Finite Element Particle Transport Using Wachspress Rational Basis Functions

Gregory Davidson, University of Michigan

My research involves computational methods for solving the linear Boltzmann transport equation. This equation describes the flow of radiation through and interactions with a background medium. Often, the transport equation is spatially discretized using a finite-element method. Advances in computer power have made deterministic transport algorithms discretized on unstructured, arbitrary polygonal and polyhedral meshes viable.


Unfortunately, the development of robust basis functions for arbitrary zone shapes has lagged behind other progress in the field. In 1975, Eugene Wachspress published a book unveiling a new class of rational basis functions that may be applied to any convex polygon or polyhedron, as well as certain polycons (zones with curved sides). We know of no one having implemented these basis functions in a finite element radiation transport algorithm.


There are new reasons to consider using Wachspress basis functions for transport problems. In a recent analysis of a family of discontinuous finite element radiation transport methods in thick diffusive regions, Adams (2000) showed that using Wachspress rational basis functions should yield solutions that are robust and accurate for many zone types.


Our research is concerned with investigating the properties of Wachspress rational basis functions, and providing numerical confirmation of Adams’s analysis showing them to be robust in the diffusive limit.

Abstract Author(s): Greg Davidson<br />Todd Palmer, Ph.D.