Session: PDE-Constrained Optimization
Chair: Zhexian Li
Cluster: nan
Talk 1: Robust Topology Optimisation of Electric Machines using Topological Derivatives
Speaker: Theodor Komann
Abstract: Designing high-performance electrical machines that remain efficient and reliable under uncertain material and operating conditions is important for industrial applications. In this paper, we present a robust topology optimization framework with PDE constraints to address this challenge. We formulate the robust optimisation problem as a min-max problem, where the inner maximisation considers the worst-case under predefined uncertainties, and the outer minimisation searchs for a design that remains robust to these variations using the topological derivative. A level set function represents the shape of the domain, allowing for arbitary perturbations. We use a theorem of Clarke to compute subgradients of the worst-case function, thereby ensuring efficient solution of the min-max problem. Finally, numerical results on a two-material permanent magnet synchronous machine illustrate both the effectiveness of the method and the improved performance of designs that account for uncertainty. Theodor Komann, TU Darmstadt Joint work with Peter Gangl and Nepomuk Krenn, RICAM Linz, and Stefan Ulbrich, TU Darmstadt
Talk 2: Generalized Nash equilibrium problems for linear hyperbolic PDE-constrained games
Speaker: Marcelo Bongarti
Abstract: The concept of Nash equilibrium is fundamental to a wide range of applications, spanning fields from particle mechanics to micro and macroeconomics. However, much of the existing literature focuses on finite-dimensional settings. In this talk, we draw on energy markets coupled with transport dynamics to motivate the study of multi-objective optimization problems with linear hyperbolic PDE constraints. We present recent results on the existence and characterization of Nash equilibria, emphasizing optimality conditions as a framework for understanding such solutions.
Talk 3: Bilinear optimal control of advection-diffusion-reaction equations
Speaker: Zhexian Li
Abstract: We consider bilinear optimal control of advection-diffusion-reaction equations, where control arises as the velocity in the advection term. Existing works approached the problem by discretizing the ADR equation in space and time and solving the resulting finite-dimensional optimization. In this study, we treat the advection term as a forcing term and the equation becomes a reaction-diffusion equation. Then, we apply the Fourier transform to reformulate the PDE as infinitely many decoupled ODEs for which optimal control can be derived. The resulting optimal control is in an integral representation in the complex plane and can be numerically solved efficiently. Numerical experiments on canonical examples show that the optimality gap under our approach is smaller than existing approaches that discretize the equation.
