Anisotropic Delaunay hypervolume meshing for space-time applications: point insertion, quality heuristics, and bistellar flips

Abstract

This paper provides a comprehensive guide to generating unconstrained, simplicial, four-dimensional (4D), hypervolume meshes for space-time applications. While several universal procedures for constructing unconstrained, d-dimensional, anisotropic Delaunay meshes are already known, many of the explicit implementation details are missing from the relevant literature for cases in which d >= 4. As a result, the purpose of this paper is to provide explicit descriptions of the key components in the 4D meshing algorithm: namely, the point-insertion process, geometric predicates, element quality heuristics, and bistellar flips. This paper represents a natural continuation of the work which was pioneered by Anderson et al. in "Surface and hypersurface meshing techniques for space-time finite element methods", Computer-Aided Design, 2023. In this previous paper, hypersurface meshes were generated using a novel, trajectory-tracking procedure. In the current paper, we are interested in generating coarse, 4D hypervolume meshes (boundary meshes) which are formed by sequentially inserting points from an existing hypersurface mesh. In the latter portion of this paper, we present numerical experiments which demonstrate the viability of this approach for a simple, convex domain. Although, our main focus is on the generation of hypervolume boundary meshes, the techniques described in this paper are broadly applicable to a much wider range of 4D meshing methods. We note that the more complex topics of constrained hypervolume meshing, and boundary recovery for non-convex domains will be covered in a companion paper.