A Nodal-Lie-Group Beam Element for Absolute Nodal Coordinate Formulations

Abstract

A new 12DOF beam element is proposed to simulate large deformation and large rotation based on the 24DOF absolute nodal coordinate formulation (ANCF) beam element proposed before. The centerline of the beam is interpolated by Hermite shape functions, and the frame of the beam is interpolated by linear shape functions. To reduce DOFs, the Lie-group method is used to normalize and orthogonalize the frame on each node of the beam. This way of using the Lie-group method keeps a linear relationship between the nodal vectors and shape functions and leads to the constant mass matrix and elastic tensors. Therefore, the generalized elastic and inertial forces do not require Gaussian integration at each time-step. To avoid singularity of the rotation, a relative rotation vector is adopted; correspondingly, the generalized-α integrator based on the Lie group is used to solve the dynamic equations. To improve the convergency speed and alleviate the shear locking and Poisson locking problems of this element, the assumed natural strain (ANS) method is adopted. To improve the calculational accuracy of axis stretching and torsion effects, the enhanced assumed strain (EAS) method is adopted. The formulas presented in this paper have been successfully tested in several static and dynamic examples of other ANCF beam elements and analytic solutions.