A procedure for advection and diffusion in thin cavities

In this paper a finite element formulation is proposed for the calculation of advection and diffusion in a thin cavity. For these kinds of systems, very high aspect ratio elements are necessary for cost-effective simulation. Locally, element dimensions, say in x and y, are comparable, whereas the dimension in the transverse direction z is orders of magnitude smaller than those for x and y. In our formulation, the three-dimensional basis functions for interpolation are constructed as a tensor product of the basis functions that span the lateral (x, y) plane of an element and those that span the transverse direction. Unknowns along the transverse direction are solved implicitly, in a line-by-line fashion, using the tridiagonal matrix algorithm, while the "out-of-line" unknowns are treated either explicitly or semi-implicitly. Several applications to material processing are discussed, as are the computationally-intensive components of three variations of our basic procedure.