An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes

[2608.15176] An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computational Engineering, Finance, and Science arXiv:2608.15176 (cs) [Submitted on 15 Aug 2026] Title: An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes Authors: Philip Caplan View a PDF of the paper titled An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes, by Philip Caplan View PDF HTML (experimental) Abstract: Boundary-conforming four-dimensional meshes are essential for being able to run spacetime numerical simulations about complex, moving three-dimensional geometries. Specifically, a mesh of pentatopes is needed in which the tetrahedral faces of this mesh conform to the boundary of the domain. In the three-dimensional setting, a common approach consists of generating a constrained Delaunay tetrahedralization. Implementations of this approach are mature, but it is unclear how it extends to the four-dimensional setting, particularly in how the local mesh operations are scheduled to recover the constraints. This paper develops a new algorithm for recovering boundary constraints which is simple to implement in any dimension. The algorithm is primarily an advancing-front approach and uses a constrained cavity operator to incrementally insert constraints into the mesh. Compared to existing advancing-front approaches, which advance from a front of $(d-1)$-simplices (faces), the proposed approach advances from a front of $(d-2)$-simplices, called ridges. Steiner vertices can be added to the boundary when the front stalls and several examples in $3d$ demonstrate the ability of this algorithm to recover a complete representation of the input surface. For the four-dimensional geometries studied here, the algorithm generally recovers at least 99% of the input tetrahedralization with this advancing ridge procedure. For some simpler domains, complete conformity with the input tetrahedralization is achieved by adding Steiner vertices, thereby demonstrating the ability to produce boundary-conforming four-dimensional meshes. The design and efficiency of the underlying cavity operator implementation is also evaluated, showing that 30 million pentatopes can be created in about 1.5 minutes, and 300 million pentatopes in about 15 minutes on a workstation laptop. Subjects: Computational Engineering, Finance, and Science (cs.CE) ; Computational Geometry (cs.CG) Cite as: arXiv:2608.15176 [cs.CE] (or arXiv:2608.15176v1 [cs.CE] for this version) https://doi.org/10.48550/arXiv.2608.15176 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Philip Caplan [ view email ] [v1] Sat, 15 Aug 2026 11:29:23 UTC (19,900 KB) Full-text links: Access Paper: View a PDF of the paper titled An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes, by Philip Caplan View PDF HTML (experimental) TeX Source view license Current browse context: cs.CE < prev | next > new | recent | 2026-08 Change to browse by: cs cs.CG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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