Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.