Simultaneous volume parametrization and adaptive tetrahedral mesh generation with the Meccano method

Abstract

We have recently introduced the meccano technique for constructing adaptive tetrahedral meshes of solids. The method requires a surface triangulation of the solid, a meccano and a tolerance that fixes the desired approximation of the solid surface. The name of the method stems from the fact that the process starts from an outline of the solid, i.e. a meccano composed by connected polyhedral pieces. The method builds a 3-D triangulation of the solid as a deformation of an appropriate tetrahedral mesh of the meccano. The main idea of the new mesh generator is to combine an automatic parametrization of surface triangulations, a local refinement algorithm for 3-D nested triangulations and a simultaneous untangling and smoothing procedure. In this paper, we present significant advances in the method. Specifically, we describe the procedure for a solid whose boundary is a surface of genus 0; i.e. a surface that is homeomorphic to the surface of a sphere. In this case, the meccano is a single cube, and we define an automatic parametrization of a solid surface triangulation to the meccano boundary such that the global mapping is the combination of six patch-mapping. A crucial consequence of our technique is the volume parametrization of a complex solid to a cube. The efficiency of the proposed technique is shown with several applications.