The Treatment of “pinching Locking” in 3d-shell Elements

We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon – that we call “pinching locking” – is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source of locking is present, we are able to obtain error estimates independent of the thickness parameter, which shows that pinching locking is effectively treated. This is also confirmed by some numerical experiments of which we give an account. Mathematics Subject Classification. 65N30, 74K25. Received: May 21, 2002. Revised: October 20, 2002. Introduction As defined and discussed in [10] (see also [6]), “3D-shell” elements are finite elements – for shell structures – which belong to the category of general shell elements (see [1,8]) and are based on a complete quadratic expansion of the displacements across the thickness (as opposed to, e.g., the Reissner–Mindlin kinematical assumption which corresponds to an incomplete linear expansion). These shell elements feature three major advantages compared to standard general shell elements: 1. The complete quadratic expansion allows the use of nodes on the outer (i.e. upper and lower) surfaces of the shell, with degrees of freedom that correspond to displacements only (rotational degrees of freedom are not required). Hence these elements have the same external characteristics (geometry, shape functions) as 3D isoparametric elements. This – in particular – makes the coupling with other elements (fluids, solids, ...) sharing the same outer surfaces straightforward, see [10]. 2. These shell elements can be used with a 3D variational formulation without modifying the constitutive equation, namely without resorting to a plane stress assumption which is rather cumbersome to handle when considering large strains and stresses. 3. We can expect that structures that undergo large strains are much better accounted for with this approach, since the quadratic kinematical assumption allows for complex transverse strains. One of the main challenges in the design of shell finite elements is their ability to resist numerical locking phenomena, see in particular [1,7,9]. This is – to a large extent – still an open problem since we do not know of