Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas

Elschner, Johannes; Schmidt, G.

doi:10.1002/(sici)1099-1476(19980925)21:14<1297::aid-mma997>3.0.co;2-c

Cited by 83 publications

(82 citation statements)

References 27 publications

(25 reference statements)

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“…We anticipate that the variational formulation will also be very suitable for numerical solution via finite element discretization, as are similar formulations for the 2D diffraction grating case [6,15,16]. Moreover, the explicit bounds we obtain should be helpful in establishing the dependence, on the wave number and the domain, of the constants in a priori error estimates for finite element schemes.…”

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confidence: 80%

Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Chandler-Wilde¹,

Monk²

2005

SIAM J. Math. Anal.

127

179

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Abstract. In this paper we study, via variational methods, the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface. The boundary ∂D is assumed to lie within a finite distance of a flat plane and the incident wave is that arising from an inhomogeneous term in the Helmholtz equation whose support lies within some finite distance of the boundary ∂D. Via analysis of an equivalent variational formulation, we provide the first proof of existence of a unique solution to a three-dimensional rough surface scattering problem for an arbitrary wave number. Our method of analysis does not require any smoothness of the boundary which can, for example, be the graph of an arbitrary bounded continuous function. An attractive feature is that all constants in a priori bounds, for example the inf-sup constant of the sesquilinear form, are bounded by explicit functions of the wave number and the maximum surface elevation. 1. Introduction. This paper is concerned with the development and analysis of a variational formulation for scattering by unbounded surfaces, in particular, with the study of what are termed rough surface scattering problems in the engineering literature. We shall use the phrase rough surface to denote surfaces which are a (usually nonlocal) perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. Such problems arise frequently in applications, for example in modeling acoustic and electromagnetic wave propagation over outdoor ground and sea surfaces, and are the subject of intensive studies in the engineering literature, with a view to developing both rigorous methods of computation and approximate, asymptotic, or statistical methods (see, e.g., the reviews and monographs by Ogilvy In this paper we will focus on a particular, typical problem of the class, which models time harmonic acoustic scattering by a sound soft rough surface. In particular, we seek to solve the Helmholtz equation with wave number k > 0, Δu + k 2 u = g, in the perturbed half-plane or half-space D ⊂ R n , n = 2, 3. We suppose that the homogeneous Dirichlet boundary condition u = 0 holds on ∂D, and a suitable radiation condition is imposed to select a unique solution to this problem. We shall give in the next section complete details about our assumptions on D and on the radiation condition, but we now note that the inhomogeneous term g might be in L 2 (D) with bounded support, or be a more general distribution. In addition the boundary ∂D may or may not be the graph of a function.

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confidence: 80%

Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Chandler-Wilde¹,

Monk²

2005

SIAM J. Math. Anal.

127

179

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“…[3,19]. For all but possibly a discrete set of frequencies, the existence and uniqueness of a solution of Eq.…”

Section: Plane Wave Incidence From Above the Grating: Variational Formentioning

confidence: 99%

“…For all but possibly a discrete set of frequencies, the existence and uniqueness of a solution of Eq. (40) has been proved and the convergence of the finite element solution has been established [3,19]. In most of the analytical studies, which use a nonlocal boundary operator (Dirichletto-Neumann map), Eq.…”

Section: Plane Wave Incidence From Above the Grating: Variational Formentioning

confidence: 99%

Finite element computation of grating scattering matrices and application to photonic crystal band calculations

Dossou

Byrne

Botten

2006

Journal of Computational Physics

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We consider the calculation of the band structure and Bloch mode basis of twodimensional photonic crystals, modelled as stacks of one-dimensional diffraction gratings. The scattering properties of each grating are calculated using an efficient finite element method (FEM) and allow the complete mode structure to be derived from a transfer matrix method. A range of numerical examples showing the accuracy, flexibility and utility of the method is presented.

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“…Ammari, Bao & Wood [3], Bao & Dobson [9], Bonnet-Bendhia & Starling [10], Elschner & Schmidt [26], Elschner, Hinder, Penzel & Schmidt [27], Elschner & Yamamoto [28] and Kirsch [33]. In the case of elastic scattering by periodic surfaces, the variational approach is established by Elschner & Hu in [24,25] for the boundary value problems of the first, second, third and fourth kind as well as for transmission problems with non-smooth interfaces in R n (n = 2, 3).…”

Section: Introductionmentioning

confidence: 99%

Elastic Scattering by Unbounded Rough Surfaces

Elschner¹,

Hu²

2012

SIAM J. Math. Anal.

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We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators.

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Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas

Cited by 83 publications

References 27 publications

Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Finite element computation of grating scattering matrices and application to photonic crystal band calculations

Elastic Scattering by Unbounded Rough Surfaces

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