| The growing demand for reliable shell finite elements has presented, over the last two decades, a formidable challenge to engineers and mathematicians alike. Very few affirmative answers have been developed for locking-free finite elements of proven reliability for thin shell problems of general geometries. In this study, we introduce the dual mixed method as a promising, alternative approach to obtaining a stable, locking-free finite element approximation of the solution of the Naghdi shell problems with small parameter t, the thickness of the shell. Our finite elements employ the energy-splitting approach proposed by Arnold and Brezzi [4] paired with the stability condition, weaker than the inf-sup condition, introduced by Bramble and Sun [19]. In the lowest order case, our analysis shows that if we use piecewise quadratics enriched by bubble functions for approximating the displacements and rotations, the second lowest order Raviart-Thomas [58] space for the shear stress and the lowest order Arnold-Winther [6] space for the membrane stress, we have an element that does not lock and that provides uniform optimal error estimate as long as h2 ≤ Ct for mesh size h. The elements that we propose satisfy the mathematical condition of stability and convergence, and they promise to provide efficient elements for practical solutions. |
