Mathematical Analysis and Simulation of Field Models in Accelerator Circuits

Abstract

Typically in electrical engineering a network modelling approach for the simulation of devices and their surrounding circuitry is taken, where each device is considered by a voltage-to-current relation. For some applications, however, this simplification does not yield the required accuracy. In these cases, refined modelling can be performed, where a spatially distributed partial differential equation modelling the required physical quantity is coupled to the classic network equations. The resulting coupled system of equations often exhibits a multiscale, multirate and even multiphysical behaviour that is tackled with involved algorithms so as to efficiently simulate it. Its structural analysis is therefore important, to numerically treat the system appropriately and to ensure that the algorithms converge properly. This thesis deals with the mathematical analysis of these type of systems as well as their simulation. The systems of equations obtained from circuits with semidiscrete refined models are typically differential algebraic equations. Their numerical and analytical difficulties is studied in the context of their differential algebraic index. For that, three generalised circuit element definitions are given, that allow the classification of the refined models. Hereby, the index of the entire coupled system can be specified by means of topological properties of the circuit. Several approximations to Maxwell’s equations are classified with the generalised element definitions to obtain the index properties of the field-circuit coupled systems. For the simulation two algorithms are studied. First the co-simulation waveform relaxation method is analysed for field-circuit coupled systems arising from magnetoquasistatic fields with eddy current effects on superconducting coils. The convergence of the algorithm is sped up by means of optimised Schwarz methodologies. Here, the information exchange between both subsystems is improved by a linear combination of the coupling conditions. To further speed up simulation time, the parallel-in-time method Parareal is analysed. The algorithm is investigated in the context of differential algebraic equations by studying its applicability to nonlinear higher index systems arising e.g. from circuit simulation. Finally, two approaches are proposed for the combination of Parareal and waveform relaxation. One of them is specifically designed for field-circuit coupled systems and yields a micro-macro-like Parareal algorithm. However, the idea behind it can be applied to other type of coupled systems. Numerical tests of field-circuit coupled systems are made to underline the results obtained from the mathematical theory as well as test the efficiency of the proposed algorithms.