Geometric Properties of the Adaptive Delaunay Tessellation

Recently, the Adaptive Delaunay Tessellation (ADT) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the ADT necessary for the computation of the nodal integration scheme. This paper gives pending proofs for the three main claims which are uniqueness of the ADT, connectedness of the ADT, and coverage of the Voronoi tiles by adjacent ADT tiles. Furthermore, this paper provides a critical assessment of the ADT for arbitrary point sets.