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

Abstract

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.

MISSILES: an Efficient Resolution of the Co-simulation Coupling Constraint on Nearly Linear Differential Systems through a Global Linear Formulation

“…The advanced interactions are not equivalently rare in practice: lots of simulation and modelling platforms can generate systems with the possibility to represent polynomial inputs for instance, yet the rollback is very seldom possible on a system for co-simulation. As the rollback is mandatory to use an iterative co-simulation method like [2] [12] [18] [3] [14] [15] or any implicit co-simulation method [13] [19], lots of industrial models cannot benefit from the advantages of such methods.…”

COSTARICA estimator for rollback-less systems handling in iterative co-simulation algorithms

“…Let's define the Laplace transforms of the inputs, outputs and states of the time-shifted linear system (18).…”

“…Thanks to all the elements introduced above, we can express the linear contribution to outputs estimator as the inverse Laplace of Y on the step size δt [N ] as this is the final time of the time-shifted system (18).…”

“…Other methods do not require systems to be disclosed: modular models can be made of interconnected, black-boxed systems. Such methods, when they correspond to advanced co-simulation algorithms [24] [15] [11] [12] [18], usually require the systems to perform advanced actions in addition to the minimal set of possible ones. Indeed, depending on the modeling and simulation platform that generates a system, the latter might be able to perform more than just the basic actions.…”

“…The rollback is the ability of a system to re-integrate itself on a time slice on which it has already undergone integration. When every system of a modular model has this capability, an iterative co-simulation algorithm can be used [20] [3] [4] [11] [12] [18] [31] [30] or the methods referred to as ICSs (implicit cou-pling schemes) in [35]. Iterative methods, when they converge, are a good way to reach a required accuracy on a wide range of models while supporting blackboxed systems.…”