Co-rotational finite element formulation used in the Koiter-Newton method for nonlinear buckling analyses

The Koiter–Newton approach is a novel reduced order modeling technique for buckling analysis of geometrically nonlinear structures. The load carrying capability of the structure is achieved by tracing the entire equilibrium path in a stepwise manner. At each step a reduced order model generated from Koiter׳s asymptotic expansion provides a nonlinear prediction for the full model, corrected by a few Newton steps. The construction of the reduced order model requires derivatives of the strain energy with respect to the degrees of freedom up to the fourth order, which is two orders more than traditionally needed for a Newton based nonlinear finite element technique. In this paper we adopt the co-rotational formulation to facilitate these complex differentiations. We extend existing co-rotational beam and shell element formulations to make them applicable for the high order derivatives of the strain energy. The geometrical nonlinearities are taken into account using derivatives of the local co-rotational frame with respect to global degrees of freedom. This is done outside the standard element routines and is thus independent of the element type. We utilize three configurations and the nonlinear rotation matrix to describe finite rotations of the shell accurately, and profit from the automatic differentiation technique to optimize the programming of high order derivatives. The performance of the proposed approach using the co-rotational formulation is demonstrated using benchmark examples of isotropic and laminated composite structures