A Resistance-Integral Natural-Coordinate Method for Diffusive Transport

Abstract

Numerical models of diffusive transport, commonly considered as approximate solvers of the partial differential equation, focus on evaluation of gradients at a point. Gradient-approximation inaccuracies manifest themselves as errors in conductances and capacitances that enter into matrices that are finally solved. In turn, these errors arise from a lack of consideration of flow geometry in the point’s vicinity. In order to improve accuracy, flow geometry may be incorporated into evaluation of conductances and capacitances by choosing segments of flow tubes as volume elements. Flow tubes, inherent in the initial conditions, can be generated using appropriate interpolation schemes that are built into contouring algorithms. The principal task is to structure contouring tools in such a way as to generate information to facilitate accurate evaluation of conductances and capacitances. The logical framework of this novel approach is founded on the contributions of Maxwell, who visualized a flow domain as a collection of flow tubes, Ohm, who introduced the notion of resistance of a finite body, and Fick, who introduced a differential equation for diffusive flow through a tube of non-uniform cross section. Computer-graphics capability to implement this approach has become available only over the past two decades. Casting aside the notion of a point, the new paradigm is to evaluate flow-resistance over finite distances. The method suggested is referred to as a Resistance-Integral Natural-Coordinate Method (RINC). Formulated from first principles, this method dispenses with the differential equation as an intermediary to formulate the numerical equations. It proceeds directly from the physical problem to its numerical representation.