Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Analysis of triangle quality measures

Authors: Philippe P. Pébay and Timothy J. Baker
Journal: Math. Comp. 72 (2003), 1817-1839
MSC (2000): Primary 32B25, 65M50; Secondary 51N20
Published electronically: January 8, 2003
MathSciNet review: 1986806
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several of the more commonly used triangle quality measures are analyzed and compared. Proofs are provided to verify that they do exhibit the expected extremal properties. The asymptotic behavior of these measures is investigated and a number of useful results are derived. It is shown that some of the quality measures are equivalent, in the sense of displaying the same extremal and asymptotic behavior, and that it is therefore possible to achieve a concise classification of triangle quality measures.

References [Enhancements On Off] (What's this?)

  • 1. T.J. BAKER, Element quality in tetrahedral meshes, Proc. 7th Int. Conf. Finite Element Methods in Flow Problems, 1018-1024, Huntsville, U.S.A., 1989.
  • 2. I. BABUSKA and A.K. AZIZ, On the angle condition in the finite element method, SIAM J. Numer. Anal. 13, 214-227, 1976. MR 56:13700
  • 3. T.J. BAKER, Deformation and quality measures for tetrahedral meshes, Proc. ECCOMAS 2000, Barcelona, Spain, September 2000.
  • 4. R.E. BANK and R.K. SMITH, Mesh smoothing using a posteriori error estimates, SIAM J. Numer. Anal. 34, 979-997, 1997. MR 98m:65162
  • 5. P.G. CIARLET and P.A. RAVIART, General Lagrange and Hermite interpolation in ${\mathbb R}^n$ with applications to finite element methods, Arch. Rational Mech. Anal. 46, 177-199, 1972. MR 49:1730
  • 6. H.S.M. COXETER, Introduction to Geometry, Wiley, New York - London, 1961. MR 23:A1251
  • 7. L.A. FREITAG and P.M. KNUPP, Tetrahedral element shape optimization via the Jacobian determinant and condition number, Proc. 8th Int. Mesh. Roundtable, South Lake Tahoe, U.S.A., October 1999.
  • 8. P.J. FREY and P.L. GEORGE, Mesh Generation, Hermes Science Publishing, Oxford & Paris, 2000. MR 2002c:65001
  • 9. P.M. KNUPP, Algebraic mesh quality metrics, SIAM J. Sci. Comput. 23, 193-218, 2001.
  • 10. A. LIU and B. JOE, On the shape of tetrahedra from bisection, Math. Comp. 63 207, 141-154, July 1994. MR 94j:65113
  • 11. V.T. PARTHASARATHY, C.M. GRAICHEN and A.F. HATHAWAY, A comparison of tetrahedral quality measures, Fin. Elem. Anal. Des. 15, 255-261, 1993.
  • 12. V.T. RAJAN, Optimality of the Delaunay triangulations in ${\mathbb R}^d$, Discrete Comput. Geom. 12, 189-202, 1994. MR 98e:52027

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 32B25, 65M50, 51N20

Retrieve articles in all journals with MSC (2000): 32B25, 65M50, 51N20

Additional Information

Philippe P. Pébay
Affiliation: Mechanical and Aerospace Engineering Department, E-Quad, Princeton University, New Jersey 08544
Email: Current E-mail address:

Timothy J. Baker
Affiliation: Mechanical and Aerospace Engineering Department, E-Quad, Princeton University, New Jersey 08544

Keywords: Triangulation, surface triangulation, triangle quality, mesh quality
Received by editor(s): July 9, 2001
Received by editor(s) in revised form: December 21, 2001
Published electronically: January 8, 2003
Additional Notes: The first author was supported in part by CNRS, UMR 5585, France.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society