六面体奇异节点到奇异曲线的局部分解
Local Decomposition of Hexahedral Singular Nodes into Singular Curves
AI总结：
本文研究六面体网格中奇异节点的拓扑结构，证明所有常见及任意边价的奇异节点均可局部分解为奇异曲线，并通过网格实验验证其可降低畸变。
摘要：
六面体（hex）网格划分是几何处理中一个长期研究的课题，伴随许多引人入胜且富有挑战性的相关问题。根据应用或关注区域的不同，六面体网格的复杂度从结构化到非结构化不等。完全结构化网格要求所有内部网格边恰好邻接四个六面体。不满足该标准的边被视为奇异边，并表明该六面体网格是非结构化的。奇异边连接成奇异曲线，这些曲线要么形成闭合环，要么终止于网格边界，要么终止于奇异节点——即两条以上奇异曲线形成的复杂交汇点。尽管所有含奇异性的六面体网格都是非结构化的，但具有更复杂奇异节点的网格往往具有更多畸变单元和更小的缩放 Jacobian 值。在本工作中，我们研究奇异节点的拓扑。我们证明，八种最常见的奇异节点都可以仅分解为奇异曲线。我们进一步证明，无论边价是多少，所有奇异节点都是局部可分解的。最后，我们在六面体网格上演示这些分解，从而降低其畸变，并将所有奇异节点转化为奇异曲线。通过这种分解，三维奇异节点难以捉摸的复杂性实际上变成了二维问题。
英文摘要：
Hexahedral (hex) meshing is a long studied topic in geometry processing with many fascinating and challenging associated problems. Hex meshes vary in complexity from structured to unstructured depending on application or domain of interest. Fully structured meshes require that all interior mesh edges are adjacent to exactly four hexes. Edges not satisfying this criteria are considered singular and indicate an unstructured hex mesh. Singular edges join together into singular curves that either form closed cycles, end on the mesh boundary, or end at a singular node, a complex junction of more than two singular curves. While all hex meshes with singularities are unstructured, those with more complex singular nodes tend to have more distorted elements and smaller scaled Jacobian values. In this work, we study the topology of singular nodes. We show that all eight of the most common singular nodes are decomposable into just singular curves. We further show that all singular nodes, regardless of edge valence, are locally decomposable. Finally we demonstrate these decompositions on hex meshes, thereby decreasing their distortion and converting all singular nodes into singular curves. With this decomposition, the enigmatic complexity of 3D singular nodes becomes effectively 2D.
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