Tesselating a Pascal-like tetrahedron for the subdivision of high order tetrahedral finite elements

Abstract

Three-dimensional $N^{th}$ order nodal Lagrangian tetrahedral finite elements ($P_N$ elements) can be generated using Pascal's tetrahedron $\mathcal{H}$ where each node in 3D element space corresponds to an entry in $\mathcal{H}$. For the purposes of visualization and post-processing, it is desirable to "subdivide" these high-order tetrahedral elements into sub-tetrahedra which cover the whole space without intersections and without introducing new exterior edges or vertices. That is, the exterior triangulation of the element should be congruent with the "natural" triangulation of the $2D$ Pascal's triangle. This work attempts to describe that process of subdivision for arbitrary $N$.