Skip to main content
SpringerLink
Log in

Mesh shape-quality optimization using the inverse mean-ratio metric

  • Full Lenght Paper
  • Published:
Mathematical Programming Submit manuscript

Abstract

Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finite-element method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program that relocates the vertex coordinates of a given mesh to optimize the average element shape quality as measured by the inverse mean-ratio metric. To solve the resulting large-scale optimization problems, we apply an efficient implementation of an inexact Newton algorithm that uses the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional functions, applying this result to components of the inverse mean-ratio metric, and showing that each block in the preconditioner is invertible. Numerical results obtained with this special-purpose code on several test meshes are presented and used to quantify the impact on solution time and memory requirements of using a modeling language and general-purpose algorithm to solve these problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armijo L. (1966). Minimization of functions having Lipschitz-continuous first partial derivatives. Pac. J. Math. 16:1–3

    MATH  MathSciNet  Google Scholar 

  2. Babuška I., Suri M. (1994). The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36:578–632

    Article  MathSciNet  Google Scholar 

  3. Baker, T.J.: Mesh movement and metamorphosis. In: Proceedings of the 10th International Meshing Roundtable, pp. 387–296. Sandia National Laboratories (2001)

  4. Bank R.E., Smith R.K. (1997). Mesh smoothing using a posteriori estimates. SIAM J. Numer. Anal. 34(3):979–997

    Article  MATH  MathSciNet  Google Scholar 

  5. Berzins, M.: Solution-based mesh quality for triangular and tetrahedral meshes. In: Proceedings of the 6th International Meshing Roundtable, pp. 427–436. Sandia National Laboratories (1997)

  6. Berzins, M.: Mesh quality – geometry, error estimates or both? In: Proceedings of the 7th International Meshing Roundtable, pp. 229–237. Sandia National Laboratories (1998)

  7. Bischof, C.H., Hovland, P.D., Norris, B.: Implementation of automatic differentiation tools. Higher-Order and Symbolic Computation (to appear) (2004)

  8. Brenner S.C., Scott L.R. (2002). The Mathematical Theory of Finite Element Methods. Springer, Berlin Heidelberg, New York

    MATH  Google Scholar 

  9. Brooke A., Kendrick D., Meeraus A. (1988). GAMS: A User’s Guide. The Scientific Press, South San Francisco

    Google Scholar 

  10. Byrd R., Hribar M.E., Nocedal J. (1999). An interior point method for large scale nonlinear programming. SIAM J. Optim. 9:877–900

    Article  MATH  MathSciNet  Google Scholar 

  11. Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 24th National Conference ACM, pp. 157–172. ACM Press (1969)

  12. Ding, C., Kennedy, K.: Improving cache performance in dynamic applications through data and computation reorganization at run time. In: Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), pp. 229–241 (1999)

  13. Dolan, E.D., Moré, J.J., Munson, T.S.: Benchmarking optimization software with COPS 3.0. Technical Memorandum ANL/MCS-TM-273. Argonne National Laboratory, Argonne (2004)

  14. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, 2nd edn. Brooks/Cole–Thomson Learning, Pacific Grove, California (2003)

  15. Freitag, L., Knupp, P.: Tetrahedral element shape optimization via the Jacobian determinant and condition number. In: Proceedings of the 8th International Meshing Roundtable, pp. Sandia National Laboratories (1999)

  16. Freitag L., Knupp P. (2002). Tetrahedral mesh improvement via optimization of the element condition number. Int. J. Numer. Methods Eng. 53:1377–1391

    Article  MATH  MathSciNet  Google Scholar 

  17. Freitag, L., Knupp, P., Munson, T., Shontz, S.: A comparison of optimization software for mesh shape-quality improvement problems. In: Proceedings of the 11th International Meshing Roundtable. Sandia National Laboratories (2002)

  18. Freitag, L., Knupp, P., Munson, T., Shontz, S.: A comparison of inexact newton and coordinate descent mesh optimization techniques. In: Proceedings of the 13th International Meshing Roundtable. Sandia National Laboratories (2004)

  19. Freitag L., Ollivier-Gooch C. (1997). Tetrahedral mesh improvement using swapping and smoothing. Int. J. Numer. Methods Eng. 40:3979–4002

    Article  MATH  MathSciNet  Google Scholar 

  20. Freitag L., Ollivier-Gooch C. (2000). A cost/benefit analysis for simplicial mesh improvement techniques as measured by solution efficiency. Int. J. Comput. Geom. Appl. 10:361–382

    MATH  MathSciNet  Google Scholar 

  21. Freitag L., Plassmann P. (2000). Local optimization-based simplicial mesh untangling and improvement. Int. J. Numer. Methods Eng. 49:109–125

    Article  MATH  Google Scholar 

  22. Griewank A. (2000). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia

    MATH  Google Scholar 

  23. Han, H., Tseng, C.: A comparison of locality transformations for irregular codes. In Proceedings of the 5th International Workshop on Languages, Compilers, and Run-time Systems for Scalable Computers, pp. 70–84, Springer Rochester, New York (2000)

  24. Kelley C.T. (2003). Solving Nonlinear Equations with Newton’s Method. SIAM, Philadelphia

    MATH  Google Scholar 

  25. Knupp P. (2000). Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities, Part I – A framework for surface mesh optimization. Int. J. Numer. Methods Eng. 48:401–420

    Article  MATH  Google Scholar 

  26. Knupp P. (2000). Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities, Part II – A framework for volume mesh optimization and the condition number of the Jacobian matrix. Int. J. Numer. Methods Eng. 48:1165–1185

    Article  MATH  Google Scholar 

  27. Lin C.-J., Moré J.J. (1999). Newton’s method for large bound-constrained optimization problems. SIAM J Optim. 9:1100–1127

    Article  MATH  MathSciNet  Google Scholar 

  28. Liu A., Joe B. (1994). Relationship between tetrahedron quality measures. BIT 34:268–287

    Article  MATH  MathSciNet  Google Scholar 

  29. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969). SIAM Classics in Applied Mathematics 10, SIAM, Philadelphia (1994)

  30. Munson, T., Hovland, P.: The FeasNewt benchmark. In: Proceedings of the 2005 IEEE International Symposium on Workload Characterization (IISWC2005). IEEE Computer Society Press (2005)

  31. Munson T.S. (2004). Mesh shape-quality optimization using the inverse mean-ratio metric: Tetrahedral proofs. Technical Memorandum ANL/MCS-TM-275. Argonne National Laboratory, Argonne

    Google Scholar 

  32. Munson T.S. (2005). Optimizing the quality of mesh elements. SIAG/Optim. News Views 16:27–34

    Google Scholar 

  33. Nocedal J., Wright S.J. (1999). Numerical Optimization. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  34. Ortega J.M., Rheinboldt W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic, San Diego

    MATH  Google Scholar 

  35. Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  36. Saad Y. (2003). Iterative Methods for Sparse Linear Systems 2nd edn. SIAM, Philadelphia

    MATH  Google Scholar 

  37. Sandia National Laboratories, Albuquerque, New Mexico. CUBIT 8.1 Mesh Generation Toolkit (2003)

  38. Shephard M., Georges M. (1991). Automatic three-dimensional mesh generation by the finite octree technique. Int. J. Numer. Methods Eng. 32:709–749

    Article  MATH  Google Scholar 

  39. Shewchuk, J.: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Proceedings of the 1st Workshop on Applied Computational Geometry, pp. 124–133, Philadelphia, Pennsylvania, May 1996. ACM

  40. Shewchuk, J.: What is a good linear element? Interpolation, conditioning, and quality measures. In: Proceedings of the 11th International Meshing Roundtable, pp. 115–126. Sandia National Laboratories (2002)

  41. Shontz, S.M., Vavasis, S.A.: A mesh warping algorithm based on weighted Laplacian smoothing. In Proceedings of the 12th International Meshing Roundtable, pp. 147–158. Sandia National Laboratories (2003)

  42. Trefethan L.N. (2000). Spectral Element Methods in MATLAB. SIAM, Philadelphia

    Google Scholar 

  43. Vanderbei R.J. (2000). LOQO user’s manual – Version 4.05. Technical report. Princeton University, Princeton

    Google Scholar 

  44. Vanderbei R.J., Shanno D.F. (1999). An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13:231–252

    Article  MATH  MathSciNet  Google Scholar 

  45. Waltz R., Nocedal J. (2003). KNITRO user’s manual – Version 3.1. Technical Report 5. Northwestern University, Evanston

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

    Todd Munson

Authors
  1. Todd Munson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Todd Munson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Munson, T. Mesh shape-quality optimization using the inverse mean-ratio metric. Math. Program. 110, 561–590 (2007). https://doi.org/10.1007/s10107-006-0014-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0014-3

Keywords

Search

Navigation