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

2026-08-15

AI Summary

This study addresses the challenge of high-dimensional boundary-constrained mesh recovery in four-dimensional spacetime simulations by proposing an advancing front algorithm based on (d−2)-simplex ridges. The method innovatively integrates a constrained cavity operator with an incremental Steiner point insertion strategy to effectively ensure geometric consistency and topological correctness of complex boundaries. Experimental results demonstrate that the algorithm achieves a boundary recovery rate exceeding 99% in four-dimensional scenarios, generating 300 million pentatopes within merely 15 minutes. These findings indicate a significant breakthrough in overcoming efficiency and robustness bottlenecks associated with high-dimensional constrained mesh generation, thereby providing reliable computational support for large-scale spacetime numerical simulations.

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.

Problem

• Boundary-conforming mesh
• Four-dimensional mesh
• Boundary constraint recovery
• Spacetime simulation

Innovation

• Advancing-ridge
• Boundary constraint recovery
• Constrained cavity operator
• 4D mesh generation
• Steiner vertices