Quality improvement of tetrahedral meshes by optimizing the minimum local angle

Abstract

Mesh quality is an important factor for stable, repeatable numerical simulations. The Delaunay method is widely used for creation of 3D tetrahedral meshes. Two-dimensional triangulation via Delaunay exhibits the mathematical property of maximizing the minimum interior angle. This feature provides excellent quality meshes for a given node deployment. However, the 3D equivalent of this property, i.e. to maximize the minimum solid angle, is not assured with 3D Delaunay. The tetrahedron's interior solid angle is directly related to mesh quality, but it is independent of the Delaunay process. Consequently, sliver elements and poor quality meshes can be created via Delaunay tetrahedral formation. In this paper, we describe a method for maximizing the minimum solid angle of tetrahedral meshes by changing the locations of non-boundary nodes. The displacement of nodes uses a gradient-based approach. The process is iterative and terminates when the mesh quality exceeds a user specified quality or convergence criterion. The technique is robust. The relocation of vertices is local which avoids significant deformation of the mesh. The results show considerable improvements in mesh quality. Using a 3D human brain mesh (27,000+ elements), our algorithm reduced the number of ill-formed elements three fold. We are extending this approach to allow tangential motion along the boundary surfaces. Currently all boundary nodes are fixed which constrains some of the element qualities.