Abstract
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} with semi-direct product Lie group structure, where mathbb{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.
Highlights
The present work aims to study a geometrically exact Cosserat rod model using a structure preserving variational integrator for constrained systems on Lie groups
For rotations being parametrized by unit quaternions, the corresponding Lie groups are multiples of the semi-direct product S3 R3, see [11, 19]
We have shortly introduced the toolbox of Lie group structured configuration spaces and the concept of velocity and derivative vectors with special attention to the Lie group of rigid body motions S3 R3, where rotations are parametrized using unit quaternions S3
Summary
The present work aims to study a geometrically exact Cosserat rod model using a structure preserving variational integrator for constrained systems on Lie groups. For rotations being parametrized by unit quaternions, the corresponding Lie groups are multiples of the semi-direct product S3 R3, 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 the last part of the present paper. Static and quasistatic solution techniques are still dominating, but the THREAD project will result in efficient and numerically stable methods for dynamic problems To this end, we consider coarse grid discretizations that preserve certain geometrical properties of the equations of motion and achieve numerical stability in that way [9]. We will use the well-known ingredients of Lie group structured configuration spaces, variational integrators, and numerical treatment of DAEs, to find a way of simulating thin beams efficiently and accurately at the same time. We analyze how the steps in arc length and the time step sizes influence the errors
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