Multisymplectic Galerkin Lie group variational integrators for geometrically exact beam dynamics based on unit dual quaternion interpolation — no shear locking

Leitz, Thomas; Almagro, Rodrigo T. Sato Martín de; Leyendecker, Sigrid

doi:10.1016/j.cma.2020.113475

Cited by 16 publications

(14 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For rotations being parametrized by unit quaternions, the corresponding Lie groups are multiples of the semi-direct product S 3 R 3 , see [11,19]. This semi-direct product (as well as the special Euclidean group) refers to rigid body motions in space and its application in beam theory helps to reduce the risk of locking phenomena [21,34], see also the last part of the present paper. Recently published paper [21] follows a very similar approach, while using higher order integration schemes but being restricted to the unconstrained case.…”

Section: Introductionmentioning

confidence: 88%

“…This semi-direct product (as well as the special Euclidean group) refers to rigid body motions in space and its application in beam theory helps to reduce the risk of locking phenomena [21,34], see also the last part of the present paper. Recently published paper [21] follows a very similar approach, while using higher order integration schemes but being restricted to the unconstrained case. In the present paper, however, the straightforward incorporation of internal and external constraints allows, e.g., enforcing the cross sections to remain normal to the centerline tangent such that transverse shearing is inhibited, see also [15,17] as well as Sect.…”

Section: Introductionmentioning

confidence: 91%

“…Since this behavior cannot be observed here even for this very coarse spatial discretization, we can conclude that shear locking is not present in our beam model. The recently published paper [21] describes a similar experiment. Furthermore, the authors of [21] prove that there is no locking in their model because…”

Section: Roll-upmentioning

confidence: 96%

“…The recently published paper [21] describes a similar experiment. Furthermore, the authors of [21] prove that there is no locking in their model because…”

Section: Roll-upmentioning

confidence: 96%

“…Notice that due to the 360°rotation of the right end, the unit quaternion p (n) K approaches the antipodal −p (n) 0 which encode the same orientation. The right plot shows the norm v (n) K 2 of the velocity vector of the right end of the beam over time t (n) of the semi-direct product structure of the unit dual quaternions used in [21], similar to the structure of S 3 R 3 used in the present paper. 7…”

Section: Roll-upmentioning

confidence: 99%

See 4 more Smart Citations

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021

View full text Add to dashboard Cite

In this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

show abstract

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 91%

Section: Roll-upmentioning

confidence: 96%

“…The recently published paper [21] describes a similar experiment. Furthermore, the authors of [21] prove that there is no locking in their model because…”

Section: Roll-upmentioning

confidence: 96%

Section: Roll-upmentioning

confidence: 99%

See 3 more Smart Citations

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021

View full text Add to dashboard Cite

show abstract

Hamilton–Pontryagin spectral-collocation methods for the orbit propagation

2021

View full text Add to dashboard Cite

Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics

Schubert,

Sato Martín de Almagro,

Nachbagauer

et al. 2023

Multibody Syst Dyn

Self Cite

View full text Add to dashboard Cite

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE.

show abstract

Multisymplectic Galerkin Lie group variational integrators for geometrically exact beam dynamics based on unit dual quaternion interpolation — no shear locking

Cited by 16 publications

References 17 publications

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

Hamilton–Pontryagin spectral-collocation methods for the orbit propagation

Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics

Contact Info

Product

Resources

About