3D POLYLLA: Polyhedral meshing algorithm based on terminal-edge regions and terminal-face regions

TL;DR

The paper extends the 2D Polylla polygonal-mesh framework to 3D by introducing terminal-face regions and two variants for converting tetrahedral meshes into polyhedral meshes suitable for simulations and the Virtual Element Method (VEM). It defines a formal set of concepts (joining criterion $J$, terminal-face regions, frontier/barrier faces, and barrier tips) and develops a three-phase process (label, traversal, repair) to construct polyhedra from tetrahedra. Experimental results across several mesh types and joining criteria show substantial tetrahedron reduction (about $70\%$) with an average of roughly $3$ tetrahedra per polyhedron and around $18$–$21\%$ of polyhedra containing barrier faces, while a Voronoi comparison provides context. The authors identify the need to refine joining strategies to balance polyhedron quality and compatibility with VEM, and outline future work toward validating the produced polyhedra in simulations. The work advances 3D polygonal/polyhedral meshing by generalizing terminal-edge concepts to terminal-face regions and preserving a three-phase workflow analogous to the 2D algorithm.

Abstract

Polylla is a polygonal mesh algorithm that generates meshes with arbitrarily shaped polygons using the concept of terminal-edge regions. Until now, Polylla has been limited to 2D meshes, but in this work, we extend Polylla to 3D volumetric meshes. We present two versions of Polylla 3D. The first version generates terminal-edge regions, converts them into polyhedra, and repairs polyhedra that are joined by only an edge. This version differs from the original Polylla algorithm in that it does not have the same phases as the 2D version. In the second version, we define two new concepts: longest-face propagation path and terminal-face regions. We use these concepts to create an almost direct extension of the 2D Polylla mesh with the same three phases: label phase, traversal phase, and repair phase.

Figures (6)

• Figure 1: Example of five terminal-faces region generated using the incircle joining criterion.
• Figure 2: Comparison of two polyhedra with a greater number of faces, derived from the same tetrahedralization, generated using two different joining criteria.
• Figure 3: Example of labeling with adjacent $3$ tetrahedrons, Red faces are the terminal-faces, and green faces are frontier-faces. Tetrahedron $t_1$ is connected to $t_2$ by face $f_1$, and tetrahedron $t_1$ is connected to $t_3$ by face $f_2$. According to the Joining criterion of the largest area, $f_1$ is the largest face of $t_1$ and $t_2$, meaning that $f_1$ is a terminal-face, thus $t_1$ is chosen a seed tetrahedron. $f_2$ is not the largest face of $t_1$ and $t_3$, thus $f_2$ is label as a frontier-face.
• Figure 4: Visualization of the DFS, the tetrahedron at the center is a seed tetrahedron, the DFS travels inside a terminal-face region until to find a frontier-face (green faces) and stores it as a face of the new polyhedron.
• Figure 5: Example of the meshes generated for the experiments, all meshes have near 5000 vertices. The cubes were cut by a plane to show the interior of the mesh.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 2.1
• Proof 1
• Definition 7
• Definition 8