Inverse and optimization problems in electromagnetics -- a finite-element method perspective

Sammanfattning: In this thesis, a selection of inverse and optimization problems are studied where the finite element method (FEM) serves as a comprehensive tool to solve electromagnetic field problems that lack an analytic solution. The inverse problems are typically formulated in terms of an optimization problem where the misfit between a measurement and the corresponding result of a computational model is minimized. The optimization problems are solved by a combination of techniques that involve gradient-based methods, stochastic methods and parameter studies.

The first contribution of the thesis is a new higher-order hybrid FEM for Maxwell's equations that combines (i) brick-shaped elements for large homogeneous regions with (ii) tetrahedrons for regions where local refinement is necessary. The tangential continuity of the electric field at the interface between the different element types is enforced in the weak sense using Nitsche's method. This yields a flexible and efficient computational method that is free of spurious solutions and features a low dispersion error. We employ a stable implicit-explicit time-stepping scheme using an implicitness parameter associated with the tetrahedrons and the hybrid interface. No late-time instabilities are observed in the solution for computations with up to 300 000 time steps.

The second contribution of this thesis deals with four inverse scattering problems:
(i) gradient-based estimation of the dielectric properties of moist micro-crystalline cellulose in terms of a Debye model;
(ii) detection and positioning of multiple scatterers inside a metal vessel using compressed sensing;
(iii) monitoring of the material perturbations in a pharmaceutical process vessel using a linearized model around an operation point that varies with the process state;
and (iv) a subspace-based classification method for the detection of intracranial bleedings in a simulated data set.

The third contribution of the thesis explores stochastic optimization for an inductive power transfer (IPT) system consisting of four magnetically coupled resonance circuits, which is intended for power transfer distances on the order of the coils' radius. A genetic algorithm is employed to compute the Pareto front that contrast the maximum efficiency and power transfer. Results are presented for both linear and non-linear circuits: (i) a time-harmonic model for magnetically coupled resonance circuits with a resistive load; and (ii) a transient model for an IPT system with square-wave excitation, rectifier, smoothing filter and battery.