Integral equation methods for scattering by infinite rough surfaces

Zhang, Bo; Chandler-Wilde, Simon N.

doi:10.1002/mma.361

Cited by 83 publications

(104 citation statements)

References 33 publications

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“…(2.13)-(2.14)], causing any method using it to have divergent round-off error near to, and failure at, each such anomaly. In the related case of periodic surface scattering, Zhang and Chandler-Wilde [47] modified the Green's function to that of a half-space, which cures this divergence (this was implemented in [2]); however, this idea fails to help in our case of disconnected obstacles. A final problem is that the quasi-periodic Green's function is often computed using lattice sums [25,26], which is natural when using fast multipole acceleration in large-scale scattering problems [36].…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green's function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green's function diverges for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.In this paper, we bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. Summing nearby images directly leaves auxiliary densities that are smooth, hence easily represented in the Fourier domain using Sommerfeld integrals. Wood's anomalies are handled analytically by deformation of the Sommerfeld contour. The resulting integral equation is of the second kind and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls are handled easily and automatically. We include an implementation and simple code example with a freely-available MATLAB toolbox.Communicated by Per Lötstedt.

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Section: Introduction and Problem Formulationmentioning

confidence: 99%

A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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“…For instance, for the case of a bounded obstacle, existence of solution for the time harmonic exterior Dirichlet or impedance scattering problem is known for a long time [13]. For the rough surface scattering problem with a Dirichlet boundary condition, corresponding results have only been achieved during the last decade, firstly by using integral equation approach [4,5,16], and more recently, by using a variational approach in [2,6]. For scattering from rough infinite layers we also refer to recent results in [11].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic models for scattering from unbounded media with high conductivity

Haddar¹,

Lechleiter²

2010

ESAIM: M2AN

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Abstract.We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of L2 solution operators. We also discuss some numerical experiments concerning these boundary conditions. Mathematics Subject Classification. 35C20, 78A40. IntroductionTime harmonic wave scattering from rough layers is an important problem in science and engineering, as it describes for instance scattering of electromagnetic waves from the ground when one models the earth as a rough stratified medium. In such a model, the moisture of soil causes absorption of the electromagnetic wave inside the ground, and thus naturally leads to a scattering problem for a rough absorbing layer. Since waves inside the absorbing part of the medium decay exponentially with respect to the distance to the layer's boundary, a lot of research has been carried out how to replace the wave scattering problem inside the absorbing layer by some easily handable absorbing boundary condition on the interface in between the absorbing layer and free space [1,8,9,14,15]. The aim of such a boundary condition is to set up an approximate scattering problem merely in the complement of the absorbing object, while still guaranteeing a reliable error bound on the solution of the approximate problem. This error bound depends on what we call the order of the boundary condition as well as on the magnitude of the absorption inside the layer. Indeed, we treat the magnitude of absorption as a parameter and expand the wave field in a power series with respect to the inverse of this parameter. Approximate boundary conditions are built after truncation of this series, the order of the conditions so obtained then corresponds to the truncation index. Truncation at order 0 simply leads to a Dirichlet boundary condition, which is naturally the formal limit condition as the absorption tends to infinity; truncation at order 1 leads to a (usual) impedance Keywords and phrases. Scattering problems, unbounded domains, asymptotic models, generalized impedance boundary conditions, high conductivity. boundary condition. This is the reason why we call the condition arising from truncation at higher order generalized impedance boundary condition (GIBC).In this paper, we analyse GIBCs for rough absorbing layers up to order 3 and shall restrict ourselves to the scalar problem (which corresponds in 2-D to the E-polarization of electromagnetic waves). While the construction of such conditions is rather analogous to the case of a bounded absorbing inhomogeneity, the error analysis is...

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“…For the two-dimensional (2D) rough surface scattering case much progress has been made in terms of deriving well-posed BIEs for a variety of acoustic, electromagnetic, and elastic wave problems [9,8,32,2]. Surprisingly, none of the analysis for the 2D case extends straightforwardly to three dimensions; indeed most of the 2D analysis appears to be unsuitable in the 3D case.…”

Section: Introductionmentioning

confidence: 99%

“…In more detail, in the 2D case bounded integral operators have been obtained by replacing the standard fundamental solution by the Dirichlet or impedance Green's function for a half-plane that contains the domain D of propagation (see, e.g., [8,32]). This modification leads to kernels of boundary integral operators that are weakly singular in their asymptotic behavior at infinity so that the integral operators are bounded on L p (Γ) for 1 ≤ p ≤ ∞ and on BC(Γ), the space of bounded continuous functions on Γ.…”

Section: Introductionmentioning

confidence: 99%

“…After these results, of significant interest in their own right, we consider the problem of acoustic scattering by the rough surface Γ in the case when the surface is sound soft (the field vanishes on Γ) and the incident field is due to a source distribution with compact support in D. We reduce this scattering problem to a second kind boundary integral equation via an ansatz for the solution as a combined single-and double-layer potential. (The analogous ansatz was used for the 2D rough surface scattering case in [32], based on the analogous approach for scattering by bounded obstacles dating back to [4].) For a flat surface, unique solvability of the BIE is shown by explicit computation of a symbol.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

Chandler-Wilde¹,

Heinemeyer²,

Potthast³

2006

SIAM J. Appl. Math.

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Abstract. For a nonlocally perturbed half-space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single-and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth (Lyapunov) we show that the integral operators are nevertheless bounded as operators on L 2 (Γ) and on L 2 (Γ) ∩ BC(Γ) and that the operators depend continuously in norm on the wave number and on Γ. We further show that for mild roughness, i.e., a surface Γ which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space L 2 (Γ) ∩ BC(Γ) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.

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Integral equation methods for scattering by infinite rough surfaces

Cited by 83 publications

References 33 publications

A new integral representation for quasi-periodic scattering problems in two dimensions

A new integral representation for quasi-periodic scattering problems in two dimensions

Asymptotic models for scattering from unbounded media with high conductivity

Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

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