Mesh optimization for the virtual element method: How small can an agglomerated mesh become?

IF 3.8 2区 物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Journal of Computational Physics Pub Date : 2024-10-31 DOI:10.1016/j.jcp.2024.113552
Tommaso Sorgente , Fabio Vicini , Stefano Berrone , Silvia Biasotti , Gianmarco Manzini , Michela Spagnuolo
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Abstract

We present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). Once the local quality of the mesh elements is analyzed through a quality indicator specific to the VEM, groups of elements are agglomerated to optimize the global mesh quality. A user-set parameter regulates the percentage of mesh elements, and consequently of faces, edges, and vertices, to be removed. This significantly reduces the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, particularly for high-order formulations. We show how the VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones. We observe that the optimization has a regularization effect over low-quality meshes, removing the most pathological elements. In such cases, these “badly-shaped” elements yield a system matrix with very large condition number, which may cause the VEM to diverge, while the optimized meshes lead to convergence. We conclude by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.
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虚拟元素法的网格优化:聚合网格可以变得多小?
我们提出了一种针对虚拟元素法(VEM)的通用多边形或多面体网格优化程序。通过虚拟元素法特有的质量指标对网格元素的局部质量进行分析后,对元素组进行聚合,以优化整体网格质量。用户可通过设置参数来调节要去除的网格元素以及面、边和顶点的百分比。这就大大减少了与使用 VEM 在网格上定义的离散问题相关的自由度总数,特别是对于高阶公式。我们展示了如何在优化网格中保持 VEM 的收敛速度,并且近似误差与原始网格相当。我们观察到,优化对低质量网格具有正则化效果,可以去除最病态的元素。在这种情况下,这些 "形状不佳 "的元素会产生一个条件数非常大的系统矩阵,这可能会导致 VEM 发散,而优化网格则会导致收敛。最后,我们将展示如何在模拟随时间变化的问题时有效地使用实际 CAD 模型的优化方法。
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来源期刊
Journal of Computational Physics
Journal of Computational Physics 物理-计算机:跨学科应用
CiteScore
7.60
自引率
14.60%
发文量
763
审稿时长
5.8 months
期刊介绍: Journal of Computational Physics thoroughly treats the computational aspects of physical problems, presenting techniques for the numerical solution of mathematical equations arising in all areas of physics. The journal seeks to emphasize methods that cross disciplinary boundaries. The Journal of Computational Physics also publishes short notes of 4 pages or less (including figures, tables, and references but excluding title pages). Letters to the Editor commenting on articles already published in this Journal will also be considered. Neither notes nor letters should have an abstract.
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