Analysis of h-Adaptive Finite Element Method in Static Problems of Cylindrical Shells: І. Wellposedness of Axisymmetric Variational Formulation

This article is devoted to establishing the correctness of boundary value problems in the statics of Timoshenko shells and the approximation of their solutions using the finite element method (FEM) with a priori guaranteed accuracy. In the first part of this study, the following results are established for this shell model: (i) sufficient (and entirely practical) conditions for the solvability of the variational formulation of the boundary value problem for a cylindrical shell under axisymmetric loading; (ii) the structure of the rigid displacement vectors of such shell; (iii) a criterion for the singular perturbation of the boundary value problem. In the second part, the following are proposed:(iv) an a posteriori error estimator (APE) for piecewise linear FEM approximations of the generalized displacement vector of the shell; (v) a local mesh refinement strategy for the efficient computation of FEM approximations with a predefined level of allowable error. The efficiency and reliability of the developed iterative h-adaptive approximation procedure are illustrated by numerical results from the analysis of model problem solutions, in particular, in comparison with traditional uniform mesh refinement.