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

TL;DR

HexOpt addresses the challenge of producing high-quality, inversion-free all-hex meshes that conform to a given triangular surface. It combines a differentiable objective ReHQJ, which blends Jacobian and scaled Jacobian metrics, with an Augmented Lagrangian constraint enforcing surface projection, solved efficiently by L-BFGS with Armijo line search. The approach yields robust, automated improvements across diverse input meshes, keeping the boundary on the target surface while substantially increasing the minimum scaled Jacobian and reducing element inversion. The work also provides open-source code and data, enabling replication and further research into theoretical guarantees and pre-processing enhancements for hex-mesh optimization.

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.

Figures (5)

• Figure 1: Hex mesh optimization process. The triangle surface $\mathcal{T}$ is shown in white. The quadrilateral surface $\mathcal{S}_\mathcal{H}$ is shown in blue. The hex core mesh $\mathcal{H}\setminus\mathcal{S}_\mathcal{H}$ is shown in yellow. The vertices are warped to minimize the objective function in Equation (\ref{['equ:AL']}). The middle stage mesh $\mathcal{H}'$ with $\mathcal{S}_{\mathcal{H'}}$ exactly fits to $\mathcal{T}$ with the minimum scaled Jacobian of 0.01. The minimum scaled Jacobian is increased by 0.01 every time with $\mathcal{S}_{\mathcal{H'}}$ exactly fitting to $\mathcal{T}$ until we cannot improve anymore, and we export the final mesh at the bottom right.
• Figure 2: ReSJ, ReHJ, and ReHQJ plots of a quadrilateral element with three points (in black) fixed at $(0, 0)$, $(1, 0)$, and $(0, 1)$, and one point (in green) free to move within the plane. $\Theta = 0.6$ is set. (a) The contour plot of function ReSJ. Singular points are encircled with green circles, and local minimum regions are marked with green squares. (b) The contour plot of function ReHJ on the same element. The functional landscape is piecewise linear in the negative Jacobian region. (c) The contour plot of function ReHQJ (the final adopted objective function) on the same element.
• Figure 3: Three types of optimization constraints: (1) the sharp corner point $x$ and its target position $x^t$ in red; (2) the sharp edge point $x$ and its target position $x^t$ in green; and (3) the face point $x$ and its target position $x^t$ in blue.
• Figure 4: Convergence plot on the exhibited eight models.
• Figure 5: Optimization results for eight models. The target surface is displayed translucently. For each model, the left/right figure shows the mesh before/after optimization. Inverted elements are highlighted in red, and non-inverted elements are shown in blue. The mesh interior is shown in yellow with some elements removed. The maximum relative distance and the scaled Jacobian histogram before and after optimization are provided. In the histograms, purple bars are intersected for better visualization due to their high frequency.