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Coiflets solutions for Föppl-von Kármán equations governing large deflection of a thin flat plate by a novel wavelet-homotopy approach. (English) Zbl 1433.65339

The homotopy analysis method provides an analtyic approximation technique to transform nonlinear partial differential equations to a collection of linear partial differential equations. The Galerkin method reduces a collection of linear partial differential equations into a linear system of equations through projection onto a set of functions. Researchers have explored the combination of these methods with Fourier bases in [R. A. Van Gorder, Numer. Algorithms 76, No. 1, 151–162 (2017; Zbl 1375.65103)]. However, Fourier bases can struggle to resolve features at small scales leading to inaccuracies. The authors investigate wavelet bases following X. Ruyi [Math. Probl. Eng. 2013, Article ID 982810, 7 p. (2013; Zbl 1299.65249)] with focus on Föppl-von Kármán equations for deflection of thin plates. Rigorous determination of convergence-controlled parameters provides solutions with better accuracy compared to analytical and numerical benchmarks particularly in the large load regime.

MSC:

65N99 Numerical methods for partial differential equations, boundary value problems
65T60 Numerical methods for wavelets
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74K20 Plates
74B20 Nonlinear elasticity
35Q74 PDEs in connection with mechanics of deformable solids
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[1] Föppl, A.: Vorlesungen über technische mechanik Bd. 3,B.G. Teubner, Leipzig (1907)
[2] von Karman, T., Festigkeitsproblem im maschinenbau, Encyk. D. Math. Wiss., 4, 311-385, (1910)
[3] Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill (1956) · Zbl 0070.41104
[4] Landau, L.D., Lifshit’S, E.M.: Theory of Elasticity. World Book Publishing Company (1999)
[5] Knightly, GH, An existence theorem for the von Kármán equations, Arch. Ration. Mech. Anal., 27, 233-242, (1967) · Zbl 0162.56303
[6] Kesavan, S., Application of Kikuchi’s method to the von Kármán equations, Numer. Math., 32, 209-232, (1979) · Zbl 0395.73054
[7] Chueshov, ID, On the finiteness of the number of determining elements for von Kármán evolution equations, Math. Methods Appl. Sci., 20, 855-865, (1997) · Zbl 0903.58032
[8] da Silva, PP; Krauth, W., Numerical solutions of the von Kármán equations for a thin plate, Int. J. Modern Phys. C, 8, 427-434, (1996)
[9] Lewicka, M.; Mahadevan, L.; Pakzad, MR, The föppl-von Kármán equations for plates with incompatible strains, Proc. R. Soc. Lond., 467, 402-426, (2011) · Zbl 1219.74027
[10] Xue, CX; Pan, E.; Zhang, SY; Chu, HJ, Large deflection of a rectangular magnetoelectroelastic thin plate, Mech. Res. Commun., 38, 518-523, (2011) · Zbl 1272.74416
[11] Ciarlet, GP; Gratie, L.; Kesavan, S., Numerical analysis of the generalized von Kármán equations, Comptes Rendus Mathematique, 341, 695-699, (2005) · Zbl 1081.74041
[12] Ciarlet, PG; Gratie, L.; Kesavan, S., On the generalized von Kármán equations and their approximation, Math. Models Methods Appl. Sci., 17, 617-633, (2007) · Zbl 1121.65118
[13] Ciarlet, PG; Gratie, L., From the classical to the generalized von Kármán and marguerre-von Kármán equations, J. Comput. Appl. Math., 190, 470-486, (2006) · Zbl 1085.74031
[14] Ciarlet, PG; Paumier, JC, A justification of the marguerre-von Kármán equations, Comput. Mech., 1, 177-202, (1986) · Zbl 0633.73069
[15] Ciarlet, PG; Gratie, L.; Sabu, N., An existence theorem for generalized von Kármán equations, J. Elast., 62, 239-248, (2001) · Zbl 1043.74030
[16] Milani, A.J., Chueshov, I., Lasiecka, I.: Von Kármán Evolution Equations. Springer, New York (2010) · Zbl 1298.35001
[17] Coman, CD, On the compatibility relation for the föppl-von Kármán plate equations, Appl. Math. Lett., 25, 2407-2410, (2012) · Zbl 1386.74084
[18] Doussouki, AE; Guedda, M.; Jazar, M.; Benlahsen, M., Some remarks on radial solutions of föppl-von Kármán equations, Appl. Math. Comput., 219, 4340-4345, (2013) · Zbl 1432.74144
[19] Gorder, RA, 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, 1-6, (2012)
[20] Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Ph.d thesis (1992)
[21] Liao, SJ, Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul., 14, 983-997, (2009) · Zbl 1221.65126
[22] Zou, K.; Nagarajaiah, S., An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation, Commun. Nonlinear Sci. Numer. Simul., 22, 780-792, (2014) · Zbl 1329.37051
[23] Gorder, RA; Vajravelu, K., Analytic and numerical solutions to the laneemden equation, Phys. Lett. A, 372, 6060-6065, (2008) · Zbl 1223.85004
[24] Varol, Y.; Oztop, HF, Control of buoyancy-induced temperature and flow fields with an embedded adiabatic thin plate in porous triangular cavities, Appl. Therm. Eng., 29, 558-566, (2009)
[25] Mastroberardino, A., Homotopy analysis method applied to electrohydrodynamic flow, Commun. Nonlinear Sci. Numer. Simul., 16, 2730-2736, (2011) · Zbl 1221.76151
[26] Gorder, RA, Relation between laneemden solutions and radial solutions to the elliptic heavenly equation on a disk, New Astron., 37, 42-47, (2015)
[27] Ablowitz, MJ; Ladik, JF, Nonlinear differential-difference equations and fourier analysis, J. Math. Phys., 17, 1011-1018, (1976) · Zbl 0322.42014
[28] Stein, EM; Weiss, G., Introduction to fourier analysis on euclidean spaces, Princeton Math. Ser., 212, 484-503, (2009)
[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)
[30] Zhou, YH; Wang, JZ, Generalized gaussian integral method for calculations of scaling function transform of wavelets and its applications, Acta Mathematica Scientia(Chinese Edition), 19, 293-300, (1999) · Zbl 0941.65015
[31] Chen, MQ; Hwang, C.; Shih, YP, The computation of wavelet-galerkin approximation on a bounded interval, Int. J. Numer. Methods Eng., 39, 2921-2944, (1996) · Zbl 0884.76058
[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, 14-26, (2013)
[33] Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. P.Noordhoff Ltd (1953) · Zbl 0052.41402
[34] Tian, J.: The Mathematical Theory and Applications of Biorthogonal Coifman Wavelet Systems. Rice University, Ph.D. thesis (1996)
[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)
[36] Katsikadelis, JT; Nerantzaki, MS, Non-linear analysis of plates by the analog equation method, Comput. Mech., 14, 154-164, (1994) · Zbl 0803.73076
[37] Azizian, ZG; Dawe, DJ, Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method, Comput. Struct., 21, 423-436, (1985) · Zbl 0592.73106
[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, 58-65, (2000) · Zbl 0974.74077
[39] Al-Tholaia, MMH; Al-Gahtani, HJ, Rbf-based meshless method for large deflection of elastic thin plates on nonlinear foundations, Eng. Anal. Bound. Elements, 51, 146-155, (2015) · Zbl 1403.74049
[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, 2136-2144, (2013) · Zbl 1344.65068
[41] Katsikadelis, JT, Large deflection analysis of plates on elastic foundation by the boundary element method, Int. J. Solids Struct., 27, 1867-1878, (1991) · Zbl 0825.73913
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