Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional perfectly reflecting surfaces using the spectral-coordinate method

DeSanto, John A.; Erdmann, G.; Hereman, Willy; Krause, B.; Misra, Manish; Swim, E.

doi:10.1088/0959-7174/11/4/306

Cited by 5 publications

(8 citation statements)

References 13 publications

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“…This method has been implemented with a variety of different bases chosen to express the unknown boundary value. Indeed, the authors of [ 30 ] proposed two different bases for the unknown boundary value to be expanded in: a local basis consisting of piecewise-constant basis functions, and a global basis consisting of the traces of the functions v in ( 2.2 ), and the authors of [ 31 ] proposed a variant of the second basis, where the complex-conjugates of the functions v , instead of the functions themselves, are used. An important point to note is that in both the null-field method and the method of DeSanto there is no flexibility in the choice of v , and thus there is no analogue of the question of how to choose λ in the global relations.…”

Section: Discussionmentioning

confidence: 99%

“…(i) The null field method for the solution of the Helmholtz equation in the exterior of a bounded obstacle (originally introduced by Waterman in [25,26]) and (ii) a method for the solution of the Helmholtz equation above a periodic rough surface introduced by DeSanto [27] and further developed by DeSanto and co-workers [28][29][30][31]. The advantage of these two methods, as well as of the method described in this paper, is that they are boundary-based discretizations that do not involve the computation of singular integrals (as opposed to the discretizations of boundary integral equations).…”

Section: Discussionmentioning

confidence: 99%

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A numerical technique for linear elliptic partial differential equations in polygonal domains

2015

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Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

A numerical technique for linear elliptic partial differential equations in polygonal domains

2015

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“…Another such BVP is the Helmholtz equation posed above an infinite rough surface, and in the case when the surface is periodic the surface is known as a diffraction grating. A method for solving scattering by diffraction gratings was introduced by DeSanto in [30] and further developed by DeSanto and co-workers in [32], [33], [31], and [4]. In [19,Section 4.2] it is shown that this method can be viewed as an implementation of the Fokas transform method to diffraction grating problems.…”

Section: The Methods Of Aziz Dorr and Kellogg For The Helmholtz Extementioning

confidence: 99%

Chapter 6: Overview of Variational Formulations for Linear Elliptic PDEs

Spence¹

2014

Unified Transform for Boundary Value Problems

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“…The authors have recently been made aware of the work of DeSanto and collaborators [16,17,15,3] who advanced an approach which is very close to the one presented by the author and Ambrose in [2]. A nice explanation of their approach and how it connects to the formulation of Fokas is given in [10].…”

Section: Introductionmentioning

confidence: 99%

A high-order perturbation of surfaces (HOPS) approach to Fokas integral equations: Vector electromagnetic scattering by periodic crossed gratings

Nicholls

Tammali

2016

Applied Numerical Mathematics

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The accurate simulation of linear electromagnetic scattering by diffraction gratings is crucial in many technologies of scientific and engineering interest. In this contribution we describe a High-Order Perturbation of Surfaces (HOPS) algorithm built upon a class of Integral Equations due to the analysis of Fokas and collaborators, now widely known as the Unified Transform Method. The unknowns in this formalism are boundary quantities (the electric field and current at the layer interface) which are an order of magnitude fewer than standard volumetric approaches such as Finite Differences and Finite Elements. With detailed numerical experiments we show the efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.

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Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional perfectly reflecting surfaces using the spectral-coordinate method

Cited by 5 publications

References 13 publications

A numerical technique for linear elliptic partial differential equations in polygonal domains

A numerical technique for linear elliptic partial differential equations in polygonal domains

Chapter 6: Overview of Variational Formulations for Linear Elliptic PDEs

A high-order perturbation of surfaces (HOPS) approach to Fokas integral equations: Vector electromagnetic scattering by periodic crossed gratings

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