An Explicitly-Interfaced Finite Element Solution of the Neutron Transport Equation.

Abstract

A numerical solution of the first-order, mono-energetic neutron transport equation is found by the finite element method. An iterative solution procedure is devised by explicit determination of the angular flux at the interfaces of segments into which the spatial domain is subdivided. This is accomplished by solving the transport equation in each segment using the incoming flux from adjoining segments as boundary conditions (natural or essential). The interface flux is subsequently updated in a Gauss-Seidel fashion. In each segment, either a phase-space finite element or a finite element-spherical harmonic approximation is used to construct a system of algebraic equations. In two-dimensional X-Y geometry, spatial finite element trial functions of low order are specified on a rectangular or a triangular mesh. In the angular domain, discontinuous trial functions are employed. The local sets of equations are solved directly. On the initial iteration, the segment coefficient matrices are factored and stored for use on subsequent iterations. The coefficient matrices obtained by the finite approximation tend to be off-diagonally dominant with decreasing spatial mesh size. Iterative solutions of this system of equations are generally unsuccessful. However, the explicitly-interfaced method yields accurate results for both fixed-source and eigenvalue problems. The discrete version of this algorithm is a block Gauss-Seidel method in which the partitioning of the global coefficient matrix reflects the physics of neutron conservation. The computational costs and storage requirements for the explicit solution procedure are substantially less than that of an entirely direct approach.