We present a new software package, ``HexOpt,'' for improving the quality of all-hexahedral (all-hex) meshes by maximizing the minimum mixed scaled Jacobian-Jacobian energy functional, and projecting the surface points of the all-hex meshes onto the input triangular mesh. The proposed HexOpt method takes as input a surface triangular mesh and a volumetric all-hex mesh. A constrained optimization problem is formulated to improve mesh quality using a novel function that combines Jacobian and scaled Jacobian metrics which are rectified and scaled to quadratic measures, while preserving the surface geometry. This optimization problem is solved using the augmented Lagrangian (AL) method, where the Lagrangian terms enforce the constraint that surface points must remain on the triangular mesh. Specifically, corner points stay exactly at the corner, edge points are confined to the edges, and face points are free to move across the surface. To take the advantage of the Quasi-Newton method while tackling the high-dimensional variable problem, the Limited-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm is employed. The step size for each iteration is determined by the Armijo line search. Coupled with smart Laplacian smoothing, HexOpt has demonstrated robustness and efficiency, successfully applying to 3D models and hex meshes generated by different methods without requiring any manual intervention or parameter adjustment.

翻译：我们提出一个新的软件包“HexOpt”，用于提升全六面体网格的质量。其方法是通过最大化最小混合缩放雅可比能量泛函，并将全六面体网格的表面点投影到输入的三角网格上。所提出的HexOpt方法以表面三角网格和体全六面体网格作为输入。我们构建了一个约束优化问题，通过使用一个新颖的函数来改善网格质量，该函数结合了雅可比和缩放雅可比度量，这些度量被校正并缩放为二次度量，同时保持表面几何形状不变。该优化问题采用增广拉格朗日法求解，其中拉格朗日项强制约束表面点必须保持在三角网格上。具体而言，角点精确保持在角上，边点被限制在边上，而面点可以在表面上自由移动。为了在应对高维变量问题时利用拟牛顿法的优势，我们采用了有限内存Broyden-Fletcher-Goldfarb-Shanno算法。每次迭代的步长由Armijo线搜索确定。结合智能拉普拉斯平滑，HexOpt已展现出鲁棒性和高效性，成功应用于由不同方法生成的3D模型和六面体网格，无需任何人工干预或参数调整。