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__Structure preserving integration of constrained multirate systems__

S. Leyendecker and S. Ober-Blöbaum

**Abstract**

Mechanical systems with dynamics on varying time scales, e.g. including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Here, small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step [1, 2]. A particular challenge is the treatment of the coupling between slow and fast dynamics, e.g. via potentials or holonomic constraints. In this talk, a multirate integrator is derived in closed form via a discrete variational principle, resulting in a symplectic and momentum preserving multirate scheme [3].

**Bibliography**

[1]
M. Arnold,
Multi-rate time integration for large scale multibody system models, in: IUTAM Symposium on Multiscale Problems in Multibody System Contacts, edited by P. Eberhard, Stuttgart, Germany (2007), pp. 1-10.

[2]
A. Stern and E. Grinspun,
Implicit-explicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), pp. 1779-1794.

[3]
S. Leyendecker and S. Ober-Blöbaum,
A variational approach to multirate integration for constrained systems, in: Multibody Dynamics, Computational Methods in Applied Sciences edited by J.C. Samin, P. Fisette, 28 (2013), pp. 97-121.