Finite number of similarity classes in Longest Edge Bisection of nearly equilateral tetrahedra

Abstract

In 1983 Adler [1] pointed out that if a tetrahedron is nearly equilateral (edge lengths within 5% of each other) and the first and second longest edges are opposite, then the iterative Longest Edge Bisection (LEB) method produces ≤37 similarity classes. The importance of nearly equilateral tetrahedra is that they generate a finite number of similarity classes during the iterative LEB, a desirable property in Finite Element computations. We prove the conjecture given by Adler and improve the bound of 5% to 22.47%. A new algorithm is introduced for the computation of similarity classes in the iterative Longest Edge Bisection (SCLEB) of tetrahedra using a compact and efficient edge-based data structure.