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Multibody Dynamics pp 97–121Cite as

A Variational Approach to Multirate Integration for Constrained Systems

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Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 28))

Abstract

The simulation of systems with dynamics on strongly varying time scales is quite challenging and demanding with regard to possible numerical methods. A rather naive approach is to use the smallest necessary time step to guarantee a stable integration of the fast frequencies. However, this typically leads to unacceptable computational loads. Alternatively, 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. In this work, a multirate integrator for constrained dynamical systems is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Being based on a discrete version of Hamilton’s principle, the resulting variational multirate integrator is a symplectic and momentum preserving integration scheme and also exhibits good energy behaviour. Depending on the discrete approximations for the Lagrangian function, one obtains different integrators, e.g. purely implicit or purely explicit schemes, or methods that treat the fast and slow parts in different ways. The performance of the multirate integrator is demonstrated by means of several examples.

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Acknowledgements

This chapter was partly developed and published in the course of the Collaborative Research Centre 614 “Self-Optimizing Concepts and Structures in Mechanical Engineering” funded by the German Research Foundation (DFG) under grant number SFB 614.

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Authors and Affiliations

  1. Chair of Applied Dynamics, University of Erlangen-Nuremberg, Konrad-Zuse-Str. 3/5, 91052, Erlangen, Germany

    Sigrid Leyendecker

  2. Computational Dynamics and Optimal Control, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Sina Ober-Blöbaum

Authors
  1. Sigrid Leyendecker
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  2. Sina Ober-Blöbaum
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Correspondence to Sigrid Leyendecker .

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Editors and Affiliations

  1. Institute for Mechanics, Materials and Civil Engineering, Universite catholique de Louvain (UCL), Place du Levant 2, Louvain-la-Neuve, 1348, Belgium

    Jean-Claude Samin

  2. Institute for Mechanics, Materials and Civil Engineering, Universite catholique de Louvain (UCL), Place du Levant 2, Louvain-la-Neuve, 1348, Belgium

    Paul Fisette

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Leyendecker, S., Ober-Blöbaum, S. (2013). A Variational Approach to Multirate Integration for Constrained Systems. In: Samin, JC., Fisette, P. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5404-1_5

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  • DOI: https://doi.org/10.1007/978-94-007-5404-1_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5403-4

  • Online ISBN: 978-94-007-5404-1

  • eBook Packages: EngineeringEngineering (R0)

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