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 (communication-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 (i.e., coupling sensitivities, coupling gradients) 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 paper, 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.