Introduction

The accurate solution of the diffusion equation is important for many varied applications. For instance, diffusion equations occur in the modeling of heat conduction, in certain formulations of fluid flow, and in radiation transport. Within the discipline of radiation transport, diffusion equations are used in single-group P₁ and Simplified Spherical Harmonics (SP_N) calculations, and in Diffusion Synthetic Acceleration (DSA) of transport iterations.

The mesh on which a diffusion problem is to be solved is often dictated by other problem constraints, such as the need to perform a Lagrangian hydrodynamics calculation in conjunction with a diffusion calculation. A mesh type which seems to be gaining prominence in three-dimensional modeling is the unstructured hexahedral mesh. This mesh consists of hexahedra and degenerate hexahedra (prisms, pyramids, tetrahedra, see Figure 1) that are connected in an arbitrary fashion.

Figure 1: Cell types handled by the model in 3-D: hexahedron, prism, pyramid and tetrahedron.

The connectivity of such a mesh must be explicitly specified. The additional complication of an unstructured mesh is balanced by the freedom to model arbitrary geometries, such as block structured meshes, and to model curved geometries with fewer distorted cells.

This paper develops a numerical method for modeling diffusion on unstructured hexahedral meshes. The method is an extension of the method described in an earlier paper by the authors (##more92 ##more92). The method in the previous paper was specifically for 2-D structured (i.e. logically rectangular) meshes. The derivation given here is applicable to 1-D, 2-D, and 3-D unstructured meshes.