A consistent finite element formulation of the geometrically non-linear Reissner-Mindlin shell model

Abstract

We present an objective, singularity-free, path independent, numerically robust and efficient geometrically non-linear Reissner-Mindlin shell finite element formulation. The formulation is especially suitable for higher order ansatz spaces. The formulation utilizes geometric finite elements presented by Sander [ 47 ] and Grohs [ 34 ] for the interpolation on non-linear manifolds. The proposed method is objective and free from artificial singularities and spurious path dependence. Due to the fact that the director field lives on the unit sphere, a special linearization procedure is required to obtain the stiffness matrix. Here, we use the simple constructions of Absil et al. [ 2 , 3 ], which yields an easy way to obtain the correct tangent operator of the potential energy. Additionally, we compare three different interpolation schemes for the shell director that can be found in the literature, where one of them is applied for the first time for the Reissner-Mindlin shell model. Furthermore, we compare the exponential map to the radial return normalization as procedure to update the nodal directors and conclude the superiority of the latter, in terms of fewer load steps. We also investigate the construction of a consistent tangent base update scheme. Path independence, efficiency and objectivity of the formulation are verified via a set of numerical examples.