A hierarchic optimisation approach towards locking-free shell finite elements

Abstract

A hierarchic optimisation approach is presented for relieving inaccuracies in conforming shell elements arising from locking phenomena. This approach introduces two sets of strain modes: (i) objective strain modes, defined in the physical coordinate system, and (ii) corrective strain modes, representing conforming strains enhanced with hierarchic strain modes. This leads to two alternative families of element, objective and corrective, both arising from minimising the difference between objective and corrective strains. Importantly, the proposed approach not only alleviates shear and membrane locking, but it also addresses locking arising from element distortion. The application of the proposed optimisation approach is demonstrated for a 9-noded quadrilateral Lagrangian shell element, where the membrane, bending and transverse shear strains are separately optimised, all within a local co-rotational framework that extends the element application to geometric nonlinear analysis. Several numerical examples, including cases with geometric and material nonlinearity, are finally presented to illustrate the effectiveness of the optimised 9-noded shell element in relieving the various sources of locking.