Local Decomposition of Hexahedral Singular Nodes into Singular Curves

TL;DR

This paper introduces a method to decompose complex 3D singular nodes in hexahedral meshes into simpler singular curves, reducing distortion and simplifying topology.

Contribution

It demonstrates that all common and complex singular nodes in hex meshes can be locally decomposed into singular curves, simplifying mesh topology.

Findings

All eight common singular nodes are decomposable into singular curves.

Decomposition reduces element distortion and improves mesh quality.

Singular nodes can be effectively transformed into simpler 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.