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Keywords:
bisection algorithm; finite element method; tetrahedral partition; mesh regularity; dynamical system; hyperbolic geometry
bisection algorithm; finite element method; tetrahedral partition; mesh regularity; dynamical system; hyperbolic geometry
Summary:
We study the dynamics of similarity classes of tetrahedra generated by the longest-edge bisection (LEB) algorithm. Building on the normalization strategy introduced by F. Perdomo, Á. Plaza (2014) 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 towards an analytic proof of the nondegeneracy property for the tetrahedral partitions generated by the LEB refinements. This property is highly desired in e.g., the finite element methods (FEMs).
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