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POLYLLA: polygonal meshing algorithm based on terminal-edge regions


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.

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  1. A terminal-edge region is formed by all triangles whose longest-edge propagation path (Lepp) [4] share the same terminal-edge.





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This research was supported by the Patagón supercomputer of Universidad Austral de Chile (FONDEQUIP EQM180042). The second author thanks to Fondecyt Project no. 1211484 and the first author to Anid doctoral scholarship 21202379.


This project was funded by Fondecyt Grant no. 1211484 and Anid doctoral scholarship 21202379.

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Appendix A: Complex geometries

Appendix A: Complex geometries

This appendix presents some examples of Polylla meshes generated using cartography maps obtained from the Library of National Congress of Chile Footnote 3, and the library PyShp Footnote 4. This library was used to read the .shp files, generate the PSLGs and store the information in .poly files. Figure 24 is a Polylla mesh generated from the PSLG of the Los Lagos Region, Fig. 25 is a mesh of the Magallanes Region and Fig. 26 is a mesh of the Budi lake.

Fig. 24
figure 24

Los Lagos Region, Chile. The .poly file contains 294,540 vertices and 294,242 segments. The initial triangulation was generated using triangle [26] with maximum area size of 100,000,000 as option. The resulting triangulation was a conforming Delaunay triangulation with 309,264 vertices, 309,305 triangles and 618,272 edges. The final Polylla mesh contains 309,264 vertices, 12,578 polygons and 321,534 edges

Fig. 25
figure 25

Magallanes Region, Chile. The .poly file contains 1,130,733 vertices and 1,130,752 segments. The initial triangulation was generated using triangle [26] with maximum area size of 100,000,000 as option. The resulting conforming Delaunay triangulation contains 1,148,594 vertices, 1,182,133 triangles and 2,324,968 edges. The final Polylla mesh contains 1,148,594 vertices, 45,300 polygons and 1,186,451 edges

Fig. 26
figure 26

Budi Lake, Araucanía Region, Chile. The .poly file contains 5794 vertices, 5794 constrained edges and 10 holes. The initial triangulation was generated using triangle [26] with maximum area size of 10,000 as option. The resulting conforming Delaunay triangulation contains 12,608 vertices, 19,406 triangles and 32,023 edges. The final Polylla mesh contains 12,608 vertices, 5169 polygons, 17,768 edges and 10 holes (Islands of the lake). Grey polygons are holes

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Salinas-Fernández, S., Hitschfeld-Kahler, N., Ortiz-Bernardin, A. et al. POLYLLA: polygonal meshing algorithm based on terminal-edge regions. Engineering with Computers (2022).

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  • Polygonal mesh
  • Terminal-edge region
  • Virtual element method
  • Delaunay triangulations