Towards FEA over Tangled Quads

In finite element analysis (FEA), a mesh is said to be tangled if it contains an element with negative Jacobian-determinant. Tangling can occur during mesh optimization and mesh morphing. Modern FEA unfortunately cannot handle such tangled meshes, i.e., it will lead to erroneous results. While significant progress has been made on untangling, there are no definitive untangling algorithms.Danczyk and Suresh recently proposed a theoretical extension to FEA such that one could accurately solve boundary value problems over such tangled meshes. However, their investigation was limited to simplicial meshes.In this paper we consider the extension of the above framework to tangled quad meshes that pose additional challenges compared to simplicial meshes. These challenges are identified, and a tangling-framework is developed to address explicit quad tangling where quads are allowed to overlap, but are required to be geometrically convex. Numerical examples illustrate the correctness of the proposed framework, opening new opportunities for meshing algorithms.