Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision

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Abstract
Let T \mathcal {T} be a tetrahedral mesh. We present a 3-D local refinement algorithm for T \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 T ∈ T \mathbf {T} \in \mathcal {T} produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, η ( T i n ) ≥ c η ( T ) \eta (\mathbf {T}_i^{n}) \geq c \eta (\mathbf {T}) , where T ∈ T \mathbf {T} \in \mathcal {T} , c c is a positive constant independent of T \mathcal {T} and the number of refinement levels, T i n \mathbf {T}_i^{n} is any refined tetrahedron of T \mathbf {T} , and η \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.

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