A control-volume finite element method for the prediction of three-dimensional diffusion-type phenomena in anisotropic media

The formulation and testing of a control-volume finite element method (CVFEM) for the prediction of three-dimensional, linear and nonlinear, diffusion-type phenomena in anisotropic media in irregular calculation domains are presented and discussed in this paper. In this CVFEM, the calculation domain is discretized into four-node tetrahedral elements. Contiguous, non-overlapping, polyhedral control volumes are then associated with each node, and the governing differential equation is integrated over these control volumes. The dependent variable is interpolated linearly in each four-node tetrahedral element. Centroidal values of the diffusion coefficients are stored and assumed to prevail over the corresponding tetrahedral element. The source term is linearized, and nodal values of its coefficients are stored and assumed to prevail over the polyhedral sub-control volumes. Using these interpolation functions, the discretized equations, which are algebraic approximations to the integral conservation equations, are derived. The discretized equations, which in general, are nonlinear and coupled, are solved using an iterative procedure. The proposed CVFEM for the solution of anisotropic diffusion- type problems appears to be the first such method that is based on tetrahedral elements and vertex-centered polyhedral control volumes. These features make it particularly attractive for amalgamation with adaptive-grid schemes and applications to problems with complex irregular geometries, such those encountered in the general areas of drying, ground-water flows, conduction in composite materials, injection molding in heterogeneous porous media, and solidification. The proposed three-dimensional CVFEM and its computer implementation were tested using several steady conduction-type problems, for which analytical solutions were constructed using a special technique. In all cases, the agreement between the numerical and analytical solutions was excellent.