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| Master Thesis | FZJ-2020-00178 |
2019
Please use a persistent id in citations: http://hdl.handle.net/2128/23846
Abstract: This master thesis is concerned with the optimization of eigenvalues of the Laplace differential operator, specifically interior Neumann eigenvalues, with respect to the shapeof the domain. Such eigenvalue problems arise in the study of acoustic scattering, whichhas applications in sonar or radar detection and medical imaging. The shape of the spacesignificantly affects the eigenvalues. Improved optimal values for some of them are reported.The main focus of the thesis is finding a description of the shape that is well suited foroptimization. The number of parameters should be low to keep the optimization spacesimple. At the same time, the range of representable shapes should be large enough toimprove upon previous results. Inspired by physics, equipotentials are used to modelthe knobbly objects found by previous researchers in a simple way.The work discusses a method of solving the eigenvalue problem. The BoundaryElement Method for boundary value problems is combined with Beyn’s method fornonlinear eigenvalue problems. The implementation of these methods is another centralissue. As the optimizer requires many evaluations, high speed is desired. The code isparallelized for efficient computation on a large cluster.The implemented solvers are tested for convergence. The parameter space is thoroughly numerically explored to facilitate optimization. Finally the results of the optimization are presented. The shape description shows a lot of promise but is not yetgeneral enough to optimize every eigenvalue.
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