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

Abstract

Based on more than three decades of rod finite element theory, this publication combines the successful contributions found in the 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 Jelenić in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SEfalse(3false)$$ SE(3) $$‐structure of the Cosserat rod kinematics. Applying a Petrov–Galerkin projection method, we propose a 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 parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.

A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

“…The Euclidean transformation matrix is an element of the special Euclidean group , which is considered here as a Lie subgroup of the general linear group with the matrix multiplication as group operation. While this article can be read with a rudimentary knowledge of matrix Lie groups, the interested reader is referred to [25, Appendix A] for a deeper understanding of the applied concepts. Direct computation readily verifies that the inverse of is Both points and base vectors can be considered as functions of time or some other parameters resulting in parameter‐dependent coordinates.…”

“…In the ‐interpolation, originally proposed by Sonneville et al [47] and recently taken up in [25], the idea from the ‐interpolation to use relative rotation vectors is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists. This leads to a strategy coupling the interpolations of centerline points and cross‐section orientations.…”

“…To simplify the expressions of the discrete virtual work, an independent approximation of the virtual rotation field is suggested in [31]. Very recently, in [25], the authors of this article took up the idea and presented a Petrov–Galerkin rod finite element formulation, in which nodal Euclidean transformation matrices are interpolated with the aid of relative twists; a strategy that originates from Sonneville et al [47] arising from the ‐structure of the Cosserat rod kinematics. Therefore, this interpolation is called here ‐interpolation.…”

“…Regularly, rod formulations are proposed together with very specific update rules for static or dynamic numerical analysis [42]. While the Petrov–Galerkin projection in combination with the ‐interpolation was already discussed in [25], the idea of the paper at hand is to present the modularity of this projection method. Indeed, we suggest an entire family of rod finite elements in which the interpolation of the ansatz functions can be exchanged more or less arbitrarily.…”

“…Eventually, in Section 4.9, the discretization results in the discrete equations of motion of the rod in the form of a first‐order ODE. The rod with the ‐interpolation has already been successfully tested against analytical solutions, see [25]. Therefore, Section 5 gives an in‐depth analysis concerning locking and convergence behavior of the different rod formulations.…”