One point quadrature shell elements: a study on convergence and patch tests

One point quadrature shell elements are being widely used in the numerical simulation of shell structures, including sheet forming, because essentially of their computational efficiency. Nowadays, the purpose of using one point quadrature shell elements is not only related to computational efficiency but also because these elements have shown to be simultaneously robust and accurate in the simulation of complex sheet metal forming processes. The main objective of this work is to study the convergence behavior of different one-point quadrature shell elements and their ability to pass the membrane and bending patch tests. For comparison purposes, two new elements include a new formulation for the membrane strain field in order to further improve the membrane behavior of the element developed in previous work of (in Cardoso et al. Comput Meth Appl Mech Eng 191:5177, 2002). The original convective membrane strains of Cardoso et al. (Comput Meth Appl Mech Eng 191:5177, 2002) (in the stabilization matrices only) are thus replaced by new membrane strains, constructed directly at the co-rotational coordinate system (located at the element's center). It is thus proved that with this new membrane formulation the elements pass now all the patch tests but, for warped (or curved) element geometries, their accuracy is not as good as the original element of (Cardoso et al. in Comput Meth Appl Mech Eng 191:5177, 2002) based on the convective coordinate system. In the numerical results presented in this paper, comprehensive comparison and discussion of these formulations are made for well known linear benchmark examples.