On the Shape of Tetrahedra from Bisection

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes T⁰,T¹,T², …, which has the following properties: (1) Each mesh Tⁿ is conforming. (2) There are a finite number of classes of similar tetrahedra in all the Tⁿ,n ≥ 0. (3) For any tetrahedron Tᵢⁿ in Tⁿ,η (Tᵢⁿ) ≥ c₁η (T), where η is a tetrahedron shape measure and c₁ is a constant. (4) δ (Tᵢⁿ) ≤ c₂(1/2)n/3δ (T), where δ (T’) denotes the diameter of tetrahedron T’ and c₂ is a constant. Estimates of c₁ and c₂ 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.