Project Details
Projekt
Shape Optimization for Maxwell's Equations Including Hysteresis Effects in the Material Laws
Subject Area
Mathematics
Term
from 2016 to 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 314066080
The main goal of this research proposal is twofold. First, we lay the analytical foundations of sharp interface shape optimization of hyperbolic problems with hysteresis phenomena, modeled for example by quasi variational inequalities. The resulting shape derivative formulations will include projection operators leading to piecewise linear reduced directional shape derivatives necessitating consideration of outer optimization schemes that work well in this setting. Second, we provide a suitable optimization algorithm called LiPsMin for such non-smooth problems, which is rigorously analyzed in the corresponding function spaces. Since the quadratic subproblems used by LiPsMin for the step computation are based on the structure of the non-smoothness, the requires also new concepts for the similarity of shapes.Being based on minimization of Lipschitzian piecewise smooth functions, LiPsMin offers a unique and novel approach with expected mesh independent global convergence, especially in comparison to the limited convergence radius of predominant semi-smooth Newton methods. Where applicable by the non-smoothness, we will augment LiPsMin by higher order shape derivative information resulting in the algorithm SLiPsMin, carrying very recent novel interpretations of shape Hessians over into the non-smooth situation. This is expected to be the transition of studying pseudo-differential operators in the smooth case to operators based on pseudo-quasi-variational inequalities.As such, this proposal will advance the topics of Area 1 of the priority program, that is modeling, problem analysis, algorithm design and convergence analysis in function spaces. Furthermore, we will also contribute to Area 2 of the priority program, i.e., the realization of algorithms and in particular model reduction when considering the interaction between physically correct hysteresis phenomena and their realization via quasi-variational inequalities. While developing the respective mathematical methods within this proposal, we intent not to underestimate the practicability of the respective mathematical foundations and with a clear focus on the hyperbolic case, the work proposed herein offers a unique outlook on possible applications.As a prototypical application, we consider an ionization tracking governed by Maxwell's equations based on a variational inequality formulation.Hence, the work proposed here is very relevant for electro-hydrodynamics, electro-kinesis, spectrometry and ozone production, enabling contact-free filtering, flow control by corona discharge ionization phenomena or very high accuracy sensors in physics. The study of type-II superconductors in accordance to Bean's model fits also in this application area, since it results in simulation problems governed by Maxwell's equations that are modified to a variational inequality. Hence, control thereof is well within the scope of applications that could be covered by the research proposed here.
DFG Programme
Priority Programmes
