### Guest lectures by the THREAD PIs Prof. Sigrid Leyendecker and Prof. Martin Arnold

**DATE**: Thursday February 4, 2021.

**Speaker: Martin Arnold**

**Time**:10:15-12

**Title**: Lie group methods for industrial multibody system simulation

**Abstract:**
Industrial multibody system simulation relies on robust and efficient time
integration methods for constrained mechanical systems with structural
damping and embedded control structures. The methods of choice are second
order methods of Newmark type with user defined algorithmic damping (HHT
method, generalized-$\alpha$ method) and higher order multistep methods of
BDF type with sophisticated algorithms for step size and order control.

Multibody system models in Lie groups are attractive from the viewpoint of modelling since they allow the representation of large rotations without singularities. But nonlinear configuration spaces are beyond the range of applicability of classical solvers in multibody system dynamics. Standard Newmark type methods and BDF may, however, be generalized to the Lie group setting exploiting the linear structure of the corresponding Lie algebra. We discuss theoretical and practical aspects of a Lie group generalized-$\alpha$ and a BLieDF multistep Lie group method for constrained mechanical systems and show that the novel Lie group methods have the same order as their classical counterparts and may be implemented by minor modifications of well approved generalized-$\alpha$ or BDF solvers.

References

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**Speaker: Sigrid Leyendecker**

**Time**:13:15-15

**Title**: Lagrangian mechanics and variational integrators

**Abstract:**
This lecture will address the derivation of the Euler-Lagrange equations from Hamilton's variational principle and discuss geometric structural properties like the conservation of momentum maps due to invariants of the Lagrangian (Noether's theorem) and symplecticity. Then, similar steps are preformed in the temporal discrete setting to derive the discrete Euler-Lagrange equations via a discrete variational principle. This leads to a time integration scheme which preserves the geometric structure of the underlying dynamical system in the sense that it is symplectic and there exists a discrete Noether theorem.