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

DOI:
https://doi.org/10.1090/S0025-5718-1994-1240660-4

MathSciNet review:
1240660

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a procedure for bisecting a tetrahedron **T** successively into an infinite sequence of tetrahedral meshes , which has the following properties: (1) Each mesh is conforming. (2) There are a finite number of classes of similar tetrahedra in all the . (3) For any tetrahedron in , where is a tetrahedron shape measure and is a constant. (4) , where denotes the diameter of tetrahedron and is a constant.

Estimates of and 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.

**[1]**A. Adler,*On the bisection method for triangles*, Math. Comp.**40**(1983), 571-574. MR**689473 (84d:51028)****[2]**G. H. Golub and C. F. Van Loan,*Matrix computations*, 2nd ed., Johns Hopkins University Press, Baltimore, MD, 1989. MR**1002570 (90d:65055)****[3]**B. Joe,*Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation*, Internat. J. Numer. Methods Engrg.**31**(1991), 987-997.**[4]**-,*Three-dimensional boundary-constrained triangulations*, Artificial Intelligence, Expert Systems, and Symbolic Computing (E. N. Houstis and J. R. Rice, eds.), Elsevier Science Publishers, 1992, pp. 215-222.**[5]**B. Kearfott,*A proof of convergence and an error bound for the method of bisection in*, Math. Comp.**32**(1978), 1147-1153. MR**0494897 (58:13677)****[6]**M.-C. Rivara,*Mesh refinement processes based on the generalized bisection of simplices*, SIAM J. Numer. Anal.**21**(1984), 604-613. MR**744176 (85i:65159)****[7]**-,*Algorithms for refining triangular grids suitable for adaptive and multigrid techniques*, Internat. J. Numer. Methods Engrg.**20**(1984), 745-756. MR**739618 (85h:65258)****[8]**-,*A grid generator based on 4-triangles conforming mesh-refinement algorithms*, Internat. J. Numer. Methods Engrg.**24**(1987), 1343-1354.**[9]**I. G. Rosenberg and F. Stenger,*A lower bound on the angles of triangles constructed by bisecting the longest side*, Math. Comp.**29**(1975), 390-395. MR**0375068 (51:11264)****[10]**M. Senechal,*Which tetrahedra fill space*?, Math. Mag.**54**(1981), 227-243. MR**644075 (83h:52020)****[11]**M. Stynes,*On faster convergence of the bisection method for all triangles*, Math. Comp.**35**(1980), 1195-1201. MR**583497 (81j:51023)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M50,
51M20,
52B10,
65N30

Retrieve articles in all journals with MSC: 65M50, 51M20, 52B10, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1994-1240660-4

Article copyright:
© Copyright 1994
American Mathematical Society