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 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.

Robust and Efficient Large Deformation Analysis of Kirchhoff–love Shells: Locking, Patch Coupling and Iterative Solution

“…Liguori, J. Kiendl, A. Reali and G. Garcea unrelated to the accuracy of the interpolation and occurs because the stresses σ σ σ g (d e ), used to evaluate the tangent stiffness matrix K e (σ σ σ g (d e ), d e ), are forced to satisfy the constitutive equations at each iteration. In [11], a strategy called Mixed Integration Point has been proposed in order to overcome these limitations in standard displacement-based finite element problems and then extended and tested in displacement-based isogeometric formulations [6,12]. The fundamental idea of the MIP Newton scheme is to relax the constitutive equations at the level of each integration point during the iterations.…”

“…It consists of a relaxation of the constitutive equations at each integration point during the Newton iterative process. Within a MIP framework, Newton's method can withstand much larger increments with a reduced number of iterations to obtain an equilibrium point compared to a standard Newton's strategy without the need of defining a stress interpolation [6,12]. Engineering models of appreciable complexity are typically modeled using multiple surface patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces.…”

Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models

“…It consists in relaxing the constitutive equations at each integration point, combining the advantages of a displacement formulation (reasonable computational cost) with the advantages of a mixed formulation (convergence in Newton more robust [38]). This method has been successfully used for Reissner beams [37], isogeometric solid shells [36], and isogeometric Kirchhoff rods [39]. The implementation is relatively simple, and we refer to [36,37] for its details.…”

On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam

“…These models are rarely examined within the analytic context and a few isogeometric numeric approaches have emerged lately, [2], [3], and [5]. Generally, IGA enables the high-accuracy calculation of curved beams and analytical results can serve for preliminary validation of these formulations, [36], [37], [38], [39], [40], [41]. Therefore, the accuracy and limits of the applicability of the analytic approach for linear analysis of the BE beam model without warping are investigated here.…”