Algorithm for reducing computational costs in problems of calculation of asymmetrically loaded shells of rotation

Abstract

The problem of calculating the shells of rotation of a variable along the meridian of rigidity under asymmetric loading is reduced o a set of systems of one-dimensional boundary value problems with respect to the amplitudes of decomposition of the required functions into trigonometric Fourier series.
 A method for reducing the number of one-dimensional boundary value problems required to achieve a given accuracy in determining the stress-strain state of the shells of rotation with a variable along the meridian wall thickness under asymmetric load. The idea of the proposed approach is to apply periodic extrapolation (prediction) of the values of the decomposition coefficients of the required functions using the results of calculations of previous coefficients of the corresponding trigonometric series, thus replacing them with some prediction values calculated by simple formulas.
 To solve this problem, we propose the joint use of Aitken-Steffens extrapolation dependences and Adams method in the form of incremental component, which is quite effective in solving the Cauchy problem for systems of ordinary differential equations and is based on Lagrange and Newton extrapolation dependences.
 The validity of the proposed approach was verified b the results of a systematic numerical experiment by predicting the values of the expansion coefficients in the Fourier series of known functions of one variable.
 The approach is quite effective in the calculation of asymmetrically loaded shells of rotation with variable along the meridian thickness, when the coefficients of decomposition of the required functions into Fourier series are functions of the longitudina lcoordinate and are calculated by solving the corresponding boundary value problem. In this case, the approach allows solving solutions of differential equations for the amplitudes of decompositionin to trigonometric series only for individual "reference" harmonics, and the amplitudes for every third harmonic can be calculated by interpolating their values for all node integration points of the corresponding boundary value problem. This significantly reduces the computational cost of obtaining the solution as a whole.
 As an example, the results of the calculation of the stress-strain state of a steel annular plate under asymmetric transverse loading are given.