Benchmark of Polygon Quality Metrics for Polytopal Element Methods

TL;DR

A benchmark to study the correlation between general 2D polygonal meshes and PEM solvers, and aims to identify weaker shape-regularity criteria under which the considered methods can reliably work.

Abstract

Polytopal Element Methods (PEM) allow to solve differential equations on general polygonal and polyhedral grids, potentially offering great flexibility to mesh generation algorithms. Differently from classical finite element methods, where the relation between the geometric properties of the mesh and the performances of the solver are well known, the characterization of a good polytopal element is still subject to ongoing research. Current shape regularity criteria are quite restrictive, and greatly limit the set of valid meshes. Nevertheless, numerical experiments revealed that PEM solvers can perform well on meshes that are far outside the strict boundaries imposed by the current theory, suggesting that the real capabilities of these methods are much higher. In this work, we propose a benchmark to study the correlation between general 2D polygonal meshes and PEM solvers. The benchmark aims to explore the space of 2D polygonal meshes and polygonal quality metrics, in order to identify weaker shape-regularity criteria under which the considered methods can reliably work. The proposed tool is quite general, and can be potentially used to study any PEM solver. Besides discussing the basics of the benchmark, in the second part of the paper we demonstrate its application on a representative member of the PEM family, namely the Virtual Element Method, also discussing our findings.

Figures (11)

• Figure 1: A PDE solved on a mesh made of snowflake-like elements (and their dual, that is elements with snowflake holes). Despite the accuracy of the solution, these elements clearly violate any known shape regularity criterion for polygonal elements.
• Figure 2: Minimal polygon shape regularity is currently expressed in terms of the ratio between the maximal ball inscribed in the kernel of the polygon ($d$), and the maximal ball inscribing the element ($D$).
• Figure 3: Our benchmark (gray shaded area) and how it relates with the PEM solver.
• Figure 4: Polygon measures we consider in our study: inscribed circle (IC); circumscribed circle (CC); polygon area (AR); kernel area (KE); minimum angle (MA); shortest edge length (SE); and minimum point to point distance (MPD). Relative metrics obtained computing ratios between these quantities are also considered. The full list is available at Table \ref{['tab:geom_metrics']}.
• Figure 6: The eight families of parametric polygons used to generate the meshes in the dataset. Each polygon starts from a rest configuration ($t=0$, left column), and progressively worsen for growing values of $t$, stressing either one or multiple quality measures listed in Table \ref{['tab:geom_metrics']}.