On the Shape of Tetrahedra from Bisection

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes ${\mathcal {T}^0},{\mathcal {T}^1},{\mathcal {T}^2}, \ldots$, which has the following properties: (1) Each mesh ${\mathcal {T}^n}$ is conforming. (2) There are a finite number of classes of similar tetrahedra in all the ${\mathcal {T}^n},n \geq 0$. (3) For any tetrahedron ${\mathbf {T}}_i^n$ in ${\mathcal {T}^n},\eta ({\mathbf {T}}_i^n) \geq {c_1}\eta ({\mathbf {T}})$, where $\eta$ is a tetrahedron shape measure and ${c_1}$ is a constant. (4) $\delta ({\mathbf {T}}_i^n) \leq {c_2}{(1/2)^{n/3}}\delta ({\mathbf {T}})$, where $\delta ({\mathbf {T’}})$ denotes the diameter of tetrahedron ${\mathbf {T’}}$ and ${c_2}$ is a constant. Estimates of ${c_1}$ and ${c_2}$ are provided. Properties (2) and (3) extend similar results of Stynes and Adler, and of Rosenberg and Stenger, respectively, for the 2-D case. The diameter bound in property (4) is better than one given by Kearfott.