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Efficient Evaluation of Doubly Periodic Green Functions in 3D Scattering, Including Wood Anomaly Frequencies

Bruno, Oscar P. and Shipman, Stephen P. and Turc, Catalin and Venakides, Stephanos (2013) Efficient Evaluation of Doubly Periodic Green Functions in 3D Scattering, Including Wood Anomaly Frequencies. . (Submitted)

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We present effcient methods for computing wave scattering by diffraction gratings that exhibit two-dimensional periodicity in three dimensional (3D) space. Applications include scattering in acoustics, electromagnetics and elasticity. Our approach uses boundary-integral equations. The quasi-periodic Green function employed is structured as a doubly infinite sum of scaled 3D free-space outgoing Helmholtz Green functions. Their source points are located at the nodes of a periodicity lattice of the grating; the scaling is effected by Bloch quasi-periodic coefficients. For efficient numerical computation of the lattice sum, we employ a smooth truncation. Super-algebraic convergence to the Green function is achieved as the truncation radius increases, except at frequency-wavenumber pairs at which a Rayleigh wave is at exactly grazing incidence to the grating. At these "Wood frequencies", the term in the Fourier series representation of the Green function that corresponds to the grazing Rayleigh wave acquires an infinite coefficient and the lattice sum blows up. A related challenge occurs at non-exact grazing incidence of a Rayleigh wave; in this case, the constants in the truncation-error bound become too large. At Wood frequencies, we modify the Green function by adding two types of terms to it. The first type adds weighted spatial shifts of the Green function to itself. The shifts are such that the spatial singularities introduced by these terms are located below the grating and therefore out of the domain of interest. With suitable choices of the weights, these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. The degree of the algebraic convergence depends on the number of the added shifts. The second-type terms are quasi-periodic plane wave solutions of the Helmholtz equation. They reinstate (with controlled coeficients now) the grazing modes, effectively eliminated by the terms of first type. These modes are needed in the Green function for guaranteeing the well-posedness of the boundaryintegral equation that yields the scattered field. We apply this approach to acoustic scattering by a doubly periodic 2D grating near and at Wood frequencies and scattering by a doubly periodic array of scatterers away from Wood frequencies.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Bruno, Oscar P.0000-0001-8369-3014
Shipman, Stephen P.0000-0001-6620-6528
Additional Information:Submitted on 4 Jul 2013. This manuscript is a preliminary version of our work, which is made available through arXiv to facilitate rapid dissemination. A more developed text is in preparation. This research was supported by grants from NSF and AFOSR (OB) and grants NSF DMS-0807325 (SPS), NSF DMS-1008076 (CT), and NSF DMS-0707488 (SV).
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
Subject Keywords:scattering, periodic Green function, lattice sum, smooth truncation, Wood frequency, Wood anomaly, boundary-integral equations, electromagnetic computation
Record Number:CaltechAUTHORS:20160219-073954516
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64597
Deposited By: Ruth Sustaita
Deposited On:19 Feb 2016 19:13
Last Modified:09 Mar 2020 13:19

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