Non-equivalent partitions of d-triangles with Steiner points

Abstract

In this paper we present lower and upper bounds for the number of equivalence classes of d-triangles with additional or Steiner points. We also study the number of possible partitions that may appear by bisecting a tetrahedron with Steiner points at the midpoints of its edges. This problem arises, for example, when refining a 3D triangulation by bisecting the tetrahedra. To begin with, we look at the analogous 2D case, and then the 1-irregular tetrahedra (tetrahedra with at most one Steiner point on each edge) are classified into equivalence classes, and each element of the class is subdivided into several non-equivalent bisection-based partitions which are also studied. Finally, as an example of the application of refinement and coarsening of 3D bisection-based algorithms, a simulation evolution problem is shown.