Coiflets solutions for Föppl-von Kármán equations governing large deflection of a thin flat plate by a novel wavelet-homotopy approach
Tóm tắt
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.
Tài liệu tham khảo
Föppl, A.: Vorlesungen über technische mechanik Bd. 3,B.G. Teubner, Leipzig (1907)
von Karman, T.: Festigkeitsproblem im maschinenbau. Encyk. D. Math. Wiss. 4, 311–385 (1910)
Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill (1956)
Landau, L.D., Lifshit’S, E.M.: Theory of Elasticity. World Book Publishing Company (1999)
Knightly, G.H.: An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27(3), 233–242 (1967)
Kesavan, S.: Application of Kikuchi’s method to the von Kármán equations. Numer. Math. 32(2), 209–232 (1979)
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)
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)
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)
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)
Ciarlet, G.P., Gratie, L., Kesavan, S.: Numerical analysis of the generalized von Kármán equations. Comptes Rendus Mathematique 341(11), 695–699 (2005)
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)
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)
Ciarlet, P.G., Paumier, J.C.: A justification of the marguerre-von Kármán equations. Comput. Mech. 1(3), 177–202 (1986)
Ciarlet, P.G., Gratie, L., Sabu, N.: An existence theorem for generalized von Kármán equations. J. Elast. 62(3), 239–248 (2001)
Milani, A.J., Chueshov, I., Lasiecka, I.: Von Kármán Evolution Equations. Springer, New York (2010)
Coman, C.D.: On the compatibility relation for the föppl-von Kármán plate equations. Appl. Math. Lett. 25(12), 2407–2410 (2012)
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)
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)
Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Ph.d thesis (1992)
Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)
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)
Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the laneemden equation. Phys. Lett. A 372(39), 6060–6065 (2008)
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)
Mastroberardino, A.: Homotopy analysis method applied to electrohydrodynamic flow. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2730–2736 (2011)
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)
Ablowitz, M.J., Ladik, J.F.: Nonlinear differential–difference equations and fourier analysis. J. Math. Phys. 17(6), 1011–1018 (1976)
Stein, E.M., Weiss, G.: Introduction to fourier analysis on euclidean spaces. Princeton Math. Ser. 212(2), 484–503 (2009)
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)
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)
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)
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)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. P.Noordhoff Ltd (1953)
Tian, J.: The Mathematical Theory and Applications of Biorthogonal Coifman Wavelet Systems. Rice University, Ph.D. thesis (1996)
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)
Katsikadelis, J.T., Nerantzaki, M.S.: Non-linear analysis of plates by the analog equation method. Comput. Mech. 14(2), 154–164 (1994)
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)
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)
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)
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)
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)