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Meshfree Methods for Partial Differential Equations II pp 231–253Cite as

Discontinuous Radial Basis Function Approximations for Meshfree Methods

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 43))

Abstract

Meshfree methods with discontinuous radial basis functions and their numerical implementation for elastic problems are presented. We study the following radial basis functions: the multiquadratic (MQ), the Gaussian basis functions and the thin-plate basis functions. These radial basis functions are combined with step function enrichments directly or with enriched Shepard functions. The formulation is coupled with level set methods and requires no explicit representation of the discontinuity. Numerical results show the robustness of the method, both in accuracy and convergence.

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References

  1. R. K. Beatson, J. B. Cherrie, and D. L. Ragozin, Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines, Siam J. Math Anal., 32 (2001), pp. 1272–1310.

    Article  MATH  MathSciNet  Google Scholar 

  2. T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45 (1999), pp. 601–620.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Belytschko, Y. Y. Lu, and L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 37 (1994), pp. 229–256.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Belytschko, N. Moës, S. Usui, and C. Parimi, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering, 50 (2001), pp. 993–1013.

    Article  MATH  Google Scholar 

  5. R. E. Carlson and T. A. Foley, The parameter r2in multiquadric interpolation, Computers and Mathematics with Applications, 21 (1991), pp. 29–42.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. C. Carr, W. R. Fright, and R. K. Beatson, Surface interpolation with radial basis functions for medical imaging, IEEE Transactions on Medical Imaging, 16 (1997), pp. 96–107.

    Article  Google Scholar 

  7. C. S. Chen, M. Ganesh, M. A. Golberg, and A. H.-D. Cheng, Multilevel compact radial functions based computational schemes for some elliptic problems, Computers and Mathematics with Applications, 43 (2002), pp. 359–378.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Chessa, H. Wang, and T. Belytschko, On the construction of blending elements for local partition of unity enriched finite elements, International Journal for Numerical Methods in Engineering, 57 (2003), pp. 1015–1038.

    Article  MATH  Google Scholar 

  9. J. Dolbow and T. Belytschko, An introduction to programming the meshless element-free Galerkin method, Archives of Computational Methods in Engineering, 5 (1998), pp. 207–241.

    MathSciNet  Google Scholar 

  10. M. Fleming, Y. A. Chu, B. Moran, and T. Belytschko, Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering, 40 (1997), pp. 1483–1504.

    Article  MathSciNet  Google Scholar 

  11. A. Gravouil, N. Moës, and T. Belytschko, Non-planar 3d crack growth by the extended finite element and level sets. part ii: level set update., International Journal for Numerical Methods in Engineering, 53 (2002), pp. 2569–2586.

    Article  Google Scholar 

  12. E. J. Kansa, A scattered data approximation scheme with application to computational fluid-dynamics-i and ii, Computers and Mathematics with Applications, 19 (1990), pp. 127–161.

    Article  MATH  MathSciNet  Google Scholar 

  13. E. J. Kansa and R. E. Carlson, Improved accuracy of multiquadric interpolation using varible shape parameters, Computers and Mathematics with Applications, 24 (1992), pp. 99–120.

    Article  MATH  MathSciNet  Google Scholar 

  14. Y. Krongauz and T. Belytschko, EFG approximation with discontinuous derivatives, International Journal for Numerical Methods in Engineering, 41 (1998), pp. 1215–1233.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. R. Liu and Y. T. Gu, A point interpolation method for two-dimensional solids, International Journal for Numerical Methods in Engineering, 50 (2001), pp. 937–951.

    Article  MATH  Google Scholar 

  16. W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory and its Appl., 4 (1988), pp. 77–89.

    MATH  MathSciNet  Google Scholar 

  17. J. M. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 39 (1996), pp. 289–314.

    Article  Google Scholar 

  18. N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46 (1999), pp. 131–150.

    Article  MATH  Google Scholar 

  19. N. Moës, A. Gravouil, and T. Belytschko, Non-planar 3d crack growth by the extended finite element and level sets. part i: Mechanical model., International Journal for Numerical Methods in Engineering, 53 (2002), pp. 2549–2568.

    Article  MATH  Google Scholar 

  20. D. Organ, M. Fleming, T. Terry, and T. Belytschko, Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Computational Mechanics, 18 (1996), pp. 1–11.

    MathSciNet  Google Scholar 

  21. S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in Computational Mathematics, 11 (1999), pp. 193–210.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Sharan, E. J. Kansa, and S. Gupta, Application of the multiquadric method for numerical solution of elliptic partial differential equations, Applied Mathematics and Computation, 84 (1997), pp. 275–302.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Stolarska, D. L. Chopp, N. Moës, and T. Belytschko, Modelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering, 51 (2001), pp. 943–960.

    Article  MATH  Google Scholar 

  24. N. Sukumar, D. L. Chopp, N. Moës, and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite element method, Computer Methods in Applied Mechanics and Engineering, 190 (2001), pp. 6183–6200.

    Article  MATH  MathSciNet  Google Scholar 

  25. S. P. Timoshenko and J. N. Goodier, Theory of Elaticity (Third ed.), New York, McGraw Hill, 1970.

    Google Scholar 

  26. G. Ventura, J. X. Xu, and T. Belytschko, A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering, 54 (2002), pp. 923–944.

    Article  MATH  Google Scholar 

  27. J. G. Wang and G. R. Liu, On the optimal shape parameters of radial basis functions used for 2-d meshless methods, Computer Methods in Applied Mechanics and Engineering, 191 (2002), pp. 2611–2630.

    Article  MATH  MathSciNet  Google Scholar 

  28. J. G. Wang and G. R. Liu, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering, 54 (2002), pp. 1623–1648.

    Article  MATH  Google Scholar 

  29. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics, 4 (1995), pp. 389–396.

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Wendland, Meshless Galerkin methods using radial basis functions, Mathematics of Computation, 68 (1999), pp. 1521–1531.

    Article  MATH  MathSciNet  Google Scholar 

  31. Z. Wu, Compactly supported positive definite radial functions, Advances in Computational Mathematics, 4 (1995), pp. 283–292.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208-3111, USA

    Jingxiao Xu & Ted Belytschko

Authors
  1. Jingxiao Xu
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  2. Ted Belytschko
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Editor information

Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Wegelerstraße 6, 53115, Bonn, Germany

    Michael Griebel  & Marc A. Schweitzer  & 

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© 2005 Springer-Verlag Berlin Heidelberg

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Xu, J., Belytschko, T. (2005). Discontinuous Radial Basis Function Approximations for Meshfree Methods. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations II. Lecture Notes in Computational Science and Engineering, vol 43. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-27099-X_13

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