A volume integral method for solving scattering problems from locally perturbed infinite periodic layers

Haddar, Houssem; Nguyen, Thi-Phong

doi:10.1080/00036811.2016.1221942

Cited by 24 publications

(28 citation statements)

References 17 publications

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“…Since n p is positive definite, then one can prove that e γκ|x| Ψ(x) ∈ L 2 (R d ) for γ > 0 sufficiently small (following the lines of the proof of Theorem 4.4 in [10]). The function u s := Φ(n p ; ·) − Ψ verifies ∆u s − κ 2 n p u s = 0 in Ω M and application of the Green formula and using the periodicity conditions of Φ(n p ; ·) imply…”

Section: The Analysis Of the New Interior Transmission Problemmentioning

confidence: 99%

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

Cakoni¹,

Haddar²,

Nguyen³

2018

Inverse Problems

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This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior transmission problem, which until now was an open problem except for the special case when the local perturbation didn't intersect the background inhomogeneities. The analysis of this new interior transmission problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.

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Section: The Analysis Of the New Interior Transmission Problemmentioning

confidence: 99%

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

Cakoni¹,

Haddar²,

Nguyen³

2018

Inverse Problems

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“…Any variational formulation of such a transformed problems hence possesses the advantage of straightforward discretization by standard techniques. Note that [HN15] analyzes the discretization of such a problem in two dimensions in a setting that is somewhat easier due to absorption, and that [FJ16] uses the Bloch transform to study scattering in an infinite wave guide.…”

Section: Introductionmentioning

confidence: 99%

The Floquet–Bloch transform and scattering from locally perturbed periodic surfaces

Lechleiter

2017

Journal of Mathematical Analysis and Applications

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We use the Floquet-Bloch transform to reduce variational formulations of surface scattering problems for the Helmholtz equation from periodic and locally perturbed periodic surfaces to equivalent variational problems formulated on bounded domains. To this end, we establish various mapping properties of that transform between suitable weighted Sobolev spaces on periodic strip-like domains and coupled families of quasiperiodic Sobolev spaces. Our analysis shows in particular that the decay of solutions to surface scattering problems from locally perturbed periodic surfaces is precisely characterized by the smoothness of its Bloch transform in the quasiperiodicity.

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“…Following this type of technique, we introduce in this paper an algorithm for solving scattering problems from local perturbations of periodic surfaces that is pretty close to the one from the recent paper [HN16]. Our convergence analysis is for various reasons different, as [HN16] for instance strongly relies on integral equations in the spatial variable. The source of inspiration for all these techniques seems to be the paper [Coa12] on wave propagation in full-space periodic media.…”

Section: Introductionmentioning

confidence: 99%

“…Based on these theoretic results, a numerical scheme has been developed to solve these kinds of scattering problems in [LZ16]. Following this type of technique, we introduce in this paper an algorithm for solving scattering problems from local perturbations of periodic surfaces that is pretty close to the one from the recent paper [HN16]. Our convergence analysis is for various reasons different, as [HN16] for instance strongly relies on integral equations in the spatial variable.…”

Section: Introductionmentioning

confidence: 99%

A Floquet--Bloch Transform Based Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

Lechleiter¹,

Zhang²

2017

SIAM J. Sci. Comput.

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Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way of implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.

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A volume integral method for solving scattering problems from locally perturbed infinite periodic layers

Cited by 24 publications

References 17 publications

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

The Floquet–Bloch transform and scattering from locally perturbed periodic surfaces

A Floquet--Bloch Transform Based Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

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