Towards high-quality, untangled meshes via a force-directed graph embedding approach

High quality meshes are crucial for the solution of partial differential equations (PDEs) via the finite element method (or other PDE solvers). The accuracy of the PDE solution, and the stability and conditioning of the stiffness matrix depend upon the mesh quality. In addition, the mesh must be untangled in order for the finite element method to generate physically valid solutions. Tangled meshes, i.e., those with inverted mesh elements, are sometimes generated via large mesh deformations or in the mesh generation process. Traditional techniques for untangling such meshes are based on geometry and/or optimization. Optimization-based mesh untangling techniques first untangle the mesh and then smoothe the resulting untangled mesh in order to obtain high quality meshes; such techniques require the solution of two optimization problems. In this paper, we study how to modify a physical, force-directed method based upon the Fruchterman-Reingold (FR) graph layout algorithm so that it can be used for untangling. The objectives of aesthetic graph layout, such as minimization of edge intersections and near equalization of edge lengths, follow the goals of mesh untangling and generating good quality elements, respectively. Therefore, by using the force-directed method, we can achieve both steps of mesh untangling and mesh smoothing in one operation. We compare the effectiveness of our method with that of the optimization-based mesh untangling method in [1] and implemented in Mesquite by untangling a suite of unstructured triangular, quadrilateral, and tetrahedral finite element volume meshes. The results show that the force-directed method is substantially faster than the Mesquite mesh untangling method without sacrificing much in terms of mesh quality for the majority of the test cases we consider in this paper. The force-directed mesh untangling method demonstrates the most promise on convex geometric domains. Further modifications will be made to the method to improve its ability to untangle meshes on non-convex domains.