Mesh Smoothing Algorithms for Complex Geometric Domains

  • Hale Erten
  • Alper Üngör
  • Chunchun Zhao

Abstract

Whenever a new mesh smoothing algorithm is introduced in the literature, initial experimental analysis is often performed on relatively simple geometric domains where the meshes need little or no element size grading. Here, we present a comparative study of a large number of well-known smoothing algorithms on triangulations of complex geometric domains. Our study reveals the limitations of some well-known smoothing methods. Specifically, the optimal Delaunay triangulation smoothing and weighted centroid of circumcenter smoothing methods are shown to have difficulty achieving smooth grading and adapting to complex domain boundary. We propose modifications and report significant improvements and behavior change in the performance of these algorithms. More importantly, we propose three new smoothing strategies and show their effectiveness in computing premium quality triangulations for complex geometric domains. While the proposed algorithms give the practitioners additional tools to chose from, our comparative study of over a dozen algorithms should guide them selecting the best smoothing method for their particular application.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alliez, P., Cohen-Steiner, D., Yvinec, M., Desbrun, M.: Variational tetrahedral meshing. ACM Transactions on Graphics 24, 617–625 (2005)CrossRefGoogle Scholar
  2. 2.
    Amenta, N., Bern, M.W., Eppstein, D.: Optimal point placement for mesh smoothing. In: ACM-SIAM Symp. Discrete Algorithms, pp. 528–537 (1997)Google Scholar
  3. 3.
    Babuška, I., Aziz, A.: On the angle condition in the finite element method. SIAM J. Numer. Analysis 13, 214–227 (1976)MATHCrossRefGoogle Scholar
  4. 4.
    Bank, R.E., Smith, R.K.: Mesh smoothing using A posteriori error estimates. SIAM Journal on Numerical Analysis 34(3), 979–997 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bern, M., Eppstein, D.: Mesh generation and optimal triangulation. Computing in Euclidean Geometry, 23–90 (1992)Google Scholar
  6. 6.
    Bern, M., Eppstein, D., Gilbert, J.R.: Provably good mesh generation. J. Comp. System Sciences 48, 384–409 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bern, M., Plassmann, P.: Mesh generation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 291–332. Elsevier, Amsterdam (1998)Google Scholar
  8. 8.
    Canann, S.A., Tristano, J.R., Blacker, T.: Opti-smoothing: An optimization-driven approach to mesh smoothing. Finite Elements in Analysis and Design 13, 185–190 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Canann, S.A., Tristano, J.R., Staten, M.L.: An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes. In: 7th Int. Meshing Roundtable, pp. 479–494 (1998)Google Scholar
  10. 10.
    Chen, L.: Mesh smoothing schemes based on optimal Delaunay triangulations. In: 13th Int. Meshing Roundtable, pp. 109–120 (2004)Google Scholar
  11. 11.
    Chen, L., Xu, J.: Optimal Delaunay triangulation, Journal of Computational Mathematics 22, 299–308 (2004)Google Scholar
  12. 12.
    Du, Q., Emelianenko, M., Ju, L.: Convergence of the Lloyd algorithm for computing centroidal Voronoi tessellations. SIAM. J. Numer. Anal. 44(1), 102–119 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41, 637–676 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Du, Q., Gunzburger, M.: Grid generation and optimization based on centroidal Voronoi tessellations. Appl. Math. Comp. 133, 591–607 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Du, Q., Wang, D.: Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations. Inter. J. Numer. Meth. Eng. 56(9), 1355–1373 (2002)MathSciNetGoogle Scholar
  16. 16.
    Erten, H., Üngör, A.: Computing no small no large angle triangulations, To be appear in Proc. Int. Symp. Voronoi Diagrams (2009)Google Scholar
  17. 17.
    Erten, H., Üngör, A.: Quality triangulations with locally optimal steiner points. SIAM Journal of Scientfic Computing 31(3), 2103–2130 (2009)CrossRefGoogle Scholar
  18. 18.
    Field, D.A.: Laplacian smoothing and Delaunay triangulations. Comm. in Applied Numer. Analysis 4, 709–712 (1988)MATHCrossRefGoogle Scholar
  19. 19.
    Freitag, L.A.: On combining laplacian and optimization-based mesh smoothing techniques. In: Trends in Unstructured Mesh Generation, pp. 37–43 (1997)Google Scholar
  20. 20.
    Lloyd, S.: Least square quatization in PCM. IEEE Trans. Inform. Theory 28, 129–137 (1982)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mukherjee, N.: A hybrid, variational 3D smoother for orphaned shell meshes. In: 11th Int. Meshing Roundtable, pp. 379–390 (2002)Google Scholar
  22. 22.
    Owen, S.: A survey of unstructured mesh generation technology. In: 7th Int. Meshing Roundtable, pp. 239–267 (1998)Google Scholar
  23. 23.
    Parthasarathy, V.N., Kodiyalam, S.: A constrained optimization approach to finite element mesh smoothing. Finite Elements in Analysis and Design 9, 309–320 (1991)MATHCrossRefGoogle Scholar
  24. 24.
    Shewchuk, J.R.: What is a good linear element? interpolation, conditioning, and quality measures. In: 11th Int. Meshing Roundtable, pp. 115–126 (2002)Google Scholar
  25. 25.
    Üngör, A.: Off-centers: A new type of steiner points for computing size-optimal quality-guaranteed Delaunay triangulations. Comput. Geom. Theory Appl. 42(2), 109–118 (2009)MATHGoogle Scholar
  26. 26.
    VanderZee, E., Hirani, A.N., Guoy, D., Ramos, E.: Well-centered planar triangulation–an iterative approach. In: 16th Int. Meshing Roundtable, pp. 121–138 (2007)Google Scholar
  27. 27.
    Wang, D., Du, Q.: Mesh optimization based on the centroidal Voronoi tessellation. Int. J. Numer. Anal. Mod. 2, 100–113 (2005)MathSciNetGoogle Scholar
  28. 28.
    Xu, H., Newman, T.S.: An angle-based optimization approach for 2d finite element mesh smoothing. Finite Elem. Anal. Des. 42(13), 1150–1164 (2006)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Zhou, T., Shimada, K.: An angle-based approach to two-dimensional mesh smoothing. In: 9th Int. Meshing Roundtable, pp. 373–384 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hale Erten
    • 1
  • Alper Üngör
    • 1
  • Chunchun Zhao
    • 1
  1. 1.Dept. of Computer & Info. Sci. & Eng.University of Florida 

Personalised recommendations