Element-Saving Hexahedral 3-Refinement Templates

TL;DR

This paper tackles automatic generation of conforming hex meshes from adaptive 3-refinement grids, addressing excessive refinement in prior 3-refinement schemes. It introduces a moderately-balanced refinement condition and two template families (vertex-based and edge-based) enabling local, template-driven replacement to achieve hex-conforming meshes with far fewer elements. The vertex-based template system enumerates 256 vertex configurations with 10 fundamental and 12 composite templates, achieving O(n) performance; the edge-based system extends to 4,096 edge configurations, using a greedy algorithm to reduce element counts. Experimental results on 202 models show orders of magnitude reduction in hex counts compared to previous 3-refinement methods, with competitive geometry fitting relative to a leading 2-refinement method. The work provides practical, convex-quad hex meshes and highlights directions for further reducing order-dependence and improving connectivity metrics.

Abstract

Conforming hexahedral (hex) meshes are favored in simulation for their superior numerical properties, yet automatically decomposing a general 3D volume into a conforming hex mesh remains a formidable challenge. Among existing approaches, methods that construct an adaptive Cartesian grid and subsequently convert it into a conforming mesh stand out for their robustness. However, the topological schemes enabling this conversion require strict compatibility conditions among grid elements, which inevitably refine the initial grid and increase element count. Developing more relaxed conditions to minimize this overhead has been a persistent research focus. State-of-the-art 2-refinement octree methods employ a weakly-balanced condition combined with a generalized pairing condition, using a dual transformation to yield exceptionally low element counts. Yet this approach suffers from critical limitations: information stored on primal cells, such as signed distance fields or triangle index sets, is lost after dualization, and the resulting dual cells often exhibit poor minimum scaled Jacobian (min SJ) with non-planar quadrilateral (quad) faces. Alternatively, 3-refinement 27-tree methods can directly generate conforming hex meshes through template-based replacement of primal cells, producing higher-quality elements with planar quad faces. However, previous 3-refinement techniques impose conditions far more strict than 2-refinement counterparts, severely over-refining grids by factors of ten to one hundred, creating a major bottleneck in simulation pipelines. This article introduces a novel 3-refinement approach that transforms an adaptive 3-refinement grid into a conforming grid using a moderately-balanced condition, slightly stronger than the weakly-balanced condition but substantially more relaxed than prior 3-refinement requirements...... (check PDF for the full abstract)

Figures (8)

• Figure 1: Grid cells refined under unbalanced, weakly-balanced, moderately-balanced, and strongly-balanced conditions. The total cell counts are 71 (100%), 92 (130%), 113 (159%), and 120 (169%), respectively, demonstrating increasing refinement.
• Figure 2: Transition templates for a grid cell (left) and a face (right) from prior work: (A) schneiders1996refining; (B) ito2009octree; and (C) elsheikh2014consistent. Vertices adjacent to deeper-level cells are marked with a black dot (annotated as 1); others are annotated as 0. In all cell transition sets on the left, the cell is subdivided into hanging-node-free hex elements, with the transition across any face of the cell conforming to one of the five types in the face transition set on the right.
• Figure 3: Transition templates for (A, B) a grid cell and (C) a face in vertex-based templates. Vertices adjacent to deeper-level cells are marked with a black dot (annotated as 1); others are annotated as 0. (A) The ten fundamental templates subdivide a cell into a conforming hex mesh, where each face transition is restricted to the six types in (C). (B) The twelve composite templates are obtained by applying local refinement consistent with the face transition constraints. Face conformity can be achieved by recursively applying templates to each sub-cell in a composite template.
• Figure 4: (A) Transition templates for a grid cell in edge-based templates. Edges adjacent to deeper-level cells are bolded and annotated as 1; others are annotated as 0. This is also marked in the 000000000000 cell, with 12 binary $r(i)$ indicating if an edge is trisected and therefore bolded (1) or not (0). In all 32 cases, the cube is subdivided into hanging-node-free hex elements, with the transition across any face of the cube conforming to one of the six types in face transition schemes in (C). For the last three cases, "[1110]" red faces denote configurations with three 1-edges and one 0-edge, while "X" red edges signify arbitrary edges (may be 0 or 1). (B) Universal transition templates for all six possible sub-cell configurations in the last three [1110][X][1110], [1110][X][X], and [X][X][X] cases, where "[X]" represents "XXXX". (C) Square face transition schemes with consistent edge patterns.
• Figure 5: Performance comparison on 202 models in the dataset of gao2019feature. All panels except the upper right show relative growth in hex element count versus the initial grid: the x- or y-axis represents ratios of final to initial hex elements for the vertex-based or edge-based method or previous 3-refinement methods schneiders1996refiningito2009octreeelsheikh2014consistent. The upper right panel displays computation time versus initial grid cell count for the proposed methods.