Modeling of quality parameter values for improving meshes

  • O. EgorovaEmail author
  • M. Savchenko
  • I. Hagiwara
  • V. Savchenko


A novel quasi-statistical approach to improve the quality of triangular meshes is presented. The present method is based on modeling of an event of the mesh improvement. This event is modeled via modeling of a discrete random variable. The random variable is modeled in a tangent plane of each local domain of the mesh. One domain collects several elements with a common point. Values of random variable are calculated by modeling formula according to the initial sampling data of the projected elements with respect to all neighbors of the domain. Geometrical equivalent called potential form is constructed for each element of the domain with a mesh quality parameter value equal to the modeled numerical value. Such potential forms create potential centers of the domain. Averaging the coordinates of potential centers of the domain gives a new central point position. After geometrical realization over the entire mesh, the shapes of triangular elements are changed according to the normal distribution. It is shown experimentally that the mean of the final mesh is better than the initial one in most cases, so the event of the mesh improvement is likely occurred. Moreover, projection onto a local tangent plane included in the algorithm allows preservation of the model volume enclosed by the surface mesh. The implementation results are presented to demonstrate the functionality of the method. Our approach can provide a flexible tool for the development of mesh improvement algorithms, creating better-input parameters for the triangular meshes and other kinds of meshes intended to be applied in finite element analysis or computer graphics.

Key words

random variable modeling randomness triangulation mesh quality parameter mesh improvement 


  1. [1]
    N. Amenta, M. Bern and D. Eppstein, Optimal point placement for mesh smoothing. J. Algorithms30 (1999), 302–322.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    B. Balendran, A direct smoothing method for surface meshing. Proceedings of the 8th International Meshing Roundtable. South Lake Tahoe, CA, USA, 1999, 189–193.Google Scholar
  3. [3]
    O. Egorova, M. Savchenko, N. Kojekine, I. Semenova, I. Hagiwaraand V. Savchenko, Improvement of mesh quality using a statistical approach. Proceedings of the 3th IASTED International conference on Vizualization. Imaging and Image Processing (VHP), Spain, 2003, 1016–1021.Google Scholar
  4. [4]
    P.J. Frey, About Surface Remeshing. Proceedings of the 9th International Meshing Roundtable. Sandia National Laboratories, 2000, 123–136.Google Scholar
  5. [5]
    M. Holder and J. Richardson, Genetic algorithms, another tool for quad mesh optimization? Proceedings of the 7th International Meshing Roundtable. Sandia National Lab., 1998, 497–504.Google Scholar
  6. [6]
    F.J. Bossen and P.S. Heckbert, A Pliant method for anisotropic mesh generation. Proceedings of the 5th International Meshing Roundtable. Pittsburgh, PA, 1996, 63–74.Google Scholar
  7. [7]
    D.A. Field, Laplacian smoothing and delaunay triangulations. J. Communications in Applied Numerical Methods,4 (1998), 709–712.CrossRefGoogle Scholar
  8. [8]
    T. Zhou and K. Shimada, An angle-based approach to two-dimensional mesh smoothing. Proceedings of the 9th International Meshing Roundtable. 2000, 373–384.Google Scholar
  9. [9]
    L.A. Freitag, On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques. AMD220, Trends in Unstructured Mesh Generation, 1997, 37–43.Google Scholar
  10. [10]
    L.A. Freitag and C. Olliver-Gooch, A comparison of tetrahedral mesh improvement techniques. Proceedings of the 5th International Roundtable. Sandia National Lab., Albuquerque NM, 1996, 87–106.Google Scholar
  11. [11]
    V. Parthasarathy and S. Kodiyalam, A constrained optimization approach to finite element mesh smoothing. J. Finite Elements in Analysis and Design,9 (1991), 309–320.zbMATHCrossRefGoogle Scholar
  12. [12]
    S.A. Canann, J.R. Tristano and M.L. Staten, An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes. Proceedings of the 7th International Meshing Roundtable, 1998, 479–494.Google Scholar
  13. [13]
    O.P. Jacquotte and G. Coussement, Structured mesh adaptation: space accuracy and interpolation methods. Computer Methods in Applied Mechanics and Engineering,101 (1992), 397–432.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    S.A. Canann, M.B. Stephenson and T.D. Blacker, Optismoothing: an optimization-driven approach to mesh smoothing. J. Finite Elements in Analysis and Design,13 (1993), 185–190.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    R. Lohner, K. Morgan and O.C. Zienkiewicz, Adaptive Grid Refinement for the Euler and Compressible Navier Stokes Equation. Proceedings of the International Conference on Accuracy Estimates and Adaptive Refinement in Finite Element Computations. Lisbon, 1984.Google Scholar
  16. [16]
    I. Babuska, O.C. Zienkiewicz, J. Gago and E.R. de A. Oliviera (eds.), Accuracy Estimates and Adaptive Refinements in Finite Element Computations. John Wiley & Sons, Chichester, 1986, 281–297.zbMATHGoogle Scholar
  17. [17]
    K. Shimada and D.C. Gossard, Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing. Proceedings of the ACM Third Symposium on Solid Modeling and Applications. 1995, 409–419.Google Scholar
  18. [18]
    K. Shimada, Anisotropic triangular meshing of parametric surfaces via close packing of ellipsoidal bubbles. Proceedings of the 6th International Meshing Roundtable. Sandia National Laboratories, 1997, 375–390.Google Scholar
  19. [19]
    H.N. Djidjev, Force-directed methods for smoothing unstructured triangular and tetrahedral meshes. Proceedings of the 9th International Meshing Roundtable. Sandia National Laboratories, 2000, 395–406.Google Scholar
  20. [20]
    G. Taubin, A signal processing approach to fair surface design. Proceedings of SIGGRAPH’ 95,29 (1995), 351–358.CrossRefGoogle Scholar
  21. [21]
    M. Desbrun, M. Meyer, P. Schrder and A.H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow. Proceedings of SIGGRAPH’ 99,33 (1999), 317–32.CrossRefGoogle Scholar
  22. [22]
    J. Warren and H. Weimer, Subdivision Methods for Geometric Design. Academic Press, 2002.Google Scholar
  23. [23]
    L. Kobbelt, S. Campagna, J. Vorsatz and H-P. Seidel, Interactive multi-resolution modeling on arbitrary meshes. Proceedings of SIGGRAPH’ 98,32 (1998), 105–114.CrossRefGoogle Scholar
  24. [24]
    J.R. Shewchuk, What is a good linear element? interpolation, conditioning, and quality measures. Proceedings of the 11th International Meshing Roundtable. Sandia National Laboratories, 2002, 115–126.Google Scholar
  25. [25]
    K.E. Jansen, M.S. Shepard, and M.W. Beall, On anisotropic mesh generation and quality control in complex flow problems. Proceedings of the 10th International Meshing Roundtable. Sandia National Laboratories, 2001, 341–349.Google Scholar
  26. [26]
    I.M. Sobol, Monte Carlo Numerical Methods. Nauka, Moscow, 1973, in Russian.Google Scholar
  27. [27]
    H. Hoppe, T. DeRose, T. Duchamp, J. McDonald and W. Stuetzle, Surface reconstruction from unorganized points. Proceedings of SIGGRAPH’ 92, 1992, 71–78.Google Scholar

Copyright information

© JJIAM Publishing Committee 2007

Authors and Affiliations

  • O. Egorova
    • 1
    Email author
  • M. Savchenko
    • 2
  • I. Hagiwara
    • 3
  • V. Savchenko
    • 4
  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.Inter Locus. Inc.TokyoJapan
  3. 3.Tokyo Institute of TechnologyTokyoJapan
  4. 4.Hosei UniversityTokyoJapan

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