Acta Mechanica Solida Sinica

, Volume 29, Issue 1, pp 1–12 | Cite as

Mesh Deformation Method Based on Mean Value Coordinates Interpolation

  • Shuli Sun
  • Shuming Lv
  • Yuan Yuan
  • Mingwu Yuan


Mesh deformation technique is widely used in many application fields, and has received a lot of attentions in recent years. This paper focuses on the methodology and algorithm of algebraic type mesh deformation for unstructured mesh in numerical discretization. To preserve mesh quality effectively, an algebraic approach for two and three dimensional unstructured mesh is developed based on mean value coordinates interpolation combined with node visibility analysis. The proposed approach firstly performs node visibility analysis to find out the visible boundary for each grid point to be moved, then evaluates the mean value coordinates of each grid point with respect to all vertices on its visible boundary. Thus the displacements of grid points can be calculated by interpolating the boundary movement by the mean value coordinates. Compared with other methods, the proposed method has good deformation capability and predictable computational cost, with no need to select parameters or functions. Applications of mesh deformation in different fields are presented to demonstrate the effectiveness of the proposed approach. The results of numerical experiments exhibit not only superior deformation capability of the method in traditional applications of fluid dynamic grid, but also great potential in modeling for large deformation analysis and inverse design problems.

Key Words

unstructured mesh mesh deformation mean value coordinates (MVC) visibility analysis domain partition 


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  1. 1.
    Farhat, C., Degand, C., Koobus, B. and Lesoinne, M., Torsional springs for two-dimensional dynamic unstructured fluid meshes. Computer Methods in Applied Mechanics and Engineering, 1998, 163: 231–245.CrossRefGoogle Scholar
  2. 2.
    Blom, F.J., Considerations on the spring analogy. International Journal for Numerical Methods in Fluids, 2000, 32: 647–668.CrossRefGoogle Scholar
  3. 3.
    Degand, C. and Farhat, C., A three-dimensional torsional spring analogy method for unstructured dynamic meshes. Computers & Structures, 2002, 80: 305–316.CrossRefGoogle Scholar
  4. 4.
    Zeng, D. and Ethier, C.R., A semi-torsional spring analogy model for updating unstructured meshes in 3D moving domains. Finite Elements in Analysis and Design, 2005, 41: 1118–1139.CrossRefGoogle Scholar
  5. 5.
    Stein, K., Tezduyar, T.E. and Benney, R., Automatic mesh update with the solid-extension mesh moving technique. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 2019–2032.CrossRefGoogle Scholar
  6. 6.
    Helenbrook, B.T., Mesh deformation using the biharmonic operator. International Journal for Numerical Methods in Engineering, 2003, 56(7): 1007–1021.CrossRefGoogle Scholar
  7. 7.
    Burg, C., Analytic study of 2D and 3D grid motion using modified Laplacian. International Journal for Numerical Methods in Fluids, 2006, 52(2): 163–197.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, X., Qin, N. and Xia, H., Fast dynamic grid deformation based on Delaunay graph mapping. Journal of Computational Physics, 2006, 211: 405–423.CrossRefGoogle Scholar
  9. 9.
    Sun, S.L., Chen, B., Liu, J.F. and Yuan, M.W., An efficient implementation scheme for the moving grid method based on Delaunay graph mapping. In: Proceedings of the 2nd International Symposium on Computational Mechanics and the 12th International Conference on the Enhancement and Promotion of Computational Methods in Engineering and Science, Hong Kong-Macau, 2009.Google Scholar
  10. 10.
    de Boer, A., van der SchootM, S. and Bijl, H., Mesh deformation based on radial basis function interpolation. Computers & Structures, 2007, 85: 784–795.CrossRefGoogle Scholar
  11. 11.
    Witteveen J.A.S. and Bijl, H., Explicit mesh deformation using inverse distance weighting interpolation. In: The 19th AIAA Computational Fluid Dynamics Conference, San Antonio, Texas, AIAA-2009-3996, 2009.Google Scholar
  12. 12.
    Witteveen, J.A.S., Explicit and robust inverse distance weighting mesh deformation for CFD. In: 48th AIAA Aerospace Sciences Meeting, Florida, Orlando, AIAA-2010-165, 2010.Google Scholar
  13. 13.
    Allen, C.B., Parallel universal approach to mesh motion and application to rotors in forward flight. International Journal for Numerical Methods in Engineering, 2007, 69: 2126–2149.CrossRefGoogle Scholar
  14. 14.
    Luke, E., Colins, E. and Blades, E., A fast mesh deformation method using explicit interpolation. Journal of Computational Physics, 2012, 231: 586–601.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sun, S.L. and Chen, B., An algebraic deformation approach for moving grid based on barycentric coordinates. In: The 2010 International Conference on Computational Intelligence and Software Engineering, Wuhan, China, 2010.Google Scholar
  16. 16.
    Sun, S.L. and Yuan, M.W., High quality mesh deformation method for large scale unstructured hybrid grid. International association for computational mechanics Expressions, 2011, 29: 9–12.Google Scholar
  17. 17.
    Floater, M.S., Mean value coordinates. Computer Aided Geometric Design, 2003, 20: 19–27.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hormann, K. and Floater, M.S., Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics, 2006, 25: 1424–1441.CrossRefGoogle Scholar
  19. 19.
    Kidner, D.B., Sparkes, A.J., Dorey, M.I., Ware, J.M. and Jones, C.B., Visibility analysis within the multiscale implicit TIN. Transactions in GIS, 2001, 5(1): 19–37.CrossRefGoogle Scholar
  20. 20.
    Kaucic, B. and Zalik, B., Comparison of viewshed algorithms on regular spaced points. International Conference on Computer Graphics and Interactive Techniques, 2002: 177–183.Google Scholar
  21. 21.
    Ju, T., Schaefer, S. and Warren, J., Mean value coordinates for closed triangular meshes. ACM Transactions on Graphics, 2005, 24(3): 561–566.CrossRefGoogle Scholar
  22. 22.
    Floater, M.S., Kós, G. and Reimers, M., Mean value coordinates in 3D. Computer Aided Geometric Design, 2005, 22: 623–631.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Knupp, P.M., Algebraic mesh quality metrics for unstructured initial meshes. Finite Elements in Analysis and Design, 2003, 39: 217–241.CrossRefGoogle Scholar
  24. 24.
    Zhang, L.P., Chang, X.H., Duan, X.P., Zhao, Z. and He, X., Applications of dynamic hybrid grid method for three-dimensional moving/deforming boundary problems. Computers & Fluids, 2012, 62: 45–63.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Dheeravongkit, A. and Shimada, K., Inverse pre-deformation of finite element mesh for large deformation analysis. Journal of Computing and Information Science in Engineering, 2004, 5(4): 338–347.CrossRefGoogle Scholar
  26. 26.
    Sun, S.L., Tang, B. and Zhang, M.L., Back calculation for cold-state geometric shape of turbine blade based on modification of nodal coordinates and mesh smoothing. Journal of Mechanical Engineering, 2014, 50(10): 143–148.CrossRefGoogle Scholar
  27. 27.
    Sun, S.L., Sui, J., Chen, B. and Yuan, M.W., An efficient mesh generation method for fractured network system based on dynamic grid deformation. Mathematical Problems in Engineering, Special issue of Advances in Finite Element Method, 2013.Google Scholar
  28. 28.
    Rendall, T. and Allen, C., Parallel efficient mesh motion using radial basis functions with application to multi-bladed rotors. International Journal for Numerical Methods in Engineering, 2010, 81(1): 89–105.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2016

Authors and Affiliations

  1. 1.LTCS, Department of Mechanics and Engineering Science, College of EngineeringPeking UniversityBeijingChina

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