Skip to main content
SpringerLink
Log in

Direct meshless local Petrov–Galerkin method for elastodynamic analysis

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This article describes a new and fast meshfree method based on a generalized moving least squares (GMLS) approximation and the local weak forms for vibration analysis in solids. In contrast to the meshless local Petrov–Galerkin method, GMLS directly approximates the local weak forms from meshless nodal values, which shifts the local integrations over the low-degree polynomial basis functions rather than over the complicated MLS shape functions. Besides, if the method is set up properly, all local integrals have the same value if all local subdomains have the same shape. These features reduce the computational costs, remarkably. The new technique is called direct meshless local Petrov–Galerkin (DMLPG) method. In DMLPG, the stiff and mass matrices are constructed by integration against polynomials. This overcomes the main drawback of meshfree methods in comparison with the finite element methods (FEM). The Newmark scheme is adapted as a time integration method, and numerical results are presented for various dynamic problems. The results are compared with the exact solutions, if available, and the FEM solutions.

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. Atluri S.N., Cho J.Y., Kim H.G.: Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations. Comput. Mech. 24, 334–347 (1999)

    Article  MATH  Google Scholar 

  2. Babuska I., Banerjee U., Osborn J., Zhang Q.: Effect of numerical integration on meshless methods. Comput. Methods Appl. Mech. Eng. 198, 27–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beissel S., Belytschko T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139, 49–74 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Carpinteri A., Ferro G., Ventura G.: The partition of unity quadrature in meshless methods. Int. J. Numer. Methods Eng. 54, 987–1006 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chiba F., Kako T.: Stability and error analyses by energy estimate for Newmarks method. Natl. Inst. Fusion Sci. 17-18, 40, 82–91 (1999)

    Google Scholar 

  7. Dai, B., Cheng, J., Zheng, B.: A moving kriging interpolation-based meshless local Petrov–Galerkin method for elastodynamic analysis. Int. J. Appl. Mech. 5, 135,001 (21 pages) (2013)

  8. Dolbow J., Belytschko T.: Numerical integration of the galerkin weak form in meshfree methods. Comput. Mech. 23, 219–230 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ghadiri Rad, M.H., Shahabian, F., Hosseini, S.M.: A meshless local Petrov-Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping. Acta Mech. p. In press (2014). doi:10.1007/s00707-014-1266-2

  10. Gu Y.T., Liu G.R.: A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput. Mech. 27, 188–198 (2001)

    Article  MATH  Google Scholar 

  11. Hoghes T., Pister K., Taylor R.: Implicit–explicit finite elements in nonlinear transient analysis. Comput. Methods Appl. Mech. Eng. 17–18, Part 1, 159–182 (1979)

    Article  MATH  Google Scholar 

  12. Idesman A., Pham D.: Finite element modeling of linear elastodynamics problems with explicit time-integration methods and linear elements with the reduced dispersion error. Comput. Methods Appl. Mech. Eng. 271, 86–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kandilas C.B.: Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies. Acta Mech. 223, 861–878 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Long S.Y., Liu K.Y., Hu D.A.: A new meshless method based on MLPG for elastic dynamic problems. Eng. Anal. Bound. Elem. 30, 43–48 (2006)

    Article  MATH  Google Scholar 

  16. Mazzia A., Pini G.: Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov–Galerkin method. J. Complex. 26, 82–101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mazzia A., Pini G., Sartoretto F.: Numerical investigation on direct MLPG for 2D and 3D potential problems. Comput. Model. Simul. Eng. 88, 183–209 (2012)

    MathSciNet  Google Scholar 

  18. Mazzia A., Pini G., Sartoretto F.: Meshless techniques for anisotropic diffusion. Appl. Math. Comput. 236, 54–66 (2014)

    Article  MathSciNet  Google Scholar 

  19. Mirzaei D.: Analysis of moving least squares approximation revisited. J. Comput. Appl. Math. 282, 237–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mirzaei, D.: A new low–cost meshfree method for two and three dimensional problems in elasticity. Appl. Math. Model. p. In press (2015)

  21. Mirzaei D.: Error bounds for GMLS derivatives approximations of sobolev functions. J. Comput. Appl. Math. 294, 93–101 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mirzaei D., Schaback R.: Direct Meshless Local Petrov–Galerkin (DMLPG) method: a generalized MLS approximation. Appl. Numer. Math. 33, 73–82 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mirzaei D., Schaback R.: Solving heat conduction problem by the Direct Meshless Local Petrov-Galerkin (DMLPG) method. Numer. Algorithms 65, 275–291 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mirzaei D., Schaback R., Dehghan M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32, 983–1000 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moosavi M.R., Khelil A.: Finite volume meshless local Petrov-Galerkin method in elastodynamic problems. Eng. Anal. Bound. Elem. 33, 1016–1021 (2009)

    Article  MATH  Google Scholar 

  26. Pecher R.: Efficient cubature formulae for MLPG and related methods. Int. J. Numer. Methods Eng. 65, 566–593 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ramezani M., Mojtabaei M., Mirzaei D.: DMLPG solution of the fractional advection–diffusion problem. Eng. Anal. Bound. Elem. 59, 36–42 (2015)

    Article  MathSciNet  Google Scholar 

  28. Sladek J., Sladek V., Van Keer R.: Meshless local boundary integral equation method for 2D elastodynamic problems. Int. J. Numer. Methods Eng. 57, 235–249 (2003)

    Article  MATH  Google Scholar 

  29. Sladek J., Sladek V., Zhang C.: Application of meshless local Petrov-Galerkin (MLPG) method to elastodynamic problems in continuously nonhomogeneous solids. CMES Comput. Model. Eng. Sci. 4, 637–648 (2000)

    MATH  Google Scholar 

  30. Soares D. Jr., Sladek V., Sladek J.: Modified meshless local Petrov–Galerkin formulations for elastodynamics. Int. J. Numer. Methods Eng. 90, 1508–1828 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Taleei A., Dehghan M.: Direct meshless local Petrov-Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic. Comput. Methods Appl. Mech. Eng. 278, 479–498 (2014)

    Article  MathSciNet  Google Scholar 

  32. Wendland H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  33. Zhang Q., Banerjee U.: Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients. Adv. Comput. Math. 37, 453–492 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, P.O.Box 81746-73441, Isfahan, Iran

    Davoud Mirzaei

  2. Department of Mechanical Engineering, Faculty of Engineering, University of Isfahan, P.O.Box 81746-73441, Isfahan, Iran

    Kourosh Hasanpour

Authors
  1. Davoud Mirzaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kourosh Hasanpour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davoud Mirzaei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirzaei, D., Hasanpour, K. Direct meshless local Petrov–Galerkin method for elastodynamic analysis. Acta Mech 227, 619–632 (2016). https://doi.org/10.1007/s00707-015-1494-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-015-1494-0

Keywords

Search

Navigation