Simulation models and research algorithms of thin shell structures deformation. Part II. Algorithms for studying shell structures

Mathematical models of a thin shell deformation, which are considered in the first part of the article, constitute either a variational problem of energy functional minimum in  terms of shell deformation or a boundary problem for differential equations of shell equilibrium. In both cases, the boundary conditions are also introduced according to the type of shell fixation. To solve the specified tasks, the different methods are considered. Using either the Ritz method for the variational problem of energy functional minimum for shell deformation or the Bubnov – Galerkin method for the boundary problem for differential equations of shell equilibrium, we will get systems of linear or nonlinear equations. The finite element method (FEM) in application to shell theory problems also leads to systems of linear equations, and the order of the equations can be very large. It is possible to use the Gauss method to solve the linear systems of algebraic equations in case  the system order is less than 10$^3$. In another case, it is necessary to use iterative methods.  For nonlinear tasks of thin shell theory, the parameter marching method is used. If the load is taken as a parameter, it is the V. V. Petrov's sequential loading method. It allows transforming the nonlinear tasks into a consistent linear solution with coefficients varying at each stage of loading. For solving nonlinear problems of shell theory, we consider the iteration method, when the nonlinear terms are transferred to the right side and successively changed at each iteration stage. In the article, it is also considered the method of quickest descent. A. L. Goldenweiser developed the special method: The asymptotic-integration method of thin shell theory, which is described in the article. If the equilibrium equation of the shell contains the discontinuous function (unit functions, delta-functions), then for this case there is a special G. N. Bialystochny's method, which is also specified in the article. Examples of the application of the described methods for solving specific problems of shell theory are also given.

elastic thin shell, mathematical model, algorithms for solving nonlinear problems, numerical methods, stability of shells