Variational tetrahedral meshing

TLDR

The energy used is the ∠1 distance between an isotropic quadratic function and its linear interpolation, which leads to well‑shaped tetrahedra. The paper introduces a Delaunay‑based variational approach to isotropic tetrahedral meshing and lays out its variational principle and geometric interpretation. The method minimizes a mesh‑dependent energy by globally updating vertex positions and connectivity, with mesh design controlled by a gradation smoothness parameter and target vertex count. The resulting meshes are shown to be high quality in a series of illustrative examples.

Abstract

In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. As this energy is known to be the ∠1 distance between an isotropic quadratic function and its linear interpolation on the mesh, our minimization procedure generates well-shaped tetrahedra. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes through a series of examples.