A variational multiscale stabilized finite element formulation for Reissner-Mindlin plates and Timoshenko beams

The theories for thick plates and beams, namely Reissner-Mindlin's and Timoshenko's theories, are well known to suffer numerical locking when approximated using the standard Galerkin finite element method for small thicknesses. This occurs when the same interpolations are used for displacement and rotations, reason for which stabilization becomes necessary. To overcome this problem, a Variational Multiscale stabilization method is analyzed in this paper. In this framework, two different approaches are presented: the Algebraic Sub-Grid Scale formulation and the Orthogonal Sub-Grid Scale formulation. Stability and convergence is proved for both approaches, explaining why the latter performs much better. Although the numerical examples show that the Algebraic Sub-Grid Scale approach is in some cases able to overcome the numerical locking, it is highly sensitive to stabilization parameters and presents difficulties to converge optimally with respect to the element size in the L2 norm. In this regard, the Orthogonal Sub-Grid Scale approach, which considers the space of the sub-grid scales to be orthogonal to the finite element space, is shown to be stable and optimally convergent independently of the thickness of the solid. The final formulation is similar to approaches developed previously, thus justifying them in the frame of the Variational Multiscale concept.