Research Article Cell Curvature and Far-Field ...

Research ArticleCell Curvature and Far-Field Superconvergence in NumericalSolutions of Electromagnetic Integral Equations

Andrew F Peterson

School of ECE Georgia Institute of Technology Atlanta GA 30332 USA

Correspondence should be addressed to Andrew F Peterson petersonecegatechedu

Received 5 January 2016 Accepted 3 April 2016

Academic Editor Toni Bjorninen

Copyright copy 2016 Andrew F PetersonThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two curved targets are used to explore far-field superconvergence effects arising in numerical solutions of the electric-field andmagnetic-field integral equations Three different orders of basis and testing functions are used to discretize these equations andthree different types of target models (flat facets quadratic-curved facets and cubic-curved facets) are employed Ideal far-fieldconvergence rates are only observed when the model curvature is one degree higher than the basis order

1 Introduction

Electromagnetic analysis has often exploited ldquovariationalrdquo orldquostationaryrdquo techniques in an attempt to obtain results thatexhibit smaller error or faster convergence than might oth-erwise be obtained for a given order of approximation Rep-resentations in terms of degree 119901 polynomials are expectedto produce error rates of 119874(ℎ119901+1) as ℎ rarr 0 where ℎ is thecharacteristic cell dimension For instance a linear represen-tation should produce119874(ℎ2) convergence Superconvergenceoccurs when the error decreases at a faster rate Galerkinmethod-of-moments (MoM) procedures which involve theuse of identical functions for expanding the unknown surfacecurrents and enforcing the integral equation sometimesexhibit superconvergence in the far-zone fields or scatteringcross section (SCS) [1ndash4] Theoretical SCS convergence rateswere established by Warnick for the electric-field integralequation (EFIE) andmagnetic-field integral equation (MFIE)[5] and are reported belowOne example of superconvergencearises with the popular Rao-Wilton-Glisson (RWG) solutionsof the EFIE [6] which usually yield SCS error rates of O(ℎ2)-O(ℎ3) in practice instead of the119874(ℎ) rate expected from basisfunctions that are only complete to degree 0

When andwhere superconvergence occurs are not alwaysunderstood It was demonstrated in [7] that mixed-order

divergence-conforming basis functions often produce super-convergent far fields with the EFIE but polynomial-completebasis functions apparently do not In a complementary man-ner polynomial-complete basis functions seem to producesuperconvergent far fields with the MFIE while mixed-orderbases do notThus theMFIE requires types of basis functionsdifferent than the EFIE for optimal SCS errors Furthermoreideal convergence rates should not be expected if targetmodels are not accurate enough for instance if flat facetsare used to model curved surfaces Convergence rates areusually slower if the problem geometry contains edges orcorners where current or charge densities may be singularunless special basis functions containing the proper behaviorare employed [8] The purpose of the following is to attemptto clarify conditions under which one can reasonably expectto achieve the ideal convergence rates of [5] Primarily weinvestigate the issue of curved-cell models and the requiredaccuracy in the testing integrals associated with the MoMprocedure for perfectly conducting targets that are spheres ortoroids

We use mixed-order divergence-conforming vector basisfunctions of the Nedelec variety [9] on curved triangularpatches to discretize the EFIE The RWG basis functions[6] are the lowest-order (119901 = 05) members of this familyHierarchical versions of these functions are described in [10]

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 9808637 7 pageshttpdxdoiorg10115520169808637

2 International Journal of Antennas and Propagation

and were implemented in a Galerkin scheme Functions oforders 119901 = 05 119901 = 15 and 119901 = 25 are considered and arereported as follows

Local Definition of the Hierarchical Divergence-ConformingVector Basis Functions Used with the EFIE in Simplex Coor-dinates (119906 V 119908) on a Standard Triangle The Edge-BasedFunctions Straddle Adjacent Cells

Edge-based base vectors (119901 = 05) are

Λ1 = (119906 minus 1) + VVΛ2 = 119906 + (V minus 1)VΛ3 = 119906 + VV

Order 119901 = 15 includes three additional edge-basedfunctions

radic3(V minus 119908)Λ1radic3(119908 minus 119906)Λ2radic3(119906 minus V)Λ3

and 2 cell-based functions

2radic3119906Λ12radic3VΛ2


radic5(3(V minus 119908)2 minus 1)2 minus (1199062)(119906 minus 2)Λ1radic5(3(119908 minus 119906)

2minus 1)2 minus (V2)(V minus 2)Λ2

radic5(3(119906 minus V)2 minus 1)2 minus (1199082)(119908 minus 2)Λ3


6radic5(V minus 119908)119906Λ16radic5(119908 minus 119906)VΛ22radic3119906(5119906 minus 3)Λ12radic3V(5V minus 3)Λ2

Inner integrals were evaluated using adaptive quadratureroutines in conjunction with singularity cancellation proce-dures to return results accurate to some number of decimalplaces (6 digits for the 119901 = 25 results) Testing integralswere evaluated with a 7-point Gauss rule (that can exactlyintegrate degree 5 functions) for 119901 = 05 and 119901 = 15and a 12-point Gauss rule that can exactly integrate degree7 polynomials for 119901 = 25 These rules were selected afternumerical experimentation showed that higher-order rulesdo not improve the accuracy or convergence rates of theresults FromWarnickrsquos analysis the error in the SCSobtainedfrom the EFIE under optimal conditions should convergeat a rate of 119874(ℎ2119901+2) where half-integer indices are usedfor 119901 to denote the mixed-order functions For instancethe SCS error associated with 119901 = 05 functions is ideallyexpected to converge as 119874(ℎ3) while that of the 119901 = 15

functions is ideally expected to converge as 119874(ℎ5) Theseare superconvergent rates since they are different from therates associated with the underlying current density namely119874(ℎ119901+05) It is noteworthy that 119874(ℎ119901+05) rates are observed

for SCS values when point-testing is used with the EFIEinstead of Galerkin testing [11]

On the other hand [7 12 13] suggest that the MFIE SCSis not superconvergent unless polynomial-complete bases areemployed So for the MFIE we use polynomial-completebases of orders 119901 = 00 (constant) 119901 = 10 (linear) and119901 = 20 (quadratic) on curved triangular patches to treatthe MFIE The 119901 = 10 and 119901 = 20 functions are of thecurl-conforming variety and belong to Nedelecrsquos polynomial-complete spaces [14] The specific basis functions used areinterpolatory and are reported as follows

Local Definition of the Interpolatory Curl-Conforming VectorBasis Functions Used with the MFIE in Simplex Coordinates(119906 V 119908) on a Standard Triangle The Edge-Based FunctionsStraddle Adjacent Cells

Order 119901 = 00 includes two constant cell-basedfunctions

Ω1 = Ω2 = V

Order 119901 = 10 includes six linear edge-basedfunctions

V119906V119906119908119908VV

Order 119901 = 20 includes twelve quadratic functions 9edge-based

V(1 minus 2V)119908(1 minus 2119908)119906(1 minus 2119906)V119908(1 minus 2119908)V119906(1 minus 2119906)V(1 minus 2V)V119908( minus V)119908119906( minus )119906V(V minus )

and 3 cell-based

V119908119908119906V119906V

International Journal of Antennas and Propagation 3

minus05

0

05

minus05

0

05minus05

0

05

Figure 1 Illustration of the triangular-cell sphere models used inthe analysis This model contains 72 flat cells

As with the EFIE adaptive quadrature and singularity can-cellation are used for the inner integrals Galerkin testing isimplemented with a 7-point Gauss rule which was demon-strated to be adequate for all three orders Under idealconditions the error in the SCS obtained from the MFIEshould converge at a rate of 119874(ℎ2119901+2) where integer indicesare used for 119901 [5] Thus the SCS error associated with 119901 = 00functions is expected to converge as 119874(ℎ2) while that of the119901 = 10 functions is expected to converge as 119874(ℎ4) Note thatthese rates are one order less than the corresponding rates forthe EFIE SCS based on the extent towhich the basis functionsare complete to a given degree but are still superconvergent

In the following we investigate the convergence rates thatcan be obtained for a sphere and toroid target as a functionof the type of curvature used with the model (flat patchesquadratic-curved patches and cubic-curved patches) Thiswork is intended to build upon and extend the previous workof the author in [7 11 15] in establishing baseline convergencerates for numerical solutions of the EFIE and MFIE

2 Results for Scattering from a Sphere

Triangular-cell models are obtained by dividing a sphereuniformly in 120579 between its north pole and its south pole anddividing it uniformly in 120601 so that the number of cell edgesalong the equator is twice the number of divisions in 120579 Eachoctant of the sphere surface is therefore a curved trianglethat is divided into smaller triangles of fairly uniform sizeFigure 1 shows one example employing 72 flat patches Forcurved patches the cells aremapped from triangles to patchesdefined by the usual 6 Lagrange points (quadratic) or 10Lagrange points (cubic) with all the points on the sphere sur-faceThemapping procedures for the divergence-conforming(EFIE) and curl-conforming (MFIE) bases are described in

p = 20 cubicp = 10 quadraticp = 00 flat facets

10minus5

10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103102

Unknowns

Figure 2 Error in the SCS versus number of unknowns for a spherewith 119896119886 = 2120587 obtained from theMFIEwithGalerkin testing Resultsare reported for flat-facetedmodels used in conjunctionwith119901 = 00basis and test functions quadratic-curvedmodels used with 119901 = 10functions and cubic-curved models used with 119901 = 20 functions

detail in [16]The surface area of each subsectional-cellmodelis normalized to that of the desired sphere

Relative accuracy is determined by evaluating the errorin the bistatic SCS at 5∘ increments in 120579 and 120601 over the entirespherical range with the 2-norm error computed accordingto

119864 =

radic(1119873)sum119873

119899=1

100381610038161003816100381611989011989910038161003816100381610038162 sin 120579119899

|SCS|max (1)

where 119890119899 denotes the error at point 119899 119899 = 1 2 119873 andwhere 120579119899 denotes the value of 120579 at point 119899 The reference isthe exact eigenfunction ldquoMie seriesrdquo expansion [1]

Consider a sphere with 119896119886 = 2120587 or a surface area ofapproximately 12571205822 Thus a density of 100 unknowns persquare wavelength of surface area is reached when the totalnumber of unknowns equals 1257 Figure 2 shows the error inSCS versus number of unknowns for results obtained fromthe MFIE Results are reported for three representations aflat-faceted model used in conjunction with 119901 = 00 basisand test functions (piecewise constant) a quadratic-curvedmodel used with 119901 = 10 functions (linear) and a cubic-curved model used with 119901 = 20 functions As expected thehigher-order bases and higher-order target models producemore accurate solutions Figure 3 shows the same data plottedversus the average edge length used in the models It isapparent that the 119901 = 00 result closely follows an 119874(ℎ2)trajectory the 119901 = 10 result follows an 119874(ℎ4) trajectory


2-no

rm er

ror i

n SC

S

10minus4

10minus3

10minus2

10minus1 100

Average edge length h

p = 20 cubicp = 20 quadratic

p = 10 quadraticp = 00 flat facets

h6h4h2

Figure 3 Error in the SCS versus the average edge lengths in themodels for a sphere with 119896119886 = 2120587 MFIE results from Figure 2 arecompared to ideal asymptotic rates of119874(ℎ2)119874(ℎ4) and119874(ℎ6) Alsoshown is the error obtained with 119901 = 20 basis functions when usedwith quadratic-curved models

and the 119901 = 20 result follows an 119874(ℎ6) trajectory for smallℎ These are the ideal rates suggested in [5] Also shown inFigure 3 is a plot of the corresponding error for the 119901 = 20case when quadratic-curved patches are used in the modelWith the quadratic-patch models the 119901 = 20 results followa slower rate near 119874(ℎ4) Similar behavior was observedby the author for other targets (not shown) suggesting theconclusion that 119901 = 20 bases must be used with cubic-patchmodels to produce optimal convergence rates

The SCS errors produced using the EFIE are presented inFigures 4 and 5 for the same sphere target Datawere obtainedusing 119901 = 05 (RWG) 119901 = 15 and 119901 = 25 mixed-orderbasis and test functions Figure 4 shows results versus thenumber of unknowns with all models represented by cubic-curved patches In common with the MFIE results once thenumber of unknowns is in a reasonable range for a problemofthis electrical size higher-order bases produce more accurateresults for a given number of unknowns Figure 5 shows EFIEresults plotted versus the average edge length used in themodels but for the case when the 119901 = 05 results are obtainedwith flat-faceted models and the 119901 = 15 results are obtainedwith quadratic-curved models The 119901 = 05 results follow an119874(ℎ3) trajectory while the 119901 = 15 results in this case are

somewhere between 119874(ℎ4) and 119874(ℎ5) The 119901 = 25 resultsappear to follow a trajectory between119874(ℎ6) and119874(ℎ7) Thusthese data are close to the ideal rates of [5] but as observedin [7 15] the ideal rates are not always obtained in practice

p = 25 cubicp = 15 cubicp = 05 cubic

Unknowns103

10minus4

10minus3

2-no

rm er

ror i

n SC

SFigure 4 Error in the SCS versus number of unknowns for a spherewith 119896119886 = 2120587 produced by the EFIE Results are reported for 119901 =05 119901 = 15 and 119901 = 25mixed-order divergence-conforming basisand test functions in conjunction with cubic-curved-patch models

The author has also investigated the use of 119901 = 25 baseswith quadratic-patch sphere models (not shown) but alwaysobserved a substantially slower convergence rate for thatcombination This suggests that 119901 = 25 bases must be usedwith cubic-patch models to approach optimal convergencerates

3 Results for Scattering from a Torus

A torus is a simple target that like a sphere avoids thecomplications introduced by edges and corners For thefollowing comparisons a torus with major radius of 3120587120582and minor radius of 15120587120582 for a total surface area of 1801205822is employed The torus is centered at the origin parallel tothe 119909-119910 plane and is illuminated by a uniform plane wave inthe 119911-direction with the electric field polarized along Forillustration Figure 6 shows a torus model containing 196 flattriangular patches

The torus is divided uniformly along both circumferencesinto curved quadrilateral cells which are each divided intotwo (curved) triangles The models employed twice as manydivisions along the larger circumference as along the smallerand are scaled to the proper surface area For comparisonpurposes a reference solution was obtained using the for-mulation described in [17] with basis functions complete topolynomial degree 5 and a total of 9216 unknowns Relativeaccuracy is determined from (1) by evaluating the error in thebistatic SCS at 5∘ increments in 120579 and 90∘ increments in 120601 Inthe figures for the torus example a density of 100 unknowns


10minus4

10minus32-

norm

erro

r in

SCS

10010minus1


p = 25 cubicp = 15 quadraticp = 05 flat

h7h5h3

Figure 5 Error in the SCS versus the average edge lengths in themodels for a sphere with 119896119886 = 2120587 EFIE results are reportedfor flat-faceted models used in conjunction with 119901 = 05 basisand test functions quadratic-curved models used with 119901 = 15functions and cubic-curved models used with 119901 = 25 functionsIdeal asymptotic rates of 119874(ℎ3) 119874(ℎ5) and 119874(ℎ7) are shown forcomparison

minus050

05

minus050

05minus02

minus01

0

01

02

Figure 6 Illustration of the triangular-cell torus models used in theanalysis This model contains 196 flat cells

per square wavelength of surface area is reached when thetotal number of unknowns equals 1800

Figure 7 shows the SCS error produced by the MFIEversus the number of unknowns for flat-faceted models with119901 = 00 functions quadratic-curved models with 119901 = 10functions and cubic-curved models with 119901 = 20 functionsFigure 8 shows the same data plotted versus the average edgelength used in the models For these results the 119901 = 00 dataexhibit a convergence rate between 119874(ℎ) and 119874(ℎ2) whilethe 119901 = 10 and 119901 = 20 data closely approximate the idealtrajectories of 119874(ℎ4) and 119874(ℎ6)

10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103

Unknowns


Figure 7 Error in the SCS versus number of unknowns for atorus with ka = 3 and kb = 6 obtained from the MFIE withGalerkin testing Results are reported for flat-facetedmodels used inconjunction with 119901 = 00 basis and test functions quadratic-curvedmodels used with 119901 = 10 functions and cubic-curved models usedwith 119901 = 20 functions

10010minus1


10minus3

10minus2

2-no

rm er

ror i

n SC

S


h6h4h2

Figure 8 Error in the SCS versus the average edge lengths in themodels for the same torus used in Figure 7 MFIE results fromFigure 7 are compared to ideal asymptotic rates of119874(ℎ2)119874(ℎ4) and119874(ℎ6)


10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3

Figure 9 Error in the SCS versus the average edge lengths in themodels for the same torus used in Figures 7 and 8 EFIE results arereported for a flat-faceted model used in conjunction with 119901 = 05basis and test functions a quadratic-curvedmodel usedwith119901 = 15functions and a cubic-curved model used with 119901 = 25 functionsIdeal asymptotic rates of 119874(ℎ3) 119874(ℎ5) and 119874(ℎ7) are shown forcomparison

Figures 9 and 10 show EFIE results for the SCS error forthe same torus plotted versus the average edge length The119901 = 05 and 119901 = 25 results converge somewhat slower thanthe ideal rates of 119874(ℎ3) and 119874(ℎ7) while the 119901 = 15 dataappear to closely follow the 119874(ℎ5) rate Figure 10 shows thatthe convergence rate for the 119901 = 25 results is slower than119874(ℎ5) if quadratic-patch models are used while it is between

119874(ℎ6) and 119874(ℎ7) when cubic-patch models are used

4 Conclusions

Far-field convergence rates are studied using two curvedtargets Numerical solutions of the EFIE and MFIE areemployed with higher-order vector basis functions and targetmodels that involve flat facets quadratic-curved patches orcubic-curved patches Superconvergence in the SCS at ratesclose to those suggested in [5] is observed in the resultsprovided that the models in use have high enough curvatureand the testing integration is done with quadrature rules ofsufficient accuracy

An interesting result of these observations is that itappears necessary to employ cubic-curved-patch modelsin order to obtain ideal convergence rates with quadraticbasis functions and the MFIE This is at odds with theclassical literature onnumericalmethodswhere it is supposedthat quadratic models should be adequate with quadratic

10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1


p = 25 cubic modelsp = 25 quadratic models

h5h7h6

Figure 10 Error in the SCS versus the average edge lengths in themodels for the same torus used in Figures 7ndash9 EFIE results arereported for quadratic-curved and cubic-curved models used with119901 = 25 functions Asymptotic rates of 119874(ℎ5) 119874(ℎ6) and 119874(ℎ7) areshown for comparison

expansions It also appears necessary to employ cubic modelswith the 119901 = 25 mixed-order basis functions (which areonly complete to degree 2) used with the EFIE to approachideal SCS convergence rates A similar conclusion applies tothe lower-order bases namely the linear 119901 = 10 functions(used with the MFIE) and 119901 = 15 functions (used withthe EFIE) require quadratic-patch models to yield near-idealconvergence rates

Competing Interests

The author declares that he has no competing interests

References

[1] R F Harrington Time-Harmonic Electromagnetic FieldsMcGraw-Hill New York NY USA 1961

[2] S Wandzura ldquoOptimality of Galerkin method for scatteringcomputationsrdquo Microwave and Optical Technology Letters vol4 no 5 pp 199ndash200 1991

[3] J R Mautz ldquoVariational aspects of the reaction in the methodof momentsrdquo IEEE Transactions on Antennas and Propagationvol 42 no 12 pp 1631ndash1638 1994

[4] A F Peterson D R Wilton and R E Jorgenson ldquoVariationalnature of galerkin and non-galerkin moment method solu-tionsrdquo IEEE Transactions on Antennas and Propagation vol 44no 4 pp 500ndash503 1996


[5] K F Warnick Numerical Analysis for Electromagnetic IntegralEquations Artech House Boston Mass USA 2008

[6] S M Rao D R Wilton and A W Glisson ldquoElectromagneticscattering by surfaces of arbitrary shaperdquo IEEE Transactions onAntennas and Propagation vol 30 no 3 pp 409ndash418 1982

[7] A F Peterson ldquoObserved baseline convergence rates andsuperconvergence in the scattering cross section obtainedfrom numerical solutions of the MFIErdquo IEEE Transactions onAntennas and Propagation vol 56 no 11 pp 3510ndash3515 2008

[8] MM Bibby A F Peterson and CM Coldwell ldquoOptimum cellsize for high order singular basis functions at geometric cor-nersrdquo Applied Computational Electromagnetics Society Journalvol 24 no 4 pp 368ndash374 2009

[9] J C Nedelec ldquoMixed finite elements in R3rdquo NumerischeMathematik vol 35 no 3 pp 315ndash341 1980

[10] R D Graglia and A F Peterson Higher-Order Techniques inComputational Electromagnetics SciTech Publishing EdisonNJ USA 2016

[11] A F Peterson ldquoBeyond RWGGalerkin solutions of the EFIEinvestigations into point-matched discontinuous and higherorder discretizationsrdquo in Proceedings of the 27th Annual Reviewof Progress in Applied Computational Electromagnetics pp 117ndash120 Williamsburg Va USA March 2011

[12] O Ergul and L Gurel ldquoInvestigation of the inaccuracy of theMFIE discretized with the RWGbasis functionsrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium vol 3 pp 3393ndash3396 Monterey Calif USA June2004

[13] O Ergul and L Gurel ldquoImproving the accuracy of the magneticfield integral equation with the linear-linear basis functionsrdquoRadio Science vol 41 no 4 2006

[14] J C Nedelec ldquoA new family of mixed finite elements in R3rdquoNumerische Mathematik vol 50 no 1 pp 57ndash81 1986

[15] A F Peterson ldquoObserved accuracy of point-tested and galerkinimplementations of the volume EFIE for dielectric targetsrdquoApplied Computational Electromagnetics Society Journal vol 30no 3 pp 255ndash260 2015

[16] A F Peterson Mapped Vector Basis Functions for Electromag-netic Integral EquationsMorganampClaypool Synthesis LecturesMorgan amp Claypool San Rafael Calif USA 2006

[17] M M Bibby C M Coldwell and A F Peterson ldquoA high ordernumerical investigation of electromagnetic scattering from atorus and a circular looprdquo IEEE Transactions on Antennas andPropagation vol 61 no 7 pp 3656ndash3661 2013

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



and were implemented in a Galerkin scheme Functions oforders 119901 = 05 119901 = 15 and 119901 = 25 are considered and arereported as follows

Local Definition of the Hierarchical Divergence-ConformingVector Basis Functions Used with the EFIE in Simplex Coor-dinates (119906 V 119908) on a Standard Triangle The Edge-BasedFunctions Straddle Adjacent Cells

Edge-based base vectors (119901 = 05) are

Λ1 = (119906 minus 1) + VVΛ2 = 119906 + (V minus 1)VΛ3 = 119906 + VV


radic3(V minus 119908)Λ1radic3(119908 minus 119906)Λ2radic3(119906 minus V)Λ3


2radic3119906Λ12radic3VΛ2


radic5(3(V minus 119908)2 minus 1)2 minus (1199062)(119906 minus 2)Λ1radic5(3(119908 minus 119906)

2minus 1)2 minus (V2)(V minus 2)Λ2

radic5(3(119906 minus V)2 minus 1)2 minus (1199082)(119908 minus 2)Λ3


6radic5(V minus 119908)119906Λ16radic5(119908 minus 119906)VΛ22radic3119906(5119906 minus 3)Λ12radic3V(5V minus 3)Λ2

Inner integrals were evaluated using adaptive quadratureroutines in conjunction with singularity cancellation proce-dures to return results accurate to some number of decimalplaces (6 digits for the 119901 = 25 results) Testing integralswere evaluated with a 7-point Gauss rule (that can exactlyintegrate degree 5 functions) for 119901 = 05 and 119901 = 15and a 12-point Gauss rule that can exactly integrate degree7 polynomials for 119901 = 25 These rules were selected afternumerical experimentation showed that higher-order rulesdo not improve the accuracy or convergence rates of theresults FromWarnickrsquos analysis the error in the SCSobtainedfrom the EFIE under optimal conditions should convergeat a rate of 119874(ℎ2119901+2) where half-integer indices are usedfor 119901 to denote the mixed-order functions For instancethe SCS error associated with 119901 = 05 functions is ideallyexpected to converge as 119874(ℎ3) while that of the 119901 = 15

functions is ideally expected to converge as 119874(ℎ5) Theseare superconvergent rates since they are different from therates associated with the underlying current density namely119874(ℎ119901+05) It is noteworthy that 119874(ℎ119901+05) rates are observed

for SCS values when point-testing is used with the EFIEinstead of Galerkin testing [11]

On the other hand [7 12 13] suggest that the MFIE SCSis not superconvergent unless polynomial-complete bases areemployed So for the MFIE we use polynomial-completebases of orders 119901 = 00 (constant) 119901 = 10 (linear) and119901 = 20 (quadratic) on curved triangular patches to treatthe MFIE The 119901 = 10 and 119901 = 20 functions are of thecurl-conforming variety and belong to Nedelecrsquos polynomial-complete spaces [14] The specific basis functions used areinterpolatory and are reported as follows

Local Definition of the Interpolatory Curl-Conforming VectorBasis Functions Used with the MFIE in Simplex Coordinates(119906 V 119908) on a Standard Triangle The Edge-Based FunctionsStraddle Adjacent Cells

Order 119901 = 00 includes two constant cell-basedfunctions

Ω1 = Ω2 = V

Order 119901 = 10 includes six linear edge-basedfunctions

V119906V119906119908119908VV

Order 119901 = 20 includes twelve quadratic functions 9edge-based

V(1 minus 2V)119908(1 minus 2119908)119906(1 minus 2119906)V119908(1 minus 2119908)V119906(1 minus 2119906)V(1 minus 2V)V119908( minus V)119908119906( minus )119906V(V minus )

and 3 cell-based

V119908119908119906V119906V


minus05

0

05

minus05

0

05minus05

0

05







10minus5

10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103102

Unknowns




119864 =

radic(1119873)sum119873

119899=1

100381610038161003816100381611989011989910038161003816100381610038162 sin 120579119899

|SCS|max (1)




2-no

rm er

ror i

n SC

S

10minus4

10minus3

10minus2

10minus1 100




h6h4h2






Unknowns103

10minus4

10minus3

2-no

rm er

ror i

n SC







10minus4

10minus32-

norm

erro

r in

SCS

10010minus1



h7h5h3


minus050

05

minus050

05minus02

minus01

0

01

02




10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103

Unknowns



10010minus1


10minus3

10minus2

2-no

rm er

ror i

n SC

S


h6h4h2



10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3




4 Conclusions



10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1



h5h7h6



Competing Interests


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus05

0

05

minus05

0

05minus05

0

05







10minus5

10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103102

Unknowns




119864 =

radic(1119873)sum119873

119899=1

100381610038161003816100381611989011989910038161003816100381610038162 sin 120579119899

|SCS|max (1)




2-no

rm er

ror i

n SC

S

10minus4

10minus3

10minus2

10minus1 100




h6h4h2






Unknowns103

10minus4

10minus3

2-no

rm er

ror i

n SC







10minus4

10minus32-

norm

erro

r in

SCS

10010minus1



h7h5h3


minus050

05

minus050

05minus02

minus01

0

01

02




10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103

Unknowns



10010minus1


10minus3

10minus2

2-no

rm er

ror i

n SC

S


h6h4h2



10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3




4 Conclusions



10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1



h5h7h6



Competing Interests


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2-no

rm er

ror i

n SC

S

10minus4

10minus3

10minus2

10minus1 100




h6h4h2






Unknowns103

10minus4

10minus3

2-no

rm er

ror i

n SC







10minus4

10minus32-

norm

erro

r in

SCS

10010minus1



h7h5h3


minus050

05

minus050

05minus02

minus01

0

01

02




10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103

Unknowns



10010minus1


10minus3

10minus2

2-no

rm er

ror i

n SC

S


h6h4h2



10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3




4 Conclusions



10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1



h5h7h6



Competing Interests


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus4

10minus32-

norm

erro

r in

SCS

10010minus1



h7h5h3


minus050

05

minus050

05minus02

minus01

0

01

02




10minus4

10minus3

10minus2

2-no

rm er

ror i

n SC

S

104103

Unknowns



10010minus1


10minus3

10minus2

2-no

rm er

ror i

n SC

S


h6h4h2



10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3




4 Conclusions



10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1



h5h7h6



Competing Interests


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus4

10minus3

10minus22-

norm

erro

r in

SCS

10010minus1



h7h5h3




4 Conclusions



10minus4

10minus3

2-no

rm er

ror i

n SC

S

10010minus1



h5h7h6



Competing Interests


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Cell Curvature and Far-Field ...

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