv...

Parallel MDOM for light transport in participating media

“…The step characteristics method [Lathrop 1969] is one of the earliest techniques which uses the short characteristics approach; it assumes source functions to be constant over a short step along the characteristic. This method has also been widely used in the neutron transport domain [Dehart 1992]. In computer graphics, Languenou et.…”

“…Based on this formulation, we can represent the outgoing radiance at each of the outgoing voxel faces as a linear combination of radiances arriving from all of the incoming faces as explained by Dehart [1992] and Languenou et. al [Languenou et al 1994].…”