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 non-reentrant polygons can therefore be used to represent any given two dimensional space. 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 one-dimensional 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.