Isogeometric analysis and shape optimization in electromagnetism

In this thesis a recently proposed numerical method for solving partial differential equations, isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular emphasis on applications to two-dimensional design problems arising in electromagnetic applications. The study is motivated by the fact that in contrast with most commonly utilized finite element approximations, IGA allows one to exactly represent geometries arising in computer aided design applications with relatively few variables using splines. The following problems coming from theoretical considerations or engineering applications are solved in the thesis utilizing IGA: finding a shape having a few prescribed eigenvalues of the Laplace operator; shape optimization of sub-wavelength micro-antennas for energy concentration; shape optimization of nano-antennas for field enhancement; economical design of magnetic density separators. From the point of view of method development, several heuristic approaches for extending a valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The parametrization approaches and a method for validating a spline parametrization are combined into an iterative algorithm for shape optimization of two dimensional electromagnetic problems. The algorithm may also be relevant for problems in other engineering disciplines. Using the methods developed in this thesis, remarkably we have obtained antennas that perform one million times better than an earlier topology optimization result. This shows a great potential of shape optimization using IGA in the area of electromagnetic antenna design in particular, and for electromagnetic