A Nyström Method for a Class of Integral Equations on the Real Line with Applications to Scattering by Diffraction Gratings and Rough Surfaces

Meier, A.; Arens, Tilo; Chandler-Wilde, Simon N.; Kirsch, Andreas

doi:10.1216/jiea/1020282209

Cited by 64 publications

(76 citation statements)

References 19 publications

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“…With reference to Table 3 in [16], we note that the algorithm [16] requires N = 128 to achieve the same accuracy as the present method does with N = 10: clearly, the combination of Floquet expansions and spectral weight-based integration provides high-order accuracy in a very effective manner. a For z Ͼ 8000 we use the linear relationship * = 210+ 3.1535z, which leads to slight overestimates of the truncation index Q = ͓*͑z͔͒ +1.…”

Section: Resultsmentioning

confidence: 74%

“…To demonstrate the rapid high-order convergence provided by our algorithm, in Table 3 we compare the results of our method to those provided by the highly accurate high-order algorithm introduced in [16] for the profile used in that contribution: f͑x͒ = d + a sin x. With reference to Table 3 in [16], we note that the algorithm [16] requires N = 128 to achieve the same accuracy as the present method does with N = 10: clearly, the combination of Floquet expansions and spectral weight-based integration provides high-order accuracy in a very effective manner.…”

Section: Resultsmentioning

confidence: 99%

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Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Bruno

Haslam

2009

J. Opt. Soc. Am. A

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We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z = f͑x͒. For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01°from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.

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Section: Resultsmentioning

confidence: 74%

Section: Resultsmentioning

confidence: 99%

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Bruno

Haslam

2009

J. Opt. Soc. Am. A

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“…If u satisÿes (27) and (28) and u| ∈BC( ) then u satisÿes the impedance problem (IP) with a = −1=2 in (3) and with every Â ∈(0; 1) in (4).…”

Section: Theorem 42mentioning

confidence: 99%

“…Results related to those contained in this paper, including a numerical analysis of a novel Nystr om discretization scheme suitable for all the integral equation formulations we propose, and a study of the stability and convergence of truncation to a ÿnite section of the integrals over the inÿnite boundary which occur in each integral equation, are discussed in References [27,28]. For the special case of a at surface, e cient boundary element techniques for the impedance problem have recently been proposed and analysed in Reference [29].…”

Section: Introductionmentioning

confidence: 99%

Integral equation methods for scattering by infinite rough surfaces

Zhang

Chandler-Wilde

2003

Math Methods in App Sciences

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SUMMARYIn this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of timeharmonic acoustic scattering of an incident ÿeld by a sound-soft, inÿnite rough surface where the total ÿeld vanishes (the Dirichlet problem) or by an inÿnite, impedance rough surface where the total ÿeld satisÿes a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double-and single-layer potential and a Dirichlet half-plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green's function, the ÿrst derived from Green's representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident ÿelds including an incident plane wave, the impedance boundary value problem for the scattered ÿeld has a unique solution under certain constraints on the boundary impedance.

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“…The general characteristics of the waves scattered by a rough surface are directional effects, slowness and attenuation, as well as possible resonances for surface gratings (corrugations). The main difficulty in getting more definite results in this problem resides in modelling conveniently the inhomogeneities and the surface roughness, such as to arrive at mathematically operational approaches [34,35] .…”

Section: Introductionmentioning

confidence: 99%

Scattering of longitudinal waves (sound) by defects in fluids. Rough surface

Apostol

2013

Open Physics

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Abstract:The classical theory of scattering of longitudinal waves (sound) by small inhomogeneities (scatterers) in an ideal fluid is generalized to a distribution of scatterers and such as to include the effect of the inhomogeneities on the elastic properties of the fluid. The results are obtained by a new method of solving the wave equation with spatial restrictions (caused by the presence of the scatterers), which can also be applied to other types of inhomogeneities (like surface roughness, for instance). A coherent forward scattering is identified for a uniform distribution of scatterers (practically equivalent with a mean-field approach), which is due to the fact that our treatment does not include multiple scattering. The reflected wave is obtained for a half-space (semi-infinite fluid) of uniformly distributed scatterers, as well as the field diffracted by a perfect lattice of scatterers. The same method is applied to a (inhomogeneous) rough surface of a semi-infinite ideal fluid. A perturbation-theoretical scheme is devised, with the roughness function as a perturbation parameter, for computing the waves scattered by the surface roughness. The waves scattered by the rough surface are both waves localized (and propagating only) on the surface (two-dimensional waves) and waves reflected back in the fluid. They exhibit directional effects, slowness, attenuation or resonance phenomena, depending on the spatial characteristics of the roughness function. The reflection coefficients and the energy carried on by these waves are calculated both for fixed and free surfaces. In some cases, the surface roughness may generate waves confined to the surface (damped, rough-surface waves).

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A Nyström Method for a Class of Integral Equations on the Real Line with Applications to Scattering by Diffraction Gratings and Rough Surfaces

Cited by 64 publications

References 19 publications

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Integral equation methods for scattering by infinite rough surfaces

Scattering of longitudinal waves (sound) by defects in fluids. Rough surface

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