Skip to main content
SpringerLink
Log in

The interpolating element-free Galerkin method for elastic large deformation problems

  • Article
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.

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. Belytschko T, Krongauz Y, Organ D, et al. Meshless methods: An overview and recent developments. Comput Methods Appl Mech Eng, 1996, 139: 3–47

    Article  Google Scholar 

  2. Cheng Y M, Li J H. A complex variable meshless method for fracture problems. Sci China Ser G-Phys Mech Astron, 2006, 49: 46–59

    Article  Google Scholar 

  3. Gao H F, Cheng Y M. A complex variable meshless manifold method for fracture problems. Int J Comput Methods, 2010, 07: 55–81

    Article  MathSciNet  Google Scholar 

  4. Li D M, Bai F N, Cheng Y M, et al. A novel complex variable element-free Galerkin method for two-dimensional large deformation problems. Comput Methods Appl Mech Eng, 2012, 233–236: 1–10

    MathSciNet  MATH  Google Scholar 

  5. Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods. Int J Numer Meth Engng, 1994, 37: 229–256

    Article  MathSciNet  Google Scholar 

  6. Zhang Z, Liew K M, Cheng Y M, et al. Analyzing 2D fracture problems with the improved element-free Galerkin method. Eng Anal Bound Elem, 2008, 32: 241–250

    Article  Google Scholar 

  7. Ren H P, Cheng Y M. The interpolating element-free Galerkin (IEFG) method for two-dimensional elasticity problems. Int J Appl Mech, 2011, 03: 735–758

    Article  Google Scholar 

  8. Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Math Comp, 1981, 37: 141

    Article  MathSciNet  Google Scholar 

  9. Cheng Y M, Chen M J. A boundary element-free method for linear elasticity. Acta Mech Sin, 2003, 35: 181–186

    Google Scholar 

  10. Zhang Z, Hao S Y, Liew K M, et al. The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng Anal Bound Elem, 2013, 37: 1576–1584

    Article  MathSciNet  Google Scholar 

  11. Zhang Z, Zhao P, Liew K M. Analyzing three-dimensional potential problems with the improved element-free Galerkin method. Comput Mech, 2009, 44: 273–284

    Article  MathSciNet  Google Scholar 

  12. Zhang Z, Li D M, Cheng Y M, et al. The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech Sin, 2012, 28: 808–818

    Article  MathSciNet  Google Scholar 

  13. Zhang Z, Wang J F, Cheng Y M, et al. The improved element-free Galerkin method for three-dimensional transient heat conduction problems. Sci China-Phys Mech Astron, 2013, 56: 1568–1580

    Article  Google Scholar 

  14. Wu Y, Ma Y Q, Feng W, et al. Topology optimization using the improved element-free Galerkin method for elasticity. Chin Phys B, 2017, 26: 080203

    Article  Google Scholar 

  15. Yu S Y, Peng M J, Cheng H, et al. The improved element-free Galerkin method for three-dimensional elastoplasticity problems. Eng Anal Bound Elem, 2019, 104: 215–224

    Article  MathSciNet  Google Scholar 

  16. Ren H P, Cheng Y M, Zhang W. Researches on the improved interpolating moving least-squares method (in Chinese). Chin J Eng Math, 2010, 27: 1021–1029

    MathSciNet  MATH  Google Scholar 

  17. Ren H P, Cheng Y M, Zhang W. An interpolating boundary elementfree method (IBEFM) for elasticity problems. Sci China-Phys Mech Astron, 2010, 53: 758–766

    Article  Google Scholar 

  18. Ren H P, Cheng Y M, Zhang W. An improved boundary element-free method (IBEFM) for two-dimensional potential problems. Chin Phys B, 2009, 18: 4065–4073

    Article  Google Scholar 

  19. Ren H P, Cheng Y M. The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Eng Anal Bound Elem, 2012, 36: 1568–1580

    Article  MathSciNet  Google Scholar 

  20. Zhao N, Ren H P. The interpolating element-free Galerkin method for 2D transient heat conduction problems. Math Probl Eng, 2014, 2014: 712834

    MathSciNet  MATH  Google Scholar 

  21. Cheng Y M, Bai F N, Peng M J. A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl Math Model, 2014, 38: 5187–5197

    Article  MathSciNet  Google Scholar 

  22. Cheng Y M, Bai F N, Liu C, et al. Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method. Int J Comput Mater Sci Eng, 2016, 5: 1650023

    Google Scholar 

  23. Wang J F, Sun F X, Cheng Y M. An improved interpolating elementfree Galerkin method with a nonsingular weight function for two-dimensional potential problems. Chin Phys B, 2012, 21: 090204

    Article  Google Scholar 

  24. Wang J F, Wang J F, Sun F X, et al. An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems. Int J Comput Methods, 2013, 10: 135004

    MathSciNet  Google Scholar 

  25. Sun F X, Wang J F, Cheng Y M. An improved interpolating elementfree Galerkin method for elasticity. Chin Phys B, 2013, 22: 120203

    Article  Google Scholar 

  26. Sun F X, Wang J F, Cheng Y M. An improved interpolating elementfree Galerkin method for elastoplasticity via nonsingular weight functions. Int J Appl Mech, 2016, 08: 1650096

    Article  Google Scholar 

  27. Liu F B, Wu Q, Cheng Y M. A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems. Int J Appl Mech, 2019, 11: 195006

    Google Scholar 

  28. Liu F B, Cheng Y M. The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems. Int J Comput Mater Sci Eng, 2018, 7: 1850023

    Google Scholar 

  29. Liu F B, Cheng Y M. The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int J Appl Mech, 2018, 10: 1850047

    Article  Google Scholar 

  30. Jun S, Liu W K, Belytschko T. Explicit reproducing kernel particle methods for large deformation problems. Int J Numer Meth Engng, 1998, 41: 137–166

    Article  Google Scholar 

  31. Chen J S, Pan C, Wu C T. Application of reproducing kernel particle method to large deformation contact analysis of elastomers. Rubber Chem Tech, 1998, 71: 191–213

    Article  Google Scholar 

  32. Liew K M, Ng T Y, Wu Y C. Meshfree method for large deformation analysis—A reproducing kernel particle approach. Eng Struct, 2002, 24: 543–551

    Article  Google Scholar 

  33. Han Z D. Rajendran A M. Atluri S N. Meshless local Petrov-Galerkin (MLPG) approaches for solving nonlinear problems with large deformations and rotations. Comput Model Eng Sci, 2005, 10: 1–12

    MathSciNet  MATH  Google Scholar 

  34. Tiago C, Pimenta P M. An EFG method for the nonlinear analysis of plates undergoing arbitrarily large deformations. Eng Anal Bound Elem, 2008, 32: 494–511

    Article  Google Scholar 

  35. Li D M, Liew K M, Cheng Y M. Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method. Comput Mech, 2014, 53: 1149–1162

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Qiang Wu, PiaoPiao Peng & YuMin Cheng

Authors
  1. Qiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. PiaoPiao Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. YuMin Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YuMin Cheng.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 11571223).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Q., Peng, P. & Cheng, Y. The interpolating element-free Galerkin method for elastic large deformation problems. Sci. China Technol. Sci. 64, 364–374 (2021). https://doi.org/10.1007/s11431-019-1583-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-019-1583-y

Keywords

Search

Navigation