Elsevier

Graphical Models

Volume 121, May 2022, 101136
Graphical Models

Untangling all-hex meshes via adaptive boundary optimization

https://doi.org/10.1016/j.gmod.2022.101136Get rights and content

Abstract

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.

Introduction

All-hex meshes are widely used in finite element method to perform physical simulation [1], [2], [3]. Simulation results rely on both average and minimum hexahedral element quality [4]. To avoid numerical instability, at least the elements of all-hex meshes cannot be inverted [5].

The all-hex mesh generation process usually contains two steps: (1) generate an initial mesh whose connectivity is optimized to fit the input mesh; and (2) update the positions of vertices to improve the mesh quality without changing the connectivity. In this paper, we focus on the second step, which is still an open and challenging research problem (Fig. 1). In practice, there are two common requirements. First, the resulting all-hex mesh is of high-quality without inverted hexahedral elements. Second, boundary surface preservation is required to obtain a high degree of similarity between the input and the optimized shapes.

These two requirements affect each other. In general, preserving boundary surfaces limits the movement of the boundary vertices, so that the interior vertices cannot be updated freely to obtain inversion-free meshes. To mitigate these mutual influences, the following pipeline is developed [7]. First, an inversion-free high-quality mesh is generated without considering the boundary preservation constraint. Second, the relaxed boundary is pulled back to reduce the distance difference from the input boundary while explicitly keeping high quality and avoiding inverted elements.

It is challenging to design a practically robust algorithm based on this pipeline, especially in the second step. The second step relies on two factors. First, as the relaxed shape after the first step is the initial shape of the second step, the distance between the relaxed and the input boundaries should be controlled to facilitate the second step. Second, the optimization methods to effectively generate inversion-free meshes and decrease the boundary distance are desired. To the best of our knowledge, only one former method uses this pipeline [7]; however, it does not carefully and holistically consider these factors. Then, it fails to generate inversion-free results for most models (Figs. 2, 15, and 16).

In this paper, we propose a novel method to untangle and optimize all-hex meshes. The algorithm follows the aforementioned pipeline. For the first factor, our first step is an adaptive boundary relaxation procedure that gradually relaxes the boundary constraint to obtain an inversion-free mesh. Since the used optimization solver [8] can eliminate most inverted elements by just moving the interior vertices, only small movements are required for all vertices to obtain an inversion-free mesh, thereby implicitly constraining the distance between the relaxed and the input boundaries. For convenience, we define the distance between boundaries as the sum of squared distance between the corresponding vertices. To achieve a high-quality result, the input boundary quad mesh is optimized to serve as the target of the relaxed boundary in the second step. For the second factor, we adaptively reduce the boundary distance to nearly zero while keeping the mesh always inversion-free based on an elegant second-order solver.

Although we cannot theoretically guarantee inversion-free all-hex meshes in every case, our method has succeeded in producing inversion-free meshes on a data set containing 1004 examples (Fig. 3). Compared to existing methods, our method is more practically robust and efficient (Figs. 1, 2, 15, and 16).

Section snippets

Related work

All-hex mesh untangling and optimization

The most common optimization approach is moving vertices to the weighted average of their neighbors. Geometric flows are also applied to improve all-hex mesh quality [9], [10]. However, there is no guarantee that the result mesh is inversion-free [3]. Several methods [11], [12] start from an inversion-free mesh and relocate vertices while avoiding inversions. In practice, many raw hex-meshing outputs contain inverted elements, limiting the utility of this

Overview

Input and goal The input is an all-hex mesh M that contains many inverted hexahedral elements. Its boundary surface is a quad mesh B. Our goal is to untangle the input mesh M to generate an inversion-free mesh N that shares the same connectivity with M. The approximation error (e.g., two-sided Hausdorff distance) between the boundaries of M and N is small. Namely, we should preserve the boundary surface B of M after the mesh optimization.

Methodology The boundary surface preservation requirement

Experiments

We have tested our algorithm on various all-hex meshes to evaluate its performance. Our method is implemented in C++, and all the experiments are performed on a desktop PC with a 4.00 GHz Intel Core i7-4790K and 16 GB of RAM. The linear systems are solved using the Intel®Math Kernel Library. Statistics and timings for all the demonstrated examples are reported in Table 1.

Without boundary relaxation We test the necessity of the boundary relaxation process. A large λ in (4) can be applied to

Conclusion

In this paper, we present a novel framework to geometrically optimize hexahedral meshes. Given an input all-hex mesh with inversions, we firstly generate a quad mesh with high quality and small difference to the input boundary mesh. Then we relex the boundary preservation constraint and optimize for an inversion-free hex mesh. At last, we minimize the difference between the boundary of inversion-free mesh and the optimized quad mesh of the first step without bringing in inverted elements. We

CRediT authorship contribution statement

Qing Huang: Methodology, Software, Writing – original draft. Wen-Xiang Zhang: Investigation, Validation. Qi Wang: Data curation, Visualization. Ligang Liu: Supervision, Project administration. Xiao-Ming Fu: Conceptualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We would like to thank the anonymous reviewers for their constructive suggestions and comments. This work is supported by the National Natural Science Foundation of China (62025207).

References (49)

  • O.V. Ushakova

    Nondegeneracy tests for hexahedral cells

    Comput. Methods Appl. Mech. Eng.

    (2011)
  • T. Blacker

    Meeting the challenge for automated conformal hexahedral meshing

    9th International Meshing Roundtable

    (2000)
  • J.F. Shepherd et al.

    Hexahedral mesh generation constraints

    Eng. Comput.

    (2008)
  • S.J. Owen

    A survey of unstructured mesh generation technology

    IMR

    (1998)
  • P.P. Pébay et al.

    New applications of the verdict library for standardized mesh verification pre, post, and end-to-end processing

    Proceedings of the 16th International Meshing Roundtable

    (2008)
  • X. Gao et al.

    A local frame based hexahedral mesh optimization

    Proceedings of the 25th International Meshing Roundtable

    (2016)
  • M. Livesu et al.

    Practical hex-mesh optimization via edge-cone rectification

    ACM Trans. Graphics (TOG)

    (2015)
  • J.-P. Su et al.

    Practical foldover-free volumetric mapping construction

    Computer Graphics Forum

    (2019)
  • J. Qian et al.

    Quality improvement of non-manifold hexahedral meshes for critical feature determination of microstructure materials

    Int. J. Numer. Methods Eng.

    (2010)
  • P. Kunnp

    A method for hexahedral mesh shape optimization

    Int. J. Numer. Methods Eng.

    (2003)
  • D. Vartziotis et al.

    Improved GETMe by adaptive mesh smoothing

    Comput. Assisted Methods Eng. Sci.

    (2017)
  • P.M. Knupp

    Hexahedral and tetrahedral mesh untangling

    Eng. Comput.

    (2001)
  • T.J. Wilson et al.

    Untangling and smoothing of quadrilateral and hexahedral meshes

    Civil-Comp Proceedings

    (2012)
  • S.P. Sastry et al.

    A comparison of gradient-and hessian-based optimization methods for tetrahedral mesh quality improvement

    Proceedings of the 18th International Meshing Roundtable

    (2009)
  • Cited by (3)

    • On why mesh untangling may not be required

      2023, Engineering with Computers
    View full text