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.

翻译：六面体网格生成是几何处理领域长期研究的课题，其中存在许多引人入胜且具有挑战性的相关问题。六面体网格根据应用领域或关注范围的不同，其复杂度可从结构化网格变化至非结构化网格。完全结构化网格要求所有内部网格边恰好与四个六面体单元相邻。不满足此条件的边被视为奇异边，标志着网格属于非结构化六面体网格。奇异边连接形成奇异曲线，这些曲线要么构成闭合环，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交汇的复杂连接点。虽然所有包含奇异性的六面体网格都是非结构化的，但那些具有更复杂奇异节点的网格往往包含更多畸变单元，且其缩放雅可比值更小。在本研究中，我们分析了奇异节点的拓扑结构。我们证明了最常见的八种奇异节点均可分解为单纯的奇异曲线。进一步地，我们表明无论边价如何，所有奇异节点在局部范围内皆可分解。最后，我们在六面体网格上验证了这些分解方法，从而降低了网格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被有效地转化为二维问题。