Mini workshop on Shape Optimization and Inverse Problems

April 12th 2024, 9AM-12PM, SP2 416-1

The mini workshop is organized by Peter Gangl und Mourad Sini.

Program

Abstracts

Florian Feppon. Dehomogenization techniques are becoming increasingly popular for generating lattice designs of compliant mechanical systems with ultra-large resolutions. These methods are based on computing a deformed periodic grid that enables the reconstruction of fine-scale designs with modulated and oriented patterns. In this presentation, I will propose an approach for extending dehomogenization methods to laminar fluid systems. I will first discuss some of the important difficulties in the development of dehomogenization methods for fluid systems due to shortcomings in the available homogenization theory. Then, I will present a simplified numerical methodology leveraging homogenized fluid models that allows the generation of fine-scale microchannel designs for two-dimensional applications. In the second part of the talk, I will discuss recent results in the homogenization theory of porous media which could help devise homogenized models more suited to industrial applications in the future. These include the derivation of higher-order models able to capture the transition from the Brinkman to the Darcy regime, and the derivation of effective boundary conditions for periodic systems featuring inlets and outlets.

Kamran Sadiq. This talk concerns an inverse source problem for the linearized Boltzmann equation in two dimensions. The medium is assumed known. The outgoing radiation is measured on an arc of the boundary. For scattering kernels dependent on the angle of scattering, we show that a source can be recovered in the convex hull of the measuring arc. The method, specific to two dimensional domains, relies on Bukgheim’s theory of A-analytic maps and it is joint work with A. Tamasan (UCF) and H. Fujiwara (Kyoto U).

Soumen Senapati. In this talk, we will discuss an inverse problem in the time-domain wave equation which amounts to simultaneously reconstruct the medium properties (such as mass density and bulk modulus) and source function. For the data, along with traditional measurement i.e. the Dirichlet trace of wave-field on the boundary, we consider an auxiliary measurement in terms of the boundary observation of wave-field generated by the medium when it is injected by small-scaled contrast agents at different interior points. Under critical scaling of these contrast agents, we derive an asymptotic profile for the auxiliary wave-field which results in simultaneous reconstruction of material properties and source. Our analysis primarily uses the time-domain Lippmann-Schwinger equation and integral equation techniques.

Nepomuk Krenn. We consider the topology optimization problem of a 2d permanent magnet synchronous machine in magnetostatic operation. This results in a multi-material design optimization problem constrained by a quasilinear PDE. The focus of this talk is on the derivation of the topological derivative which indicates the point wise sensitivity of the problem subject to infinitesimal material changes. The asymptotic analysis leads to an auxiliary exterior problem which has to be solved again point wise. We present an efficient way to precompute the topological derivative and emphasize the additional challenges created by the presence of permanent magnets. The topological derivative is then used to update a level set representation of the design to perform the optimization.

Peter Gangl. While first order shape optimization methods are widely used in engineering applications, they typically take a large number of iterations until a locally optimal design is found. Therefore, second order shape optimization methods are often employed, which, however, come with the challenges of having to invert a shape Hessian with a large kernel and the issue of only local convergence of Newton's method. Homotopy methods have been developed for solving general systems of nonlinear equations. They are based on the idea of smoothly connecting the problem to be solved with a much simpler problem, whose solution is known or readily computed. Then the problem of interest is gradually approached from the simpler problem, e.g., by a predictor-corrector scheme. In this talk, we solve shape optimization problems by homotopy methods, employing predictors based on first and higher order shape derivatives and a shape-Newton method as a corrector. As a by-product, we illustrate that the proposed method can be applied to efficiently approximate Pareto fronts of multi-objective optimization problems.