An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics

TLDR

The paper introduces a new finite element formulation for geometrically exact rod models in three‑dimensional dynamic elastic regimes. The method uses direct finite element interpolation of director fields to construct time‑stepping algorithms that preserve the Hamiltonian system’s conservation laws, and is demonstrated through numerical simulations. The formulation yields an objective strain approximation and automatically conserves linear and angular momentum, resulting in a more accurate dynamic simulation of rods. © 2002 John Wiley & Sons, Ltd.

Abstract

Abstract We present in this paper a new finite element formulation of geometrically exact rod models in the three‐dimensional dynamic elastic range. The proposed formulation leads to an objective (or frame‐indifferent under superposed rigid body motions) approximation of the strain measures of the rod involving finite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod's cross‐section. Furthermore, the proposed framework allows the development of time‐stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties. Copyright © 2002 John Wiley & Sons, Ltd.