The continuous method of solution continuation with respect to the best parameter in the calculation of shell structures

The paper considers the numerical solution process of the strength and stability problems of thin-walled shell structures taking into account the geometric nonlinearity, transverse shifts, and material orthotropy. Similar problems have great importance in mechanical engineering, aerospace industry, and building sector. Numerical simulation of these problems using the Ritz method is reduced to solving the systems of nonlinear algebraic equations regarding the increments of the desired functions. However, the numerical solution of the systems is related to a number of difficulties associated with the presence on the solution set curve of limiting singular points or bifurcation points in which the Jacobi matrix degenerates. The paper aims to develop a computational methodology making it possible to overcome the indicated difficulties for the problems considered. For this purpose, we used the method of solution continuation with respect to the parameter developed in the works of M. Lahaye, D. Davidenko, I. Vorovich, E. Riks, E. Grigolyuk, V. Shalashilin, E. Kuznetsov, and other scientists. For the system of algebraic or transcendental equations, the solution of which is a one-parameter family of curves, the method of solution continuation is as follows. The problem original parameter is replaced with a new one, the use of which enables to overcome the singular points contained on the solution set curve. Three variants of the solution continuation method were described: Lahaye's method, Davidenko's method, and the best parameterization method. Their advantages and disadvantages were shown. The effectiveness of the best parameterization for solving the strength and stability problems of shell structures was shown using the example of the calculation of double curvature shallow shells rectangular in plan. Verification of the proposed approach was carried out. The results obtained show that the use of the technique based on the combination of the Ritz method and the method of solution continuation with respect to the best parameter allows investigation of the strength and stability of the shallow shells, overcoming the singular points of the “load-deflection” curve, obtaining the values of the upper and lower critical loads, and detecting the bifurcation points and investigate the supercritical behavior of the structure. These results are essential in shell structure calculation, for which there are the effects of snapping and buckling observed in various applications.