Local Decomposition of Hexahedral Singular Nodes into Singular Curves

TL;DR

This work shows that all eight of the most common singular nodes are decomposable into just singular curves, and demonstrates these decompositions on hex meshes, thereby decreasing their distortion and converting all singular nodes into singular curves.

Abstract

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.

Figures (14)

• Figure 1: (Left) The singular graph of a hex mesh of a sphere is shown. Singular edges are colored red, singular nodes are large black circles and singular vertices are small blue vertices. (Right) A close-up view of a singular node. Faces adjacent to the singular node are displayed in purple. A yellow sphere is overlayed on top of the singular node. Its intersection with the local hex mesh partitions the sphere into triangular regions.
• Figure 2: (Left) Red quads indicate a sheet inside of a hex mesh of an ellipsoid. Red curves depict its singular graph. The sheet is manifold with boundary on the boundary of the hex mesh. (Right) Blue quads indicate all faces of the inflated sheet.
• Figure 3: (Left) The yellow sphere triangulation indicates the structure of a singular node. Red quads indicate a sheet intersecting the singular node. The sheet intersects the sphere triangulation on a cycle of red curves that divide the sphere triangulation into two disks. (Right) Two singular nodes are visualized from inflation of the red sheet on the left. Sphere triangulations of the resulting two nodes are shown. Blue quad faces indicate newly created faces from the inflation. Blue edges indicate newly created edges in each sphere triangulation.
• Figure 4: From left to right the (4,0,0), (2,2,2), and (0,4,4) singular nodes are depicted. Top to bottom indicates steps to decompose each singular node. Red quads indicate the sheet to be inflated. Blue quads indicate faces of the newly inflated hexes. Red(Green) edges are valence 3(5) singularities. One sheet inflation is sufficient to decompose each of these nodes.
• Figure 5: The (1,3,3) singular node is decomposed into two valence 5 and one valence 3 curve via two sheet inflations.

Theorems & Definitions (2)

• Proposition 1
• proof