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Part 1 of this paper provides a comprehensive guide to generatingunconstrained, simplicial, four-dimensional (4D), hypervolume meshes. While ageneral procedure for constructing unconstrained n-dimensional Delaunay meshesis well-known, many of the explicit implementation details are missing from therelevant literature for cases in which n >= 4. This issue is especiallycritical for the case in which n = 4, as the resulting meshes have importantspace-time applications. As a result, the purpose of this paper is to provideexplicit descriptions of the key components in a 4D mesh-generation algorithm:namely, the point-insertion process, geometric predicates, element qualitymetrics, and bistellar flips. This paper represents a natural continuation ofthe work which was pioneered by Anderson et al. in "Surface and hypersurfacemeshing techniques for space-time finite element methods", Computer-AidedDesign, 2023. In this previous paper, hypersurface meshes were generated usinga novel, trajectory-tracking procedure. In the current paper, we are interestedin generating coarse, 4D hypervolume meshes (boundary meshes) which are formedby sequentially inserting points from an existing hypersurface mesh. In thelatter portion of this paper, we present numerical experiments whichdemonstrate 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 widerrange of 4D meshing methods. We note that the more complex topics ofconstrained hypervolume meshing, and boundary recovery for non-convex domainswill be covered in Part 2 of the paper.

Space-time hypervolume meshing part 1: Point insertion, geometric predicates, and bistellar flips