Introducing the Target-Matrix Paradigm for Mesh Optimization via Node-Movement

  • Patrick Knupp
Conference paper


A general-purpose algorithm for mesh optimization via nodemovement, known as the Target-Matrix Paradigm, is introduced. The algorithm is general purpose in that it can be applied to a wide variety of mesh and element types, and to various commonly recurring mesh optimization problems such as shape improvement, and to more unusual problems like boundary-layer preservation with sliver removal, high-order mesh improvement, and edge-length equalization. The algorithm can be considered to be a direct optimization method in which weights are automatically constructed to enable definitions of application-specific mesh quality. The high-level concepts of the paradigm have been implemented in the Mesquite mesh-improvement library, along with a number of concrete algorithms that address mesh quality issues such as those shown in the examples of the present paper.


Sandia National Laboratory Initial Mesh Mesh Quality Triangle Mesh Mesh Type 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brewer, M., Diachin, L., Knupp, P., Melander, D.: The Mesquite Mesh Quality Improvement Toolkit. In: Proceedings of the 12th International Meshing Roundtable, Santa Fe NM, pp. 239–250 (2003)Google Scholar
  2. 2.
    Castillo, J.E.: A discrete variational grid generation method. SIAM J. Sci. Stat. Comp. 12(2), 454–468 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Tinoco-Ruiz, J., Barrera-Sanchez, P.: Area functionals in Plane Grid Generation. In: Cross, M., et al. (eds.) Numerical Grid Generation in Computational Field Simulations, Greenwhich, UK, pp. 293–302 (1998)Google Scholar
  4. 4.
    Kennon, S., Dulikravich, G.: Generation of computational grids using optimization. AIAA Journal 24(7), 1069–1073 (1986)zbMATHCrossRefGoogle Scholar
  5. 5.
    Freitag, L.: On combining Laplacian and optimization-based mesh smoothing techniques. In: ASME 1997 Trends in Unstructured Mesh Generation, AMD, vol. 220, pp. 37–43 (1997)Google Scholar
  6. 6.
    Zhou, T., Shimada, K.: An angle-based approach to two-dimensional mesh smoothing. In: Proceedings of the 9th International Meshing Roundtable, pp. 373–384 (2000)Google Scholar
  7. 7.
    Brackbill, J., Saltzman, J.: Adaptive zoning for singular problems in two dimensions. J. Comp. Phys. 46, 342–368 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Steinberg, S., Roache, P.: Variational grid generation. Num. Meth. for P.D.E. 2, 71–96 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Liseikin, V.: On a variational method of generating adaptive grids on n-dimensional surfaces. Soviet Math. Docl. 44(1), 149–152 (1992)MathSciNetGoogle Scholar
  10. 10.
    Winslow, A.: Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh. J. Comp. Phys. 2, 149–172 (1967)MathSciNetGoogle Scholar
  11. 11.
    Knupp, P., Luczak, R.: Truncation error in grid generation. Num. Meth. P.D.E. 11, 561–571 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Frey, P., George, P.: Mesh Generation: Application to Finite Elements. Wiley, Chichester (2008)zbMATHGoogle Scholar
  13. 13.
    Dvinsky, A.: Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comp. Phys. 95, 450–476 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Thompson, J., Warsi, Z., Mastin, C.: Automatic numerical generation of body-fitted curvilinear coordinate systems. J. Comp. Phys. 24, 274–302 (1977)zbMATHCrossRefGoogle Scholar
  15. 15.
    Liseikin, V.: A computational differential geometry approach to grid generation. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  16. 16.
    Knupp, P.: Formulation of a Target-Matrix Paradigm for Mesh Optimization, SAND2006-2730J, Sandia National Laboratories (2006)Google Scholar
  17. 17.
    Knupp, P.: Local 2D Metrics for Mesh Optimization in the Target-matrix Paradigm, SAND2006-7382J, Sandia National Laboratories (2006)Google Scholar
  18. 18.
    Knupp, P., van der Zee, E.: Convexity of Mesh Optimization Metrics Using a Target-matrix Paradigm, SAND2006-4975J, Sandia National Laboratories (2006)Google Scholar
  19. 19.
    Knupp, P.: Analysis of 2D Rotational-invariant Non-barrier Metrics in the Target-Matrix Paradigm, SAND2008-8219P, Sandia National Laboratories (2008)Google Scholar
  20. 20.
    Knupp, P.: Measuring Quality Within Mesh Elements, SAND2009-3081J, Sandia National Laboratories (2009)Google Scholar
  21. 21.
    Knupp, P.: Label-invariant Mesh Quality Metrics. In: Proceedings of the 18th International Meshing Roundtable, pp. 139–155. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Knupp, P.: Target-matrix Construction Algorithms, SAND2009-7003P, Sandia National Laboratories (2009)Google Scholar
  23. 23.
    Knupp, P., Kraftcheck, J.: Surface Mesh optimization in the Target-Matrix Paradigm (manuscript)Google Scholar
  24. 24.
    Knupp, P.: Tradeoff-coefficient and Binary Metric Construction Algorithms within the Target-Matrix Paradigm (manuscript)Google Scholar
  25. 25.
    Knupp, P.: Algebraic Mesh Quality Measures. SIAM J. Sci. Comput. 23(1), 193–218 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Freitag, L., Knupp, P.: Tetrahedral mesh improvement via optimization of the element condition number. Intl. J. Numer. Meth. Engr. 53, 1377–1391 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Knupp, P., Margolin, L., Shashkov, M.: Reference-Jacobian Optimization-based Rezone Strategies for Arbitrary Lagrangian Eulerian Methods. J. Comp. Phys. 176(1), 93–128 (2002)zbMATHCrossRefGoogle Scholar
  28. 28.
    Knupp, P.: Updating Meshes on Deforming Domains. Communications in Numerical Methods in Engineering 24, 467–476 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
  30. 30.
    Luo, X., Shephard, M., Lee, L., Ge, L., Ng, C.: Moving Curved Mesh Adaptatio for Higher Order Finite Element Simulations. Engr. w/Cmptrs, February 27 (2010)Google Scholar
  31. 31.
    Knupp, P., Voshell, N., Kraftcheck, J.: Quadratic Triangle Mesh Untanglng and Optimization via the Target-matrix Paradigm. In: CSRI Summer Proceedings, SAND2010-3083P, Sandia National Laboratories (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Patrick Knupp
    • 1
  1. 1.Sandia National Laboratories 

Personalised recommendations