Finite Element Response Matrix Method for the Solution of the Transport Equation (Coarse Mesh).

Abstract

A response matrix method is applied to the time independent, mono-energetic transport equation. The finite element method serves as the mathematical basis for the method. The spatial domain is divided into several smaller regions on which the finite element method is applied. The resulting matrix is explicitly inverted and transformed into a set of four response matrices that describe the surface-to-surface, volume-to-surface, surface-to-volume, and volume-to-volume transport of particles within the region. An optional projection from a fine local mesh to a coarse global mesh may be performed. This allows the global response matrices to account for internal variations of flux or internal material heterogeneities that would not be resolved by the global mesh. By coupling the individual regions together, a two-tier iterative procedure results by which a global spatial and angular distribution of particles can be found. The inner iteration gives the interface flux distribution; the outer iteration gives the volumetric flux distribution and is used to find the fission source and eigenvalue for multiplying materials. The method has been successfully applied to one- and two-dimensional geometries. The local-global projection scheme works very well in one dimension, but additional work is needed to improve the two-dimensional projection scheme. For some one-dimensional geometries the new method is seen to be more efficient than the conventional finite element method. In two dimensions, however, the improvement in computational efficiency is dramatic.