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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)

Abstract

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.

Keywords

Mesh motion Centroidal Voronoi tessellation Finite Volume method 

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References

  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. http://foam.sourceforge.net/docs/cpp/
  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

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