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

TL;DR

The paper develops a rigorously explicit framework for fully unstructured, anisotropic Delaunay hypervolume meshing in four dimensions, aimed at space-time applications and boundary-conforming meshes derived from hypersurface data. It provides concrete methods for 4D point insertion, metric-weighted geometric predicates, and three pentatope quality heuristics, together with a comprehensive set of 4D bistellar flips (Pachner moves) and new bounding-tesseract subdivisions. Numerical experiments on a convex hypercylinder demonstrate second-order convergence of hypervolume error and effective quality improvement via flips on random point clouds, validating the approach and its practical viability. The work establishes foundational 4D anisotropic Delaunay techniques and provides directions for future 4D boundary recovery and constrained hypervolume meshing in non-convex domains, representing a first explicit framework for space-time 4D meshing with metric-aware predicates and 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.

Figures (33)

• Figure 1: Illustration of the bounding tesseract sectioned into pentatopes (1-7), for the subdivision strategy with $N_b = 23$.
• Figure 2: Illustration of the bounding tesseract sectioned into pentatopes (8-15), for the subdivision strategy with $N_b = 23$.
• Figure 3: Illustration of the bounding tesseract sectioned into pentatopes (16-23), for the subdivision strategy with $N_b = 23$.
• Figure 4: 2D Illustration of point insertion cavity operator. The top left illustrates finding the base element. The top right shows the initial cavity and the normals of its boundary edges. The dashed line indicates a boundary edge that is invisible to point $p$. The bottom left shows the reshaped cavity with all of its boundary edges having visibility to point $p$. Finally, the bottom right shows the reconnection of the cavity boundary to point $p$.
• Figure 5: Normalized difference between the output of the $inhypersphere_{M}$ predicate computed using the standard approach in conjunction with the Cholesky decomposition, and the alternative approach.

Theorems & Definitions (13)

• Remark 2.1
• Theorem 6.1
• proof
• Corollary 6.2
• proof
• Corollary 6.3
• proof
• Remark 6.4
• Remark 6.5
• Remark 6.6