Spacetime meshing of stratified spaces for spacetime discontinuous Galerkin methods in arbitrary spatial dimensions

Abstract

We introduce the spacetime discontinuous Galerkin method and motivate the need for supporting spacetime meshing on meshes comprised of multiple manifolds. We first discuss preliminary concepts behind simplices, simplicial complexes, and the generalization to oriented simplicies. Using these ideas, we define stratified spaces and how they can be used to model a mesh comprised of multiple oriented manifolds. We construct a graphical representation called a Stratified Mesh and use this representation to construct a collection of data structures, the main result being the StratifiedMesh data structure. Next we define a set of support algorithms based on the various data structures discussed. This leads us to review the fundamentals of the TentPitcher algorithm and its relationship to spacetime discontinuous Galerkin methods both theoretically and in the literature. The TentPitcher algorithm is then extended to work on stratified meshes in E^d x R for arbitrary spatial dimension d. We then briefly discuss a parametrization for tentpole vertices that generalizes the baseline TentPitcher, vertex smoothing, and tilted tentpoles. Following that, we discuss at a high level the generic software architecture and techniques used build completely new spacetime meshing software that handles stratified meshes. Visualizations of various examples from the software conclude the work, with examples of single manifold 2d x time, single manifold 3d x time, and a multiple manifold example in 2d x time.