A comparison of acceleration procedures for the two-dimensional periodic Green's function

Mathis, A.W.; Peterson, Andrew F.

doi:10.1109/8.489309

Cited by 53 publications

(66 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Its application in evaluating GFs for multilayered media is treated in [3][4][5][6]. The Ewald method is extended in [7][8][9] to 2D problems with 1D periodicity (i.e., a planar array of line sources). An analogous treatment of the GF for an array of line sources is reported in [10] where a more general view is also addressed, as well as in [11] where Schlö milch series that may arise in diffraction theory are efficiently evaluated.…”

Section: Introductionmentioning

confidence: 99%

Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

Capolino

Wilton

Johnson

2007

Journal of Computational Physics

View full text Add to dashboard Cite

The Ewald method is applied to accelerate the evaluation of the Green's function (GF) of an infinite equispaced linear array of point sources with linear phasing. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. It is shown analytically that the choice of the standard ''optimal'' Ewald splitting parameter E 0 causes overflow errors at high frequencies (period large compared to the wavelength), and convergence rates are analyzed. A recipe for selecting the Ewald splitting parameter is provided.

show abstract

Section: Introductionmentioning

confidence: 99%

Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

Capolino

Wilton

Johnson

2007

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…The formula representations obtained in the above mentioned studies, i.e., the GF for geometries such as the 2D-periodic array of point sources [46], and the 1D-periodic array of line sources [51], [52], [53], [54], cannot be applied to this new geometry. The standard purely spectral sum representation for the GF cannot be used for this new particular configuration because it diverges when the observation point lies on the axis of the linear array, thus further proving the utility of the Ewald method, which exhibits a Gaussian convergence.…”

Section: The Green's Function For a 1d Array Of Point Sourcesmentioning

confidence: 99%

“…Its application in evaluating GFs for multilayered media is treated in [47], [48], [49], [50]. The Ewald method is extended in [51], [52], [53] to 2D problems with 1D periodicity (i.e., a planar array of line sources). An analogous treatment of the GF for an array of line sources is reported in [54] where a more general view is also addressed, as well as in [55] where a Schlömilch series that may arise in diffraction theory is efficiently evaluated.…”

Section: The Green's Function For a 1d Array Of Point Sourcesmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of electromagnetic scattering by nearly periodic structures: an LDRD report.

Johnson¹,

Warne²,

Jorgenson³

et al. 2006

View full text Add to dashboard Cite

In this LDRD we examine techniques to analyze the electromagnetic scattering from structures that are nearly periodic. Nearly periodic could mean that one of the structure's unit cells is different from all the others -a defect. It could also mean that the structure is truncated, or butted up against another periodic structure to form a seam. Straightforward electromagnetic analysis of these nearly periodic structures requires us to grid the entire structure, which would overwhelm today's computers and the computers in the foreseeable future. In this report we will examine various approximations that allow us to continue to exploit some aspects of the structure's periodicity and thereby reduce the number of unknowns required for analysis. We will use the Green's Function Interpolation with a Fast Fourier Transform (GIFFT) to examine isolated defects both in the form of a source dipole over a meta-material slab and as a rotated dipole in a finite array of dipoles. We will look at the numerically exact solution of a one-dimensional seam. In order to solve a two-dimensional seam, we formulate an efficient way to calculate the Green's function of a 1d array of point sources. We next formulate ways of calculating the far-field due to a seam and due to array truncation based on both array theory and high-frequency asymptotic methods. We compare the high-frequency and GIFFT results. Finally, we use GIFFT to solve a simple, two-dimensional seam problem.3

show abstract

“…Various authors give recommendations for choosing this parameter (Jordan et al, 1986;Mathis & Peterson, 1996;Kustepeli & Martin, 2000;Capolino et al, 2005). The rule of thumb appears to be to choose a constant parameter that gives good numerical accuracy reliably.…”

Section: Introductionmentioning

confidence: 99%

Analysing Ewald's method for the evaluation of Green's functions for periodic media

Arens

Sandfort

Schmitt

et al. 2011

IMA Journal of Applied Mathematics

View full text Add to dashboard Cite

Expressions for periodic Green's functions for the Helmholtz equation in two and three dimensions are derived via Ewald's method. The decay rate of the series occurring in these expressions is analysed and rigorous estimates for the remainder are derived when the series are truncated and replaced by finite sums. The effect of choosing a control parameter occurring in Ewald's expressions is discussed and some recommendations for its choice are given based on the aforementioned estimates. We present various numerical examples for the resulting method for evaluating the Green's functions. The results can also be carried over to evaluating the partial derivatives.

show abstract

A comparison of acceleration procedures for the two-dimensional periodic Green's function

Cited by 53 publications

References 10 publications

Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

Analysis of electromagnetic scattering by nearly periodic structures: an LDRD report.

Analysing Ewald's method for the evaluation of Green's functions for periodic media

Contact Info

Product

Resources

About