Discrete differential geometry and its role in computational modeling of defects and inelasticity

Abstract

In this paper we discuss the geometry and mechanics of discrete manifolds—so called simplicial complexes that are essentially triangulated regions—motivated by the works on discrete exterior calculus and differential geometry in computer graphics. We show that the classical ideas related to the geometry of continuous manifolds, such as metric, curvature, and affine connections take on a much simpler and more intuitive aspect when discussed in the context of such discrete triangulated manifolds. We introduce the notion of a dual mesh to describe “dual” variables (technically differential forms). We use the dual mesh to show that the geometry of the defects such as microcracks, dislocations and incoherent boundaries as well as balance laws and constitutive relations can be introduced directly, without the need for “discretization” from a continuum. This opens up the possibility of direct simulations of these bodies without the need for a continuous counterpart. We end by demonstrating how the second law of thermodynamics can be used in such a situation including discrete non-local systems.