Advertisement

Discrete Mesh Optimization on GPU

  • Daniel ZintEmail author
  • Roberto Grosso
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 127)

Abstract

We present an algorithm called discrete mesh optimization (DMO), a greedy approach to topology-consistent mesh quality improvement. The method requires a quality metric for all element types that appear in a given mesh. It is easily adaptable to any mesh and metric as it does not rely on differentiable functions. We give examples for triangle, quadrilateral, and tetrahedral meshes and for various metrics. The method improves quality iteratively by finding the optimal position for each vertex on a discretized domain. We show that DMO outperforms other state of the art methods in terms of convergence and runtime.

References

  1. 1.
    V. Aizinger, C. Dawson, A discontinuous galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Res. 25(1), 67–84 (2002)CrossRefGoogle Scholar
  2. 2.
    N. Amenta, M. Bern, D. Eppstein, Optimal point placement for mesh smoothing. J. Algorithms 30(2), 302–322 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    I. Babuška, A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13(2), 214–226 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    T.J. Baker, Mesh movement and metamorphosis. Eng. Comput. 18(3), 188–198 (2002)CrossRefGoogle Scholar
  5. 5.
    R.E. Bank, R.K. Smith, Mesh smoothing using a posteriori error estimates. SIAM J. Numer. Anal. 34(3), 979–997 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    R.E. Bank, A Software Package for Solving Elliptic Partial Differential Equations–Users Guide 7.0. Frontiers in Applied Mathematics, vol. 15 (SIAM, Philadelphia, 1998)Google Scholar
  7. 7.
    T.D. Blacker, M.B. Stephenson, Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32(4), 811–847 (1991)zbMATHCrossRefGoogle Scholar
  8. 8.
    T.D. Blacker, M.B. Stephenson, S. Canann, Analysis automation with paving: a new quadrilateral meshing technique. Adv. Eng. Softw. Work. 13(5–6), 332–337 (1991)zbMATHCrossRefGoogle Scholar
  9. 9.
    F.J. Blom, Considerations on the spring analogy. Int. J. Numer. Methods Fluids 32(6), 647–668 (2000)zbMATHCrossRefGoogle Scholar
  10. 10.
    M.L. Brewer, L.F. Diachin, P.M. Knupp, T. Leurent, D.J. Melander, The mesquite mesh quality improvement toolkit, in IMR (2003)Google Scholar
  11. 11.
    S.A. Canann, Y.-C. Liu, A.V. Mobley, Automatic 3d surface meshing to address today’s industrial needs. Finite Elem. Anal. Des. 25(1–2), 185–198 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S.A. Canann, J.R. Tristano, M.L. Staten et al., An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes, in IMR (1998), pp. 479–494. CiteseerGoogle Scholar
  13. 13.
    V.F. De Almeida, Domain deformation mapping: application to variational mesh generation. SIAM J. Sci. Comput. 20(4), 1252–1275 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ch. Farhat, C. Degand, B. Koobus, M. Lesoinne, Torsional springs for two-dimensional dynamic unstructured fluid meshes. Comput. Methods Appl. Mech. Eng. 163(1–4), 231–245 (1998)zbMATHCrossRefGoogle Scholar
  15. 15.
    D.A. Field, Laplacian smoothing and delaunay triangulations. Int. J. Numer. Methods Biomed. Eng. 4(6), 709–712 (1988)zbMATHGoogle Scholar
  16. 16.
    M.S. Floater, Parametrization and smooth approximation of surface triangulations. Comput. Aided Geom. Des. 14(3), 231–250 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    L.A. Freitag, On combining laplacian and optimization-based mesh smoothing techniques. ASME Applied mechanics division-publications-amd, vol. 220 (1997), pp. 37–44Google Scholar
  18. 18.
    L.A. Freitag, P.M. Knupp, Tetrahedral mesh improvement via optimization of the element condition number. Int. J. Numer. Methods Eng. 53(6), 1377–1391 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    L. Freitag, P. Plassmann, M Jones, An efficient parallel algorithm for mesh smoothing. Technical report, Argonne National Laboratory, IL (1995)Google Scholar
  20. 20.
    L. Freitag, M. Jones, P. Plassmann, A parallel algorithm for mesh smoothing. SIAM J. Sci. Comput. 20(6), 2023–2040 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    L.A. Freitag, P. Plassmann et al., Local optimization-based simplicial mesh untangling and improvement. Int. J. Numer. Methods Eng. 49(1), 109–125 (2000)zbMATHCrossRefGoogle Scholar
  22. 22.
    P.-L. George, H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements (Hermés Science, Paris, 1998)zbMATHGoogle Scholar
  23. 23.
    C. Georgiadis, P.-A. Beaufort, J. Lambrechts, J.-F. Remacle, High quality mesh generation using cross and asterisk fields: application on coastal domains. arXiv preprint arXiv:1706.02236 (2017)Google Scholar
  24. 24.
    L.R. Herrmann, Laplacian-isoparametric grid generation scheme. J. Eng. Mech. Div. 102(5), 749–907 (1976)Google Scholar
  25. 25.
    T.R. Jensen, B. Toft, Graph Coloring Problems, vol. 39 (Wiley, New York, 2011)zbMATHGoogle Scholar
  26. 26.
    R.E. Jones, Qmesh: a self-organizing mesh generation program. Technical report, Sandia Laboratories, Albuquerque, NM (1974)Google Scholar
  27. 27.
    J. Kim, A multiobjective mesh optimization algorithm for improving the solution accuracy of pde computations. Int. J. Comput. Methods 13(01), 1650002 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    P.M. Knupp, Winslow smoothing on two-dimensional unstructured meshes. Eng. Comput. 15(3), 263–268 (1999)zbMATHCrossRefGoogle Scholar
  29. 29.
    P.M. Knupp, 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(8), 1165–1185 (2000)zbMATHGoogle Scholar
  30. 30.
    P.M. Knupp, Algebraic mesh quality metrics. SIAM J. Sci. Comput. 23(1), 193–218 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    P. Knupp, Updating meshes on deforming domains: an application of the target-matrix paradigm. Int. J. Numer. Methods Biomed. Eng. 24(6), 467–476 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    M. Křížek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29(2), 513–520 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    J. Park, S.M. Shontz, Two derivative-free optimization algorithms for mesh quality improvement. Procedia Comput. Sci. 1(1), 387–396 (2010)CrossRefGoogle Scholar
  34. 34.
    P.-O. Persson, Mesh size functions for implicit geometries and pde-based gradient limiting. Eng. Comput. 22(2), 95–109 (2006)CrossRefGoogle Scholar
  35. 35.
    P.-O. Persson, G. Strang, A simple mesh generator in matlab. SIAM Rev. 46(2), 329–345 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    R. Rangarajan, On the resolution of certain discrete univariate max–min problems. Comput. Optim. Appl. 68(1), 163–192 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    R. Rangarajan, A.J. Lew, Provably robust directional vertex relaxation for geometric mesh optimization. SIAM J. Sci. Comput. 39(6), A2438–A2471 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    M. Rumpf, A variational approach to optimal meshes. Numer. Math. 72(4), 523–540 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    J. Shewchuk, What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures. Preprint. University of California at Berkeley, 73:137 (2002)Google Scholar
  40. 40.
    Shewchuk. Stellar: A tetrahedral mesh improvement program, 05-23-2018. Available from: https://people.eecs.berkeley.edu/~jrs/stellar/input_meshes.zip
  41. 41.
    S.M. Shontz, S.A. Vavasis, A mesh warping algorithm based on weighted laplacian smoothing, in IMR (2003), pp. 147–158Google Scholar
  42. 42.
    H. Xu, T.S. Newman, 2D FE quad mesh smoothing via angle-based optimization, in International Conference on Computational Science (Springer, Berlin, 2005), pp. 9–16zbMATHGoogle Scholar
  43. 43.
    K. Xu, X. Gao, G. Chen, Hexahedral mesh quality improvement via edge-angle optimization. Comput. Graph. 70, 17–27 (2018)CrossRefGoogle Scholar
  44. 44.
    P.D. Zavattieri, E.A. Dari, G.C. Buscaglia, Optimization strategies in unstructured mesh generation. Int. J. Numer. Methods Eng. 39(12), 2055–2071 (1996)zbMATHCrossRefGoogle Scholar
  45. 45.
    T. Zhou, K. Shimada, An angle-based approach to two-dimensional mesh smoothing, in IMR (2000), pp. 373–384Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Computer Graphics ChairUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations