Moving Meshes to Fit Large Deformations Based on Centroidal Voronoi Tessellation (CVT)

  • Witalij Wambold
  • Günter BärwolffEmail author
  • Hartmut Schwandt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9155)


The essential criterion for stability and fast convergence of CFD-solvers (CFD - computational fluid dynamics) is a good quality of the mesh. Based on results of [30] in this paper we use the so-called centroidal Voronoi tessellation (CVT) not only for mesh generation and optimization. The CVT is applied to develop a new mesh motion method. The CVT provides an optimal distribution of generating points with respect to a cell density function. For a uniform cell density function the CVT results in high-quality isotropic meshes. The non-uniform cases lead to a trade-off between isotropy and fulfilling cell density function constraints. The idea of the proposed approach is to start with the CVT-mesh and apply for each time step of transient simulation the so-called Lloyd’s method in order to correct the mesh as a response to the boundary motion. This leads to the motion of the whole mesh as a reaction to movement. Furthermore, each step of Lloyd’s method provides a further optimization of the underlying mesh, thus the mesh remains close to the CVT-mesh. Experience has shown that it is usually sufficient to apply a few iterations of the Lloyd’s method per time step in order to achieve high-quality meshes during the whole transient simulation. In comparison to previous methods our method provides high-quality and nearly isotropic meshes even for large deformations of computational domains.


Mesh motion Centroidal Voronoi tessellation Finite Volume method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    de Foy, B., Dawes, W.: Unstructured pressure-correction solver based on a consistent discretization of the Poisson equation. International journal for numerical methods in fluids 34, 463–478 (1999)CrossRefGoogle Scholar
  2. 2.
    Farhat, C., Degand, C., Koobus, B., Lesoinne, M.: Torsional springs for twodimensional dynamic unstructured fluid meshes. Computer Methods in Applied Mechanics and Engineering 163(1–4), 231–245 (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bottassoa, C.L., Detomib, D., Serra, R.: The ball-vertex method: a new simple spring analogy method for unstructured dynamic meshes. Computer Methods in Applied Mechanics and Engineering 194(39–41), 4244–4264 (2005)CrossRefGoogle Scholar
  4. 4.
    Degand, C., Farhat, C.: A three-dimensional torsional spring analogy method for unstructured dynamic meshes. Computers & Structures 80(3–4), 305–316 (2002)CrossRefGoogle Scholar
  5. 5.
    Eymard, R., Herard, J.-M.: Finite Volumes for Complex Applications V. Wiley (2008)Google Scholar
  6. 6.
    Zeng, D., Ethier, C.R.: A semi-torsional spring analogy model for updating unstructured meshes in 3D moving domains. Finite Elements in Analysis and Design 41(11–12), 1118–1139 (2005)CrossRefGoogle Scholar
  7. 7.
    Wang, D., Qiang, D.: Mesh optimization based on the centroidal voronoi tessellation. International Journal of Numerical Analysis and Modeling 2, 100–113 (2005)MathSciNetGoogle Scholar
  8. 8.
    Yan, D.-M., Wang, W., Levy, B., Liu, Y.: Efficient Computation of Clipped Voronoi Diagram for Mesh Generation. Computer-Aided Design 45, 843–852 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lien, F.-S.: A pressure-based unstructured grid method for all-speed flows. International journal for numerical methods in fluids 33, 355–375 (1999)CrossRefGoogle Scholar
  10. 10.
    Markou, G.A., Mouroutis, Z.S., Charmpis, D.C., Papadrakakis, M.: The ortho-semi-torsional (OST) spring analogy method for 3D mesh moving boundary problems. Computer Methods in Applied Mechanics and Engineering 196(4–6), 747–765 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Jasak, H., Tukovic, Z.: Automatic mesh motion for the unstructured finite volume method (November 2006)Google Scholar
  12. 12.
    Donea, J., Huerta, A., Ponthot, J.Ph., Rodriguez-Ferran, A.: In: Encyclopedia of Computational Mechanics, Chapter 14, Arbitrary Lagrangian-Eulerian Methods (2004)Google Scholar
  13. 13.
    Chen, L.: Mesh smoothing schemes based on optimal delaunay triangulations. Math Department, The Pennsylvania State University, State CollegeGoogle Scholar
  14. 14.
    Ebeida, M.S., Mitchell, S.A.: Uniform random Voronoi meshes. In: Proceedings of the 20th International Meshing Roundtable, Paris, France, pp. 273–290. Sandia National Laboratories, Albuquerque (2011)Google Scholar
  15. 15.
    OpenFOAM C++ Documentation.
  16. 16.
    Alliez, P., Cohen-Steiner, D., Yvinec, M., Desbrun, M.: Variational Tetrahedral Meshing. ACM Transactions on Graphics. Proceedings of ACM SIGGRAPH 2005 24, 617–625 (2005)Google Scholar
  17. 17.
    Qiang, D., Wang, D.: Anisotropic centroidal voronoi tessellations and their applications. SIAM Journal on Scientific Computing 26(3), 737–761 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Qiang, D., Wang, D.: Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellation. International journal for numerical methods in engineering 56, 1355–1373 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Qiang, D., Emelianenko, M.: Acceleration schemes for computing centroidal Voronoi tessellations. Numerical linear algebra with applications 0, 1–19 (2005)Google Scholar
  20. 20.
    Qiang, D., Emelianenko, M., Lili, J.: Convergence of the Lloyd algorithm for computing centroidal voronoi tessellations. SIAM Journal Numerical Analysis 44(1), 102–119 (2006)zbMATHCrossRefGoogle Scholar
  21. 21.
    Qiang, D., Gunzburger, M.D., Lili, J.: Constrained Centroidal Voronoi Tessellations For Surfaces. SIAM Journal on Scientific Computing 24(5), 1488–1506 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Qiang, D., Faber, V., Gunzburger, M.: Centroidal Voronoi Tessellations: Applications and Algorithms. SIAM REVIEW 41(4), 637–676 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Löhner, R., Yang, C.: Improved ALE mesh velocities for moving bodies. Communications in Numerical Methods in Engineering 12, 599–608 (1996)zbMATHCrossRefGoogle Scholar
  24. 24.
    Rycroft, C.H.: Voro++: a three-dimensional Voronoi cell library in C++ (2009)Google Scholar
  25. 25.
    Jakobsson, S., Amoignon, O.: Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization. Computers & Fluids 36, 1119–1136 (2007)zbMATHCrossRefGoogle Scholar
  26. 26.
    Menon, S., Schmidt, D.P.: Conservative interpolation on unstructured polyhedral meshes: An extension of the supermesh approach to cell-centered finite-volume variables. Computer Methods in Applied Mechanics and Engineering 200, 2797–2804 (2011)zbMATHCrossRefGoogle Scholar
  27. 27.
    Arabi, S., Camarero, R., Guibault, F.: Unstructured meshes for large body motion using mapping operators. Mathematics and computers in simulation 106, 26–43 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhang, X., Zhou, D., Bao, Y.: Mesh motion approach based on spring analogy method for unstructured meshes. Journal of Shanghai Jiaotong University 15, 138–146 (2010)CrossRefGoogle Scholar
  29. 29.
    Zhou, X., Li, S.: A new mesh deformation method based on disk relaxation algorithm with pre-displacement and post-smoothing. Journal of Computational Physics 235, 199–215 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wambold, W., Bärwolff, G.: New mesh motion solver for large deformations based on CVT. Procedia Engineering 82, 390–402 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Witalij Wambold
    • 1
    • 2
  • Günter Bärwolff
    • 3
    Email author
  • Hartmut Schwandt
    • 3
  1. 1.Component DevelopmentVolkswagen AGSalzgitterGermany
  2. 2.Institute of MathematicsTU BerlinBerlinGermany
  3. 3.Institute of MathematicsTechnische Universität BerlinBerlinGermany

Personalised recommendations