# Isogeometrical Analysis based Shape Optimization

## DI Rainer Schneckenleitner

**Nov. 28, 2017, 4:15 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.