Untangling all-hex meshes via adaptive boundary optimization

We propose a novel method to untangle and optimize all-hex meshes. Central to this algorithm is an adaptive boundary optimization process that significantly improves practical robustness. Given an all-hex mesh with many inverted hexahedral elements, we first optimize a high-quality quad boundary mesh with a small approximation error to the input boundary. Since the boundary constraints limit the optimization space to search for the inversion-free meshes, we then relax the boundary constraints to generate an inversion-free all-hex mesh. We develop an adaptive boundary relaxation algorithm to implicitly restrict the shape difference between the relaxed and input boundaries, thereby facilitating the next step. Finally, an adaptive boundary difference minimization is developed to effectively and efficiently force the distance difference between the relaxed boundary and the optimized boundary of the first step to approach zero while avoiding inverted elements. We demonstrate the efficacy of our algorithm on a data set containing 1004 all-hex meshes. Compared to previous methods, our method achieves higher practical robustness.