Current status of closed-form Green's functions in layered media composed of natural and artificial materials

Aksun, M.I.; Alparslan, Aytac; Michalski, Krzysztof A.

doi:10.1109/iceaa.2009.5297398

Cited by 7 publications

(11 citation statements)

References 15 publications

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“…in terms of the total field ϕ = u| Γ and its normal derivative ψ = ∂u ∂n on Γ, where letting G j (x, y) = iH (1) 0 (k j |x − y|)/4, j = 1, 2 denote the free-space Green function for the Helmholtz equation with wavenumber k j , the single-and double-layer potentials in equation ( 2) are defined by…”

Section: Windowed Green Function Methodsmentioning

confidence: 99%

“…A variety of methods have been provided for the solution of problems of scattering by obstacles in presence of layered media. Amongst the most effective such approaches we mention 1) Methods which evaluate Sommerfeld integrals on the basis of path-integration in the complex plane [17,7,6,18] (such approaches require numerical evaluation of integrals of functions that oscillate, grow exponentially in a bounded section of the integration path and, depending on the relative position of the source and observation points to the interface between the two media, may decay slowly at infinity); 2) The complex images method reviewed in [1] (a discussion indicating certain instabilities and inefficiencies in this method is presented in [7, section 5.5]); and 3) The steepest descent method [9,10] which, provided the steepest descent path is known, reduces the Sommerfeld integral to an integral of an exponentially decaying function (unfortunately, however, the determination of steepest descent paths for each observation point can be challenging and expensive). As is well known, in any case, all of these methods entail significant computational costs [6].…”

Section: Introductionmentioning

confidence: 99%

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Windowed Green Function Method for Layered-Media Scattering

Bruno

Lyon

Peźrez-Arancibia

et al. 2016

SIAM J. Appl. Math.

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This paper introduces a new Windowed Green Function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in presence of dielectric or conducting half-planes. The WGF method, which is based on use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function, does not require evaluation of expensive Sommerfeld integrals. The proposed approach is fast, accurate, flexible and easy to implement. In particular, straightforward modifications of existing (accelerated or unaccelerated) solvers suffice to incorporate the WGF capability. The mathematical basis of the method is simple: the method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media. Numerical experiments demonstrate that both the near-and far-field errors resulting from the proposed approach decrease faster than any negative power of the window size. In the examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than a corresponding method based on the layer-Green-function.

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Section: Windowed Green Function Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Windowed Green Function Method for Layered-Media Scattering

Bruno

Lyon

Peźrez-Arancibia

et al. 2016

SIAM J. Appl. Math.

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“…Following [4], in this section we introduce rapidly-convergent windowed versions of the integral formulation (14). In order to do so we utilize the 1) given by…”

Section: Windowed Integral Equationsmentioning

confidence: 99%

“…The Sommerfeld integrals amount to singular Fourier integrals [4,5] whose evaluation is generally quite challenging. A wide range of approaches have been proposed for evaluation of these quantities [6][7][8][9][10][11][12][13][14] but, as is known, all of these methods entail significant computational costs [2,7,11,12].…”

Section: Introductionmentioning

confidence: 99%

Windowed Green function method for the Helmholtz equation in the presence of multiply layered media

Bruno

Pérez‐Arancibia

2017

Proc. R. Soc. A.

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This paper presents a new methodology for the solution of problems of two-and threedimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in presence an arbitrary number of penetrable layers. Relying on use of certain slow-rise windowing functions, the proposed Windowed Green Function approach (WGF) efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multi-layer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (SIAM J. Appl. Math. 76(5), p. 1871, 2016), is fast, accurate, flexible, and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on use of Sommerfeld integrals.

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“…This technique leads to single closed‐form expressions of the spatial Green's functions that are accurate in the whole range of values of ρ. Recently, Alparslan et al [18, 19] have been proposed three‐level DCIM approaches to obtain the closed‐form Green's functions over all ranges from the source in planar stratified media, giving proper emphasis to capture the signature of each singularity in the spectral domain. The three‐level DCIM algorithms can take into account the possible wave constituents of a dipole in stratified media, such as spherical, cylindrical, and lateral waves, to obtain accurate closed form approximations of Green's functions.…”

Section: Introductionmentioning

confidence: 99%

A robust method for the computation of new closed-form Green's functions for microstrip structures

Firouzeh

Moini

Sadeghi

et al. 2011

Int J RF and Microwave Comp Aid Eng

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A robust approach for the computation of new closed-form Green's functions is considered to calculate the symmetrical microstrip Green's functions. In this method, the surface-wave poles are first extracted using a recursive contour integration method. Then, the remainder is approximated by a series of complex exponentials using the Prony's method or the generalized pencil-of-function method (GPOF) along the extended rooftop shaped path in k q -plane. Subsequently, an analytical identity is employed to obtain the new spatial-domain Green's functions. It is observed that this method can evaluate the spatial-domain Green's functions accurately and efficiently for both near and far fields. In addition, there is no erroneous results in the near-field region when z = z 0 and q ! 0.

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Current status of closed-form Green's functions in layered media composed of natural and artificial materials

Cited by 7 publications

References 15 publications

Windowed Green Function Method for Layered-Media Scattering

Windowed Green Function Method for Layered-Media Scattering

Windowed Green function method for the Helmholtz equation in the presence of multiply layered media

A robust method for the computation of new closed-form Green's functions for microstrip structures

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