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Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics

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Abstract

Recently, the radial point interpolation meshfree method has gained popularity owing to its advantages in large deformation and discontinuity problems, however, the accuracy of this method depends on many factors and their influences are not fully investigated yet. In this work, three main factors, i.e., the shape parameters, the influence domain size, and the nodal distribution, on the accuracy of the radial point interpolation method (RPIM) are systematically studied and conclusive results are obtained. First, the effect of shape parameters (R, q) of the multi-quadric basis function on the accuracy of RPIM is examined via global search. A new interpolation error index, closely related to the accuracy of RPIM, is proposed. The distribution of various error indexes on the R-q plane shows that shape parameters q ∈ [1.2, 1.8] and R ∈ [0, 1.5] can give good results for general 3-D analysis. This recommended range of shape parameters is examined by multiple benchmark examples in 3D solid mechanics. Second, through numerical experiments, an average of 30–40 nodes in the influence domain of a Gauss point is recommended for 3-D solid mechanics. Third, it is observed that the distribution of nodes has significant effect on the accuracy of RPIM although it has little effect on the accuracy of interpolation. Nodal distributions with better uniformity give better results. Furthermore, how the influence domain size and nodal distribution affect the selection of shape parameters and how the nodal distribution affects the choice of influence domain size are also discussed.

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References

  1. BELYTSCHKO T, LU Y Y, GU L. Element-free Galerkin methods [J]. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256.

    Article  MathSciNet  MATH  Google Scholar 

  2. LIU W K, JUN S, ZHANG Y F. Reproducing kernel particle methods [J]. International Journal for Numerical Methods in Fluids, 1995, 20(8/9): 1081–1106.

    Article  MathSciNet  MATH  Google Scholar 

  3. ATLURI S N, ZHU T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics [J]. Computational Mechanics, 1998, 22(2): 117–127.

    Article  MathSciNet  MATH  Google Scholar 

  4. LIU G R, GU Y T. A point interpolation method for two-dimensional solids [J]. International Journal for Numerical Methods in Engineering, 2001, 50(4): 937–951.

    Article  MATH  Google Scholar 

  5. CHEN W. New RBF collocation methods and kernel RBF with applications [M]// GRIBEL M, SCHWEITZER M A, Eds. Meshfree Methods for Partial Differential Equations. Vol. 1, Springer Verlag, 2000.

    Google Scholar 

  6. CHEN W, FU Z J. Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations [J]. Journal of Marine Science and Technology, 2009, 17(3): 157–163.

    Google Scholar 

  7. XIE H, NOGAMI T, WANG J G. A radial boundary node method for two-dimensional elastic analysis [J]. Engineering Analysis with Boundary Elements, 2003, 27: 853–862.

    Article  MATH  Google Scholar 

  8. WANG J G, LIU G R. A point interpolation meshless method based on radial basis functions [J]. International Journal for Numerical Methods in Engineering, 2002, 54(11): 1623–1648.

    Article  MATH  Google Scholar 

  9. LIU G R, GU Y T. A matrix triangularization algorithm for the polynomial point interpolation method [J]. Computer Methods in Applied Mechanics and Engineering, 2003, 192(19): 2269–2295.

    Article  MATH  Google Scholar 

  10. LIU G R, ZHANG G Y, GU Y T, WANG Y Y. A meshfree radial point interpolation method (RPIM) for three-dimensional solids [J]. Computational Mechanics, 2005, 36(6): 421–430.

    Article  MathSciNet  MATH  Google Scholar 

  11. WANG J G, LIU G R, WU Y G. A point interpolation method for simulating dissipation process of consolidation [J]. Computer Methods in Applied Mechanics and Engineering, 2001, 190(45): 907–5922.

    Article  Google Scholar 

  12. WANG J G, LIU G R, LIN P. Numerical analysis of Biot’s consolidation process by radial point interpolation method [J]. International Journal of Solids and Structures, 2002, 39(6): 1557–1573.

    Article  MATH  Google Scholar 

  13. WANG J G, ZHANG B, NOGAMI T. Wave-induced seabed response analysis by radial point interpolation meshless method [J]. Ocean Engineering, 2004, 31(1): 21–42.

    Article  Google Scholar 

  14. WANG J G, LIU G R. On the optimal shape parameters of radial basis functions used for 2-D meshless methods [J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(23/24): 2611–2630.

    Article  MathSciNet  MATH  Google Scholar 

  15. LIU G R, GU Y T. An introduction to meshfree methods and their programming [M]. Dordrecht: Springer, 2005.

    Google Scholar 

  16. BUHMANN M D. Radial basis functions: Theory and implementations [M]. Cambridge: Cambridge University Press, 2003: 57–73.

    Book  Google Scholar 

  17. WENDLAND H. Scattered data approximation[M]. Cambridge: Cambridge University Press, 2005: 101–125.

    Google Scholar 

  18. HARDY R L. Multiquadric equations of topography and other irregular surfaces [J]. J Geophys Res, 1971, 76(8): 1905–1915.

    Article  Google Scholar 

  19. HARDY R L. Theory and applications of multiquadric-biharmonic method: 20 Years of discovery 1968–1988 [J]. Computers & Mathematics with Applications, 1990, 19(88): 163–208.

    Article  MathSciNet  MATH  Google Scholar 

  20. FRANKE R. Scatter data interpolation: Test of some methods [J]. Mathematics of Computation, 1982, 38(157): 181–200.

    MathSciNet  MATH  Google Scholar 

  21. RIPPA S. An algorithm for selecting a good value for the parameter c in radial basis function interpolation [J]. Advances in Computational Mathematics, 1999, 11(2): 193–210.

    Article  MathSciNet  MATH  Google Scholar 

  22. FORNBERG B, ZUEV J. The Runge phenomenon and spatially variable shape parameters in RBF interpolation [J]. Computers & Mathematics with Applications, 2007, 54(3): 379–398.

    Article  MathSciNet  MATH  Google Scholar 

  23. SARRA S A, STURGILL D. A random variable shape parameter strategy for radial basis function approximation methods [J]. Engineering Analysis with Boundary Elements, 2009, 33(11): 1239–1245.

    Article  MathSciNet  MATH  Google Scholar 

  24. XIAO J R, McCARTHY M A. A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions [J]. Computational Mechanics, 2003, 31(3): 301–315.

    MathSciNet  MATH  Google Scholar 

  25. LI L, ZHU J, ZHANG S. A hybrid radial boundary node method based on radial basis point interpolation [J]. Engineering Analysis with Boundary Elements, 2009, 33(11): 1273–1283.

    Article  MathSciNet  MATH  Google Scholar 

  26. XIAO J R, GAMA B A, GILLESPIE J W, KANSA E. Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions [J]. Engineering Analysis with Boundary Elements. 2005, 29(2): 95–106.

    Article  MATH  Google Scholar 

  27. DINIS L M J S, JORGE R M N, BELINHA J. Analysis of 3D solids using the natural neighbour radial point interpolation method [J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196(13/16): 2009–2028.

    Article  MATH  Google Scholar 

  28. QIAN Wei-chang. Elastic mechanics [M]. Beijing: Science Press, 1956. (in Chinese)

    Google Scholar 

  29. TIMOSHENKO S P, GOODIER J N. Theory of elasticity [M]. New York: McGraw-Hill, 1970.

    Google Scholar 

  30. LIU G R, GU Y T. Boundary meshfree methods based on the boundary point interpolation methods [J]. Engineering Analysis with Boundary Elements, 2004, 28(5): 475–487.

    Article  MathSciNet  MATH  Google Scholar 

  31. DU Q, FABER V, GUNZBURGER M. Centroidal Voronoi tessellations: Applications and algorithms [J]. SIAM Review, 1999, 41(4): 637–676.

    Article  MathSciNet  MATH  Google Scholar 

  32. LIU Yan, JIE Yu-xin. Simulated-annealing-based algorithm for generating point sets for meshfree methods [J]. Journal of Tsinghua University: Science and Technology, 2008, 48(6): 959–962. (in Chinese)

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Hydroscience and Engineering (Department of Hydraulic Engineering, Tsinghua University), Beijing, 100084, China

    Chong Peng  (彭翀), Hui-na Yuan  (袁会娜), Bing-yin Zhang  (张丙印) & Yan Zhang  (张琰)

Authors
  1. Chong Peng  (彭翀)
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  2. Hui-na Yuan  (袁会娜)
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  3. Bing-yin Zhang  (张丙印)
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  4. Yan Zhang  (张琰)
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Corresponding author

Correspondence to Hui-na Yuan  (袁会娜).

Additional information

Foundation item: Project(2010CB732103) supported by the National Basic Research Program of China; Project(51179092) supported by the National Natural Science Foundation of China; Project(2012-KY-02) supported by the State Key Laboratory of Hydroscience and Engineering, China

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Peng, C., Yuan, Hn., Zhang, By. et al. Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics. J. Cent. South Univ. 20, 3229–3246 (2013). https://doi.org/10.1007/s11771-013-1847-6

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  • DOI: https://doi.org/10.1007/s11771-013-1847-6

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