1,206 research outputs found

    Three-dimensional quasi-periodic shifted Green function throughout the spectrum--including Wood anomalies

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    This work presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near "Wood anomaly frequencies". At these frequencies, one or more grazing Rayleigh waves exist, and the lattice sum for the quasi-periodic Green function ceases to exist. We present a modification of this sum by adding two types of terms to it. The first type adds linear combinations of "shifted" Green functions, ensuring that the spatial singularities introduced by these terms are located below the grating and therefore outside of the physical domain. With suitable coefficient choices these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. Convergence of arbitrarily high order can be obtained by including sufficiently many shifts. The second type of added terms are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on the this new Green function for the problem of scattering by doubly periodic three-dimensional surfaces at and around Wood frequencies. We believe this is the first solver in existence that is applicable to Wood-frequency doubly periodic scattering problems. We demonstrate the proposed approach by means of applications to problems of acoustic scattering by doubly periodic gratings at various frequencies, including frequencies away from, at, and near Wood anomalies

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

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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

    Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders---including Wood anomalies

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    This paper presents a full-spectrum Green function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section-with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both TE and TM polarized illumination. The proposed method, which, for definiteness is demonstrated here for arrays of perfectly conducting particles under TE polarization, is based on use of the shifted Green-function method introduced in the recent contribution (Bruno and Delourme, Jour. Computat. Phys. pp. 262--290 (2014)). A certain infinite term arises at Wood anomalies for the cylinder-array problems considered here that is not present in the previous rough-surface case. As shown in this paper, these infinite terms can be treated via an application of ideas related to the Woodbury-Sherman-Morrison formulae. The resulting approach, which is applicable to general arrays of obstacles even at and around Wood-anomaly frequencies, exhibits fast convergence and high accuracies. For example, a few hundreds of milliseconds suffice for the proposed approach to evaluate solutions throughout the resonance region (wavelengths comparable to the period and cylinder sizes) with full single-precision accuracy

    Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space

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    This paper, Part I in a two-part series, presents (i) A simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain "Wood frequencies" at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on use of smooth windowing functions, gives rise to lattice sums which converge superalgebraically fast--that is, faster than any power of the number of terms used--in sharp contrast with the extremely slow convergence exhibited by the corresponding sums in absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) A proof presented in this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach

    Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

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    This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction -- used to properly impose the Rayleigh radiation condition -- yield a robust second-kind BIE that produces super-algebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nystr\"om and boundary element methods, and it leads to linear systems suitable for iterative linear-algebra solvers as well as standard fast matrix-vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodolog

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

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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

    Wave-Scattering by Periodic Media

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    This thesis presents a full-spectrum, well-conditioned, Green-function methodology for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies, where most existing methods 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, the thorough understanding of RW-anomalies this thesis provides 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 thesis 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. A variety of computational examples are presented which demonstrate the flexibility of the overall approach, including, in particular, a problem of diffraction by a double-helix structure, for which numerical simulations did not previously exist, and for which the scattering pattern presented in this thesis closely resembles those obtained in crystallography experiments for DNA molecules.</p
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