Multibody System Dynamics

, Volume 36, Issue 1, pp 1–36 | Cite as

Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques

  • Bernhard SchweizerEmail author
  • Daixing Lu
  • Pu Li


Based on the stabilized index-2 formulation for multibody systems, a semi-implicit co-simulation approach for solver coupling with algebraic constraints has been presented by Schweizer and Lu (Multibody Syst. Dyn., 2014) for the case that constant approximation is used for extrapolating/interpolating the coupling variables. In the manuscript at hand, this method is generalized to the case that higher-order approximation is employed. Direct application of higher-order polynomials for extrapolating/interpolating the coupling variables fails. Using linear approximation polynomials, artificial oscillations in the Lagrange multipliers of the kinematical differential equations are observed. For quadratic and higher-order polynomials, the co-simulation becomes unstable. In this work, the key idea to obtain stable solutions without artificial oscillations is to apply a relaxation technique. A detailed stability and convergence analysis is presented in the paper for the case of higher-order approximation. In this context, the influence of the relaxation parameter on the stability and convergence behavior is investigated. Applicability and robustness of the stabilized index-2 co-simulation method incorporating higher-order approximation polynomials is demonstrated with different numerical examples. Using piecewise constant approximation polynomials for the coupling variables produces discontinuous accelerations and reaction forces in the subsystems at the macrotime points, which may entail problems for the subsystem integrator. With higher-order approximation polynomials, the coupling variables and in consequence the accelerations and reaction forces in the subsystems become continuous.


Co-simulation Solver coupling Subcycling Implicit Semi-implicit Algebraic constraints Stability Convergence Relaxation technique 


  1. 1.
    Ambrosio, J., Pombo, J., Rauter, F., Pereira, M.: A memory based communication in the co-simulation of multibody and finite element codes for pantograph-catenary interaction simulation. In: Bottasso, C.L. (ed.) Multibody Dynamics: Computational Methods and Applications, pp. 231–252. Springer, Berlin (2009) Google Scholar
  2. 2.
    Ambrosio, J., Pombo, J., Pereira, M., Antunes, P., Mosca, A.: A computational procedure for the dynamic analysis of the catenary-pantograph interaction in high-speed trains. J. Theor. Appl. Mech. 50(3), 681–699 (2012) Google Scholar
  3. 3.
    Anderson, K.S.: An order-\(n\) formulation for the motion simulation of general multi-rigid-body tree systems. Comput. Struct. 46(3), 547–559 (1993) zbMATHCrossRefGoogle Scholar
  4. 4.
    Anderson, K., Duan, S.: A hybrid parallelizable low-order algorithm for dynamics of multi-rigid-body systems: Part I, Chain systems. Math. Comput. Model. 30(9–10), 193–215 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arnold, M.: Numerical methods for simulation in applied dynamics. In: Arnold, M., Schiehlen, W. (eds.) Simulation Techniques for Applied Dynamics. Springer, Berlin (2009) CrossRefGoogle Scholar
  6. 6.
    Arnold, M., Clauss, C., Schierz, T.: Error analysis and error estimates for co-simulation in FMI for model exchange and co-simulation in V2.0. Arch. Mech. Eng. 60(1), 75–94 (2013) Google Scholar
  7. 7.
    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, IMSD 2010, Lappeenranta, Finland, 25–27 May (2010) Google Scholar
  8. 8.
    Busch, M., Schweizer, B.: An explicit approach for controlling the macro-step size of co-simulation methods. In: Proceedings of the 7th European Nonlinear Dynamics, ENOC 2011, Rome, Italy, 24–29 July (2011) Google Scholar
  9. 9.
    Busch, M., Schweizer, B.: Coupled simulation of multibody and finite element systems: an efficient and robust semi-implicit coupling approach. Arch. Appl. Mech. 82(6), 723–741 (2012) zbMATHCrossRefGoogle Scholar
  10. 10.
    Carstens, V., Kemme, R., Schmitt, S.: Coupled simulation of flow-structure interaction in turbomachinery. Aerosp. Sci. Technol. 7, 298–306 (2003) zbMATHCrossRefGoogle Scholar
  11. 11.
    Cuadrado, J., Cardenal, J., Morer, P., Bayo, E.: Intelligent simulation of multibody dynamics: Space-state and descriptor methods in sequential and parallel computing environments. Multibody Syst. Dyn. 4, 55–73 (2000) zbMATHCrossRefGoogle Scholar
  12. 12.
    Daniel, W.J.T.: Analysis and implementation of a new constant acceleration subcycling algorithm. Int. J. Numer. Methods Eng. 40, 2841–2855 (1997) zbMATHCrossRefGoogle Scholar
  13. 13.
    Daniel, W.J.T.: A study of the stability of subcycling algorithms in structural dynamics. Comput. Methods Appl. Mech. Eng. 156, 1–13 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Daniel, W.J.T.: A partial velocity approach to subcycling structural dynamics. Comput. Methods Appl. Mech. Eng. 192, 375–394 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Datar, M., Stanciulescu, I., Negrut, D.: A co-simulation framework for full vehicle analysis. In: Proceedings of the SAE 2011 World Congress, SAE Technical Paper 2011-01-0516, 12–14 April, Detroit, Michigan, USA (2011). Google Scholar
  16. 16.
    Datar, M., Stanciulescu, I., Negrut, D.: A co-simulation environment for high-fidelity virtual prototyping of vehicle systems. Int. J. Veh. Syst. Model. Test. 7, 54–72 (2012) CrossRefGoogle Scholar
  17. 17.
    Dörfel, M.R., Simeon, B.: Analysis and acceleration of a fluid-structure interaction coupling scheme. Numer. Math. Adv. Appl., 307–315 (2010) Google Scholar
  18. 18.
    Eberhard, P., Gaugele, T., Heisel, U., Storchak, M.: A discrete element material model used in a co-simulated Charpy impact test and for heat transfer. In: Proceedings 1st Int. Conference on Process Machine Interactions, Hannover, Germany, 3–4 September (2008) Google Scholar
  19. 19.
    Eich, E.: Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30(5), 1467–1482 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. Teubner, Stuttgart (1998) zbMATHCrossRefGoogle Scholar
  21. 21.
    Friedrich, M., Ulbrich, H.: A parallel co-simulation for mechatronic systems. In: Proceedings of the 1st Joint International Conference on Multibody System Dynamics, IMSD 2010, Lappeenranta, Finland, 25–27 May (2010) Google Scholar
  22. 22.
    Gear, C.W., Gupta, G.K., Leimkuhler, B.J.: Automatic integration of the Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12&13, 77–90 (1985) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Gonzalez, F., Gonzalez, M., Cuadrado, J.: Weak coupling of multibody dynamics and block diagram simulation tools. In: Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2009, San Diego, California, USA, August 30–September 2 (2009) Google Scholar
  24. 24.
    Gonzalez, F., Naya, M.A., Luaces, A., Gonzalez, M.: On the effect of multirate co-simulation techniques in the efficiency and accuracy of multibody system dynamics. Multibody Syst. Dyn. 25(4), 461–483 (2011) zbMATHCrossRefGoogle Scholar
  25. 25.
    Gu, B., Asada, H.H.: Co-simulation of algebraically coupled dynamic subsystems without disclosure of proprietary subsystem models. J. Dyn. Syst. Meas. Control 126, 1–13 (2004) CrossRefGoogle Scholar
  26. 26.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 3rd edn. Springer, Berlin (2009) Google Scholar
  27. 27.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (2010) Google Scholar
  28. 28.
    Haug, E.J.: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Needhom Heights (1989) Google Scholar
  29. 29.
    Hippmann, G., Arnold, M., Schittenhelm, M.: Efficient simulation of bush and roller chain drives. In: Proceedings of ECCOMAS Thematic Conference on Multibody Dynamics, Madrid, Spain, 21–24 June (2005) Google Scholar
  30. 30.
    Kübler, R., Schiehlen, W.: Two methods of simulator coupling. Math. Comput. Model. Dyn. Syst. 6, 93–113 (2000) zbMATHCrossRefGoogle Scholar
  31. 31.
    Kübler, R., Schiehlen, W.: Modular simulation in multibody system dynamics. Multibody Syst. Dyn. 4, 107–127 (2000) zbMATHCrossRefGoogle Scholar
  32. 32.
    Lacoursiere, C., Nordfeldth, F., Linde, M.: A partitioning method for parallelization of large systems in realtime. In: Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics, IMSD 2014, ACMD 2014, Bexco, Busan, Korea, June 30–July 3 (2014) Google Scholar
  33. 33.
    Laflin, J.J., Anderson, K.S., Khan, I.M., Poursina, M.: Advances in the application of the divide-and-conquer algorithm to multibody system dynamics. J. Comput. Nonlinear Dyn., 9 (2014). doi: 10.1115/1.4026072
  34. 34.
    Lehnart, A., Fleissner, F., Eberhard, P.: Using SPH in a co-simulation approach to simulate sloshing in tank vehicles. In: Proceedings SPHERIC4, Nantes, France, 27–29 May (2009) Google Scholar
  35. 35.
    Liao, Y.G., Du, H.I.: Co-simulation of multi-body-based vehicle dynamics and an electric power steering control system. Proc. Inst. Mech. Eng. K, J. Multibody Dyn. 215, 141–151 (2001) CrossRefGoogle Scholar
  36. 36.
    Lubich, Ch.: Extrapolation integrators for constraint multibody systems. Impact Comput. Sci. Eng. 3, 213–234 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Malczyk, P., Fraczek, J.: Evaluation of parallel efficiency in modeling of mechanisms using commercial multibody solvers. Arch. Mech. Eng. LVI(3), 237–249 (2009) Google Scholar
  38. 38.
    Meynen, S., Mayer, J., Schäfer, M.: Coupling algorithms for the numerical simulation of fluid-structure-interaction problems. In: ECCOMAS 2000: European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona (2000) Google Scholar
  39. 39.
    Mraz, L., Valasek, M.: Parallelization of multibody system dynamics by additional dynamics. In: Proceedings of the ECCOMAS Thematic Conference Multibody Dynamics 2013, Zagreb, Croatia, 1–4 July (2013) Google Scholar
  40. 40.
    Naya, M., Cuadrado, J., Dopico, D., Lugris, U.: An efficient unified method for the combined simulation of multibody and hydraulic dynamics: Comparison with simplified and co-integration approaches. Arch. Mech. Eng. LVIII, 223–243 (2011) CrossRefGoogle Scholar
  41. 41.
    Negrut, N., Melanz, D., Mazhar, H., Lamb, D., Jayakumar, P.: Investigating through simulation the mobility of light tracked vehicles operating on discrete granular terrain. SAE Int. J. Passeng. Cars, Mech. Syst. 6, 369–381 (2013). doi: 10.4271/2013-01-1191 CrossRefGoogle Scholar
  42. 42.
    Pombo, J., Ambrosio, J.: Multiple pantograph interaction with catenaries in high-speed trains. J. Comput. Nonlinear Dyn. 7(4), 041008 (2012) CrossRefGoogle Scholar
  43. 43.
    Schäfer, M., Yigit, S., Heck, M.: Implicit partitioned fluid-structure interaction coupling. In: ASME, PVP2006-ICPVT11-93184, Vancouver, Canada (2006) Google Scholar
  44. 44.
    Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1, 149–188 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Simeon, B.: Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach. Springer, Berlin (2013) CrossRefGoogle Scholar
  46. 46.
    Schmoll, R., Schweizer, B.: Co-simulation of multibody and hydraulic systems: comparison of different coupling approaches. In: Proceedings of ECCOMAS Thematic Conference on Multibody Dynamics 2011, Brussels, Belgium, 4–7 July (2011) Google Scholar
  47. 47.
    Schweizer, B., Lu, D.: Predictor/corrector co-simulation approaches for solver coupling with algebraic constraints. ZAMM—J. Appl. Math. Mech. (2014). doi: 10.1002/zamm.201300191 Google Scholar
  48. 48.
    Schweizer, B., Lu, D.: Co-simulation methods for solver coupling with algebraic constraints: Semi-implicit coupling techniques. In: Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics, IMSD 2014, ACMD 2014 Bexco, Busan, Korea, June 30–July 3 (2014) Google Scholar
  49. 49.
    Schweizer, B., Lu, D.: Stabilized index-2 co-simulation approach for solver coupling with algebraic constraints. Multibody Syst. Dyn. (2014). doi: 10.1007/s11044-014-9422-y Google Scholar
  50. 50.
    Schweizer, B., Lu, D.: Semi-implicit co-simulation approach for solver coupling. Arch. Appl. Mech. (2014). doi: 10.1007/s00419-014-0883-5 Google Scholar
  51. 51.
    Schweizer, B., Li, P., Lu, D.: Explicit and implicit co-simulation methods: Stability and convergence analysis for different solver coupling approaches. J. Comput. Nonlinear Dyn. (2014). doi: 10.1115/1.4028503 Google Scholar
  52. 52.
    Schweizer, B., Li, P., Lu, D., Meyer, T.: Stabilized implicit co-simulation methods: Solver coupling based on constitutive laws. Arch. Appl. Mech. (2015). doi: 10.1007/s00419-015-0999-2 Google Scholar
  53. 53.
    Schweizer, B., Li, P., Lu, D., Meyer, T.: Stabilized implicit co-simulation method: Solver coupling with algebraic constraints for multibody systems. J. Comput. Nonlinear Dyn. (2015). doi: 10.1115/1.4030508 Google Scholar
  54. 54.
    Schweizer, B., Li, P., Lu, D.: Implicit co-simulation methods: stability and convergence analysis for solver coupling with algebraic constraints. ZAMM—J. Appl. Math. Mech. (2015). doi: 10.1002/zamm.201400087 Google Scholar
  55. 55.
    Spreng, F., Eberhard, P., Fleissner, F.: An approach for the coupled simulation of machining processes using multibody system and smoothed particle hydrodynamics algorithms. Theor. Appl. Mech. Lett. 3(1), 8–013005 (2013) CrossRefGoogle Scholar
  56. 56.
    Tomulik, P., Fraczek, J.: Simulation of multibody systems with the use of coupling techniques: a case study. Multibody Syst. Dyn. 25(2), 145–165 (2011) CrossRefMathSciNetGoogle Scholar
  57. 57.
    Verhoeven, A., Tasic, B., Beelen, T.G.J., ter Maten, E.J.W., Mattheij, R.M.M.: BDF compound-fast multirate transient analysis with adaptive stepsize control. J. Numer. Anal. Ind. Appl. Math. 3(3–4), 275–297 (2008) zbMATHGoogle Scholar
  58. 58.
    Zierath, J., Woernle, C.: Development of a Dirichlet-to-Neumann algorithm for contact analysis in boundary element systems and its application to MBS-BEM co-simulation. In: Samin, J.C., Fisette, P. (eds.) MULTIBODY DYNAMICS 2011, ECCOMAS Thematic Conference, Brussels, Belgium, 4–7 July (2011) Google Scholar
  59. 59.
    Zierath, J., Woernle, C.: Contact modelling in multibody systems by means of a boundary element co-simulation and a Dirichlet-to-Neumann algorithm. In: Oñate, E. (Series ed.) Computational Methods in Applied Sciences. Springer, Berlin (2012) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Institute of Applied DynamicsTechnical University DarmstadtDarmstadtGermany

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