Shape Optimization for Maxwell's Equations Including Hysteresis Effects in the Material Laws
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 SLiRsMin, 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.
Sabrina Fiege, Andreas Griewank, Andrea Walther: An Algorithm for Nonsmooth Optimization by Successive Piecewise Linearization, A. Math. Program., 2018 (SPP1962-007).
Sabrina Fiege, Andreas Griewank, Kshitij Kulshreshtha, Andrea Walther: Algorithmic Differentiation for Piecewise Smooth Functions: A Case Study for Robust Optimization, Optimization Methods and Software, 2018 (SPP1962-009).
Olga Ebel, Stephan Schmidt, Andrea Walther: Solving Non-Smooth Semi-Linear Optimal Control Problems with Abs-Linearization (SPP1962-093, 10/2018, [bib])
Andreas Griewank, Andrea Walther: Beyond the Oracle: Opportunities of Piecewise Differentiation (SPP1962-086, 10/2018, [bib])
Andreas Griewank, Andrea Walther: Relaxing Kink Qualifications and Proving Convergence Rates in Piecewise Smooth Optimization (SPP1962-041, 11/2017, [bib])
Andrea Walther, Andreas Griewank: Characterizing and Testing Subdifferential Regularity for Piecewise Smooth Objective Functions (SPP1962-038, 11/2017, [bib])
Sabrina Fiege, Andreas Griewank, Kshitij Kulshreshtha, Andrea Walther: Algorithmic Differentiation for Piecewise Smooth Functions: A Case Study for Robust Optimization (SPP1962-009, 03/2017, [bib])
Sabrina Fiege, Andreas Griewank, Andrea Walther: An Algorithm for Nonsmooth Optimization by Successive Piecewise Linearization (SPP1962-007, 02/2017, [bib])
Modeling, problem analysis, algorithm design and convergence analysisThe focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.
Realization of algorithms, adaptive discretization and model reductionAs the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.
Incorporation of parameter dependencies and robustnessIn many applications the robustness of solutions with respect to a given parameter range (uncertainty set) is highly relevant. Correspondingly, in this research area of the SPP, bi- or multilevel optimization approaches will be studied in order to robustify problem solutions against uncertain parameters.
Project Related News
Oct 26, 2018 : New preprint submitted
Olga Ebel submitted the preprint SPP1962-093, Solving Non-Smooth Semi-Linear Optimal Control Problems with Abs-Linearization
Oct 18, 2018 : New preprint submitted
Andrea Walther submitted the preprint SPP1962-086, Beyond the Oracle: Opportunities of Piecewise Differentiation