Subject: The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems
Author: Dipl.-Ing. Michael Kolmbauer
Supervision: O.Univ.-Prof. Dr. Ulrich Langer
External referee: Prof. Dr. Fredi Tröltzsch (Technische Universität Berlin)

3D-visualization of the solution y at t=0

This thesis deals with the simulation and control of time-dependent, but time-periodic eddy current problems in unbounded domains in R3. In order to discretize such problems in the full space-time cylinder, we use a non-standard space-time discretization method, namely, the multiharmonic finite element and boundary element method. This discretization technique yields large systems of linear algebraic equations, whereas the fast solution of these systems determines the efficiency of this method. Here, suitable preconditioners are needed in order to ensure efficient and parameter-robust convergence rates of the applied iterative method. Therefore, the main focus of this thesis lies on the construction and analysis of robust and efficient preconditioning strategies for the resulting systems of linear equations.

The basic idea of the multiharmonic approach is to use a Fourier series approximation as a discretization technique in time. This allows to switch from the time domain to the frequency domain, and therefore, to replace the solution of a time-dependent problem by the solution of a system of time-independent problems for the Fourier coefficients. Due to the infinite exterior domain on the one hand, and inhomogeneities and the possible presence of sources in the interior domains on the other hand, the symmetric finite element – boundary element coupling method is used to discretize the Fourier coefficients in space.

The main challenge in the construction of efficient and parameter-robust preconditioners for the resulting frequency domain equations is indicated by the full range of crucial model, regularization and discretization parameters, that imping on the convergence rate of any iterative method. We use matrix and operator interpolation techniques to construct parameter-robust, block-diagonal preconditioners in a straightforward fashion. Furthermore, we prove rigorous bounds for the condition numbers of the preconditioned frequency domain equations, that are independent of all involved model, regularization, and discretization parameters. Numerical examples illustrate the robustness of these block-diagonal preconditioners.

1D-visualization of the x component of the state y((0.5,0.5,0.5),.), lambda=1e-8

In order to obtain efficient preconditioners, the diagonal blocks have to be replaced by efficient (and robust) preconditioners. The individual diagonal blocks rely on the solution of standard H1 or H(curl) problems, for which efficient and robust preconditioners are already available. First, we consider and analyze the eddy current problem. We start by investigating the pure multiharmonic finite element approach on a bounded domain, before we tackle the case of unbounded domains in terms of the multiharmonic finite element – boundary element coupling method. For both, we construct block-diagonal preconditioners for various types of variational formulations.

Second, we consider the eddy current optimal control problem with distributed control. Again we apply the multiharmonic finite element – boundary element coupling method to discretize in time and space. After deriving the optimality system, efficient and parameter-robust preconditioners are constructed for the resulting system of linear equations.

A challenging topic in optimal control problems is the incorporation of various constraints imposed on the state or control variables in the optimization procedure. These constraints render the resulting system of equations nonlinear. There, the semi-smooth Newton linearization leads to problems, where our constructed block-diagonal preconditioners can also be useful.

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