Comparison of the absolute nodal coordinate and geometrically exact formulations for beams

The modeling of flexibility in multibody systems has received increase scrutiny in recent years. The use of finite element techniques is becoming more prevalent, although the formulation of structural elements must be modified to accommodate the large displacements and rotations that characterize multibody systems. Two formulations have emerged that have the potential of handling all the complexities found in these systems: the absolute nodal coordinate formulation and the geometrically exact formulation. Both approaches have been used to formulate naturally curved and twisted beams, plate, and shells. After a brief review of the two formulations, this paper presents a detailed comparison between these two approaches; a simple planar beam problem is examined using both kinematic and static solution procedures. In the kinematic solution, the exact nodal displacements are prescribed and the predicted displacement and strain fields inside the element are compared for the two methods. The accuracies of the predicted strain fields are found to differ: The predictions of the geometrically exact formulation are more accurate than those of the absolute nodal coordinate formulation. For the static solution, the principle of virtual work is used to determine the solution of the problem. For the geometrically exact formulation, the predictions of the static solution are more accurate than those obtained from the kinematic solution; in contrast, the same order of accuracy is obtained for the two solution procedures when using the absolute nodal coordinate formulation. It appears that the kinematic description of structural problems offered by the absolute nodal coordinate formulation leads to inherently lower accuracy predictions than those provided by the geometrically exact formulation. These observations provide a rational for explaining why the absolute nodal coordinate formulation computationally intensive.