An efficient algorithm for 3D adaptive meshing

Abstract

Many physical phenomena in science and engineering can be modeled by partial differential equations and solved by means of the finite element method. Such a method uses as computational spatial support a mesh of the domain where the equations are formulated. The ‘mesh quality’ is a key-point for the accuracy of the numerical simulation. One can show that this quality is related to the shape and the size of the mesh elements. In the case where the element sizes are not specified in advance, a quality mesh is a regular mesh (whose elements are almost equilateral). This problem is a particular case of a more general mesh generation problem whose purpose is to construct meshes conforming to a prespecified isotropic size field associated with the computational domain. Such meshes can be seen as ‘unit meshes’ (whose elements are of unit size) in an appropriate non-Euclidean metric. In this case, a quality mesh of the domain is a unit mesh as regular as possible. In this paper, we are concerned with the generation of such a mesh and we propose a method to achieve this goal. First, the boundary of the domain is meshed using an indirect scheme via parametric domains and then the mesh of the three-dimensional (3D) domain is generated. In both cases, an empty mesh is first constructed, then enriched by field points, and finally optimized. The field points are defined following an algebraic or an advancing-front approach and are connected using a generalized Delaunay type method. To show the overall meshing process, we give an example of a 3D domain encountered in a classical computational fluid dynamics problem.