Shape optimization problems arise in many different scientific and engineering areas, e.g., in mechanical engineering or electrical engineering. For many practical problems, the underlying object of interest is represented by B-splines or NURBS due to computer-aided design software. Many properties of such objects of interest depend on the solution of a partial differential equation (PDE). So far the B-spline or NURBS based computer model is usually decomposed into finite elements for the analysis. Additionally, this has the consequence that usually the boundary of the model has to be approximated with polygonal subdomains. In 2005, a new idea came up for such problems, called isogeometric analysis (IgA). In IgA, we use the same basis functions (B-splines or NURBS) for both representing the geometry and approximating the solution of the PDE under consideration. Thus, there is no approximation of the geometry of the object of interest constructed by means of some computer-aided design program. Although the finite element method (FEM) is a well established method for shape optimization this new idea seems to be beneficial because on the one hand no conversion of the models is necessary, which can be computationally very costly. On the other hand, because geometry conversion is not needed, we have an exact representation of the domain.
In this thesis, we will investigate IgA for shape optimization problems subject to PDE constraints. We will show that the IgA approach has its justification in PDE constrained shape optimization processes. First we are going to investigate a linear model problem in IgA with a well established standard algorithm, and then we will apply a relatively new optimization algorithm to this linear model problem. Finally, we are going to treat an electric motor in the IgA framework. We compare our results with other results obtained with standard FEM to confirm the correctness of our obtained results.