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Wave Enhancement through Optimization of Boundary Conditions

by H. Ammari and O. Bruno and K. Imeri and N. Nigam

(Report number 2019-34)

Abstract
It is well-known that changing boundary conditions for the Laplacian from Dirichlet to Neumann can result in significant changes to the associated eigenmodes, while keeping the eigenvalues close. We present a new and efficient approach for optimizing the transmission signal between two points in a cavity at a given frequency, by changing boundary conditions. The proposed approach makes use of recent results on the monotonicity of the eigenvalues of the mixed boundary value problem and on the sensitivity of the Green's function to small changes in the boundary conditions. The switching of the boundary condition from Dirichlet to Neumann can be performed through the use of the recently modeled concept of metasurfaces which are comprised of coupled pairs of Helmholtz resonators. A variety of numerical experiments are presented to show the applicability and the accuracy of the proposed new methodology.

Keywords: Zaremba eigenvalue problem, boundary integral operators, mixed boundary conditions, metasurfaces.

@Techreport{ABIN19_838,
  author = {H. Ammari and O. Bruno and K. Imeri and N. Nigam},
  title = {Wave Enhancement through Optimization of Boundary Conditions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-34.pdf },
  year = {2019}
}

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