Analysis of the h-Adaptive Finite Element Method in Static Problems of Cylindrical Shells: II. Piecewise Linear Approximations and an A Posteriori Error Estimator

Abstract

In this paper, we extend the first part of our study, devoted to the analysis of the well-posedness of the axisymmetric variational formulation of the static boundary value problem for a Timoshenko cylindrical shell and to the identification of criteria for its singular perturbation. The present contribution focuses on the development of a finite element algorithm for computing piecewise linear approximations of the generalized displacement vector of the shell. To avoid computationally expensive numerical integration procedures, we derive explicit algebraic expressions for the contributions of individual finite elements to the resulting block tridiagonal system of linear algebraic equations associated with the finite element method (FEM). This representation significantly simplifies the implementation while preserving the accuracy of the method. Furthermore, an elementwise defined a posteriori error estimator (APEE) is proposed for the constructed FEM approximations. The corresponding error indicators are based on quadratic shape functions with coefficients characterizing the approximation residuals evaluated at the centers of finite elements. These coefficients are proportional to the square of the element length and include factors that emulate the values of the second derivatives of the displacement components at the element centroid. The remainder of the paper is devoted to numerical experiments for a fully clamped cylindrical shell exhibiting pronounced boundary layers in the vicinity of its ends. A detailed convergence analysis of the FEM approximations on uniformly refined meshes, performed in equivalent norms of the space of admissible displacements, demonstrates the reliability and efficiency of the proposed a posteriori error estimator, as well as its ability to reproduce the true FEM errors with high accuracy