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Technische Universität Darmstadt

1. Li, Pu. On the Numerical Stability of Co-Simulation Methods.

Degree: PhD, Institute of Applied Dynamics (AD), 2017, Technische Universität Darmstadt

To couple two or more subsystem solvers in time domain, co-simulation methods are used in many fields of application. In the framework of mechanical systems, there exist mainly two ways to couple different subsystems, namely coupling either by constitutive laws or by algebraic constraint equations. In this work, the numerical stability and the convergence behavior of co-simulation methods is analyzed. For the stability analysis, a test model has to be defined. Following the stability definition for numerical time integration schemes, namely Dahlquist’s stability theory, a linear test model is used. The co-simulation test model applied here is a two-mass oscillator, where the two masses are connected by a spring-damper element or by a rigid link. Discretizing the test model with a co-simulation method, recurrence equations can be derived, which describe the time discrete co-simulation solution. Applying an applied-force coupling approach, the stability behavior of the linear two-mass oscillator is characterized by 7 independent parameters. In order to compare different co-simulation approaches, 2D stability plots are convenient. Therefore, 5 of the 7 parameters are fixed so that the spectral radius can be depicted as a function of the remaining 2 parameters. The results presented show that implicit coupling schemes exhibit a significantly better numerical stability behavior than explicit schemes. Furthermore, enhanced stability behavior can be achieved by extending the coupling conditions, i.e., by taking into account derivatives and integrals of the constitutive equations. Especially, a very good stability behavior may be obtained with the D-extended force/force-coupling approach in combination with quadratic approximation functions. The analysis of the numerical stability of co-simulation methods with algebraic constraints is the second subject of this work. 5 independent parameters have to be introduced for the corresponding test model. The dimensionless real and imaginary part of the eigenvalue of subsystem 1 are used as axes in 2D stability plots; the other 3 parameters are held constant. Three classical methods for constraint stabilization, namely the Baumgarte stabilization technique, the weighted multiplier approach and the projection technique, are discussed for different approximation orders. Alternatively, co-simulation approaches on index-2 and on index-1 level are discussed, where the Lagrange multiplier is discretized between the macro-time points (extended multiplier approach). As a result, the coupling conditions and their time derivatives can simultaneously be fulfilled at the macro-time points. Different multibody models are used in order to demonstrate the application of the above mentioned co-simulation techniques. Advisors/Committee Members: Schweizer, Bernhard (advisor), Ams, Alfons (advisor).

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APA (6th Edition):

Li, P. (2017). On the Numerical Stability of Co-Simulation Methods. (Doctoral Dissertation). Technische Universität Darmstadt. Retrieved from

Chicago Manual of Style (16th Edition):

Li, Pu. “On the Numerical Stability of Co-Simulation Methods.” 2017. Doctoral Dissertation, Technische Universität Darmstadt. Accessed July 20, 2018.

MLA Handbook (7th Edition):

Li, Pu. “On the Numerical Stability of Co-Simulation Methods.” 2017. Web. 20 Jul 2018.


Li P. On the Numerical Stability of Co-Simulation Methods. [Internet] [Doctoral dissertation]. Technische Universität Darmstadt; 2017. [cited 2018 Jul 20]. Available from:

Council of Science Editors:

Li P. On the Numerical Stability of Co-Simulation Methods. [Doctoral Dissertation]. Technische Universität Darmstadt; 2017. Available from: