Modular Time Integration of Block-Structured Coupled Systems Without Algebraic Loops

Conference paper
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

In March 2013, a Workshop on coupled Descriptor Systems was held at Castle Eringerfeld, Geseke (Germany). There, the author of this short communication presented a key note lecture on “Efficient time integration of block-structured descriptor systems” with results that have recently been published in Arnold et al. (Arch Mech Eng LX:75–94 2013) and are given also in Chap.  6 below. In the present short communication we give a compact introduction to this material and focus on basic aspects of the convergence analysis. For a more detailed discussion the interested reader is referred to Chap.6 or the abovementioned paper.

Keywords

Co-simulation Modular time integration Algebraic loops Convergence analysis 

References

  1. 1.
    Arnold, M.: Multi-rate time integration for large scale multibody system models. In: Eberhard, P. (ed.) IUTAM Symposium on Multiscale Problems in Multibody System Contacts, Stuttgart, pp. 1–10. Springer (2007)Google Scholar
  2. 2.
    Arnold, M.: Stability of sequential modular time integration methods for coupled multibody system models. J. Comput. Nonlinear Dyn. 5, 031003 (2010)CrossRefGoogle Scholar
  3. 3.
    Arnold, M., Burgermeister, B., Führer, C., Hippmann, G., Rill, G.: Numerical methods in vehicle system dynamics: state of the art and current developments. Veh. Syst. Dyn. 49, 1159–1207 (2011)CrossRefGoogle Scholar
  4. 4.
    Arnold, M., Clauß, C., Schierz, T.: Error analysis and error estimates for co-simulation in FMI for Model Exchange and Co-Simulation v2.0. Arch. Mech. Eng. LX, 75–94 (2013)Google Scholar
  5. 5.
    Arnold, M., Günther, M.: Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT Numer. Math. 41, 1–25 (2001)CrossRefMATHGoogle Scholar
  6. 6.
    Bartel, A., Brunk, M., Günther, M., Schöps, S.: Dynamic iteration for coupled problems of electric circuits and distributed devices. SIAM J. Sci. Comput. 35, B315–B335 (2013)CrossRefMATHGoogle Scholar
  7. 7.
    Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Clarendon Press, Oxford (1995)MATHGoogle Scholar
  8. 8.
    Busch, M.: Zur effizienten Kopplung von Simulationsprogrammen. PhD thesis, Universität Kassel, Fachbereich Maschinenbau (2012)Google Scholar
  9. 9.
    Busch, M., Schweizer, B.: Numerical stability and accuracy of different co-simulation techniques: analytical investigations based on a 2-DOF test model. In: Proceedings of the 1st Joint International Conference on Multibody System Dynamics, Lappeenranta, 25–27 May 2010Google Scholar
  10. 10.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT, Cambridge (2001)MATHGoogle Scholar
  11. 11.
    Deuflhard, P., Hairer, E., Zugck, J.: One–step and extrapolation methods for differential– algebraic systems. Numer. Math. 51, 501–516 (1987)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Deuflhard, P., Hohmann, A.: Numerical Analysis in Modern Scientific Computing: An Introduction, 2nd edn. Number 43 in Texts in Applied Mathematics. Springer, New York (2003)Google Scholar
  13. 13.
    FMI: The Functional Mockup Interface. https://www.fmi-standard.org/
  14. 14.
    Kübler, R.: Modulare Modellierung und Simulation mechatronischer Systeme. Fortschritt-Berichte VDI Reihe 20, Nr. 327. VDI–Verlag GmbH, Düsseldorf (2000)Google Scholar
  15. 15.
    Kübler, R, Schiehlen, W.: Two methods of simulator coupling. Math. Comput. Model. Dyn. Syst. 6, 93–113 (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Schierz, T., Arnold, M.: Stabilized overlapping modular time integration of coupled differential-algebraic equations. Appl. Numer. Math. 62, 1491–1502 (2012)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Schierz, T., Arnold, M., Clauß, C.: Co-simulation with communication step size control in an FMI compatible master algorithm. In: Otter, M., Zimmer, D. (eds.) Proceedings of the 9th International Modelica Conference, Munich, 3–5 Sept 2012Google Scholar
  18. 18.
    Tseng, F.C., Hulbert, G.M.: A gluing algorithm for network-distributed multibody dynamics simulation. Multibody Syst. Dyn. 6, 377–396 (2001)CrossRefMATHGoogle Scholar
  19. 19.
    Veitl, A., Gordon, T., van de Sand, A., Howell, M., Valášek, M., Vaculín, O., Steinbauer, P.: Methodologies for coupling simulation models and codes in mechatronic system analysis and design. In: Proceedings of the 16th IAVSD–Symposium on Dynamics of Vehicles on Roads and Tracks. Pretoria. Supplement to Vehicle System Dynamics, vol. 33, pp. 231–243. Swets & Zeitlinger (1999)Google Scholar
  20. 20.
    Walter, W.: Ordinary Differential Equations. Number 182 in Graduate Texts in Mathematics. Springer, New York (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of MathematicsMartin Luther University Halle-WittenbergHalle (Saale)Germany

Personalised recommendations