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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the shape of tetrahedra from bisection

Authors: Anwei Liu and Barry Joe
Journal: Math. Comp. 63 (1994), 141-154
MSC: Primary 65M50; Secondary 51M20, 52B10, 65N30
MathSciNet review: 1240660
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Abstract: 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.

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Article copyright: © Copyright 1994 American Mathematical Society