Skip to main content
SpringerLink
Log in

An adaptive procedure based on combining finite elements with meshless methods

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

This paper presents a new method for combining finite elements with meshless methods, which increases the accuracy of computational solutions in a coarse mesh by adding nodes in the domain of interest. The present method shares the features of the finite element and meshless methods such as (a) the meshless interpolation of the MLS type is employed; (b) integration domains are consistent with support domains; and (c) essential boundary conditions can be applied directly. In the present method, a ground mesh with triangular or quadrilateral elements is constructed to define polygonal support domains, and then additional nodes are placed arbitrarily in a domain without the reconstruction of a mesh. The method is very useful in an adaptive calculation, because nodes can be easily added or removed without any remeshing process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. W. K. Liu, S. Jun and Y. F. Zhang, Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids, 20 (1995) 1081–1106.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Babuska and J. M. Melenk, The partition of unity method, International Journal for Numerical Methods in Engineering, 40 (1997) 727–758.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. A. Duarte and J. T. Oden, Hp-cloud-a meshless method to solve boundary-value problems, Computer Methods in Applied Mechanics and Engineering, 139 (1996) 237–262.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. N. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 22 (1998) 117–127.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. R. Idelsohn and E. Onate, To mesh or not to mesh. That is the question…, Computer Methods in Applied Mechanics and Engineering, 195 (2006) 4681–4696.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Sukumar, B. Moran, A. Y. Semenov and V. Belikov, Natural neighbour Galerkin methods, International Journal for Numerical Methods in Engineering, 50 (2001) 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. R. Idelsohn, E. Onate, N. Calvo and F. D. Pin, The meshless finite element method, International Journal for Numerical Methods in Engineering, 58 (2003) 893–912.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. R. Liu and Z. H. Tu, An adaptive procedure based on background cells for meshless methods, Computer Methods in Applied Mechanics and Engineering, 191 (2002) 1923–1943.

    Article  MATH  Google Scholar 

  10. J. T. Oden, C. A. Duarte and O. C. Zienkiewicz, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering, 153 (1998) 117–126.

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Strouboulis, K. Copps and I. Babuska, The generalized finite element method, Computer Methods in Applied Mechanics and Engineering, 190 (2001) 4081–4193.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. A. Duarte, D. J. Kim and D. M. Quaresma, Arbitrarily smooth generalized finite element approximations, Computer Methods in Applied Mechanics and Engineering, 196 (2006) 33–56.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Hao, W. K. Liu and T. Belytschko, Moving particle finite element method with global smoothness, International Journal for Numerical Methods in Engineering, 59 (2004) 1007–1020.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Lu, S. Li, D. C. Simkins, W. K. Liu and J. Cao, Reproducing kernel element method Part III: Generalized enrichment and applications, Computer Methods in Applied Mechanics and Engineering, 193 (2004) 989–1011.

    Article  MATH  MathSciNet  Google Scholar 

  15. W. K. Liu, R. A. Uras and Y. Chen, Enrichment of the finite element method with the reproducing kernel particle method, Journal of Applied Mechanics ASME, 64 (1998) 861–870.

    Article  Google Scholar 

  16. A. Huerta and S. Fernández-Méndez, Enrichment and coupling of the finite element and meshless methods, International Journal for Numerical Methods in Engineering, 48 (2000) 1615–1636.

    Article  MATH  Google Scholar 

  17. H.-G. Kim and S. N. Atluri, Arbitrary placement of secondary nodes, and error control, in the mesh less local Petrov-Galerkin (MLPG) method, Computer Modeling in Engineering and Sciences, 1(3) (2000) 11–32.

    MATH  MathSciNet  Google Scholar 

  18. S. N. Atluri, H.-G. Kim and J. Y. Cho, A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods, Computational Mechanics, 24 (1999) 348–372.

    Article  MATH  Google Scholar 

  19. E. A. Malsch and G. Dasgupta, Shape functions for polygonal domains with interior nodes, International Journal for Numerical Methods in Engineering, 61 (2004) 1153–1172.

    Article  MATH  Google Scholar 

  20. T. Belytschko, D. Organ and Y. Krongauz, A coupled finite element-element-free Galerkin method, Computational Mechanics, 17 (1995) 186–195.

    MATH  MathSciNet  Google Scholar 

  21. J. Dolbow and T. Belytschko, Numerical integration of the Galerkin weak form in meshless methods, Computational Mechanics, 23 (1999) 219–230.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Wandzura and H. Xiao, Symmetric quadrature rules on a triangle, Computational Applied Mathematics, 45 (2003) 1829–1840.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. L. Anderson, Fracture mechanics fundamentals and applications, CRC Press (1991).

  24. X. R. Wu and A. J. Carlsson, Weight functions and stress intensity factor solutions, Pergamon Press (1991).

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Seoul National University of Technology, Seoul, 139-743, Korea

    Hyun-Gyu Kim

  2. The Graduate School of Energy & Environment, Seoul National University of Technology, Seoul, 139-743, Korea

    Hye-Young Kim

Authors
  1. Hyun-Gyu Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hye-Young Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyun-Gyu Kim.

Additional information

This paper was recommended for publication in revised form by Associate Editor Maenghyo Cho

Hyun-Gyu Kim received his B.S. degree from Seoul National University in 1990. He then received M.S. and Ph.D. degrees from KAIST in 1993 and 1998, respectively. Dr. Kim is currently a Professor at the Department of Mechanical Engineering in Seoul National University of Technology, Korea. His research interests include multi-physics coupling analysis, interfacing non-matching meshes, development of special elements, and inverse problems.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, HG., Kim, HY. An adaptive procedure based on combining finite elements with meshless methods. J Mech Sci Technol 23, 2224–2235 (2009). https://doi.org/10.1007/s12206-009-0510-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-009-0510-5

Keywords

Search

Navigation