Subject: Sensitivity-Based Topology and Shape Optimization with Application to Electrical Machines
Author: Dipl.-Ing. Peter Gangl
Supervision: O.Univ.-Prof. Dr. Ulrich Langer
External referee: Univ.-Prof. Dr. Fredi Tröltzsch (TU Berlin)

This thesis deals with topology and shape optimization methods for finding optimal geometries of devices from electrical engineering. As a model problem, we consider the design optimization of an electric motor. Here, the performance of the motor depends on the electromagnetic fields in its interior, which, among other factors, also depend on the geometry of the motor via the solution to Maxwel's equations. In our model, we use a special regime of Maxwell's equations, namely the partial differential equation (PDE) of nonlinear magnetostatics, and consider a two-dimensional setting of the electric motor. Thus, we are facing a PDE-constrained optimization problem where the unknown is the geometry of a given part of the motor.

An important tool for solving shape optimization problems is the shape derivative, i.e., the sensitivity of the domain-dependent objective function with respect to a smooth variation of a boundary or material interface. We derive the shape derivative for the optimization problem at hand, which involves a nonlinear PDE constraint, by means of a Lagrangian approach. We employ the shape derivative to obtain an improved design of the electric motor. One shortcoming of the class of shape optimization methods is that they can only vary boundaries or interfaces of given designs and cannot alter their topology, i.e., they cannot introduce holes or new components.

Using topology optimization methods, also the connectivity of a domain can change during the optimization procedure. In this thesis, we focus on topology optimization approaches based on topological sensitivities. On the one hand, we consider the sensitivities of the objective function with respect to a local variation of the material. On the other hand we rigorously derive the topological derivative, i.e., the sensitivity of a domain-dependent objective function with respect to the introduction of a hole in the interior of the domain. The latter approach is particularly involved in this case due to the nonlinear PDE constraint. The information provided by these sensitivities can be used for determining optimal designs whose topology may be different from the topology of the initial design.

In both classes of methods, we start with an initial geometry consisting of several materials and successively update the material interfaces in the course of the optimization procedure. The update is based on topological or shape sensitivities, which depend on the solutions to two PDEs (the state equation and the adjoint equation of the optimization problem). These PDEs are approximately solved by means of the finite element method on a triangular grid in each iteration. In order to obtain accurate solutions to these PDEs, the evolving interface should be resolved by the finite element discretization. We introduce a local mesh adaptation strategy which modifies the mesh only in a neighborhood of the interface and show optimal order of convergence as the mesh size approaches zero.

Finally, we combine the three components mentioned above and apply it to the optimization of electric motors. In a first step, we perform topology optimization in order to find the optimal connectivity of the design. In a second step, we use shape optimization together with the proposed mesh adaptation strategy as a post-processing in order to get smoother designs.

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