Nonlinear dynamics of three-dimensional curved geometrically exact beams by a quadrature element formulation

Nonlinear dynamic analyses of three-dimensional (3D) curved geometrically exact beams were carried out using a quadrature element (QEM) formulation. Three types of time integration algorithms were considered, namely, the energy and momentum conserving algorithm, and Newmark-Wilson algorithm using vector-like and ‘mixed’ parameterizations. Both the geometry and the field variables of the 3D curved beams were expressed by the same QEM formulation as in isogeometric analysis (IGA). Numerical tests through benchmark examples in literatures and new examples presented in this work showed that the present formulations for the three types of algorithms are stable, accurate and flexible, and thus are promising in engineering applications. Especially, the present Newmark-Wilson algorithm using vector-like parameterization based on the weak-form QEM were shown to have excellent performance in most time, although its energy is not conserve. The Newmark algorithm using ‘mixed’ parameterization based on the weak-form QEM were shown to be able conserve the energy in most time and has similar performance as the energy and momentum conserving algorithm.