Analysis of a bilinear finite element for shallow shells. II: Consistency error

Abstract

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound O ( h + t − 1 h 1 + s ) O(h + t^{-1}h^{1+s}) can be obtained if the contribution of transverse shear is neglected. Here t t is the thickness of the shell, h h the mesh spacing, and s s a smoothness parameter. In the bending-dominated case, the uniformly optimal bound O ( h ) O(h) is achievable but requires that membrane and transverse shear strains are of order O ( t 2 ) O(t^2) as t → 0 t \rightarrow 0 . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound O ( h ) O(h) .