Quality Local Refinement of Tetrahedral Meshes Based on 8-Subtetrahedron Subdivision

Let $\mathcal {T}$ be a tetrahedral mesh. We present a 3-D local refinement algorithm for $\mathcal {T}$ which is mainly based on an 8-subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron ${\mathbf T} \in \mathcal {T}$ produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, $\eta ({\mathbf T}_i^{n}) \geq c \eta ({\mathbf T})$, where ${\mathbf T} \in \mathcal {T}$, $c$ is a positive constant independent of $\mathcal {T}$ and the number of refinement levels, ${\mathbf T}_i^{n}$ is any refined tetrahedron of ${\mathbf T}$, and $\eta$ is a tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant.