Element-Saving Hexahedral 3-Refinement Templates

2025-12-16

AI Summary

This work addresses the pervasive over-refinement problem in adaptive 3D volumetric domain conversion to high-quality conformal hexahedral meshes. We propose a novel 3-subdivision template method grounded in a moderately-balanced condition—a concept newly introduced into the 3-subdivision framework. This condition relaxes traditional strict compatibility constraints while preserving original geometry (e.g., planar quadrilateral faces), attributes (signed distance, triangle indices), and topological integrity, thereby significantly suppressing redundant element generation. Integrating primal-space template replacement, conformal topological repair, and minimum scaled Jacobian (min-SJ)-driven quality optimization, our method reduces element count by over 90% compared to conventional 3-subdivision, achieving mesh complexity comparable to optimal 2-subdivision. Crucially, it maintains high hexahedral quality and data fidelity throughout the process.

Technology Category

• Computer Vision: 3D Computer Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

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)

Problem

• Develop relaxed topological conditions for hexahedral mesh conversion
• Address element loss and quality issues in dual transformation methods
• Reduce over-refinement in 3-refinement techniques for simulation efficiency

Innovation

• 3-refinement with moderately-balanced condition for hex meshes
• Template-based primal cell replacement for planar quad faces
• Reduced over-refining compared to prior 3-refinement methods