We present HexOpt , a novel software package designed to optimize hexahedral mesh quality through a Rectified Hybrid Quadratic Jacobian energy functional and geometry-aware mapping. HexOpt accepts a triangular surface mesh and a volumetric hexahedral mesh as inputs. To overcome critical challenges such as element size-dependent gradient biases, prolonged optimization times, and erroneous boundary projection, we formulate a constrained optimization problem. The Rectified Hybrid Quadratic Jacobian energy is set as the objective function, while equality constraints enforce precise alignment between the quadrilateral boundary and the input geometry. These constraints fix vertices at geometric corners, restrict edge points to edges, and map face points to triangular faces. The optimization is solved via the Augmented Lagrangian method, with the Limited-Broyden–Fletcher–Goldfarb–Shanno method and the Armijo line search method driving the iterative process. The geometry-aware mapping employs normal-guided closest-point projection and mean value Tutte embedding to ensure geometric fidelity. HexOpt demonstrates robust performance across diverse 3D models and hexahedral meshes generated by various methods. It achieves consistent quality improvements without requiring manual intervention or parameter tuning compared with the initial state. Experimental results validate its computational efficiency and robustness in complex geometries, underscoring its utility as a reliable post-processing tool for hexahedral mesh optimization. We provide the HexOpt source code, input/output meshes (around 100 examples), and statistical data in the repository: https://github.com/CMU-CBML/HexOpt .

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.