Mathematical model of physically non-linear Kirchhoff plates: Investigation and analysis of effective computational iterative methods

Abstract

Non-linear partial derivative equations describe mathematical models of mechanics of plates and shells. Solving such equations analytically and numerically is a huge problem, and it is possible to solve it using Fourier’s idea, the method of separation of variables, which goes by the name of Fourier. The combination of Fourier and Bubnov–Galerkin methods revolutionized applied mathematics and numerical methods. In this paper, the symbiosis of the Fourier and Bubnov–Galerkin methods was given for the problems of plate and shell theory: the Bubnov–Galerkin method, Kantorovich–Vlasov method, variational iteration methods, Vaindiner and Agranovskii–Baglai–Smirnov methods, applied to each of these methods, which gives practically exact solutions. A rationale for these methods is for the problems of plate theory concerning non-linearities. A mathematical model was constructed to analyse the stress–strain state of elastic–plastic problems in plate theory, according to the deformation theory of plasticity. The above methods solved the problems in elastic–plastic formulation for different types of boundary conditions and loads. It was noted that the procedure of the Agranovskii–Baglai–Smirnov method in combination with the Bubnov–Galerkin methods in higher approximations, Kantorovich–Vlasov method and variational iteration method makes it possible to obtain a practically exact solution of the problem. It was shown that the variational iteration method is the fastest. It should also be noted that this method does not require a system of approximating functions that satisfy the boundary conditions.