Microstructures, physical processes, and discrete differential forms

Presented is a new mathematical framework for analysis of physical processes in solids with complex internal structures. Unlike the classical description of solids as continua, the solids here are treated as assemblies of discrete, finite entities that represent different microstructural elements. Scalar quantities and momenta are defined as discrete differential forms on such assemblies, and their balances are formulated using topological and metric operations with such forms. The new description is background-independent, i.e., it does not rely on structures external to the solid. The resulting boundary value problems are given by matrix equations with constraints. Contrary to the familiar numerical methods, these equations do not approximate continuum problems but represent the physics on discrete assemblies exactly. The method provides a unique modelling capability: elements of materials' internal structures with different dimensions may have different physical properties. For example, in a polycrystalline assembly, the substance diffusivity inside a crystal (bulk, 3D) can be different from its diffusivity along a grain boundary (surface, 2D) and from its diffusivity along a triple junction (curve, 1D), or these microstructural elements can have different mechanical properties.