An efficient implicit time stepping scheme for nonlinear dynamics analysis of 3D geometrically exact beam using exponential mapping

Abstract

In this article, we propose a new integrator scheme for a 3D geometrically exact beam. This scheme can conserve or decay energy, which is needed in dealing with stiff equations and long-term computations. This scheme is based on the midpoint rule, which is applied to both kinematics variables and strains. The scheme enforces the orthogonality of the rotation matrix at the midpoint, which provides an advantage against the previously proposed version of the midpoint scheme. The incremental rotation vector and exponential mapping are implemented to define the corresponding representation of the rotation matrix. Several numerical simulations are presented to demonstrate that the proposed scheme is more efficient than its predecessors in terms of overall stability and computational robustness.

Keywords:

• Dynamic
• geometrically exact beam
• energy conservation
• energy decay
• exponential mapping