Schneiders金字塔的44单元网格：限制六角网格问题的难度
A 44-element mesh of Schneiders' pyramid: bounding the difficulty of hex-meshing problems
摘要：
本文表明，约束规划技术可以成功地用于解决具有挑战性的六角啮合问题。施奈德金字塔是一种基于正方形的金字塔，其小平面通过在边缘中点和小平面质心处添加顶点而细分为三个或四个四边形。本文证明了Schneiders金字塔不存在内部顶点数小于18的六面体网格和17个六面体网格，并引入了一个有效的44个六面体网格。我们还构建了已知最小的八角形纺锤体网格，有40个六面体和42个内部顶点。这些结果是通过一个通用的算法，计算六面体网格共形到一个给定的四边形表面边界。Schneiders金字塔的下界是通过详尽地列出内部顶点多达17个并且与金字塔具有相同边界的六面体网格来获得的。我们的44个单元网格是通过修改先前的88个六面体解决方案获得的。使用局部简化六面体组的算法减少了元素的数量。给定这样一个组的边界，我们的算法是用来找到一个网格的内部，有更少的元素比初始细分。将生成的网格解开以获得有效的六面体网格。
英文摘要：
This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders' pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders' pyramid has no hexahedral meshes with fewer than 18 interior vertices and 17 hexahedra, and introduce a valid mesh with 44 hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders' pyramid is obtained by exhaustively listing the hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid hexahedral mesh.
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