Mathematical studies in rigorous grating theory

Bao, Gang; Dobson, David C.; Cox, J. Allen

doi:10.1364/josaa.12.001029

Cited by 165 publications

(150 citation statements)

References 22 publications

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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.

128

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.

128

179

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“…Scattering theory in biperiodic structures has many important applications in micro-optics, where the biperiodic structures are also termed as 2D gratings or crossed gratings. We refer to Dobson and Friedman [27], Abboud [1], Bao [7], Bao, Dobson, and Cox [11], Bao and Dobson [10], Bao and Zhou [13], Li [33], Yachin and Yasumoto [49], and Chang, Li, Chu, and Opsal [20] for the existence, uniqueness, and numerical approximations of solutions to 2D grating problems. An introduction to grating problems can be found in Petit [41].…”

Section: ×3mentioning

confidence: 99%

An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures

Bao

2010

Math. Comp.

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Abstract. An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.

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“…the classical monograph [1]). The diffraction problems admit variational formulations in a bounded periodic cell which were introduced in [9,10]. In the following k denotes the piecewise constant function ω(µ 0 ) 1/2 .…”

Section: Variational Formulation Of the Direct Scattering Problemmentioning

confidence: 99%

“…Multiplying the differential equation (2.1) by some smooth function, applying Green's formula, and taking into account the transmission conditions at the material interfaces and the outgoing wave condition on ± , it can be shown (cf. [10,15]) that the diffraction problem for TE polarization is equivalent to the variational equation…”

Section: Fig 21) We Assume Thatmentioning

confidence: 99%

“…In the case of binary structures, whose surface profile is given by a piecewise constant function, the mathematical complexities are amplified by singularities of the solutions. Recently, a new variational approach was proposed (see [9][10] and the references therein), which appears to be well adapted for the analytical and numerical treatment of very general diffraction structures as well as complex materials and allows straightforward extensions to diffraction problems for conical mounting and crossed gratings [11][12]. In particular, this approach is the basis for the convergence analysis of finite element solution methods provided in [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

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