PhD Defence: Sensitivity-Based Topology and Shape Optimization with Application to Electrical Machines

Sensitivity-Based Topology and Shape Optimization with Application to Electrical Machines

This thesis deals with methods of topology and shape optimization and is motivated by a design optimization problem from electrical engineering which is constrained by the quasilinear PDE of two-dimensional magnetostatics.

We compute the topological derivative of the objective function, i.e., its sensitivity with respect to a topological perturbation of the domain, and compare it to other topological sensitivities used in the literature. Moreover, we compute the shape derivative, which is the sensitivity of a domain-dependent functional with respect to a smooth variation of its boundary. These sensitivities are used to obtain optimal designs using gradient descent methods.

In the course of the optimization algorithm, the material interface between different subdomains evolves and has to be taken into account in order to obtain accurate finite element solutions. We present a local mesh modification strategy for triangular meshes which yields smoother designs and show optimal order of convergence of the obtained finite element solution to the true solution.

Finally, we combine these three techniques and apply them to a design optimization problem for an electric motor.