Skip to main content
SpringerLink
Log in

Influence of coupling approximation on the numerical stability of explicit co-simulation

  • Original Article
  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

Co-simulation allows the parallelization of several subsystems in a distributed manner with data exchanges at discrete user-defined communication points (macro-time points). Between two macro-time points, subsystems are integrated in a local circumstance using specified solvers. By adopting the technique, the computational efficiency can be enhanced greatly without exposure of inner information. As a weak coupling, communication between subsystems is usually approximated by extrapolated or interpolated polynomials that may induce instability. Implicit or semi-implicit co-simulation with iteration of macro-time steps can improve stability behavior significantly. However, reinitialization may be unfeasible for realtime simulation or not supported by most commercial tools. Therefore, stabilized non-iterative co-simulation has been more popular that can be achieved either by appropriate communication techniques or by interface error estimations. However, the influence of the coupling approximations on numerical stability is not well understood. In this paper, the numerical stability of explicit co-simulation will be studied using different approximations of coupling techniques. The influence of communication schemes (parallel and sequential coupling scheme), least square as well as relaxation techniques using an applied force coupling approach will be investigated quantitatively with the help of a linear two-mass oscillator. A quarter car model is used to validate the stability analysis of the linear oscillator. The results indicate that by adopting appropriate coupling techniques, stability improvement can be accomplished. Higher order of extrapolation (interpolation) introduces damping to the distributed system while relaxation and least square approaches might trigger artificial oscillations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

c̄i(i = 1,2,c):

Dimensionless stiffness coefficient of subsystem 1, 2 and coupling element

d̄i(i = 1,2,c):

Dimensionless damping coefficient of subsystem 1, 2 and coupling element

mi(i = 1,2):

Mass of subsystem 1, 2

α m21 :

Mass ratio of subsystem 2 w.r.t. subsystem 1

αc21,αcc1 :

Stiffness ratio of subsystem 2 and coupling element w.r.t subsystem 1

αd21,αdc1 :

Damping ratio of subsystem 2 and coupling element w.r.t subsystem 1

zi(i = 1,2):

State vector of subsystem 1 and 2

uN, uN−1, uN−2 :

Vector of coupling variables at macro-time point TN, TN−1 and TN−2

f(ū):

Dimensionless coupling force in a force-force decomposition

Ai(i = 1,2):

Coefficient matrix of subsystem 1 and 2

fkex(t̄):

Extrapolated polynomials of order k

fkin(t̄):

Interpolated polynomials of order k

fmLS(t̄):

Extrapolated polynomials of order m using least square techniques

f1Re(t̄):

Linear extrapolated polynomial using relaxation

Re(λ̄1):

Real part of the eigenvalue of subsystem 1

Im(λ̄1):

Imaginary part of the eigenvalue of subsystem 1

\(\alpha_{Re(\overline{\lambda}_i)}\)(i = 2,c):

Real part ratio of the eigenvalue of subsystem 2 and coupling system w.r.t subsystem 1

\(\alpha_{Im(\overline{\lambda}_i)}\)(i = 2,c):

Imaginary part ratio of the eigenvalue of subsystem 2 and coupling system w.r.t subsystem 1

ε glo :

Global error of co-simulation method

References

  1. W. G. Dettmer and D. Perić, A new staggered scheme for fluid-structure interaction, International Journal for Numerical Methods in Engineering, 93 (1) (2013) 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Rahikainen, X. González, M. Á. Naya, X. Sopanen and A. Mikkola, On the cosimulation of multibody systems and hydraulic dynamics, Multibody System Dynamics (2020).

    Google Scholar 

  3. M. A. Naya, X. Cuadrado, X. Dopico and U. Lugris, An efficient unified method for the combined simulation of multibody and hydraulic dynamics: Comparison with simplified and co-integration approaches, Archive of Mechanical Engineering, 58 (2) (2011) 223–243.

    Article  Google Scholar 

  4. A. Recuero, X. Serban, X. Peterson, X. Sugiyama, X. Jayakumar and D. Negrut, A high-fidelity approach for vehicle mobility simulation: Nonlinear finite element tires operating on granular material, Journal of Terramechanics, 72 (2017) 39–54.

    Article  Google Scholar 

  5. M. O. Neal and T. Belytschko, Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems, Computers & Structures, 31 (6) (1989) 871–880.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Lelarasmee, X. E. Ruehli and A. L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1 (3) (1982) 131–145.

    Article  Google Scholar 

  7. J. Z. Wang, X. D. Ma and G. M. Hulbert, A gluing algorithm for distributed simulation of multibody systems, Nonlinear Dynamics, 34 (1-2) (2003) 159–188.

    Article  MATH  Google Scholar 

  8. F. C. Tseng and G. M. Hulbert, A gluing algorithm for network-distributed multibody dynamics simulation, Multibody System Dynamics, 6 (4) (2001) 377–396.

    Article  MATH  Google Scholar 

  9. J. Kraft, X. Meyer and B. Schweizer, Reduction of the computation time of large multibody systems with co-simulation methods, Cham (2019) 131–152.

    Google Scholar 

  10. M. Busch and B. Schweizer, Coupled simulation of multibody and finite element systems: An efficient and robust semi-implicit coupling approach, Archive of Applied Mechanics, 82 (6) (2012) 723–741.

    Article  MATH  Google Scholar 

  11. A. Bautista, X. Montesinos and P. Pintado, Dynamic interaction between pantograph and rigid overhead lines using a coupled FEM — multibody procedure, Mechanism and Machine Theory, 97 (2016) 100–111.

    Article  Google Scholar 

  12. B. Schweizer and D. Lu, Semi-implicit co-simulation approach for solver coupling, Archive of Applied Mechanics, 86 (12) (2014) 1739–1769.

    Article  Google Scholar 

  13. T. Meyer, X. Li, X. Lu and B. Schweizer, Implicit co-simulation method for constraint coupling with improved stability behavior, Multibody System Dynamics (2018).

    Google Scholar 

  14. R. Kübler and W. Schiehlen, Two methods of simulator coupling, Mathematical and Computer Modelling of Dynamical Systems, 6 (2) (2000) 93–113.

    Article  MATH  Google Scholar 

  15. B. Schweizer and D. Lu, Predictor/corrector co-simulation approaches for solver coupling with algebraic constraints, Journal of Applied Mathematics and Mechanics, 95 (9) (2014) 911–938.

    MATH  MathSciNet  Google Scholar 

  16. M. Arnold, X. Burgermeister, X. Führer, G. Hippmann and G. Rill, Numerical methods in vehicle system dynamics: State of the art and current developments, Vehicle System Dynamics, 49 (7) (2011) 1159–1207.

    Article  Google Scholar 

  17. M. Arnold, Stability of sequential modular time integration methods for coupled multibody system models, Journal of Computational and Nonlinear Dynamics, 5 (3) (2010) 1–9.

    Article  Google Scholar 

  18. M. Benedikt and E. Drenth, Relaxing stiff system integration by smoothing techniques for non-iterative co-simulation, Cham (2019) 1–25.

    Google Scholar 

  19. M. Benedikt, X. Watzenig and A. Hofer, Modelling and analysis of the non-iterative coupling process for co-simulation, Mathematical and Computer Modelling of Dynamical Systems, 19 (5) (2013) 451–470.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Gomes, X. Thule, X. Deantoni, X. G. Larsen and H. Vangheluwe, Co-simulation: The past, future, and open challenges, Cham (2018) 504–520.

    Google Scholar 

  21. C. Gomes, X. Thule, X. Broman, X. G. Larsen and H. Vangheluwe, Co-simulation: A survey, ACM Computing Surveys, 51 (3) (2018).

    Google Scholar 

  22. M. Busch, Performance improvement of explicit cosimulation methods through continuous extrapolation, Cham (2019) 57–80.

    Google Scholar 

  23. M. Busch, Continuous approximation techniques for co-simulation methods: Analysis of numerical stability and local error, ZAMM - Journal of Applied Mathematics and Mechanics, 96 (2016) 1061–1081.

    Article  MathSciNet  Google Scholar 

  24. J. Rahikainen, X. González and M. Á. Naya, An automated methodology to select functional co-simulation configurations, Multibody System Dynamics, 48 (1) (2020) 79–103.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Zhao, X. Wang and H. Zhang, A regression-based collaborative filtering recommendation approach to time-stepping multi-solver co-simulation, IEEE Access (2019) 1–13.

    Google Scholar 

  26. M. Benedikt and F. R. Holzinger, Automated configuration for non-iterative co-simulation, 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE) (2016) 1–7.

    Google Scholar 

  27. A. Peiret, X. Kövecses, F. González and M. Teichmann, Interface models for multirate co-simulation of nonsmooth multi-body systems, Proceedings of the ASME Design Engineering Technical Conference (2019).

    Google Scholar 

  28. F. González, S. Arbatani, X. Mohtat and J. Kövecses, Energyleak monitoring and correction to enhance stability in the co-simulation of mechanical systems, Mechanism and Machine Theory, 131 (2019) 172–188.

    Article  Google Scholar 

  29. A. Peiret, X. González, J. Kövecses and M. Teichmann, Multibody system dynamics interface modelling for stable multirate co-simulation of multiphysics systems, Mechanism and Machine Theory, 127 (2018) 52–72.

    Article  Google Scholar 

  30. S. Sadjina, X. T. Kyllingstad, X. Skjong and E. Pedersen, Energy conservation and power bonds in co-simulations: Non-iterative adaptive step size control and error estimation, Engineering with Computers, 33 (3) (2017) 607–620.

    Article  Google Scholar 

  31. S. Genser and M. Benedikt, A pre-step stabilization method for non-iterative co-simulation and effects of interface-jacobians identification, Advances in Intelligent Systems and Computing (2018).

    Google Scholar 

  32. P. Li, X. Yuan, X. Lu, X. Meyer and B. Schweizer, Improved explicit co-simulation methods incorporating relaxation techniques, Archive of Applied Mechanics, 90 (1) (2020) 17–46.

    Article  Google Scholar 

  33. B. Schweizer, X. Lu and P. Li, Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques, Multibody System Dynamics, 36 (1) (2015) 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  34. T. Sweafford and H.-S. Yoon, Co-simulation of dynamic systems in parallel and serial model configurations, Journal of Mechanical Science and Technology, 27 (12) (2013) 3579–3587.

    Article  Google Scholar 

  35. T. Moshagen, On meeting energy balance errors in co-simulations, Mathematical and Computer Modelling of Dynamical Systems, 25 (2) (2019) 139–166.

    Article  MathSciNet  Google Scholar 

  36. F. González, M. Naya, X. Luaces and M. González, On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics, Multibody System Dynamics, 25 (4) (2011) 461–483.

    Article  MATH  Google Scholar 

  37. P. Li, X. Lu and B. Schweizer, On the stability of explicit and implicit co-simulation approaches, ECCOMAS Thematic Conference on Multibody Dynamics (2015).

    Google Scholar 

Download references

Acknowledgments

This work is supported by National Nature Science Foundation of China (No. 11902237, No. 11872289), China Postdoctoral Science Foundation (No. 2018M643622).

Author information

Authors and Affiliations

  1. School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Pu Li & Qi Yuan

  2. Shaanxi Engineering Laboratory of Turbomachinery and Power Equipment, Xi’an, 710049, China

    Pu Li & Qi Yuan

Authors
  1. Pu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Yuan.

Additional information

Recommended by Editor No-cheol Park

Pu Li is a Post Doctor of the School of Energy and Power Engineering at Xi’an Jiaotong University, China. He received his Ph.D. in Mechanical Engineering from Technical University Darmstadt. His research interests include co-simulation and rotor dynamics.

Qi Yuan is a Professor of the School of Energy and Power Engineering, Xi’an Jiaotong University. His research interests include structural strength and vibration of turbomachinery, wind turbine and biofluid mechanics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, P., Yuan, Q. Influence of coupling approximation on the numerical stability of explicit co-simulation. J Mech Sci Technol 34, 2289–2298 (2020). https://doi.org/10.1007/s12206-020-0504-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-020-0504-x

Keywords

Search

Navigation