Hybrid
CFIE-EFLE
Solution ofComposite Geometries with Coexisting
Open and Closed Surfaces tOzgurErguland LeventGurel* DepartmentofElectricalandElectronics Engineering BilkentUniversity,TR-06800,Bilkent,Ankara,Turkey
E-mail:lgurel@bilkent.edu.tr,ergul@ee.bilkent.edu.tr
Abstract
The combined-fieldintegral equation (CFIE) is employed to formulatethe
electro-magnetic scattering and radiationproblems ofcomposite geometrieswithcoexisting
open and closedconducting surfaces. Conventional formulations oftheseproblems
withtheelectric-fieldintegral equation (EFIE) leadtoinefficient solutionsdue to the
ill-conditioning ofthematrixequations and theinternal-resonanceproblems. The hybrid CFIE-EFIE techniqueintroduced in this paper, based on theapplication of the CFIEon theclosedsurfacesand EFIE on the opensurfaces,significantlyimprovestheefficiency ofthesolution.
I. INTRODUCTION
Inthis paper,weconsider the solutionof the scattering and radiation problems involving both open and closed conducting surfacesas shown inFig. l(a). These problems often arise in thesimulations ofpracticalelectromagnetic scenarios, where the thin and thick conducing parts of the objects are modelled with open and closed surfaces, respectively. The formulationof these problems arecustomarily achieved by the electric-field integral equation (EFIE) duetothe presence of the open surfaces. However, the EFIE is prone to internal-resonance problems and also leads toill-conditioned matrix equations that decrease theperformances of the iterative solvers, especially when the problem size is large.
For the electromagnetic modelling of problems involving closed conducting surfaces, thecombined-fieldintegral equation(CFGE)ispreferred mainly because it is free of the internalresonanceproblems [1].Inaddition, although this equation is simply the linear combination ofthe EFIE and themagnetic-fieldintegral equation (MFIE), i.e.,
CFIE=aEFIE +(1-ca)MFIE, (1)
it generatesconsiderably better-conditioned matrix equations comparedtoboth the EFIE and MEIE[2]. This crucial property of theCFGE becomes useless for theproblems of opensurfaces,wherethe EFIE becomes the inevitable choice. Inconsolation, the EFIE is generally observedtobeperformingbetterforproblems of solely opensurfaces, having rapid convergence of the iterativesolutions, especially when preconditioned properly. Inthispaper,weinvestigate the solution of thecomposite-geometry problems involving both open and closedconductingsurfaces, where theEFJE solutionsbecome inefficient duetothepresence oftheclosed parts. Weimprove the solution of these problems by tThis work was supportedby the Turkish Academy of Sciences in the framework of the Young
Scientist AwardProgram(LGITUBA-GEBIP/2002-1-12),by the Scientific and Technical Research Council of Turkey(TUBITAK) under Research Grant103E008,andby contracts from ASELSAN and SSM.
0-7803-8883-6/05/$20.00 @2005IEEE
takinga hintfrom the improvement by the CFIE in the solution of the scattering problems involving only closed surfaces. The proposed technique is based of the application of a hybrid CFIE-EF1IE formulation, leading to better-conditioned matrix equations and consequently more efficient iterative solutions. In the implementations using the hybrid technique, the open parts of the problems are still formulated with the EFIE, while the CFE is enforced on the closed parts to improvethe conditioning. An example involving aradiation problem will be given to demonstrate the overall improvement by the hybrid formulation compared to the use of the pureEFLEon thesame problem.
II. HYBRIDCFIE-EFIEFORMULATION
Forconducting surfaces, numerical application of theEFIE andMFEEleads tomatrix equationsas
N
EZ:
4
an=VmE,
m=1.N,
(2)n=1
wherethe matrix elements, namely the interactions between the basis functions
bn(r)
and testingfunctionstm(r),arederived as
Z =
drtm(r)
jdr'bn(r')g(r,r')
Is
j
drtm(r)
Ldr'bn(r')
[VV'g(r,r')]
(3)
for the EFIE, and asZI=
jdrtm(r)
*bn(r)
-drtm(r)
*fn
xjdr'bn(r')
xV'g(r, r')
(4)
for theMFIE. To form the CFIE system, the matrix elements in Eqs. (3) and (4) are linearly combined asZmn
=amZmmn
+(1-am)
-(5)
Different from (1), where a is a constant, the parameter am to weight the EFIE and MFIE contributions in the CFIE is flexible in (5). Such a definition provides the freedom ofchoosing different linear combinations for different testing functions. Consequently, it becomespossibletoemploy theCFIEwith Am#40 for the testingfunctions located on the closed parts of thegeometry while setting am=1 to use the EFIE on the open parts. As aresult, by the inclusion of theCFIEinteractions into the matrix equation, we obtain better-conditioned formulations leading to faster converging iterative solutions compared tothe pureEFIE.
III. RESULTS
To demonstrate theimprovement by the use of the hybrid CFIE-EFIE formulation, we presentthe results ofa radiation problem involving a dipole antenna placed over a perfectlyconducting rectangularboxwithdimensions Ax5Ax5A asdepicted in Fig.1(a). The dipoleantenna is modelled by astrip ofzero thickness with length A and width A/10.Thecompositestructure is triangulated with a mesh size of A/10 corresponding to 16,124triangleson the box and only 20 triangles on thestrip. The feed of the antenna is simulatedbyadelta-gap source located at the center of the strip as shown in Fig.l(b).
Local electric field withamplitude 1/d is defined inside theinfinitelynarrow opening between thetrianglessothat
V =mV +
(1am)V4
=Vm=
dimO
k7
Is
drtm(r) *K.[m,
el,
0 m e (6)Consequently, only asingle element of the excitation vector
vg,
i.e., m = e will be nonzero. The resulting radiation problem is solved withthe multi-level fastmultipole algorithm [3] employing theRao-Wilton-Glisson [4] functionsas the basis andtesting functions defined on the triangular domains, leading to a linear system with 24,205 unknowns.Fig. 2(a) demonstrates the iteration counts for the solution of the problem with the conjugate gradient squared (CGS) algorithm with respect to the value of am in (5) applied on the closed parts of the geometry, i.e.,ontherectangular box. The dashed andsolid curves representthe number of iterations required toreach 10-3 and 10-6 residualerrors, respectively, while employing a block-diagonal preconditioner (BDP) with 561,221 nonzero elementstoaccelerate the iterative solution. Bothcurves areobservedto be minimized when am isabout0.2-0.3,withsignificantimprovementin the convergence comparedtothe pure EFIE solution of the problem (am=1 Vm),which isnotshown inthefigure due to the extremely high iteration counts.
Toobtain convergent solutions with the EFIEformulation, strongpreconditioners are required as presented inFig. 2(b), where the iterative solver doesnot converge with aBDP and thus needs to be accelerated witha near-field preconditioner (NFP) with 5,267,535 nonzero elements obtained by keeping all of the near-field interactions in the impedance matrix. However,evenwith this strongpreconditioner thatrequires extensive useofthe memory and leadstosignificantincrease theprocessing time, the residual error doesnot drop under10-6 until the 1400th iteration. Theimprovement by thehybrid formulationis demonstrated in Fig. 3(a), where the convergence characteristics of the EFIE, thehybridMFIE-EFIE (am =0on the closedpart), and the hybrid CFIE-EFIE (am =0.2 onthe closedpart) formulations aredepictedon thesamegraph. With the sameBDpreconditioner, the convergence issignificantlyimproved by theuseof the CFIE onthe closed partof the geometry. Finally,Fig. 3(b)shows thenormalized radarcross section(RCS/A2indB) values on the z-x plane with respectto
9,
where we observe the accuracyof thehybridCFIE-EFIEcomparedtothe referenceEFIE solution.IV. CONCLUSION
Anovelhybrid
CFIPE-EFLE
formulation is presented for the solution of the composite-geometryproblems withcoexisting openand closed conducting surfaces. With the ap-plicationof thetechniquetothe radiation and scattering problems involving composite geometries, theefficiency of the solutions canbe significantly improved comparedto using onlytheEFIEfornulationonthe whole geometry.REFERENCES
[1] J. R. Mautz and R. F.Harrington,"H-field, E-field,and combined field solutions forconductingbodies ofrevolution," AEU,vol.32,no.4, pp.157-164, Apr. 1978.
[2] L.Gureland0.Ergiil,"Comparisonof FMMimplementationsemploying differentformulationsand iterativesolvers," 2003IEEEAP-SInternationalSymposium, Columbus,Ohio, vol. 1, pp.19-22,June 2003.
[3] W. C.Chew,J.-M.Jin,E.Michielssen,andJ.Song, Fast andEfficient AlgorithmsinComputational Electromagnetics. Boston,MA: ArtechHouse,2001.
[4] S. M.Rao,D. R.Wilton, and A. W. Glisson, "Electromagneticscatteringbysurfaces ofarbitraryshape,'
IEEE Trans. Antennas Propagat.,vol. AP-30,no.3,pp.409-418,May 1982.
Z
(a) (b)
Fig. 1. (a)Compositegeometry involvingadipoleantenna (open surface)of length A locatedover a
perfectly conducting rectangular box (closed surface) with the dimensionsofAx5Ax5A.(b) Delta-gap sourcelocatedatthecenteroftheantenna.
2 E 2 9 a
(a)
Iteratins(b)Fig.2. (a) Iterationcountsforthe solution of the radiation problemin Fig. 1(a) withtheCGS algorithm
withrespect tothe value of A_m. (b) Convergence characteristics of the EFIE formulation using the BDP and NFPtoaccelerate the iterative solution.
10' Serations (a)
-2C
45 90 (b) 135Fig. 3. (a) Comparison ofconvergencecharacteristics of the EFIE, the MFIE-EFIE, and the CFIE-EFIE
formulations using BDP. (b) Normalizedradarcross section(RCS/A2 indB)onthez-xplane withrespect
toO. 292 -30i -40_1---EFIE -5( -CFIE-EFIE|
