Exploring Mesh Generation and Quality Enhancement with Open Source Codes

Abstract

The finite element method is a very reliable and precise technique for solving partial differential equations in three-dimensional domains, with relevant applications in several areas. However, 3D simulations by FEM require computer programs for solid modeling and automatic mesh generation and there are few examples of open source codes available and dedicated to these tasks. Unfortunately, these open source codes are not usually conceived to operate together in an integrated fashion, showing distinct life cycles and different origins, which may result in contradictory specifications. In this study, a method is proposed to integrate solid modeling and automatic mesh generation with focus on open source codes and how the quality of FEM simulations can be improved by the improvement of the mesh. The method was structured in desired features for the solid construction and in integration strategies for an automatic mesh generation. The approach was tested in nontrivial domains and with known relevance for studies focused on computational electromagnetics. Meshes were generated with millions of tetrahedral elements and the results were compared to the quality values commonly discussed in literature focused on FEM. Complex geometries were meshed in a few seconds, with consistent values of aspect ratios (more than 90% of the tetrahedral elements were constructed with values at most 5) and dihedral angles (the values were bounded between 5.9 to 166.7°C, with a peak value around 90°C). Finally, in order to show the relation among highly refined meshes and quality criteria which can be explored by proposed method, the Laplace's equation was simulated by FEM in order to analyze the equipotential lines of a parallel-plate capacitor. The results show how the quality of a simulation can be improved, especially concerning the increasing number of tetrahedra in the mesh with proper aspect ratio.