Coiflets solutions for Föppl-von Kármán equations governing large deflection of a thin flat plate by a novel wavelet-homotopy approach

Original Paper
  • 21 Downloads

Abstract

In this paper, a novel technique incorporated the homotopy analysis method (HAM) with Coiflets is developed to obtain highly accurate solutions of the Föppl-von Kármán equations for large bending deflection. The characteristic scale transformation is introduced to nondimensionalize the governing equations. The results are obtained for the transformed nondimensional equations, which are in very excellent agreement with analytical ones or numerical benchmarks performing good efficiency and validity. Besides, we notice the nonlinearity of the Föppl-von Kármán equations is closely connected with the load and length-width ratio of the plate. For the case of the plate suffering tremendous loads, the traditional linear theory does not work, while our Coiflets solutions are still very accurate. It is expected that our proposed approach not only keeps the outstanding merits of the HAM technique for handling strong nonlinearity, but also improves on the computational efficiency to a great extent.

Keywords

Coiflets Galerkin method Wavelet Foppl-von Karman equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Föppl, A.: Vorlesungen über technische mechanik Bd. 3,B.G. Teubner, Leipzig (1907)Google Scholar
  2. 2.
    von Karman, T.: Festigkeitsproblem im maschinenbau. Encyk. D. Math. Wiss. 4, 311–385 (1910)Google Scholar
  3. 3.
    Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill (1956)Google Scholar
  4. 4.
    Landau, L.D., Lifshit’S, E.M.: Theory of Elasticity. World Book Publishing Company (1999)Google Scholar
  5. 5.
    Knightly, G.H.: An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27(3), 233–242 (1967)CrossRefMATHGoogle Scholar
  6. 6.
    Kesavan, S.: Application of Kikuchi’s method to the von Kármán equations. Numer. Math. 32(2), 209–232 (1979)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chueshov, I.D.: On the finiteness of the number of determining elements for von Kármán evolution equations. Math. Methods Appl. Sci. 20(10), 855–865 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    da Silva, P.P., Krauth, W.: Numerical solutions of the von Kármán equations for a thin plate. Int. J. Modern Phys. C 8(2), 427–434 (1996)CrossRefGoogle Scholar
  9. 9.
    Lewicka, M., Mahadevan, L., Pakzad, M.R.: The föppl-von Kármán equations for plates with incompatible strains. Proc. R. Soc. Lond. 467(2126), 402–426 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Xue, C.X., Pan, E., Zhang, S.Y., Chu, H.J.: Large deflection of a rectangular magnetoelectroelastic thin plate. Mech. Res. Commun. 38(7), 518–523 (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Ciarlet, G.P., Gratie, L., Kesavan, S.: Numerical analysis of the generalized von Kármán equations. Comptes Rendus Mathematique 341(11), 695–699 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G., Gratie, L., Kesavan, S.: On the generalized von Kármán equations and their approximation. Math. Models Methods Appl. Sci. 17(04), 617–633 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ciarlet, P.G., Gratie, L.: From the classical to the generalized von Kármán and marguerre–von Kármán equations. J. Comput. Appl. Math. 190(1-2), 470–486 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ciarlet, P.G., Paumier, J.C.: A justification of the marguerre-von Kármán equations. Comput. Mech. 1(3), 177–202 (1986)CrossRefMATHGoogle Scholar
  15. 15.
    Ciarlet, P.G., Gratie, L., Sabu, N.: An existence theorem for generalized von Kármán equations. J. Elast. 62(3), 239–248 (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Milani, A.J., Chueshov, I., Lasiecka, I.: Von Kármán Evolution Equations. Springer, New York (2010)MATHGoogle Scholar
  17. 17.
    Coman, C.D.: On the compatibility relation for the föppl-von Kármán plate equations. Appl. Math. Lett. 25(12), 2407–2410 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Doussouki, A.E., Guedda, M., Jazar, M., Benlahsen, M.: Some remarks on radial solutions of föppl-von Kármán equations. Appl. Math. Comput. 219(9), 4340–4345 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Van Gorder, R.A.: Analytical method for the construction of solutions to the föppl-von kármán equations governing deflections of a thin flat plate. Int. J. Non-Linear Mech. 47(3), 1–6 (2012)CrossRefGoogle Scholar
  20. 20.
    Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Ph.d thesis (1992)Google Scholar
  21. 21.
    Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zou, K., Nagarajaiah, S.: An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 780–792 (2014)MathSciNetMATHGoogle Scholar
  23. 23.
    Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the laneemden equation. Phys. Lett. A 372(39), 6060–6065 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Varol, Y., Oztop, H.F.: Control of buoyancy-induced temperature and flow fields with an embedded adiabatic thin plate in porous triangular cavities. Appl. Therm. Eng. 29(2–3), 558–566 (2009)CrossRefGoogle Scholar
  25. 25.
    Mastroberardino, A.: Homotopy analysis method applied to electrohydrodynamic flow. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2730–2736 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Van Gorder, R.A.: Relation between laneemden solutions and radial solutions to the elliptic heavenly equation on a disk. New Astron. 37, 42–47 (2015)CrossRefGoogle Scholar
  27. 27.
    Ablowitz, M.J., Ladik, J.F.: Nonlinear differential–difference equations and fourier analysis. J. Math. Phys. 17(6), 1011–1018 (1976)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Stein, E.M., Weiss, G.: Introduction to fourier analysis on euclidean spaces. Princeton Math. Ser. 212(2), 484–503 (2009)Google Scholar
  29. 29.
    Wang, J.Z.: Generalized theory and arithmetic of orthogonal wavelets and applications to researches of mechanics including piezoelectric smart structures. Lanzhou University, Ph.d thesis (2001)Google Scholar
  30. 30.
    Zhou, Y.H., Wang, J.Z.: Generalized gaussian integral method for calculations of scaling function transform of wavelets and its applications. Acta Mathematica Scientia(Chinese Edition) 19(3), 293–300 (1999)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Chen, M.Q., Hwang, C., Shih, Y.P.: The computation of wavelet-galerkin approximation on a bounded interval. Int. J. Numer. Methods Eng. 39(17), 2921–2944 (1996)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Xing, R.: Wavelet-based homotopy analysis method for nonlinear matrix system and its application in burgers equation. Math. Problems Eng. 2013,(2013-6-25) 2013 (5), 14–26 (2013)Google Scholar
  33. 33.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. P.Noordhoff Ltd (1953)Google Scholar
  34. 34.
    Tian, J.: The Mathematical Theory and Applications of Biorthogonal Coifman Wavelet Systems. Rice University, Ph.D. thesis (1996)Google Scholar
  35. 35.
    Liu, X.J.: A Wavelet Method for Uniformly Solving Nonlinear Problems and Its Application to Quantitative Research on Flexible Structures with Large Deformation. Lanzhou University, Ph.d thesis (2014)Google Scholar
  36. 36.
    Katsikadelis, J.T., Nerantzaki, M.S.: Non-linear analysis of plates by the analog equation method. Comput. Mech. 14(2), 154–164 (1994)CrossRefMATHGoogle Scholar
  37. 37.
    Azizian, Z.G., Dawe, D.J.: Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method. Comput. Struct. 21(3), 423–436 (1985)CrossRefMATHGoogle Scholar
  38. 38.
    Wang, W., Ji, X., Tanaka, M.: A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates. Comput. Mech. 26(1), 58–65 (2000)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Al-Tholaia, M.M.H., Al-Gahtani, H.J.: Rbf-based meshless method for large deflection of elastic thin plates on nonlinear foundations. Eng. Anal. Bound. Elements 51, 146–155 (2015)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhao, Y., Lin, Z., Liao, S.: An iterative ham approach for nonlinear boundary value problems in a semi-infinite domain. Comput. Phys. Commun. 184 (9), 2136–2144 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Katsikadelis, J.T.: Large deflection analysis of plates on elastic foundation by the boundary element method. Int. J. Solids Struct. 27(15), 1867–1878 (1991)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration(CISSE), State Key Laboratory of Ocean Engineering, School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations