Element-Saving Hexahedral 3-Refinement Templates

arXiv:2512.14862v5 Announce Type: replace Abstract: Conforming hex meshes are widely regarded as an effective computational domain for simulation because of their nice 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, topological conversion schemes require strict compatibility conditions that inevitably increase element count. State-of-the-art 2-refinement octree methods employ weakly-balanced and generalized pairing conditions to yield low element counts, but suffer from critical limitations: primal cell information is lost after dualization, and resulting dual cells often exhibit non-planar quad faces. Alternatively, 3-refinement 27-tree methods directly generate conforming hex meshes through template-based replacement, producing higher-quality elements with planar faces, but previous techniques impose far stricter conditions, severely over-refining grids by factors of ten to one hundred. This article introduces a novel 3-refinement approach using a moderately-balanced condition, slightly stronger than weakly-balanced but substantially more relaxed than prior 3-refinement requirements. The key insight is that recursively applying local refinements can isolate and reduce complex configurations to simpler cases covered by a fundamental template set. Two open-sourced variants are provided: one optimized for speed, and another trading some computational cost for marginally reduced element counts. Compared to previous 3-refinement methods, they significantly reduce final hex element counts while preserving min SJ values and guaranteeing convex polyhedral cells; relative to 2-refinement state-of-the-art, they also achieve a lower Hausdorff ratio using slightly fewer elements.