New numerical results for the optimization of Neumann eigenvalues. (English) Zbl 1448.65275
Constanda, Christian (ed.), Computational and analytic methods in science and engineering. Selected papers based on the presentations at the 19th international conference on computational and mathematical methods in science and engineering, CMMSE’19, Rota, Spain, June 30 – July 6, 2019. Cham: Birkhäuser. 1-20 (2020).
Summary: 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.
For the entire collection see [Zbl 1446.74005].
For the entire collection see [Zbl 1446.74005].
MSC:
| 65R15 | Numerical methods for eigenvalue problems in integral equations |
| 65K10 | Numerical optimization and variational techniques |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
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