A topology optimization problem is a problem where the variables allow to define optimally the structure of certain parts of an object. This type of optimization is essential in many applications, such as the design of the structure of aircraft wings or electric motors. In our case, we are interested in optimizing the design of a 3D magnetic circuit which resembles a Hall effect thruster. This circuit contains a variable area, where, at any point, the presence of matter or vacuum is described by a level set function, the value of which is decided by the optimization process. The objective function to be optimized is defined from the magnetic field calculated in a target area. This magnetic field must correspond as well as possible to a fixed objective field in order to ensure proper thruster operation. This optimization problem is a PDE constrained optimization problem, where the PDEs in this case are Maxwell equations (of the magnetostatic).

One of the difficulties encountered in manufacturing a so-optimized magnetostatic structures, is that they are not necessarily mechanically stable. In order to take this mechanical constraint into account, we have developed a level set-based topology optimization algorithm which relies on gradient information and on numerical simulations of both the mechanical deformation and the magnetostatic behavior of the structure. By comparing the designs obtained with those from magnetostatic optimization alone, our approach proves some effectiveness in obtaining robust designs.

Shape optimization is concerned with identifying shapes, or subsets of $\mathbb{R}$, behaving in an optimal way with respect to a given physical system. It has been an active field of research for the past decades and is used for example in engineering. Many relevant problems in the area of shape optimization involve a constraint in the form of a partial differential equation (PDE). Theory and algorithms in shape optimization can be based on techniques from differential geometry, e.g., a Riemannian manifold structure can be used to define the distances of two shapes. Thus, shape spaces are of particular interest in shape optimization.

In this talk, we apply the differential-geometric structure of Riemannian shape spaces to the theory of classical PDE constrained shape optimization problems. We also present a space containing shapes in $\mathbb{R}^2$ that can be identified with a Riemannian product manifold but at the same time admits piecewise smooth curves as elements. We present algorithms to solve PDE constrained (multi-)shape optimization problems and give numerical results of these algorithms.

We present a novel approach to denoising and inpainting problems for surface meshes.
The purpose of these problems is to remove noise or fill in missing parts while preserving important features such as sharp edges.
A discrete variant of the total variation of the unit normal vector field serves as a regularizing functional to achieve these goals.
In order to solve the resulting problem, we use a version of the split Bregman (ADMM) iteration adapted to the problem.
A new formulation of the total variation regularizer, as well as the use of an inexact Newton method for the shape optimization step, bring significant speed-ups compared to earlier methods.
Numerical examples are included, demonstrating the performance of our algorithm with some complex 3D geometries.

In the field of electric machine design, optimizing geometry and parameters is crucial and the consideration of uncertainties like manufacturing tolerances is important. This work presents a robust optimization framework, which is applied to design a 3-phase, 6-pole Permanent Magnet Synchronous Motor (PMSM) and addresses these uncertainties. Our approach uses Isogeometric Analysis (IGA) for the two-dimensional construction of the geometry and the discretization of the governing state equation. To optimize the performance of a PMSM, we combine shape and parameter optimization, which allows for a larger design space. To robustify the optimal design against uncertainty we apply a robust optimization technique that uses a min-max formulation. To solve this bilevel problem, we work with the value functions of the lower level maximization problems and apply Danskin's Theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower level problems with gradient based optimization. In the end we present numerical results, that show the efficiency of our approach.