Safronova I. Models and algorithms for accelerating the convergence of iterative processes in the problems of calculation and optimization of shell elements of structures.

The dissertation is devoted to the development of methods for calculating shell structures with inhomogeneous parameters. An effective algorithm for accelerating the convergence of iterative processes arising in solving nonlinear problems of calculation and selection of optimal parameters of shell elements of structures of modern technology (in particular, annular plates, sensitive membranes of sinusoidal profile, dynamometric washers, bellows) meridian shape and for accelerating the variable stiffness at large displacements was developed. The approach is based on the usage of the author's methods of joint application of the relaxing multiplier method (linear extrapolation), Lagrange and Newton polynomials (in the form of the Adams method) and the Aitken-Stefansson iterative process. The essence of the approach is to reduce the number of stages of the iterative process of solving a sequence of linear boundary value problems by periodically extrapolating the values of linear components based on successful previous steps, instead of performing the entire volume of calculations on the k-th step. Effective algorithms are constructed and the results of solving nonlinear boundary value problems for various mathematical models of describing the behavior of such shells with irregular parameters are presented. The reliability of the approach is confirmed by the results of special experimental studies. For the case of asymmetrically loaded shells of rotation of a variable along the stiffness meridian, the developed algorithm is used to reduce the number of solutions of boundary value problems for systems of ordinary differential equations with variable coefficients by predicting the values of coefficients of Fourier series. For a variable in two directions of rigidity the problem is solved by applying the discrete-continuum method of lines, when in the circumferential direction a finite-difference approach is used, in the meridional - the problem of integrating one-dimensional boundary value problems, and the developed algorithm for convergence acceleration tasks. The results of the dissertation, which are presented in the form of a description of algorithms, graphs and tables of numerical calculations and experimental data, can be directly used to reduce computational costs in calculation problems and select optimal parameters for a wide range of shell mechanics problems. Keywords: convergence acceleration, numerical algorithms, variable stiffness, flexible shell elements, large displacements, experimental studies