Geometrically Exact Finite Element Formulations for Curved Slender Beams: Kirchhoff-Love Theory vs. Simo-Reissner Theory

TL;DR

This paper develops and compares novel geometrically exact Kirchhoff-Love finite element formulations for slender beams with existing Simo-Reissner models, demonstrating improved accuracy and solver performance for highly slender beam problems.

Contribution

It introduces four new Kirchhoff-Love finite element variants with strong or weak constraint enforcement and different triad parametrizations, fulfilling key large-deformation and objectivity requirements.

Findings

Kirchhoff-Love elements show lower discretization errors.

Proposed formulations exhibit better nonlinear solver performance.

Shear-free Kirchhoff-Love models are advantageous for highly slender beams.

Abstract

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and Simo-Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a C1-continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large-deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes or preservation of objectivity and path-independence will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff-Love beam elements proposed in this work are the first ones of this type that fulfill all the considered requirements. On the contrary, Simo-Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff-Love formulations can provide considerable numerical advantages when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff-Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo-Reissner element formulations from the literature.