Discrete exterior calculus for meshes with concyclic polygons

Abstract

Discrete exterior calculus (DEC) is a numerical method for solving partial differential equations on meshes with applications in computer graphics, numerics, and physical simulations. It discretizes PDEs in a way such that important integral theorems hold exactly instead of being approximated. The drawback of the method is that it has strong requirements on the mesh. While the narrow range of admissible meshes mentioned in the original work could be widened, current methods still exclude an essential category of meshes, i.e., meshes with concyclic triangle pairs. Such meshes are common, as many synthetic meshes, e.g., triangulations of CAD models, handcrafted 3D models, and even results of surface meshing algorithms, contain such triangle pairs. Our paper describes an approach that allows us to use meshes with arbitrary triangulations that may contain concyclic triangle pairs by defining DEC operators for concyclic polygons.