A conforming locking-free approximation for a Koiter shell

As in the Naghdi framework, membrane locking is expected for bending-dominated Koiter shell when the thickness decreases. Inspired by Arnold and Brezzi (1997), we design a locking-free mixed finite element method for the Koiter shell. This method is implemented, in terms of the displacement variables, as the minimization of an altered energy over a conforming finite element space. We approximate the tangential displacements by continuous piecewise polynomials augmented by bubbles and the transversal displacements by the consistent HCT (Hsieh-Clough-Tocher) element. The membrane stresses, derived from a partial integration of the membrane energy, is approximated by discontinuous piecewise polynomials. We establish optimal error estimates independent of the thickness under some restrictions which prove that the mixed solution is locking-free. We confirm our theoretical predictions with some numerical tests, in particular, we consider a hemicylindrical shell and an hyperbolic paraboloid shell.