Improved conformal adaptation for hexahedral meshes

Abstract

To improve the quality of the results of a numerical simulation with a finite element method, mesh refinement is an effective solution. Various techniques have been implemented for many years. In the h-refinement method, the first point consists in selecting some meshes of the domain which are too large. These meshes are divided by cutting their edges. This division produces new meshes of the same category: the division of triangles produces triangles, the division of hexahedra produces hexahedra, etc. The connection between two zones with different levels of refinement must be taken into account if the numerical method needs a conformal mesh for the resolution: in this case, a node cannot be isolated at the middle of an edge. If the mesh is made of triangle or tetrahedra, this connection can be implemented with specific triangles or tetrahedra. But if the initial mesh is made of quadrangles or hexahedra, the solution is not simple. It cannot be achieved with quadrangles or hexahedra. In a first attempt, for every hexahedron at the interface between the zones with different levels of refinement a solution was proposed [1, 2]. A quadrangular face could be intact, regularly split or only one of its edges is cut. In this case, the quadrangle is split into 3 triangles. That rule induces 4 situations for the hexahedron and it is cut, using pyramids and tetrahedra. This technique is effective and allows mesh adaptation. Nevertheless, there is a drawback: in some circumstances, the refinement could spread a lot. The improvement was found by authorizing the splitting of 2 edges of a quadrangle, defining a new refinement of it with 3 quadrangles. Based on this new rule, 47 new patterns are designed to achieve the splitting of the hexahedron. Doing that, the refinement is improved: a conformal mesh is produced and the number of created meshes and nodes is as low as possible.