POLYLLA: Polygonal meshing algorithm based on terminal-edge regions

TL;DR

Polylla is a new polygonal meshing algorithm that constructs meshes from triangulations by identifying terminal-edge regions, producing simpler, faster meshes with comparable numerical performance to Voronoi-based meshes.

Contribution

The paper introduces Polylla, a novel, efficient polygonal meshing algorithm that generates meshes with fewer polygons and comparable accuracy to Voronoi meshes, validated through numerical experiments.

Findings

Polylla meshes have fewer polygons than Voronoi meshes.

Polylla algorithm is simpler and faster than constrained Voronoi algorithms.

Numerical performance of VEM on Polylla meshes is comparable to Voronoi meshes.

Abstract

This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations. Each polygon is built from a terminal-edge region surrounded by edges that are not the longest-edge of any of the two triangles that share them. The algorithm is termed Polylla and is divided into three phases. The first phase consists of labeling each edge of the input triangulation according to its size; the second phase builds polygons (simple or not) from terminal-edges regions using the label system; and the third phase transforms each non simple polygon into simple ones. The final mesh contains polygons with convex and non convex shape. Since Voronoi based meshes are currently the most used polygonal meshes, we compare some geometric properties of our meshes against constrained Voronoi meshes. Several experiments were run to compare the shape and size of polygons, the number of final mesh points and polygons. For the same input, Polylla meshes contain less polygons than Voronoi meshes, and the algorithm is simpler and faster than the algorithm to generate constrained Voronoi meshes. Finally, we have validated Polylla meshes by solving the Laplace equation on an L-shaped domain using the Virtual Element Method (VEM). We show that the numerical performance of the VEM using Polylla meshes and Voronoi meshes is similar.