Coupled system of Maxwell equations and circuit equations in electro-magnetism

This thesis is devoted to modeling electro-magnetic coupling, the so-called crosstalk phenomenon, in circuit simulation, where a new modeling approach via bilateral coupling of the Maxwell equations with circuit equations is considered. Using the bilaterally coupled model allows full simulation of the crosstalk phenomenon in the evolution of time, when the disturbances generated by the excitation current are themselves causing the excitation current to be influenced. In fact, the additional coupling relation corresponding to the bilateral model is beneficial for crosstalk modeling since it models the sensitivity of the phenomenon to the distance of the elements as one of the main characteristics of the crosstalk. The uncoupled circuit equations have been already completely analyzed by many people. Therefore, in this thesis, the partial differential equation (PDE) part of the bilateral model as well as the bilateral coupling relations with respect to the underlying differential-algebraic equations (DAEs) are studied. In the first part, three-dimensional (3D) modeling of the electro-magnetic coupling in a general framework leading to partial differential-algebraic equations (PDAEs) is presented. The involved set of PDEs is viewed as an operator DAE on Banach spaces in an abstract setting. This permits converting one of the main considered issues with DAEs, sensitivity to perturbation, from the PDEs to the DAEs. It is shown that the operator DAE is of a high index by considering the semi-discretization in space. Therefore, the regularization of high index DAE operators before time integration is performed by the minimal extension technique on an operator level, where the introduced new reformulation of the system is necessary. In fact, the minimal extension is a technique of reducing the index and is applied to the spatial discretized DAE operators to obtain a lower index with better properties from the numerical point of view. Additionally, the spatial semi-discretization of the bilateral coupling relations is presented to obtain a DAE set. The presented model is in a general framework and satisfying from a theoretical point of view but does not sufficiently fulfil numerical aspects since coupling is normally involved with imposing some assumptions (or equivalently making some decisions) on the geometry of the elements. In fact, the recoupling from the circuit equations to the Maxwell equations requires approximation of an extra system of equations within an unbounded domain, where this forces the scattering problem to be involved. To skip the unbounded scattering problem, a new two-dimensional (2D) model for the electrical field recoupling is presented in the second part. Regarding the magnetic field coupling, it is shown that it suffices to consider one component of the magnetic field with respect to the Helmholtz decomposition and not the whole field. This component is obtained by approximation of a new system of equations, which can be computed efficiently during postprocessing. It is beneficial since post-processing is done independently of our simulation at each time step. Furthermore, the system structure is mathematically easier to deal with due to the presence of the Laplace operator and the absence of the curl operator. The remaining part of this thesis is concerned with the simulation of the crosstalk within a 2D electro-magnetic system, where the linear, isotropic, and