Three-Dimensional Elasticity and Thin Walled Structures
Solutions of elastic thin-walled and beam structures constitute a very significant part of numerical simulations conducted with the finite element method (FEM). The approaches which are most often used in such computations are based on dimesionally reduced models of a beam (solid-or thin-walled), a plate or a shell. The dimensional reduction of the independent variable for these models is achieved by accepting appropriate hypotheses concerning variation of the displacements and stresses in the direction of the eliminated variable. The assumptions are justified due to the significant domination of one or two dimensions of the structure over the remaining ones. Using the dimensionally reduced models becomes cumbersome and of disputable reliability if one considers different kinds of models within one structure or when the real-life structure departs essentially from the ideal model. This happens, for instance, when the wall thickness changes discontinuously, when a local enforcement by stiffening ribs is used, or because of presence of openings, connecting rivets, and so on. In such situations, the analyst might be tempted to exclusively use solid elements of linear elasticity, with optimal flat/elongated shape and anisotropic order of approximation selected in such a way that the computational effort would be comparable to dimensionally reduced methods. One of the early works that expressed the hope for such computations was the article of Szabo and Sharaman  in 1988. They used, with success, solid elements with high order in directions tangent to the surface of a shell to solve shells of wall-thickness h to radius r ratio up to h/r = 0.01. Since then, the methodologies referred to as hierarchical models gained popularity. The approach is based on the variational statement of linear elasticity applied to the displacement field in the natural curvilinear coordinates associated with the normal and tangential directions of the shell. The tangential and normal displacements in these techniques are expanded into specially selected polynomial shaped functions. Various aspects of hierarchical models for thin-walled structures were subject of many works including articles of Babusˇka and Li [22, 103, 23, 24], Stein et al. , Zboin´ski [177, 179, 178], Oden and Cho [121, 46, 47], Basar et al. , Babusˇka
and Schwab , Schwab , Ainsworth , Ainsworth and Arnold [5, 4], Arnold and Falk , Rank et al. , and others. The development of hierarchical modeling for elliptic problems in thin domains was pioneered in 1981 by Vogelius and Babusˇka in their articles [166, 167, 168].