Fast and Robust Hexahedral Mesh Optimization via Augmented Lagrangian, L-BFGS, and Line Search

TL;DR

HexOpt is a robust and efficient software package that optimizes all-hexahedral meshes by maximizing a quality functional while preserving surface geometry, using advanced constrained optimization techniques.

Contribution

The paper introduces HexOpt, a novel mesh optimization method combining augmented Lagrangian, L-BFGS, and line search to improve mesh quality without manual tuning.

Findings

Successfully improves mesh quality on various 3D models.

Demonstrates robustness and efficiency across different mesh generation methods.

No manual intervention or parameter tuning required.

Abstract

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.