On the shape of tetrahedra from bisection

Source abstract

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes T 0 , T 1 , T 2 , … {\mathcal {T}^0},{\mathcal {T}^1},{\mathcal {T}^2}, \ldots , which has the following properties: (1) Each mesh T n {\mathcal {T}^n} is conforming. (2) There are a finite number of classes of similar tetrahedra in all the T n , n ≥ 0 {\mathcal {T}^n},n \geq 0 . (3) For any tetrahedron T i n {\mathbf {T}}_i^n in T n , η ( T i n ) ≥ c 1 η ( T ) {\mathcal {T}^n},\eta ({\mathbf {T}}_i^n) \geq {c_1}\eta ({\mathbf {T}}) , where η \eta is a tetrahedron shape measure and c 1 {c_1} is a constant. (4) δ ( T i n ) ≤ c 2 ( 1 / 2 ) n / 3 δ ( T ) \delta ({\mathbf {T}}_i^n) \leq {c_2}{(1/2)^{n/3}}\delta ({\mathbf {T}}) , where δ ( T ′ ) \delta ({\mathbf {T’}}) denotes the diameter of tetrahedron T ′ {\mathbf {T’}} and c 2 {c_2} is a constant. Estimates of c 1 {c_1} and c 2 {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.