Geometrically Exact Finite Element Formulations for Curved Slender Beams: Kirchhoff-Love Theory vs. Simo-Reissner Theory (1609.00119v1)

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.

• The paper compares geometrically exact finite element formulations for curved slender beams based on Kirchhoff-Love vs. Simo-Reissner theories, addressing computational challenges like locking.
• Numerical results demonstrate that Kirchhoff-Love formulations offer lower spatial error, better nonlinear solver performance, and increased efficiency for beams with high slenderness ratios.
• The findings have significant implications for accurate and efficient modeling in fields like biomechanics and aerospace engineering by mitigating locking effects in complex geometries.

Essay on Geometrically Exact Finite Element Formulations for Curved Slender Beams

The paper "Geometrically Exact Finite Element Formulations for Curved Slender Beams: Kirchhoff-Love Theory vs. Simo-Reissner Theory" presents a detailed comparative paper of finite element formulations for modeling slender beams within the contexts of Kirchhoff-Love and Simo-Reissner theories. This work stands out in its approach to anisotropic slender beams with complex initial curvatures, addressing essential computational and mechanical principles such as objectivity, path-independence, and the elimination of locking effects—issues pertinent to the accuracy and efficiency of simulations for mechanical systems.

Overview of Finite Element Formulations

The paper proposes four novel finite element variants for slender beam simulations, all based on a geometrically exact application of the Kirchhoff-Love theory. This formulation diverges from the more common Simo-Reissner theory by emphasizing its shear-free qualities, which are particularly advantageous for high beam slenderness ratios. By exploring two different rotation interpolation schemes—strong and weak Kirchhoff constraint enforcement—as well as nodal triad parametrizations, a nuanced mixture of geometrically and dynamically exact solutions is offered.

The Kirchhoff-Love formulations are significant for their capability to represent general 3D, large-deformation problems accurately. The authors demonstrate that these shear-free formulations can offer lower spatial discretization error levels, improved nonlinear solver performance, and smoother geometric representation compared to shear-deformable Simo-Reissner elements, especially in the range of high beam slenderness.

Numerical Results and Implications

The numerical findings of the paper emphasize the advantages of choosing Kirchhoff-Love type formulations over Simo-Reissner types. The results confirmed that Kirchhoff-Love formulations could achieve strong convergence rates and reduce computational overhead by minimizing degrees of freedom required for simulation. Notably, numerical tests involving slender beams exhibited improved solver efficiency and accuracy, critical for applications from industrial webbings to DNA strand analysis in biophysics.

The implications are profound in fields demanding accuracy in beam modeling where high slenderness ratios are involved. The formulation's capacity to remove locking effects enhances practical scenarios where traditional methods fail to efficiently model dynamic and curved geometries. Practically, this translates to substantial efficiency gains in computational mechanics applications with potential impacts extending to advanced simulations in biomechanics, material science, and aerospace engineering.

Future Directions

Given these advancements, future research suggested by this paper should explore further integration of these finite element formulations into real-world engineering applications. Expanding the scope of the Kirchhoff-Love models to accommodate dynamic problems with variable cross-section shapes and irregular initial geometries appears promising. There is also potential in employing these methodologies in simulations that require precise modeling of contact interactions within complex mechanical systems, thus broadening their applicability in virtual testing environments.

Additionally, leveraging the benefits of these formulations could be foundational in advancing simulations involving DNA mechanics or cytoskeletal dynamics where accurate modeling of highly slender biopolymer networks remains challenging.

In conclusion, the paper establishes a robust paradigm for slender beam modeling, championing the Kirchhoff-Love theory’s advantages in efficacy and computational performance over classical Simo-Reissner approaches. Its contributions serve as pivotal benchmarks for modern computational methods in applied mechanics and engineering simulations.