The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Abstract

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin's approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton's iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.

Mathematical Models with Buckling and Contact Phenomena for Elastic Plates: A Review

“…As it is mentioned above, a classical solution for Equations ( 2) and (6) has not yet been found. However, the buckling and postbuckling behavior of the solution of the nonlinear model in Equations ( 2) and (6) has intensively been studied numerically by many authors, e.g., Caloz and Rappaz [42], Dossou and Pierre [49], Chien et al [44], Chien et al [52], Muradova [46,47], and Matkowsky and Putnick [43]. The existing techniques for treating the nonlinear mechanical model are mainly based on finite element analysis, finite difference approximations, and spectral and pseudo-spectral methods.…”

“…In the works [46,47], the Fourier transform is proposed for Equations ( 2) and ( 6). The solution is expanded into double Fourier series and the partial sums are considered.…”

“…The coefficients in the series are good guides for following branches of the solution and the trigonometrical functions reflect the shape of the eigenfunctions. A detailed bifurcation analysis, based on spectral approach, for the simply supported, partially clamped, and totally clamped plates is presented in [47]. The buckling loads are computed and trained by the neural network for prediction in the paper [54].…”

“…The classical linear and nonlinear plate equations are the main keys for creating various plate models, including delamination of composite plates (Vasiliev and Morozov [27], Stavroulakis and Panagiotopoulos [28], Storakers and Andersson [29], Xue et al [30], and Haghani et al [31]), contact problems (Ohtake et al [32,33], Borisovich et al [34], and Malekzadeh and Setoodeh [35], Muradova and Stavroulakis [36][37][38][39], Muradova et al [40], Fichera [41]), analysis of buckled plates (Ciarlet and Rabier [3], Caloz and Rappaz [42], Matkowsky; Putnick [43], Chien et al [44], Chien et al [45], Muradova [46][47][48], Dossou and Pierre [49]), etc.…”

Hybrid control of vibrations of a smart von Kármán plate

“…where the global basis functions ω i j , ϕ i j are chosen to match the boundary conditions (see [24]) and N is a natural number. For N → ∞, the partial sums tend to the analytical solution of (1), (2).…”