Part 1 of this paper provides a comprehensive guide to generating unconstrained, simplicial, four-dimensional (4D), hypervolume meshes. While a general procedure for constructing unconstrained n-dimensional Delaunay meshes is well-known, many of the explicit implementation details are missing from the relevant literature for cases in which n >= 4. This issue is especially critical for the case in which n = 4, as the resulting meshes have important space-time applications. As a result, the purpose of this paper is to provide explicit descriptions of the key components in a 4D mesh-generation algorithm: namely, the point-insertion process, geometric predicates, element quality metrics, 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 Part 2 of the paper.

翻译：本文第一部分系统阐述了无约束单纯形四维超体积网格的生成方法。尽管构建无约束n维Delaunay网格的通用流程已广为人知，但现有文献对n≥4情况下的具体实现细节存在明显缺失。这一问题在n=4时尤为关键，因为由此生成的网格在时空领域具有重要应用。为此，本文旨在明确阐述四维网格生成算法中的关键组成部分：点插入过程、几何谓词、单元质量度量及双星翻转。本文是对Anderson等人在《Computer-Aided Design》(2023)发表的"Surface and hypersurface meshing techniques for space-time finite element methods"中开创性工作的自然延续。在先期研究中，超曲面网格通过新型轨迹追踪方法生成。本文则聚焦于通过从现有超曲面网格逐点插入构建粗粒度四维超体积网格（边界网格）。论文后半部分通过数值实验验证了该方法在简单凸域中的可行性。尽管本研究重点在于超体积边界网格生成，但所述技术可广泛适用于更丰富的四维网格剖分方法。需要指出的是，约束超体积网格剖分及非凸域边界恢复等更复杂的论题将在本文第二部分进行探讨。