EXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems

Abstract

A strategy for the Hybridizable Discontinuous Galerkin (HDG) solution of problems with voids, inclusions, free surfaces, and material interfaces is proposed. It is based on an eXtended Finite Element (X-FEM) philosophy with a level-set description of interfaces where the computational mesh is not required to fit the interface (i.e. the boundary). This reduces the cost of mesh generation and, in particular, avoids continuous remeshing for evolving interfaces. Differently to previous proposals for the HDG solution with unfitting meshes, the computational mesh covers the domain in our approach, avoiding extrapolations and ensuring the robustness of the method. The local problem in elements not cut by the interface and the global problem are discretized as usual in HDG. A modified local problem is considered for elements cut by the interface. At every cut element, an auxiliary trace variable on the boundary is introduced, which is eliminated afterwards using interface conditions, keeping the original unknowns and the structure of the local problem solver. The solution is enriched with Heaviside functions in case of bimaterial problems; in case of problems with voids, inclusions, or free surfaces no such enrichment is required. Numerical experiments demonstrate how X-HDG keeps the optimal convergence, superconvergence, and accuracy of HDG with no need of adapting the computational mesh to the interface boundary.