A mixed Petrov-Galerkin Cosserat rod finite element formulation

TL;DR

This paper introduces a novel mixed Petrov-Galerkin finite element method for Cosserat rods that avoids singularities and locking, improving computational efficiency and robustness through a total Lagrangian approach with quaternion-based orientation parametrization.

Contribution

It presents a new singularity-free, locking-free finite element formulation for Cosserat rods using mixed Petrov-Galerkin approach with quaternion orientation parametrization.

Findings

Enhanced computational robustness and efficiency demonstrated on benchmark examples.

Reduction in load steps and iterations needed for convergence.

Effective mitigation of shear locking effects.

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.