良好中心三角剖分
Well-Centered Triangulation
AI总结：
本文提出了一种基于优化的启发式算法，通过最小化代价函数并移动内部顶点，将二维或三维三角剖分转化为良好中心三角剖分，并证明了其最优性质及与Delaunay和最小最大角三角剖分的关系。
摘要：
由良好中心单纯形组成的网格具有正交对偶网格（对偶Voronoi图）的良好性质，这对于某些偏好这种原始-对偶网格对的数值算法很有用。我们证明了良好中心网格也具有最优性质，并且与Delaunay和最小最大角三角剖分有关。我们提出了一种迭代算法，通过最小化代价函数并移动内部顶点，同时保持网格连通性和边界顶点固定，试图将给定的二维或三维三角剖分转化为良好中心剖分。该代价函数是我们提出的任意维度下良好中心性新表征的直接结果。我们的算法是第一个基于优化的良好中心性启发式算法，也是第一个同时适用于二维和三维的算法。我们展示了将算法应用于小型和大型二维网格的结果，其中一些具有复杂边界，并获得了立方体的良好中心四面体化。我们还提供了数值证据，表明我们的算法保持了梯度，并改善了由已知最佳先前方法创建的锐角三角剖分的最大和最小角度。
英文摘要：
Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.
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