On the Orbits of Similarity Classes of Tetrahedra Generated by the Longest-Edge Bisection Algorithm

TL;DR

This paper investigates the shape evolution of tetrahedra under the longest-edge bisection algorithm using a novel normalized geometric representation, revealing finite orbits and non-degeneracy properties crucial for finite element methods.

Contribution

It introduces a canonical normalized space for tetrahedra and analyzes the dynamical behavior of the LEB algorithm, showing finite orbits and stability properties.

Findings

The orbit of the Sommerville tetrahedron contains only 4 similarity classes.

Small perturbations lead to finite orbits, indicating stability.

Numerical evidence suggests LEB does not produce degenerating tetrahedra.

Abstract

In this work, we study the dynamics of similarity classes of tetrahedra generated by the longest-edge bisection (LEB) algorithm. Building on the normalization strategy introduced by Perdomo and Plaza for triangles, we construct a canonical representation of tetrahedra in a normalized space embedded in the product of the hyperbolic half-plane and the hyperbolic half-space model. This representation allows us to define the left and right refinement maps, $\Phi_L$ and $\Phi_R$, acting on the space of normalized tetrahedral shapes, and to study their iterative orbits as discrete dynamical systems. Using these maps, we show that the orbit of the space-filling Sommerville tetrahedron contains only 4 similarity classes, 3 of which form an attractive cycle corresponding to the orbit of the path tetrahedron. We also show that small perturbations of elements in those orbits still lead to finite orbits. In addition, we study small perturbations of the regular tetrahedron and show that their orbits are also finite. Extensive numerical exploration of orbits for the other types of tetrahedra suggests that the LEB algorithm does not produce degenerating tetrahedra. Our framework provides a geometric and dynamical foundation for analyzing the shape evolution of tetrahedral meshes and offers a possible route toward an analytic proof of the non-degeneracy property for the tetrahedral partitions generated by the LEB refinements. This property is highly desired in e.g. the finite element methods (FEMs).