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)

翻译：在数值模拟中，符合要求的六面体（六面体）网格因其优越的数值特性而备受青睐，然而，将任意三维体自动分解为符合要求的六面体网格仍然是一项艰巨的挑战。在现有方法中，先构建自适应笛卡尔网格再将其转换为符合要求的网格的方法因其鲁棒性而脱颖而出。然而，实现这种转换的拓扑方案要求网格单元之间满足严格的兼容性条件，这不可避免地会细化初始网格并增加单元数量。开发更宽松的条件以最小化这种开销一直是持续的研究重点。最先进的二维细化八叉树方法采用弱平衡条件与广义配对条件相结合，并通过对偶变换产生极低的单元数量。然而，这种方法存在关键局限性：存储在原始单元上的信息（如符号距离场或三角形索引集）在对偶化后会丢失，并且生成的对偶单元通常表现出较差的最小缩放雅可比行列式（min SJ），且具有非平面四边形（四边形）面。另一方面，三维细化27叉树方法可以通过基于模板替换原始单元直接生成符合要求的六面体网格，产生具有平面四边形面的更高质量单元。然而，以往的三维细化技术施加的条件比二维细化对应方法严格得多，导致网格严重过度细化，细化因子高达十倍至一百倍，成为模拟流程中的主要瓶颈。本文介绍了一种新颖的三维细化方法，该方法采用适度平衡条件将自适应三维细化网格转换为符合要求的网格，该条件比弱平衡条件稍强，但比先前的三维细化要求宽松得多......（完整摘要请查阅PDF）