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