A mixed Petrov-Galerkin Cosserat rod finite element formulation

TL;DR

The paper tackles singularity and locking issues in Cosserat rod finite element formulations by introducing a total Lagrangian mixed Petrov–Galerkin approach that uses nodal centerline positions and unit quaternions to describe cross-section orientation.A quaternion-based interpolation coupled with a Petrov–Galerkin projection yields an objective, singularity-free kinematic representation and a mixed Hellinger–Reissner functional that introduces independent resultant forces and moments to mitigate locking.The formulation supports constrained theories such as Kirchhoff–Love via compliance-based constitutive laws and demonstrates superior numerical robustness, requiring fewer load steps and iterations than displacement-based methods across six benchmark problems.Numerical results show complete elimination of membrane and shear locking, improved convergence rates, and sustained performance even for highly slender rods, indicating strong potential for real-time or control-oriented applications when extended to dynamics.

Abstract

This paper presents a total Lagrangian mixed Petrov-Galerkin finite element formulation that provides a computationally efficient approach for analyzing Cosserat rods that is free of singularities and locking. To achieve a singularity-free orientation parametrization of the rod, the nodal kinematical unknowns are defined as the nodal centerline positions and unit quaternions. We apply Lagrange interpolation to all nodal kinematic coordinates, and in combination with a projection of non-unit quaternions, this leads to an interpolation with orthonormal cross-section-fixed bases. To eliminate locking effects such as shear locking, the variational Hellinger-Reissner principle is applied, resulting in a mixed approach with additional fields composed of resultant contact forces and moments. Since the mixed formulation contains the constitutive law in compliance form, it naturally incorporates constrained theories, such as the Kirchhoff-Love theory. This study specifically examines the influence of the additional internal force fields on the numerical performance, including locking mitigation and robustness. Using well-established benchmark examples, the method demonstrates enhanced computational robustness and efficiency, as evidenced by the reduction in required load steps and iterations when applying the standard Newton-Raphson method.

Figures (28)

• Figure 1: Setup of the experiment
• Figure 2: Tip displacement
• Figure 3: Deformed configurations
• Figure 5: 45° bent experiment: Spatial convergence rates for the different slenderness ratios and different formulations of the internal virtual work. For the rod, discretized with polynomial degree $p$ and $n_\mathrm{el}$ elements, the number of nodes is $N = (p n_\mathrm{el} + 1)$. The used error measure $e^{100}_{{\bm{\theta}}}$ is a combination of position and orientation error at $100$ points along the rod. The shown rates are for $\mathcal{Q}^1$ (), $\mathcal{Q}^2$ () and ${SE}3$ (), interpolations. The additional lines are proportional to $N^{-2}$ () and $N^{-3}$ ().
• Figure 6: Quaternion interpolation with polynomial degree $p=1$ and $n_\mathrm{el}=8$ elements ($N=9$ nodes)