一种用于恢复d维网格中边界(d-1)-单形的推进脊方法
An advancing-ridge approach for recovering boundary $(d-1)$-simplices in $d$-dimensional meshes
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中文总结 AI 辅助
本文提出一种适用于任意维度、易于实现的推进脊算法，可恢复d维网格的边界约束，三维示例及四维测试表明其能高效生成边界贴合的四维网格。
中文摘要 AI 辅助
边界贴合的四维网格对于运行关于复杂运动三维几何的时空数值模拟至关重要，具体而言，需要生成五胞体网格，且该网格的四面体面需贴合于定义域边界。在三维场景中，常用方法是生成约束Delaunay四面体化，该方法的实现已较为成熟，但尚不清楚其如何扩展至四维场景，尤其是如何调度局部网格操作以恢复约束。本文开发了一种适用于任意维度、易于实现的新边界约束恢复算法，该算法本质上是推进前沿方法，使用约束空腔算子逐步将约束插入网格。与现有从(d-1)-单形（面）前沿推进的推进前沿方法不同，所提方法从称为脊的(d-2)-单形前沿推进；当前沿停滞时，可在边界添加Steiner顶点。多个三维示例表明，该算法能恢复输入曲面的完整表示。对于本文研究的四维几何，该推进脊过程通常能恢复至少99%的输入四面体化；对于一些更简单的定义域，通过添加Steiner顶点可实现与输入四面体化的完全贴合，证明其能生成边界贴合的四维网格。本文还评估了底层空腔算子实现的设计与效率，结果显示，在工作站级笔记本电脑上，约1.5分钟可生成3000万个五胞体，约15分钟可生成3亿个五胞体。
英文摘要
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.
