Implicit Co-Simulation and Solver-Coupling: Efficient Calculation of Interface-Jacobian and Coupling Sensitivities/Gradients

We consider implicit co-simulation and solver-coupling methods, where different subsystems are coupled in time domain in a weak sense. Within such weak coupling approaches, a macro-time grid is introduced. Between the macro-time points, the subsystems are integrated independently. The subsystems only exchange information at the macro-time points. To describe the connection between the subsystems, coupling variables have to be defined. For many implicit co-simulation and solver-coupling approaches an Interface-Jacobian is required. The Interface-Jacobian describes, how certain subsystem state variables at the interface depend on the coupling variables. Concretely, the Interface-Jacobian contains partial derivatives of the state variables of the coupling bodies with respect to the coupling variables. Usually, these partial derivatives are calculated numerically by means of a finite difference approach. A calculation of the coupling gradients based on finite differences may entail problems with respect to the proper choice of the perturbation parameters and may therefore cause problems due to ill-conditioning. A second drawback is that additional subsystem integrations with perturbed coupling variables have to be carried out. In this manuscript, analytical approximation formulas for the Interface-Jacobian are derived, which may be used alternatively to numerically calculated gradients based on finite differences. Applying these approximation formulas, numerical problems with ill-conditioning can be circumvented. Moreover, efficiency of the implementation may be increased, since parallel simulations with perturbed coupling variables can be omitted. The derived approximation formulas converge to the exact gradients for small macro-step sizes.

Hybrid simulation using implicit solver coupling with HLA and FMI

Hybrid systems such as Cyber Physical System (CPS) are becoming increasingly popular, mainly due to the involvement of information technology in different aspects of life. For analysis and verification of hybrid system models, simulation is used extensively. As parts of a common hybrid system may belong to different domains of study, it is sometimes beneficial to use specialized simulation packages (SPs) for each domain. In this case, parts of a system are simulated in different SPs. The idea may seem simple, but coupling more than one simulation component presents challenges related to numerical stability. The presented article suggests an implicit solver coupling technique enhanced to facilitate simulation of hybrid models using multiple simulation components. The technique is developed using two of the most popular simulation interoperability standards, namely, the High Level Architecture and the Functional Mock-up Interface. By using these standards, the developed algorithm will be useful for a large number of practitioners and researchers. The developed algorithm is described using a generic distributed computation model, which makes it reproducible even without using the standards. For the verification of results, the algorithm is tested on two case studies. The results are compared to a monolithic simulator and the proximity of results initiates the validity of the developed algorithm.

Explicit and Implicit Cosimulation Methods: Stability and Convergence Analysis for Different Solver Coupling Approaches

The numerical stability and the convergence behavior of cosimulation methods are analyzed in this manuscript. We investigate explicit and implicit coupling schemes with different approximation orders and discuss three decomposition techniques, namely, force/force-, force/displacement-, and displacement/displacement-decomposition. Here, we only consider cosimulation methods where the coupling is realized by applied forces/torques, i.e., the case that the coupling between the subsystems is described by constitutive laws. Solver coupling with algebraic constraint equations is not investigated. For the stability analysis, a test model has to be defined. Following the stability definition for numerical time integration schemes (Dahlquist's stability theory), a linear test model is used. The cosimulation test model applied here is a two-mass oscillator, which may be interpreted as two Dahlquist equations coupled by a linear spring/damper system. Discretizing the test model with a cosimulation method, recurrence equations can be derived, which describe the time discrete cosimulation solution. The stability of the recurrence equations system represents the numerical stability of the cosimulation approach and can easily be determined by an eigenvalue analysis.