Mixed and nonconforming finite element approximations of Reissner-Mindlin plates

摘要

We consider a mixed finite element approximation of the Reissner-Mindlin plate model, which plays an important role in the simulation of plates and shells of small to moderate thickness. On triangles and quadrilaterals, low-order finite elements are proposed: the rotation is approximated by conforming isoparametric (bi)linear elements and the transverse displacement by constant elements and the shear stress by the lowest-order Raviart-Thomas elements. We prove that the new method satisfies the K-ellipticity and the Inf-Sup condition in the abstract framework of the Babuska-Brezzi theory for mixed problems. Consequently, the method is uniformly stable and uniformly optimally convergent (independent of the thickness of the plate). Moreover, a local postprocessing approach and the implementation by either the augmented Lagrangian algorithm or the hybridization method are discussed, including equivalence to some nonconforming methods. In particular, a new nonconforming method is deduced, which can accommodate arbitrary regular quadrilaterals.