Diffusion in multi-dimensional solids using Forman’s combinatorial differential forms

Abstract

The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining metric-dependent operators: discrete metrics, inner products, Hodge star operators, and codifferentials on forms. These are used to build an intrinsic description of physical processes dependent on scalar variables, and the corresponding boundary value problems. The description does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. The result is a discrete formulation of physical balance laws, not another method for discretising continuum problems. Importantly, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.