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On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

Bruno, Oscar P. and Fernandez-Lado, Agustin G. (2020) On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies. Journal of Computational Physics, 410 . Art. No. 109352. ISSN 0021-9991.

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This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach.

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Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2020 Elsevier Inc. Received 28 August 2019, Revised 6 February 2020, Accepted 17 February 2020, Available online 2 March 2020. This work was supported by NSF and AFOSR through contracts DMS-1714169 and FA9550-15-1-0043, and by the NSSEFF Vannevar Bush Fellowship under contract number N00014-16-1-2808. Declaration of Competing Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-15-1-0043
National Security Science and Engineering Faculty FellowshipN00014-16-1-2808
Vannever Bush Faculty FellowshipUNSPECIFIED
Record Number:CaltechAUTHORS:20200302-143817282
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Official Citation:Oscar P. Bruno, Agustin G. Fernandez-Lado, On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies, Journal of Computational Physics, Volume 410, 2020, 109352, ISSN 0021-9991, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101662
Deposited By: Tony Diaz
Deposited On:02 Mar 2020 22:42
Last Modified:17 Mar 2020 19:21

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