Mesh Optimization for the Virtual Element Method: How Small Can an Agglomerated Mesh Become?

TL;DR

The optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem, by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.

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. The resulting discretization is significantly lighter: we can remove up to 80$\%$ of the mesh elements, based on a user-set parameter, thus reducing the number of faces, edges, and vertices. This results in a drastic reduction of the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, in particular, 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. This regularization effect is evident in cases where the original meshes 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.

Figures (10)

• Figure 1: Optimization pipeline: (a) input mesh (black), with elements $E_1, E_2$ and their shared edge $f$, and mesh dual graph (blue), with nodes $n_{E_1}, n_{E_2}$ and arc $a_f$; (b) weights assignment: $w(n_{E_1})$ is the quality of the green polygon, $w(n_{E_2})$ is the quality of the purple polygon and $w(a_f)$ is the quality of the red polygon; (c) input mesh colored with respect to METIS labeling: nodes $n_{E_1}$ and $n_{E_2}$ are assigned the same label $\lambda$; (d) agglomeration of the elements with the same label: elements $E_1$ and $E_2$ become a single element $E_\Lambda$.
• Figure 2: First mesh of datasets $\textit{Tri}_{}$ and $\textit{Quad}_{}$ optimized with $\mathcal{K}=40$ and $\mathcal{K}=20$.
• Figure 3: First mesh of datasets $\textit{Tet}_{}$ and $\textit{Hex}_{}$ optimized with $\mathcal{K}=40$ and $\mathcal{K}=20$.
• Figure 4: Convergence of datasets $\textit{Tri}_{20}$, $\textit{Quad}_{20}$ (continuous lines) compared to their original datasets $\textit{Tri}_{}$, $\textit{Quad}_{}$ (dotted line). We measure $\mathcal{E}_{L^2}$ (top figures) and $\mathcal{E}_{H^1}$ (bottom figures) with respect to the number of DOFs.
• Figure 5: Convergence of datasets $\textit{Tet}_{20}$ and $\textit{Hex}_{20}$ (continuous lines) compared to their original datasets (dotted lines). We measure $\mathcal{E}_{L^2}$ (top figures) and $\mathcal{E}_{H^1}$ (bottom figures) with respect to the number of DOFs.