Abstract
We present new numerical results for shape optimization problems of interior Neumann eigenvalues. This field is not well understood from a theoretical standpoint. The existence of shape maximizers is not proven beyond the first two eigenvalues, so we study the problem numerically. We describe a method to compute the eigenvalues for a given shape that combines the boundary element method with an algorithm for nonlinear eigenvalues. As numerical optimization requires many such evaluations, we put a focus on the efficiency of the method and the implemented routine. The method is well suited for parallelization. Using the resulting fast routines and a specialized parametrization of the shapes, we found improved maxima for several eigenvalues.
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Acknowledgements
The authors gratefully acknowledge the computing time provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC).
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Abele, D., Kleefeld, A. (2020). New Numerical Results for the Optimization of Neumann Eigenvalues. In: Constanda, C. (eds) Computational and Analytic Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-48186-5_1
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DOI: https://doi.org/10.1007/978-3-030-48186-5_1
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