Configuration spaces with Lie group structure display kinematical nonlinearities of mechanical systems. In Lie group time integration, this nonlinear structure is also considered at the time-discrete level using nonlinear updates of the configuration variables. For practical implementation purposes, these update formulae have to be adapted to each specific Lie group setting that may be characterized from the algorithmic viewpoint by group operation, exponential map, tilde, and tangent operator. In this paper, we discuss these practical aspects for the time integration of a geometrically exact Cosserat rod model with rotational degrees-of-freedom being represented by unit quaternions. Shearing and longitudinal extension of the Cosserat rod may be neglected using suitable constraints that result in a differential-algebraic equation (DAE) formulation of the beam structure. The specific structure of unconstrained systems and constrained systems is exploited by tailored algorithms for the corrector iteration of the generalized-α Lie group integrator.

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.

Co-simulation is a simulation technique for time dependent coupled problems in engineering that restricts the data exchange between subsystems to discrete communication points in time. In the present paper we follow the block-oriented framework in the recently established industrial interface standard FMI for Model Exchange and Co-Simulation v2.0 and study local and global error of co-simulation algorithms for systems with force-displacement coupling. A rather general convergence result for the co-simulation of coupled systems without algebraic loops shows zero-stability of co-simulation algorithms with forcedisplacement coupling and proves that order reduction of local errors does not affect the order of global errors. The theoretical investigations are illustrated by numerical tests in the novel FMI-compatible co-simulation environment SNiMoWrapper.

In rigid body mechanics, a body's configuration consisting of orientation and position have to be described, as well as angular velocity and velocity. There are several different ways to do this, leading to different configuration spaces. Popular choices include SO(3) × R 3 , SE(3) or unit dual quaternions. All these configuration spaces possess a Lie group structure. We present the novel approach S 3 R 3 , which is a configuration space similar to SE(3), but describes the orientation in terms of unit quaternions instead of rotation matrices. Furthermore we show its relation to and advantages over other configuration spaces.The configuration of a rigid body can be described in a configuration space G. It is well-known [6], that it forms a Lie group (G, •). The velocity of an object with configurationwhere v(t) ∈ R n describes the velocity of the object with configuration q(t) ∈ G, see [3].When large rotations are present, they are often described using SO(3) and translations usually by R 3 . The angular velocity is typically described with respect to the body-fixed frame, but the velocity can be described with respect to the inertial system (fixed frame of reference) or with respect to the body-fixed frame (moving frame of reference). Both are standard approaches [7]. The former leads to the direct product SO(3) × R 3 and the latter leads to the semidirect product SO(3) R 3 SE(3).It is well-known that unit quaternions S 3 are a singularity-free description of orientations, that can outperform SO(3) in numerical computations. When we want to describe the configuration of a rigid body using S 3 , we have again the choice to describe the velocities with respect to the inertial frame or with respect to the body-fixed frame. Again, the former leads to a direct product S 3 × R 3 . Our focus, however, is on the latter approach, which leads to the Lie-group structured semi-direct product configuration space S 3 R 3 . It is comparable to the approach of unit dual quaternions, as described below. We will then illustrate the benefits of the configuration space S 3 R 3 in Lie group time integration by numerical test results for a classical benchmark problem. Rotations with unit quaternionsIt is known [2] that rotations can be described by using the set of unit quaternions S 3 = {p ∈ R 4 : p 2 = 1} which is a Lie group together with the quaternion multiplication * : R 4 → R 4 . Let p = p 0 p , q = q 0 q ∈ S 3 , z = 0 z ∈ T e S 3 and w ∈ R 3 with scalar parts p 0 , q 0 , 0 ∈ R and vector partsp,q,ž ∈ R 3 . All of the following formulae for quaternions are well-known:

No abstract