Finite element analysis of shells with layers

Abstract

It is well established that thin shell structures frequently feature narrow bands of strain concentration and localized displacement irregularities referred to as boundary and internal layers. It is crucial to capture these layers properly as they can be sources of structural failures. Unfortunately, while the absence of analytical solutions to most shell problems of practical interest has spawned the development of a variety of finite element formulations over the years, largely speaking these schemes were proposed without a rigorous and comprehensive testing procedure available. We are now faced with a wealth of existing formulations and little way to assess their ability to model boundary and internal layers. Most of the difficulties in assessing the performance of shell finite elements stem from the use of mixed formulations. These are necessary to alleviate the locking phenomenon present in bending-dominated problems when displacement-based formulations are used. We develop a new error measure approach that is physically-based and can be used to assess the performance of mixed-interpolated shell finite element formulations. We apply this approach to the MITC (Mixed Interpolation of Tensorial Components) family of shell elements, a widely-used mixed formulation. We focus in particular on the performance of these elements when employed to analyze problems featuring layers, and specifically we assess the effect of mesh refinement in the regions where layers are present.

(cont.) We demonstrate that the MITC elements are consistent with the basic shell model and find that local mesh refinement allows us to obtain optima order convergence of the MITC solution to the solution of the mathematical model even in the presence of layers. The proposed error measure can be easily extended to other mixed-formulated finite elements used for problems such as incompressible materials and fluids, beams or plates.