Hybrid Integration Scheme for the Evaluation of Strongly Singular and Near-Singular Integrals in Surface Integral Equations

Rivero, J.; Vipiana, F.; Wilton, Donald R.; Johnson, William A.

doi:10.1109/tap.2019.2920333

Cited by 14 publications

(15 citation statements)

References 23 publications

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“…We adopt the pragmatic view that well-designed quadrature schemes should avoid sampling the potential at such points, and hence we assign a value there, mostly for plotting purposes only; a logical choice for lim P→0 u/P, for example, would be its angular average over points in the infinitesimal disk in the plane d = 0 and centered at P = 0. Thus the line reading '' u/P P=0 := 0 '' in (15) implies that, although lim P→0 u/P is indeterminate at, P = 0 we assign it the value zero there. Finally, we observe in passing that removing the static part from a dynamic kernel can be especially important in computing potential gradients, as it essentially isolates all anomalous behavior to terms in (15), leaving the potential gradient integral with a difference kernel more suitable for numerical integration [15].…”

Section: A Gradient Of Electrostatic Potential Unit Sourcementioning

confidence: 99%

“…Thus the line reading '' u/P P=0 := 0 '' in (15) implies that, although lim P→0 u/P is indeterminate at, P = 0 we assign it the value zero there. Finally, we observe in passing that removing the static part from a dynamic kernel can be especially important in computing potential gradients, as it essentially isolates all anomalous behavior to terms in (15), leaving the potential gradient integral with a difference kernel more suitable for numerical integration [15].…”

Section: A Gradient Of Electrostatic Potential Unit Sourcementioning

confidence: 99%

“…where I q and ∇I q are defined in (9) and (15), respectively. Combining (15) and (19), it is easily verified that the normal component of the gradient potential (19) as T is approached from above or below has the well-known classical limit [4], [18], [19]:…”

Section: Synthesis Of Parent Triangle Potentials and Gradients Frmentioning

confidence: 99%

“…Despite the advantages of cancellation methods, they are generally more complex to implement than subtraction methods. And recently, for highly singular integrals, a hybrid approach combining subtraction of a leading singularity with singularity cancellation applied to the difference integral has proved useful [15].…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of Static Potential Integrals on Triangular Domains

et al. 2020

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Static potential integrals for constant and linear sources on triangles are derived in a straightforward way. The new representations, as presented, are robust with respect to machine evaluation in important limiting cases. The potential integrals comprise up to six functions, each dependent on the relative position and orientation (with respect to an observation point) of a vertex and edge, respectively, of the source triangle. Gradients of the potentials are derived by differentiation, thus preserving relations between the representations. Each such vertex function reveals any anomalous functional behavior near its associated vertex or edge, which is useful information for devising test integral schemes. Potential plots in the source plane of an equilateral triangle illustrate such behavior, as do similar plots for each vertex function and gradient components near their associated edge and vertex. INDEX TERMS Integral equations, moment methods, numerical analysis, singular integrals, numerical integration.

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Section: A Gradient Of Electrostatic Potential Unit Sourcementioning

confidence: 99%

Section: A Gradient Of Electrostatic Potential Unit Sourcementioning

confidence: 99%

Section: Synthesis Of Parent Triangle Potentials and Gradients Frmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Evaluation of Static Potential Integrals on Triangular Domains

et al. 2020

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“…Both MRW [21,25,28] and DE [27,29,30] quadrature rules have been successfully used for the determination of integrals with singularities that arise in the application of the MoM for multilayer problems. In more recent works, results obtained by MRW [31][32][33] or by DE [34,35] quadrature rules are used as comparison patterns. However, the performance of these two types of quadrature rules have never been compared to each other in an MoM context.…”

Section: Description Of the Problemmentioning

confidence: 99%

Comparison between Specialized Quadrature Rules for Method of Moments with NURBS Modelling Applied to Periodic Multilayer Structures

et al. 2020

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A comparison between Ma-Rokhlin-Wandzura (MRW) and double exponential (DE) quadrature rules for numerical integration of method of moments (MoM) matrix entries with singular behavior is presented for multilayer periodic structures. Non Uniform Rational B-Splines (NURBS) modelling of the layout surfaces is implemented to provide high-order description of the geometry. The comparison is carried out in order to show that quadrature rule is more suitable for MoM matrix computation in terms of sampling, accuracy of computation of MoM matrix, and CPU time consumption. The comparison of CPU time consumption shows that the numerical integration with MRW samples is roughly 15 times faster than that numerical integration using DE samples for results with similar accuracies. These promising results encourage to carry out a comparison with results obtained in previous works where a specialized approach for the specific analysis of split rings geometries was carried out. This previous approach uses spectral MoM version with specific entire domain basis function with edge singularities defined on split ring geometry. Thus, the previous approach provides accurate results with low CPU time consumption to be compared. The comparison shows that CPU time consumption obtained by MRW samples is similar to the CPU time consumption required by the previous work of specific analysis of split rings geometries. The fact that similar CPU time consumptions are obtained by MRW quadrature rules for modelling of general planar geometries and by the specialized approach for split ring geometry provides an assessment for the usage of the MRW quadrature rules and NURBS modelling. This fact provides an efficient tool for analysis of reflectarray elements with general planar layout geometries, which is suitable for reflectarray designs under local periodicity assumption where a huge number of periodic multilayer structures have to be analyzed.

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Characterization and integration of the singular test integrals in the method‐of‐moments implementation of the electric‐field integral equation

Freno

Johnson

Zinser

et al. 2021

Engineering Analysis with Boundary Elements

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The method-of-moments implementation of the electric-field integral equation yields many code-verification challenges due to the different sources of numerical error and their possible interactions. Matters are further complicated by singular integrals, which arise from the presence of a Green's function. In this paper, we provide approaches to separately assess the numerical errors arising from the use of basis functions to approximate the solution and the use of quadrature to approximate the integration. Through these approaches, we are able to verify the code and compare the error from different quadrature options.

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Hybrid Integration Scheme for the Evaluation of Strongly Singular and Near-Singular Integrals in Surface Integral Equations

Cited by 14 publications

References 23 publications

Evaluation of Static Potential Integrals on Triangular Domains

Evaluation of Static Potential Integrals on Triangular Domains

Comparison between Specialized Quadrature Rules for Method of Moments with NURBS Modelling Applied to Periodic Multilayer Structures

Characterization and integration of the singular test integrals in the method‐of‐moments implementation of the electric‐field integral equation

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