# Sensitivity-Based Topology and Shape Optimization for Electrical Machines

## Dipl.-Ing. Peter Gangl

**Dec. 9, 2014, 3:30 p.m. S2 059**

In many engineering applications one is interested in designing products whose geometry is optimal with respect to some criteria. Such optimal design problems can be solved by means of topology optimization and shape optimization methods. In this talk we will give an overview over existing topology and shape optimization methods, draw some connections between them and show applications to the optimization of an electric motor.

We will consider more in detail an approach that is based on topological sensitivities. The topological derivative of a domain-dependent functional $\mathcal J(\Omega)$ at a spatial point $x$ represents its sensitivity with respect to the insertion of an infinitesimally small hole around that point. Its sign indicates in which areas of the design domain introducing a hole would lead to a decrease of the objective functional $\mathcal J$. Topological derivatives are rather well understood in the case where the optimization problem is constrained by a linear partial differential equation (PDE). In this talk, we will derive the topological derivative for the case of a nonlinear PDE constraint and show its application to the optimization of an electric motor.

Moreover, we will address the same problem by means of shape optimization where the geometry is modified by moving a material interface along a velocity

field which guarantees a decrease in the objective functional. We will derive the shape derivative of the nonlinear problem and show numerical results.