Explicit Co-simulation Approach with Improved Numerical Stability

Abstract

For coupling different solvers, explicit co-simulation approaches are frequently applied, especially when proprietary software tools have to be coupled without full solver access. Explicit solver coupling approaches are usually much simpler to implement than implicit methods. A major drawback of explicit coupling approaches is their reduced numerical stability behavior. Applying classical explicit co-simulation techniques, the coupling variables are approximated by extrapolation/interpolation polynomials in order to carry out the subsystem integration. Typically, Lagrange polynomials are applied for generating the approximation polynomials, using the coupling variables at the macro-time points as sampling points. In this manuscript, an explicit coupling approach is presented, which shows an improved numerical stability behavior. The key idea of the proposed method is to apply polynomial approximation to predict the acceleration variables of the coupling bodies. By integrating the predicted accelerations, one obtains predicted state variables for the coupling bodies, which are used to predict the coupling variables by making use of the constitutive equations of the coupling element. Compared to classical coupling techniques, where the coupling variables are directly approximated, the approach presented here based on approximated accelerations exhibits a significantly improved numerical stability behavior. Moreover, the numerical error is markedly reduced. The co-simulation method is introduced for flexible multibody systems. However, the proposed approach can generally be applied to couple arbitrary mechanical systems. Also, the coupling technique may be used to couple non-mechanical systems, e.g. electrical or hydraulic systems.

Appendices

Appendix 1: Stability and Convergence Plots for Linear Approximation Polynomials

In this appendix, stability and convergence plots are collected for the case that linear approximation polynomials $(k = 1)$ are used for approximating the acceleration variables. The stability plots for the case $k = 1$, see Fig. 9.26, resemble the corresponding plots for $k = 0$ in Sect. 9.2.4. Compared with its classical counterpart based on quadratic approximation polynomials for the coupling variables, see Fig. 8 in Ref. [71], the integrated acceleration approach for $k = 1$ shows a significantly better numerical stability behavior.

Fig. 9.26

Stability plots for the integrated acceleration co-simulation approach based on linear approximation polynomials $(\varvec{k} = {\mathbf{1}})$: Jacobi and Gauss-Seidel type using force/force-, force/displacement- and displacement/displacement-decomposition

A convergence analysis exhibits the improved convergence behavior for the case that linear approximation polynomials are used, see Fig. 9.27. For $k = 1$, the global errors $\varepsilon_{pos,glo}$ and $\varepsilon_{vel,glo}$ converge with ${\mathcal{O}}(H^{3} )$; the local errors $\varepsilon_{pos,loc}$ converge with ${\mathcal{O}}(H^{5} )$ and $\varepsilon_{vel,loc}$ with ${\mathcal{O}}(H^{4} )$. Note that the same convergence behavior—however with larger error magnitudes—is achieved with the classical approaches using quadratic approximation polynomials for the coupling variables, see Fig. 12 in Ref. [71].

Fig. 9.27

Convergence plots for the integrated acceleration co-simulation approach $(\varvec{k} = {\mathbf{1}})$: global and local error of the position and velocity variables over the macro-step size H

Fig. 9.28

Co-simulation test model: displacement/force-decomposition approach

Appendix 2: Stability Plots for Displacement/Force-Coupling (Gauss-Seidel Type)

While force/force- and displacement/displacement-coupling represent symmetrical decomposition techniques, force/displacement-coupling is an unsymmetrical decomposition approach. Considering the co-simulation test model, one has therefore to distinguish, whether subsystem 1 or subsystem 2 is the force-excited single-mass oscillator (“force/displacement-coupling (F/D)” or “displacement/force-coupling (D/F)”), see Figs. 9.3 and 9.28. Using a parallel integration scheme (Jacobi type), the D/F co-simulation test model can be represented by the F/D co-simulation test model simply by changing the parameters $m_{1}$, $c_{1}$, $d_{1}$ and $m_{2}$, $c_{2}$, $d_{2}$. For instance, F/D-coupling with $m_{1} = 2 \cdot m_{2} = m,c_{1} = 2 \cdot c_{2} = c,d_{1} = 2 \cdot d_{2} = d,c_{c} ,d_{c}$ is equivalent with D/F-coupling using $2 \cdot m_{1} = m_{2} = m,2 \cdot c_{1} = c_{2} = c,2 \cdot d_{1} = d_{2} = d,c_{c} ,d_{c}$ provided that the subsystems are integrated in parallel.

For the sequential Gauss-Seidel scheme, however, the D/F test model cannot be represented by the F/D test model by simply changing the model parameters. Using the same test model parameters, D/F-coupling and F/D-coupling may show a different numerical stability behavior. Figure 9.29 shows stability plots for the integrated acceleration approach using F/D- and D/F-decomposition (Gauss-Seidel type, $k = 0$). Two different parameter sets for the test model are investigated. It is interesting to notice that the F/D-coupling approach shows an improved stability behavior for both parameter sets, especially for the case that subsystem 2 is much stiffer than subsystem 1 $(\alpha_{{\Lambda i21}} = 5)$.

Fig. 9.29

Stability plots for the integrated acceleration co-simulation approach based on force/displacement- and displacement/force-decomposition (Gauss-Seidel type, $\varvec{k} = 0$) for two different test model parameters