Improved explicit co-simulation methods incorporating relaxation techniques

  • Pu LiEmail author
  • Qi YuanEmail author
  • Daixing Lu
  • Tobias Meyer
  • Bernhard Schweizer


Explicit co-simulation is an increasingly popular technique for solver coupling of multidisciplinary systems in a distributed environment. Compared to implicit approaches, explicit coupling schemes are simpler to implement, since explicit methods allow the synchronization of different subdomains at user-defined discrete communication points without a reinitialization of the subsystem solvers. However, a typical disadvantage of explicit schemes is the reduced stability behavior, which is introduced by errors resulting from the approximation of the coupling variables. In the manuscript at hand, improved explicit co-simulation methods incorporating relaxation techniques are considered for the case that the coupling of the subsystems is realized by constitutive laws. We discuss three decomposition techniques, namely force/force coupling, force/displacement coupling and displacement/displacement coupling. Two novel explicit coupling approaches are analyzed. Within the first approach, classical co-simulation methods based on Lagrange polynomials for extrapolating the coupling variables are modified by incorporating a straightforward relaxation approach using only the current and previous coupling variables. Applying the second approach, predicted coupling variables are generated by using a constant or linear acceleration approach for the coupling bodies. With the predicted, current and previous coupling variables, special approximation polynomials are generated also using a relaxation approach. The numerical stability and convergence characteristics will be analyzed for two explicit co-simulation approaches. It will be shown that stability and convergence can be notably improved by a proper choice of the relaxation parameters. The stabilized co-simulation techniques will be verified using several numerical examples.


Explicit co-simulation Relaxation techniques Numerical stability Convergence behavior 



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


  1. 1.
    Peiret, A., González, F., Kövecses, J., Teichmann, M.: Multibody system dynamics interface modelling for stable multirate co-simulation of multiphysics systems. Mech. Mach. Theory 127, 52–72 (2018). CrossRefGoogle Scholar
  2. 2.
    Van der Auweraer, H., Anthonis, J., De Bruyne, S., Leuridan, J.: Virtual engineering at work: the challenges for designing mechatronic products. Eng. Comput. 29, 389–408 (2013). CrossRefGoogle Scholar
  3. 3.
    Liboni, G., Deantoni, J., Portaluri, A., Quaglia, D., Simone, R.D.: Beyond time-triggered co-simulation of cyber-physical systems for performance and accuracy improvements. In: Proceedings of the Rapido’18 Workshop on Rapid Simulation and Performance Evaluation: Methods and Tools, pp. 1–8. ACM, Manchester (2018)Google Scholar
  4. 4.
    Neema, H., Gohl, J., Lattmann, Z., Sztipanovits, J., Karsai, G., Neema, S., Bapty, T., Batteh, J., Tummescheit, H., Sureshkumar, C.: Model-based integration platform for FMI co-simulation and heterogeneous simulations of cyber-physical systems. In: Proceedings of the 10th International Modelica Conference, March 10–12 2014, Lund, Sweden, pp. 235–245. Linköping University Electronic Press (2014)Google Scholar
  5. 5.
    Baumann, P., Mikelsons, L., Baumann, M.: Comparison of methods for integrating real-time systems in a co-simulation framework. In: IUTAM Symposium on Co-Simulation and Solver Coupling. Darmstadt, Germany (2017)Google Scholar
  6. 6.
    Stettinger, G., Horn, M., Benedikt, M., Zehetner, J.: Model-based coupling approach for non-iterative real-time co-simulation. In: European Control Conference (ECC) 2014, pp. 2084–2089 (2014)Google Scholar
  7. 7.
    Schaäfer, M., Yigit, S., Heck, M.: Implicit partitioned fluid-structure interaction coupling. In: ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference, pp. 105–114. American Society of Mechanical Engineers (2006)Google Scholar
  8. 8.
    Andersson, H., Nordin, P., Borrvall, T., Simonsson, K., Hilding, D., Schill, M., Krus, P., Leidermark, D.: A co-simulation method for system-level simulation of fluid-structure couplings in hydraulic percussion units. Eng. Comput. 33, 317–333 (2017). CrossRefGoogle Scholar
  9. 9.
    Alioli, M., Morandini, M., Masarati, P.: Coupled multibody-fluid dynamics simulation of flapping wings. In: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V07BT10A014–V007BT010A014. American Society of Mechanical Engineers (2013)Google Scholar
  10. 10.
    Ambrósio, J., Pombo, J., Pereira, M.: Optimization of high-speed railway pantographs for improving pantograph-catenary contact. Theor. Appl. Mech. Lett. 3, 013006 (2013). CrossRefGoogle Scholar
  11. 11.
    Massat, J.-P., Laurent, C., Bianchi, J.-P., Balmes, E.: Pantograph catenary dynamic optimisation based on advanced multibody and finite element co-simulation tools. Veh. Syst. Dyn. 52, 338–354 (2014)CrossRefGoogle Scholar
  12. 12.
    Busch, M., Schweizer, B.: Coupled simulation of multibody and finite element systems: an efficient and robust semi-implicit coupling approach. Arch. Appl. Mech. 82, 723–741 (2012). CrossRefzbMATHGoogle Scholar
  13. 13.
    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. 60, 75–94 (2013)CrossRefGoogle Scholar
  14. 14.
    Abel, A., Blochwitz, T., Eichberger, A., Hamann, P., Rein, U.: Functional mock-up interface in mechatronic gearshift simulation for commercial vehicles. In: Proceedings of the 9th International MODELICA Conference, pp. 775-780. Linköping University Electronic Press (2012)Google Scholar
  15. 15.
    González, F., Naya, M., Luaces, A., González, M.: On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics. Multibody Syst. Dyn. 25, 461–483 (2011). CrossRefzbMATHGoogle Scholar
  16. 16.
    González, F., González, M., Mikkola, A.: Efficient coupling of multibody software with numerical computing environments and block diagram simulators. Multibody Syst. Dyn. 24, 237–253 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    González, F., González, M., Cuadrado, J.: Weak coupling of multibody dynamics and block diagram simulation tools. In: ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 93–102. American Society of Mechanical Engineers (2009)Google 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). CrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, J.Z., Ma, Z.D., Hulbert, G.M.: A gluing algorithm for distributed simulation of multibody systems. Nonlinear Dyn. 34, 159–188 (2003). CrossRefzbMATHGoogle Scholar
  20. 20.
    Malczyk, P., Fraczek, J.: A divide and conquer algorithm for constrained multibody system dynamics based on augmented Lagrangian method with projections-based error correction. Nonlinear Dyn. 70, 871–889 (2012). MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schweizer, B., Li, P., Lu, D.: Explicit and implicit cosimulation methods: stability and convergence analysis for different solver coupling approaches. J. Comput. Nonlinear Dyn. 10, 051007–051012 (2015). CrossRefGoogle Scholar
  22. 22.
    Schweizer, B., Lu, D.: Semi-implicit co-simulation approach for solver coupling. Arch. Appl. Mech. 86, 1739–1769 (2014)CrossRefGoogle Scholar
  23. 23.
    Schweizer, B., Lu, D.: Predictor/corrector co-simulation approaches for solver coupling with algebraic constraints. J. Appl. Math. Mech. 95, 911–938 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Meyer, T., Li, P., Lu, D., Schweizer, B.: Implicit co-simulation method for constraint coupling with improved stability behavior. Multibody Syst. Dyn. (2018). MathSciNetCrossRefGoogle Scholar
  25. 25.
    Arnold, M.: Stability of sequential modular time integration methods for coupled multibody system models. J. Comput. Nonlinear Dyn. 5, 1–9 (2010). CrossRefGoogle Scholar
  26. 26.
    Schierz, T., Arnold, M.: Stabilized overlapping modular time integration of coupled differential-algebraic equations. Appl. Numer. Math. 62, 1491–1502 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Benedikt, M., Stettinger, G., Horn, M.: Sliding mode control for constraint stabilization in multi-body system dynamic analysis. In: IEEE Conference on Control Applications (CCA), 2014, pp. 1557–1562 (2014)Google Scholar
  28. 28.
    Benedikt, M., Watzenig, D., Hofer, A.: Modelling and analysis of the non-iterative coupling process for co-simulation. Math. Comput. Model. Dyn. Syst. 19, 451–470 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sicklinger, S., Belsky, V., Engelmann, B., Elmqvist, H., Olsson, H., Wuchner, R., Bletzinger, K.U.: Interface Jacobian-based Co-Simulation. Int. J. Numer. Methods Eng. 98, 418–444 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cadeau, T., Magoules, F.: Coupling the Parareal algorithm with the waveform relaxation method for the solution of differential algebraic equations. In: Proceedings of the 10th International Symposium on Distributed Computing and Applications to Business, Engineering and Science (DCABES 2011), pp. 15–19 (2011).
  31. 31.
    Lelarasmee, E., Ruehli, A.E., Sangiovanni-Vincentelli, A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 1, 131–145 (1982). CrossRefGoogle Scholar
  32. 32.
    Mohtat, A., Arbatani, S., Kövecses, J.: Enhancing the stability of co-simulation via an energy-leak monitoring and dissipation framework. In: IUTAM Symposium on Co-Simulation and Solver Coupling. Darmstadt, Germany (2017)Google Scholar
  33. 33.
    Sadjina, S., Kyllingstad, L.T., Skjong, S., Pedersen, E.: Energy conservation and power bonds in co-simulations: non-iterative adaptive step size control and error estimation. Eng. Comput. 33, 607–620 (2017). CrossRefGoogle Scholar
  34. 34.
    Gu, B., Asada, H.H.: Co-simulation of algebraically coupled dynamic subsystems without disclosure of proprietary subsystem models. J. Dyn. Syst. Measure. Control-Trans. ASME 126, 1–13 (2004). CrossRefGoogle Scholar
  35. 35.
    Gensor, S., Benedikt, M.: Model-based pre-step stabilization method for non-iterative co-simulation. In: The 5th Joint International Conference on Multibody System Dynamics. Lisboa, Portugal (2018)Google Scholar
  36. 36.
    Haid, T., Stettinger, G., Watzenig, D., Benedikt, M.: A model-based corrector approach for explicit co-simulation using subspace identification. In: The 5th Joint International Conference on Multibody System Dynamics. Lisboa, Portugal (2018)Google Scholar
  37. 37.
    Schweizer, B., Lu, D., Li, P.: Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques. Multibody Syst. Dyn. 36, 1–36 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Li, P., Lu, D., Schmoll, R., Schweizer, B.: Explicit Co-simulation Approach with Improved Numerical Stability, pp. 153–201. Springer, Cham (2019)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Energy and Power Engineering, Institute of TurbomachineryXi’an Jiaotong UniversityXi’anChina
  2. 2.Shaanxi Engineering Laboratory of Turbomachinery and Power EquipmentXi’anChina
  3. 3.Department of Mechanical Engineering, Institute of Applied DynamicsTechnical University DarmstadtDarmstadtGermany

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