Approximation of shell layers using bilinear elements on anisotropically refined rectangular meshes

In this paper, we study numerical locking effects in the finite element approximation of shell layers. The focus is on four-node degenerated elements that are perhaps the most widely used in practical computations. Our approach is based on simplified reformulations of the original 3D elements in the context of a classical shallow shell model, where the deformation of the shell is described in terms of the three displacements of the middle surface and two dimensionless rotations related to transverse shear deformations. Within that model we compute finite element approximations of layers generated by smooth line segments such as the boundary line. Our numerical results show that approximation failure occurs when bilinear degenerated elements are used on anisotropically refined meshes and that the accuracy depends substantially on the specific geometry of the shell. By analyzing the layers as functions of the thickness of the shell, we obtain theoretical error estimates in the energy norm framework that predict error amplification for small values of the thickness. The error estimates are then verified by conducting further numerical experiments using overrefined meshes.