Engineering with Computers

, Volume 22, Issue 2, pp 61–74 | Cite as

A comparison of two optimization methods for mesh quality improvement

  • Lori Freitag Diachin
  • Patrick Knupp
  • Todd Munson
  • Suzanne Shontz
Original Article


We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.


Mesh quality improvement Mesh optimization Mesh smoothing 



The initial version of the analytic gradient for the inverse mean-ratio metric for tetrahedral elements was provided by Paul Hovland (Argonne National Laboratory). The clipped cube mesh image was provided by Carl Ollivier-Gooch (University of British Columbia). The work of the first, second, and third authors was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contracts W-7405-Eng-48 (UCRL-CONF-205150), DE-AC-94AL85000, and W-31-109-Eng-38, respectively. Part of the work of the fourth author was performed while a member of the Center for Applied Mathematics at Cornell University, supported by Sandia National Laboratories, Cornell University, the National Physical Science Consortium, and NSF grant ACI-0085969.


  1. 1.
    Bank R, Smith B (1997) Mesh smoothing using a posteriori error estimates. SIAM J Num Anal 34:979–997CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Canann SA, Stephenson MB, Blacker T (1993) Optismoothing: an optimization-driven approach to mesh smoothing. Fin Elem Anal Des 13:185–190CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Parthasarathy VN, Kodiyalam S (1991) A constrained optimization approach to finite element mesh smoothing. Fin Elem Anal Des 9:309–320CrossRefzbMATHGoogle Scholar
  4. 4.
    Zavattieri P, Dari E, Buscaglia G (1996) Optimization strategies in unstructured mesh generation. Int J Num Methods Eng 39:2055–2071CrossRefzbMATHGoogle Scholar
  5. 5.
    Castillo J (1991) A discrete variational grid generation method. SIAM J Sci Stat Comp 12:454–468CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Freitag L, Knupp P (2002) Tetrahedral mesh improvement via optimization of the element condition number. Int J Numer Methods Eng 53:1377–1391CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Anderson D (1990) Grid cell volume control with an adaptive grid generator. Appl Math Comp 35:209–217CrossRefzbMATHGoogle Scholar
  8. 8.
    Knupp P (2001) Algebraic mesh quality metrics. SIAM J Sci Comp 23:193–218CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Knupp P (1999) Matrix norms and the condition number. Proceedings of 8th International Meshing Roundtable, pp 13–22Google Scholar
  10. 10.
    Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, BelmontzbMATHGoogle Scholar
  11. 11.
    Nocedal J, Wright SJ (1999) Numerical optimization. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  12. 12.
    Munson TS (2004) Mesh shape-quality optimization using the inverse mean-ratio metric. Preprint ANL/MCS-P1136-0304, Argonne National Laboratory, ArgonneGoogle Scholar
  13. 13.
    Munson TS (2004) Mesh shape-quality optimization using the inverse mean-ratio metric: tetrahedral proofs. Technical memorandum ANL/MCS-TM-275, Argonne National Laboratory, ArgonneGoogle Scholar
  14. 14.
    Armijo L (1966) Minimization of functions having lipschitz-continuous first partial derivatives. Pac J Math 16:1–3zbMATHMathSciNetGoogle Scholar
  15. 15.
    Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, PhiladelphiazbMATHGoogle Scholar
  16. 16.
    Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, PhiladelphiazbMATHGoogle Scholar
  17. 17.
    Sandia National Laboratories (2003) Albuquerque, New Mexico. CUBIT 80.1 Mesh Generation ToolkitGoogle Scholar
  18. 18.
    Ollivier-Gooch CF (1998–2002) GRUMMP—Generation and refinement of unstructured mixed-element meshes in parallel.

Copyright information

© Springer-Verlag London Limited 2006

Authors and Affiliations

  • Lori Freitag Diachin
    • 1
  • Patrick Knupp
    • 2
  • Todd Munson
    • 3
  • Suzanne Shontz
    • 4
  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA
  2. 2.Sandia National LaboratoriesAlbuquerqueUSA
  3. 3.Argonne National LaboratoryArgonneUSA
  4. 4.University of MinnesotaMinneapolisUSA

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