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

TL;DR

This paper presents a method for subdividing high-order tetrahedral finite elements based on Pascal's tetrahedron, enabling consistent and intersection-free subdivision suitable for visualization and post-processing.

Contribution

It introduces a systematic approach to subdivide arbitrary Nth-order tetrahedral elements into sub-tetrahedra using Pascal's tetrahedron structure.

Findings

Subdivision process is applicable to any order N.

Ensures congruence with natural Pascal's triangle triangulation.

Facilitates visualization and post-processing of high-order 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 $N$.