Hierarchic finite elements for thin plates and shells

We consider the numerical approximation of thin plate and shell structures. The plate model is described following the Reissner-Mindlin assumptions while the shell is described using the Naghdi formulation. It is well known that the numerical approximation with standard finite elements suffers of the so-called locking phenomenon, i.e., the numerical solution degenerates as the thickness of the structure becomes smaller. Plates exhibit shear locking and shells show both shear and membrane locking. Several techniques to avoid the numerical locking have been proposed. Here we solve the problems using a family of high order hierarchic finite elements. We present several numerical results that show the robustness of the finite elements, able to avoid in many circumstances the locking behavior.