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Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method

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Abstract

Using the complex variable moving least-squares (CVMLS) approximation, a complex variable element-free Galerkin (CVEFG) method for two-dimensional elastoplastic large deformation problems is presented. This meshless method has higher computational precision and efficiency because in the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. For two-dimensional elastoplastic large deformation problems, the Galerkin weak form is employed to obtain its equation system. The penalty method is used to impose essential boundary conditions. Then the corresponding formulae of the CVEFG method for two-dimensional elastoplastic large deformation problems are derived. In comparison with the conventional EFG method, our study shows that the CVEFG method has higher precision and efficiency. For illustration purpose, a few selected numerical examples are presented to demonstrate the advantages of the CVEFG method.

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References

  1. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless method: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  Google Scholar 

  2. Ji X, Zang YL, Cheng YM (1997) Advances in boundary element methods and the general package. Tongji University Press, Shanghai

    Google Scholar 

  3. Lancaster P, Salkauskas K (1981) Surface generated by moving least squares methods. Math Comput 37:141–158

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheng YM, Li JH (2006) A complex variable meshless method for fracture problems. Sci China G 49(1):46–59

    Article  MATH  Google Scholar 

  5. Liew KM, Feng C, Cheng YM, Kitipornchai S (2007) Complex variable moving least-squares method: a meshless approximation technique. Int J Numer Methods Eng 70:46–70

    Article  MATH  MathSciNet  Google Scholar 

  6. Liew KM, Cheng YM, Kitipornchai S (2006) Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems. Int J Numer Methods Eng 65(8):1310–1332

    Article  MATH  Google Scholar 

  7. Liew KM, Cheng YM, Kitipornchai S (2005) Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform. Int J Numer Methods Eng 64(12):1610–1627

    Article  MATH  Google Scholar 

  8. Liew KM, Cheng YM, Kitipornchai S (2007) Analyzing the 2D fracture problems via the enriched boundary element-free method. Int J Solids Struct 44(11–12):4220–4233

    Article  MATH  Google Scholar 

  9. Peng MJ, Cheng YM (2009) A boundary element-free method (BEFM) for two-dimensional potential problems. Eng Anal Boundary Elements 33(1):77–82

    Article  MATH  MathSciNet  Google Scholar 

  10. Liew KM, Sun YZ, Kitipornchai S (2007) Boundary element-free method for fracture analysis of 2-D anisotropic piezoelectric solids. Int J Numer Methods Eng 69:729–749

    Article  MATH  Google Scholar 

  11. Sun YZ, Liew KM (2012) Analyzing interaction between coplanar square cracks using an efficient boundary element-free method. Int J Numer Methods Eng 91:1184–1198

    Article  MathSciNet  Google Scholar 

  12. Zhang Z, Liew KM, Cheng YM, Lee YY (2008) Analyzing 2D fracture problems with the improved element-free Galerkin method. Eng Anal Boundary Elements 32(3):241–250

    Article  MATH  Google Scholar 

  13. Zhang Z, Zhao P, Liew KM (2009) Improved element-free Galerkin method for two-dimensional potential problems. Eng Anal Boundary Elements 33:547–554

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Zhang Z, Li Dongming, Cheng YM, Liew KM (2012) The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech Sinica 28(3):808–818

    Article  MathSciNet  Google Scholar 

  16. Cheng YM, Peng MJ, Li JH (2005) The complex variable moving least-square approximation and its application. Chin J Theor Appl Mech 37(6):719–723

    MathSciNet  Google Scholar 

  17. Liew KM, Cheng YM (2009) Complex variable boundary element-free method for two-dimensional elastodynamic problems. Comput Methods Appl Mech Eng 198:3925–3933

    Article  MATH  MathSciNet  Google Scholar 

  18. Peng MJ, Liu P, Cheng YM (2009) The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems. Int J Appl Mech 1(2):367–385

    Article  Google Scholar 

  19. Peng MJ, Li Dongming, Cheng YM (2011) The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems. Eng Struct 33:127–135

    Article  Google Scholar 

  20. Li Dongming, Peng MJ, Cheng YM (2011) The complex variable element-free Galerkin (CVEFG) method for elastic large deformation problems. Sci Sinica Phys Mech Astron 41(8):1003–1014

    Article  Google Scholar 

  21. Cheng YM, Wang JF, Li RX (2012) The complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems. Int J Appl Mech 4(4):1250042-1–1250042-23

    Google Scholar 

  22. Cheng YM, Li RX, Peng MJ (2012) Complex variable element-free Galerkin (CVEFG) method for viscoelasticity problems. Chin Phys B 21(9):090205-1–090205-13

    Google Scholar 

  23. Cheng YM, Wang JF, Bai FN (2012) A new complex variable element-free Galerkin method for two-dimensional potential problems. Chin Phys B 21(9):090203-1–090203-10

    Google Scholar 

  24. Wang JF, Cheng YM (2012) A new complex variable meshless method for transient heat conduction problems. Chin Phys B 21(12):120206-1–120206-9

    Google Scholar 

  25. Bai FN, Li Dongming, Wang JF, Cheng YM (2012) An improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional elasticity problems. Chin Phys B 21(2):020204-1–020204-10

  26. Li Dongming, Bai FN, Cheng YM, Liew KM (2012) A novel complex variable element-free Galerkin method for two-dimensional large deformation problems. Comput Methods Appl Mech Eng 233–236:1–10

    MathSciNet  Google Scholar 

  27. Wang JF, Cheng YM (2013) A new complex variable meshless method for advection-diffusion problems. Chin Phys B 22(3):030208-1–030208-7

    Google Scholar 

  28. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  29. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139:195–227

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu WK, Jun S (1998) Multiple-scale reproducing kernel particle methods for large deformation problem. Int J Numer Methods Eng 41:1339–1362

    Article  MATH  MathSciNet  Google Scholar 

  31. Li S, Hao W, Liu WK (2000) Mesh-free simulations of shear banding in large deformation. Int J Solids Struct 37:7185–7206

    Article  MATH  Google Scholar 

  32. Liew KM, Ng TY, Wu YC (2002) Meshfree method for large deformation analysis—a reproducing kernel particle approach. Eng Struct 24:543–551

    Article  Google Scholar 

  33. Sladek J, Sladek VA (2003) A meshless method for large deflection of plates. Comput Mech 30(2):155–163

    Article  MATH  MathSciNet  Google Scholar 

  34. Ren J, Liew KM, Meguid SA (2002) Modelling and simulation of the superelastic behaviour of shape memory alloys using the element-free Galerkin method. Int J Mech Sci 44:2393–2413

    Article  MATH  Google Scholar 

  35. Liew KM, Ren J, Kitipornchai S (2004) Analysis of the pseudoelastic behavior of a SMA beam by the element-free Galerkin method. Eng Anal Boundary Elements 28:497–507

    Article  MATH  Google Scholar 

  36. Liew KM, Ren J, Reddy JN (2005) Numerical simulation of thermomechanical behaviours of shape memory alloys via a non-linear mesh-free Galerkin formulation. Int J Numer Methods Eng 63:1014–1040

    Article  MATH  Google Scholar 

  37. Han ZD, Rajendran AM, Atluri SN (2005) Meshless local Petrov-Galerkin (MLPG) approaches for solving nonlinear problems with large deformations and rotations. Comput Model Eng Sci 10(1):1–12

    MathSciNet  Google Scholar 

  38. Liew KM, Peng LX, Kitipornchai S (2007) Geometric non-linear analysis of folded plate structures by the spline strip kernel particle method. Int J Numer Methods Eng 71:1102–1133

    Article  MATH  Google Scholar 

  39. Liew KM, Peng LX, Kitipornchai S (2007) Nonlinear analysis of corrugated plates using a FSDT and a meshfree method. Comput Methods Appl Mech Eng 196:2358–2376

    Article  MATH  Google Scholar 

  40. Naffa M, Al-Gahtani HJ (2007) RBF-based meshless method for large deflection of thin plates. Eng Anal Boundary Elements 31(4):311–317

    Article  MATH  Google Scholar 

  41. Zhao X, Liew KM (2009) Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method. Comput Methods Appl Mech Eng 198:2796–2811

    Article  MATH  Google Scholar 

  42. Lei ZX, Liew KM, Yu JL (2013) Large deflection analysis of functionally graded carbon nanotube reinforced composite plates by the element-free kp-Ritz method. Comput Methods Appl Mech Eng 256:189–199

    Google Scholar 

  43. Li S, Hao W, Liu WK (2000) Numerical simulations of large deformation of thin shell structures using meshfree methods. Comput Mech 25:102–116

    Article  MATH  Google Scholar 

  44. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  45. Hill R (1950) The mathematical theory of plasticity. Oxford University Press, New York

    MATH  Google Scholar 

  46. Bathe KJ (1996) Finite element procedures. Prentice-Hall, Englewood Cliffs

    Google Scholar 

Download references

Acknowledgments

The work described in this paper was supported by grants from the China National Natural Science Foundation (Grant No. 51378448 and 11171208).

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Authors and Affiliations

  1. Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR

    D. M. Li & K. M. Liew

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai , 200072, China

    Y. M. Cheng

Authors
  1. D. M. Li
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  2. K. M. Liew
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  3. Y. M. Cheng
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Correspondence to K. M. Liew or Y. M. Cheng.

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Li, D.M., Liew, K.M. & Cheng, Y.M. Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method. Comput Mech 53, 1149–1162 (2014). https://doi.org/10.1007/s00466-013-0954-4

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  • DOI: https://doi.org/10.1007/s00466-013-0954-4

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