# Isogeometrical Analysis based Shape Optimization

## DI Rainer Schneckenleitner

**Nov. 28, 2017, 3:15 p.m. S2 416-1**

Shape optimization problems arise in many different scientific and engineering areas e.g. mechanical engineering, electrical engineering or chemistry. For many practical problems the underlying object of interest is represented by B-splines or NURBS curves 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). The idea in IgA is that the domain for the analysis remains the same as for the geometry of the object of interest constructed with 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 there need not be a conversion, we have an exact representation of the domain. In this thesis we will investigate IgA for shape optimization problems subject to PDEs.

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 put an electric motor into the IgA framework. We compare our results with other results obtained with standard FEM to confirm the correctness of our obtained results.