The solution of some problems in the theory of reticulate shells

Abstract

THE results of a numerical solution of a boundary value problem for a strongly elliptic system of second-order equations are described. The difference equations of the mesh method for the original problem are solved by a two-step iterative method with Chebyshev parameters and by the alternating directions method as an internal iterative process. The results of numerical experiments are quoted, and compared with results obtained by other methods. The original problem describes the displacement of a flexible reticulate shell, loaded by internal pressure and external forces distributed over the shell surface. Structures of this kind are used in pneumatic tyres. The computational results relate to loading of the shell over a part of its surface. The present paper deals with the numerical solution of boundary value problems for systems of equations of a strongly elliptic type, describing the equilibrium of flexible reticulate shells (toroidal shells of revolution and cylindrical shells), loaded by external forces and by internal pressure (see [1] ). Such problems arise in the design of structures which are armoured by extensible threads, e.g., tyres. The unknowns are the components of the shell displacements. While the boundary conditions and external loads are common to problems for shells of this kind, the coefficients of the equation of equilibrium depend on the type of shell: they are constants in the case of a cylindrical shell, and variables along the meridian in the toroidal case. Since the equations retain the same structure and type, it is possible to use a unified algorithm for their solution. The systems in question are the systems of Euler equations for the variational problem of minimizing a functional, where the functional represents the total energy of deformation by the internal pressure and the external forces on the shell. As the approximate method of solution we used a finite-difference method (mesh method), in which the system of equations is the system of Euler equations for minimizing a discrete functional on a mesh, approximating the initial functional. The systems of difference equations thus obtained were solved by a two-step iterative method with Chebyshev parameters (see [2,3]) and by the alternating directions method as an internal iterative process. The numerical results quoted may be of interest, not only from the point of view of solving such problems in reticulate shell theory, but also from the point of view of checking experimentally iterative methods for solving the difference analogues of strongly elliptic systems. The iterative methods employed are analyzed theoretically in [2, 3]. The effectiveness of the methods depends on the choice of difference operators, spectrally equivalent to the difference analogue of the initial differential operator. Such problems were investigated for the equations of equilibrium of cylindrical shells in [4, 5]. Methods similar to those of [4, 5] may also be used for shells of revolution. However, in the present paper the required equivalence conditions, which determine the iteration parameters and hence the convergence rate of the iterations, were found experimentally, since the coefficients of the equations can themselves be found approximately for a shell of revolution by solving a Cauchy problem for a system of ordinary differential equations [6].