Interpolation schemes for geometrically exact beams: A motion approach

Abstract

This paper focuses on the interpolation of the kinematic fields describing the configuration of geometrically exact beams, namely, the position and rotation fields. Two kinematic representations are investigated: the classical approach that treats the displacement and rotation fields separately and the motion approach that treats those two fields as a unit. The latter approach is found to be more consistent with the kinematic description of beams. Then, two finite element interpolation strategies are presented and contrasted. The first interpolates the displacement and rotation fields separately, whereas the second interpolates both fields as a unit, in a manner consistent with the motion approach. The performance of both strategies is evaluated in light of the fundamental requirements for the convergence of the finite element method: the ability to represent rigid-body motions and constant strain states. It is shown that the traditional uncoupled interpolation scheme for the position field approximates that based on the motion approach and that the coupling induced by the interpolation of motion yields superior convergence rates for the representation of constant strain states. This property is known to lead to finite elements that are less prone to the locking phenomenon.Several issues with these early techniques have been documented in the literature. Crisfield and Jelenić [7,8] were the first to point out that the strain measures used in the formulation should be objective, that is, should be invariant under the addition of a rigid-body motion to the configuration of the system. They showed that several commonly used discretization schemes for rotation fields do not satisfy this requirement: interpolated strain measures do not remain invariant under a superimposed rigid-body motion. Typically, the use of mesh and load step size refinements alleviate these problems. Romero and Armero [9] developed interpolation schemes that guarantee objectivity and the same goal was achieved by Betsch and Steinmann [10] who used a redundant set of generalized coordinates to represent the configuration of the system. Another recurring issue of these early schemes is the locking phenomenon. The requirement for the convergence of the finite element method is that the interpolated field be able to capture constant strain states accurately; see Zienkiewicz [11]. Unfortunately, when using the polynomial interpolation functions found in finite element textbooks for interpolating beam problems, it is not possible to represent constant or vanishing shear strain distributions, for instance, resulting in the well-known shear locking phenomenon. Typically, reduced integration, higher-order interpolation schemes, or both, are implemented to mitigate this problem.The deficiencies of these early schemes limit their performance and indicate that fundamental underlying concepts are not treated properly. The goal of this paper is to present a fresh look at beam formulation. The choice of an interpolation strategy is rooted in the kinematic de...

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