Convergence of Waveform Relaxation for Coupled DAEs Describing Field/Circuit Coupled Systems
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Lumped circuit models are employed in electrical engineering to describe the behavior of certain devices. However, these models neglect spatially distributed electromagnetic phenomena, which, for some applications, is a too restrictive modeling assumption. In these cases, field/circuit coupling can be performed, where fully spatially distributed electromagnetic field elements are coupled into a surrounding lumped circuit. Here, these field models can be understood as a type of generalization of the classical circuit elements. Cosimulation methods are well suited to solve field/circuit coupled systems since they allow for dedicated solvers and time scales for the different subsystems. A well-established family of iterative cosimulation methods is waveform relaxation. While waveform relaxation is convergent for coupled ordinary differential-equations on finite time intervals, it can diverge for coupled differential-algebraic equations unless an additional contractivity condition is satisfied. Computing the contractivity condition, however, is usually costly. Here, we build on recent results concerning the classification of field models as so-called generalized circuit elements and circuit-topological convergence criteria for waveform relaxation on coupled circuits. Notably, we present an easy-to-check topological convergence criterion for coupled field/circuit systems, where the field can be classified as a generalized circuit element and the circuit is modeled by modified nodal analysis. If the convergence criterion is not satisfied, we have to expect waveform relaxation to diverge, or to converge comparatively slowly. For the case of slow convergence, accelerating strategies such as the optimized waveform relaxation method can be employed to speed-up simulation time. In addition to the topological convergence criterion, we present simulation results which confirm our theoretical findings for both convergent and divergent cases.