Isogeometric analysis for nonlinear planar Kirchhoff rods: Weighted residual formulation and collocation of the strong form

Abstract High-order shape functions used in isogeometric analysis allow the direct solution not only to the weighted residual formulation of the strong form, opening the door to new integration schemes (e.g. reduced Gauss–Lobatto quadrature, integration at superconvergent sites) but also to collocation approaches (e.g. using Greville or superconvergent collocation points). The goal of the present work is to compare these different methods through the application to the planar Kirchhoff rod, a fourth-order rotation-free formulation used to model slender beams under large deformations. Robustness of the geometrically-nonlinear solution is improved using the Mixed Integration Point (MIP) Newton, a method that combines the advantages of displacement and mixed formulations. Based on the observations of the convergence plots, convergence order estimates are provided for each discretization method. Advantages and weaknesses in terms of robustness and computation cost are also discussed for a set of benchmarks.