A total Lagrangian, objective and intrinsically locking-free Petrov-Galerkin SE(3) Cosserat rod finite element formulation

TL;DR

This paper introduces a new finite element formulation for Cosserat rods that is objective, locking-free, and intrinsically consistent, utilizing SE(3) transformations and a Petrov-Galerkin approach for improved accuracy and robustness.

Contribution

It extends the interpolation of nodal orientations to Euclidean transformations using SE(3) and develops a Petrov-Galerkin formulation that avoids locking and preserves objectivity.

Findings

The formulation is locking-free in numerical tests.

Objectivity is preserved after discretization.

Effective in both static and dynamic simulations.

Abstract

Based on more than three decades of rod finite element theory, this publication unifies all the successful contributions found in literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jeleni\'c in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov-Galerkin projection method, we propose a novel rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parametrization in terms of a minimal number of nodal unknowns are demonstrated by conclusive numerical examples in both statics and dynamics.