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Mathematical Programming

, Volume 110, Issue 3, pp 561–590 | Cite as

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

  • Todd Munson
Full Lenght Paper

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.

Keywords

Hessian Matrix Ideal Element Conjugate Gradient Method Sandia National Laboratory Tetrahedral Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Armijo L. (1966). Minimization of functions having Lipschitz-continuous first partial derivatives. Pac. J. Math. 16:1–3MATHMathSciNetGoogle Scholar
  2. 2.
    Babuška I., Suri M. (1994). The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36:578–632CrossRefMathSciNetGoogle Scholar
  3. 3.
    Baker, T.J.: Mesh movement and metamorphosis. In: Proceedings of the 10th International Meshing Roundtable, pp. 387–296. Sandia National Laboratories (2001)Google Scholar
  4. 4.
    Bank R.E., Smith R.K. (1997). Mesh smoothing using a posteriori estimates. SIAM J. Numer. Anal. 34(3):979–997MATHCrossRefMathSciNetGoogle Scholar
  5. 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)Google Scholar
  6. 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)Google Scholar
  7. 7.
    Bischof, C.H., Hovland, P.D., Norris, B.: Implementation of automatic differentiation tools. Higher-Order and Symbolic Computation (to appear) (2004)Google Scholar
  8. 8.
    Brenner S.C., Scott L.R. (2002). The Mathematical Theory of Finite Element Methods. Springer, Berlin Heidelberg, New YorkMATHGoogle Scholar
  9. 9.
    Brooke A., Kendrick D., Meeraus A. (1988). GAMS: A User’s Guide. The Scientific Press, South San FranciscoGoogle Scholar
  10. 10.
    Byrd R., Hribar M.E., Nocedal J. (1999). An interior point method for large scale nonlinear programming. SIAM J. Optim. 9:877–900MATHCrossRefMathSciNetGoogle Scholar
  11. 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)Google Scholar
  12. 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)Google Scholar
  13. 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)Google Scholar
  14. 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)Google Scholar
  15. 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)Google Scholar
  16. 16.
    Freitag L., Knupp P. (2002). Tetrahedral mesh improvement via optimization of the element condition number. Int. J. Numer. Methods Eng. 53:1377–1391MATHCrossRefMathSciNetGoogle Scholar
  17. 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)Google Scholar
  18. 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)Google Scholar
  19. 19.
    Freitag L., Ollivier-Gooch C. (1997). Tetrahedral mesh improvement using swapping and smoothing. Int. J. Numer. Methods Eng. 40:3979–4002MATHCrossRefMathSciNetGoogle Scholar
  20. 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. 10:361–382MATHMathSciNetGoogle Scholar
  21. 21.
    Freitag L., Plassmann P. (2000). Local optimization-based simplicial mesh untangling and improvement. Int. J. Numer. Methods Eng. 49:109–125MATHCrossRefGoogle Scholar
  22. 22.
    Griewank A. (2000). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, PhiladelphiaMATHGoogle Scholar
  23. 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)Google Scholar
  24. 24.
    Kelley C.T. (2003). Solving Nonlinear Equations with Newton’s Method. SIAM, PhiladelphiaMATHGoogle Scholar
  25. 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. 48:401–420MATHCrossRefGoogle Scholar
  26. 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. 48:1165–1185MATHCrossRefGoogle Scholar
  27. 27.
    Lin C.-J., Moré J.J. (1999). Newton’s method for large bound-constrained optimization problems. SIAM J Optim. 9:1100–1127MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Liu A., Joe B. (1994). Relationship between tetrahedron quality measures. BIT 34:268–287MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969). SIAM Classics in Applied Mathematics 10, SIAM, Philadelphia (1994)Google Scholar
  30. 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)Google Scholar
  31. 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, ArgonneGoogle Scholar
  32. 32.
    Munson T.S. (2005). Optimizing the quality of mesh elements. SIAG/Optim. News Views 16:27–34Google Scholar
  33. 33.
    Nocedal J., Wright S.J. (1999). Numerical Optimization. Springer, Berlin Heidelberg New YorkMATHCrossRefGoogle Scholar
  34. 34.
    Ortega J.M., Rheinboldt W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic, San DiegoMATHGoogle Scholar
  35. 35.
    Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, PrincetonMATHGoogle Scholar
  36. 36.
    Saad Y. (2003). Iterative Methods for Sparse Linear Systems 2nd edn. SIAM, PhiladelphiaMATHGoogle Scholar
  37. 37.
    Sandia National Laboratories, Albuquerque, New Mexico. CUBIT 8.1 Mesh Generation Toolkit (2003)Google Scholar
  38. 38.
    Shephard M., Georges M. (1991). Automatic three-dimensional mesh generation by the finite octree technique. Int. J. Numer. Methods Eng. 32:709–749MATHCrossRefGoogle Scholar
  39. 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. ACMGoogle Scholar
  40. 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)Google Scholar
  41. 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)Google Scholar
  42. 42.
    Trefethan L.N. (2000). Spectral Element Methods in MATLAB. SIAM, PhiladelphiaGoogle Scholar
  43. 43.
    Vanderbei R.J. (2000). LOQO user’s manual – Version 4.05. Technical report. Princeton University, PrincetonGoogle Scholar
  44. 44.
    Vanderbei R.J., Shanno D.F. (1999). An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13:231–252MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Waltz R., Nocedal J. (2003). KNITRO user’s manual – Version 3.1. Technical Report 5. Northwestern University, EvanstonGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA

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