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

Jérôme Michaud, Sergey Korotov

TL;DR

This work advances a geometric dynamical framework for the longest-edge bisection refinement of tetrahedra by mapping tetrahedra to a canonical space 𝒯 ⊂ 𝔥^2 × 𝔥^3 and formulating left/right refinement maps that generate discrete orbits. It proves that the space-filling Sommerville tetrahedron has a four-element orbit, with three elements forming an attractive cycle including the path tetrahedron, and shows that small perturbations produce finite, clustered orbits; nearly regular tetrahedra also yield finite orbits with varying lengths. Through symbolic normalization and numerical exploration, the study provides evidence against degeneracy in many LEB-generated partitions and identifies both finite and potentially infinite-orbit cases, while outlining a path toward a rigorous non-degeneracy proof and regularity results for FEM mesh refinement. The results connect prior edge-length based classifications with a hyperbolic-geometric dynamical perspective, offering insights into mesh regularity and stability for adaptive tetrahedral refinement. Overall, the paper establishes a concrete framework to analyze and potentially prove non-degeneracy properties of LEB-generated tetrahedral partitions, with implications for finite element methods.

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, $Φ_L$ and $Φ_R$, 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).

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

TL;DR

This work advances a geometric dynamical framework for the longest-edge bisection refinement of tetrahedra by mapping tetrahedra to a canonical space 𝒯 ⊂ 𝔥^2 × 𝔥^3 and formulating left/right refinement maps that generate discrete orbits. It proves that the space-filling Sommerville tetrahedron has a four-element orbit, with three elements forming an attractive cycle including the path tetrahedron, and shows that small perturbations produce finite, clustered orbits; nearly regular tetrahedra also yield finite orbits with varying lengths. Through symbolic normalization and numerical exploration, the study provides evidence against degeneracy in many LEB-generated partitions and identifies both finite and potentially infinite-orbit cases, while outlining a path toward a rigorous non-degeneracy proof and regularity results for FEM mesh refinement. The results connect prior edge-length based classifications with a hyperbolic-geometric dynamical perspective, offering insights into mesh regularity and stability for adaptive tetrahedral refinement. Overall, the paper establishes a concrete framework to analyze and potentially prove non-degeneracy properties of LEB-generated tetrahedral partitions, with implications for finite element methods.

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, and , 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).

Paper Structure

This paper contains 11 sections, 5 theorems, 10 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Normalizing and visualizing the similarity classes of tetrahedra
  3. The canonical space of tetrahedra
  4. Normalization procedure
  5. LEB in the space of normalized tetrahedra
  6. On orbits produced by the LEB refinement algorithms
  7. The case of the Sommerville tetrahedron
  8. Orbits of small perturbations of the Sommerville and path tetrahedra
  9. Orbits of small perturbations of the regular tetrahedron
  10. On the behaviour of LEB on general tetrahedra
  11. Conclusions and future work

Key Result

Theorem 2.1

The normalization map is surjective and constant on similarity classes. Moreover, for two tetrahedra $T_1$ and $T_2$ that are not shape similar, $T_1 \not\sim T_2$, then $N(T_1)\neq N(T_2)$.

Figures (12)

  • Figure 1: An illustration for the canonical space of triangles. The set $\mathcal{B}$, of admissible positions for the third vertex, is colored in green.
  • Figure 2: Left: an example of a canonical tetrahedron. The red point is the normalized representation of the canonical base and the black point is the remaining vertex. All the edges of the tetrahedron are also displayed. Right: same figure, but only the edge carrying information is presented.
  • Figure 3: The orbit of the Sommerville tetrahedron.
  • Figure 4: Visualization of the orbit of the Sommerville tetrahedron. The first three panels show different views of the same results. The bottom right panel shows only the positions of the base vertices.
  • Figure 5: The results of Example \ref{['ex:Som-mod']} illustrated. Graph shows how the normalized tetrahedra listed in the tables at the top and marked by nodes in the diagram map onto each other under the action of $\Phi_L$ (red) and $\Phi_R$ (blue). All the tetrahedra involved are grouped according to how they cluster. The red node $\tau_1$ corresponds to the modification of the Sommerville tetrahedron, and it has no incoming edges.
  • ...and 7 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.1
  • proof
  • ...and 29 more