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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Institute of Applied Dynamics, Technical University Darmstadt, Otto-Berndt-Strasse 2, 64287, Darmstadt, Germany

    Bernhard Schweizer, Daixing Lu & Pu Li

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  1. Bernhard Schweizer
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  2. Daixing Lu
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  3. Pu Li
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Correspondence to Bernhard Schweizer.

Appendices

Appendix A: Stability plots

1.1 A.1 Stability plots: relaxation parameter \(\omega_{R} =1\)

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figure 15

Stability plots for the stabilized index-2 co-simulation method: Relaxation parameter \({\omega}_{R} =1\), polynomial degrees \(k=0,1,2\)

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1.2 A.2 Stability plots: relaxation parameter \(\omega_{R} =0.7\)

Fig. 16
figure 16

Stability plots for the stabilized index-2 co-simulation method: Relaxation parameter \({\omega}_{R} =0.7\), polynomial degrees \({k}=0,1,2\)

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1.3 A.3 Stability plots: relaxation parameter \(\omega_{R} =0.5\)

Fig. 17
figure 17

Stability plots for the stabilized index-2 co-simulation method: Relaxation parameter \({\omega}_{R} =0.5\), polynomial degrees \({k}=0,1,2\)

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Appendix B: Convergence plots

Fig. 18
figure 18

Convergence plots for the stabilized index-2 co-simulation method (local and global errors of the position and velocity variables): Relaxation parameters \({\omega}_{R} =1\), \(\omega_{R} =0.7\), \(\omega_{R}=0.5\); polynomial degrees \({k}=0,1,2\)

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Schweizer, B., Lu, D. & Li, P. Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques. Multibody Syst Dyn 36, 1–36 (2016). https://doi.org/10.1007/s11044-015-9464-9

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