Several finite element formulations used in the analysis of large rotation and large deformation problems employ independent interpolations for the displacement and rotation fields. As explained in this paper, three rotations defined as field variables can be sufficient to define a space curve that represents the element centerline. The frame defined by the rotations can differ from the Frenet frame of the space curve defined by the same rotation field and, therefore, such a rotation-based representation can provide measure of twist shear deformations and captures the rotation of the beam about its axis. However, the space curve defined using the rotation interpolation has a geometry that can significantly differ from the geometry defined by an independent displacement interpolation. Furthermore, the two different space curves defined by the two different interpolations can differ by a rigid body motion. Therefore, in these formulations, the uniqueness of the kinematic representation is an issue unless nonlinear algebraic constraint equations are used to establish relationships between the two independent displacement and rotation interpolations. Nonetheless, significant geometric and kinematic differences between two independent space curves cannot always be reduced by using restoring elastic forces. Because of the nonuniqueness of such a finite element representation, imposing continuity on higher derivatives such as the curvature vector is not straight forward as in the case of the absolute nodal coordinate formulation (ANCF) that defines unique displacement and rotation fields. ANCF finite elements allow for imposing curvature continuity without increasing the order of the interpolation or the number of nodal coordinates, as demonstrated in this paper. Furthermore, the relationship between ANCF finite elements and the B-spline representation used in computational geometry can be established, allowing for a straight forward integration of computer aided design and analysis.
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October 2010
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Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations
Ahmed A. Shabana
Ahmed A. Shabana
Department of Mechanical and Industrial Engineering,
e-mail: shabana@uic.edu
University of Illinois at Chicago
, 842 West Taylor Street, Chicago, IL 60607-7022
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Ahmed A. Shabana
Department of Mechanical and Industrial Engineering,
University of Illinois at Chicago
, 842 West Taylor Street, Chicago, IL 60607-7022e-mail: shabana@uic.edu
J. Comput. Nonlinear Dynam. Oct 2010, 5(4): 044501 (5 pages)
Published Online: July 28, 2010
Article history
Received:
July 20, 2009
Revised:
December 11, 2009
Online:
July 28, 2010
Published:
July 28, 2010
Citation
Shabana, A. A. (July 28, 2010). "Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations." ASME. J. Comput. Nonlinear Dynam. October 2010; 5(4): 044501. https://doi.org/10.1115/1.4001909
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