Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula

Polimeridis, Athanasios G.; Mosig, J. R.

doi:10.1109/tap.2010.2046866

Cited by 41 publications

(55 citation statements)

References 32 publications

(54 reference statements)

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“…Trigonometry tells us that a product of sines/cosines can be easily transformed into a sum of similar functions. Hence, in this case the integral (22) can be transformed into a sum of integrals of the basic type (12), to which WA applies. The situation is somewhat more delicate with other oscillating functions and must be dealt with case by case.…”

Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

“…This has been recently accomplished for the singular integrals arising in the Galerkin discretization of the Integral Equation formulations used to model the current density in the surface of metallic scatterers [22].…”

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

“…Among the multiple possibilities for solving the inner integral, the source domain can be decomposed to push the singularity to the edges of the new subdomains and DE can be used [22]. Less recognized is the obvious fact that, once evaluated, the inner integral (32) will show some type of singularity if the domains P and Q touch at a point, share an edge or are identical [22]. In all these cases, DE should be a preferred choice to evaluate the outer integral (33).…”

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

“…However, the idea was quickly extended to multidimensional singular integrals and successfully applied to the double surface integrals arising in the Integral-Equation treatment of metallic scatterers when discretized with a Galerkin's method [22]- [25]. A final development of paramount relevance for this paper is the transformation, obtained by T. Ooura, of the DE quadrature into a form that can be applied to Fourier transforms of slowly decaying functions [26], [27].…”

Section: Historical Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Weighted Averages and Double Exponential Algorithms

Mosig

2013

IEICE Trans. Commun.

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SUMMARYThis paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

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Section: Extension To Products Of Oscillating Functionsmentioning

confidence: 99%

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

Section: Historical Introductionmentioning

confidence: 99%

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Weighted Averages and Double Exponential Algorithms

Mosig

2013

IEICE Trans. Commun.

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“…This explains why the standard (e.g., Gauss-Legendre) 2D quadratures currently available in the literature fail to produce accurate results. This drawback can be alleviated via the usage of numerical rules tailored to integrate functions with endpoint singularities, as indicated in [26,27]. Here, these rules are applied to the previous analytical potential integral within a SIE-MoM context, and this is the second relevant contribution of this paper.…”

Section: Introductionmentioning

confidence: 99%

On the Analytic-Numeric Treatment of Weakly Singular Integrals on Arbitrary Polygonal Domains

Lopez-Pena¹,

Polimeridis

Mosig³

2011

PIER

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Abstract-An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided.

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Accurate 2.5‐D boundary element method for conductive media

2014

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The solution of the time-harmonic Maxwell equations using a boundary element method, for 2-D geometries illuminated by arbitrary 3-D excitations, gives rise to numerical difficulties if highly conductive media are present. In particular, the interaction integrals arising in the method of moments involve kernels that strongly oscillate in space and, at the same time, decay exponentially. We present an accurate method to tackle these issues over a very broad conductivity range (from lossy dielectric to conductor skin-effect regime), for both magnetic and nonmagnetic conductors. Important applications are the modal analysis of waveguides with nonperfect conductors, scattering problems, and shielding problems with enclosures with arbitrary permeability and conductivity and 3-D noise sources.

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Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula

Cited by 41 publications

References 32 publications

Weighted Averages and Double Exponential Algorithms

Weighted Averages and Double Exponential Algorithms

On the Analytic-Numeric Treatment of Weakly Singular Integrals on Arbitrary Polygonal Domains

Accurate 2.5‐D boundary element method for conductive media

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