Monte Carlo Integration Technique for Method of Moments Solution of EFIE in Scattering Problems

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1 J. Electromagnetc Analyss & Applcatons, 2009, 1: do: /jemaa Publshed Onlne December 2009 ( Monte Carlo Integraton Technque for Method of Moments Soluton of EFIE n Scatterng Problems Mrnal MISHRA, sha GUPTA Department of Electroncs and Communcaton Engneerng, Brla Insttute of Technology, Mesra, Inda Emal: mrnal.mshra@gmal.com, ngupta@btmesra.ac.n Receved May 8 th, 2009; revsed July 10 th, 2009; accepted July 18 th, ABSTRACT An ntegraton technque based on use of Monte Carlo Integraton s proposed for Method of Moments soluton of Electrc Feld Integral Equaton. As an example numercal analyss s carred out for the soluton of the ntegral equaton for unknown current dstrbuton on metallc plate structures. The entre doman polynomal bass functons are employed n the MOM formulaton whch leads to only small number of matrx elements thus savng sgnfcant computer tme and storage. It s observed that the proposed method not only provdes soluton of the unknown current dstrbuton on the surface of the metallc plates but s also capable of dealng wth the problem of sngularty effcently. Keywords: Scatterng, EFIE, Method of Moments, Monte Carlo Integraton 1. Introducton The Method of Moments (MoM) [1] s one of the wdely used numercal technques employed for the soluton of Integral Equatons. The MoM s based upon the transformaton of an ntegral equaton, nto a matrx equaton. However, the applcaton of the spatal-doman MoM to the soluton of ntegral equaton s qute tme consumng. The matrx-fll tme would be sgnfcantly mproved f these ntegrals can be evaluated effcently. The MoM employs expanson of the unknown functon nsde the ntegral n terms of known bass functons wth unknown coeffcents to be determned. Pont matchng technque or Galerkn s technque commonly employed n MoM results n a system of lnear equatons equal n number to that of unknown coeffcents. Ths leads to a matrx equaton for the coeffcents. The matrx thus obtaned s called the moment matrx. The unknown coeffcents can then be obtaned by matrx nverson. The MoM method nvolves two approaches, the sub doman [2,3] and the entre doman [4,5] approaches, essentally based on the two knds of bass functons employed for the expanson of the unknown functon on the metal surface. The entre doman bass functons extend over the whole regon occuped by the structure, whereas the sub doman bass functons are defned to exst over a secton of the structure and have a zero value over the rest of ts porton. The choce of the type of bass functon depends upon the sze and shape of the metallc structure n the problem. The advantage wth the sub doman bass functons s that due to ther flexblty to be defned over small polygonal domans of varyng szes. The whole structure under nvestgaton can be modeled as consstng of large number of such polygonal sub domans, thus makng possble the analyss of complcated shaped structures. The dsadvantage wth these bass functons s that they are lmted to electrcally small and moderately large structures, as the number of sub domans requred to model large structures accurately becomes very large. Ths results n the moment matrx of a large sze ncreasng the computaton costs n terms of memory and CPU tme. The entre doman bass functons, on the other hand, requre a very few number of expanson terms. These are also capable of analyzng electrcally large structures and the soluton obtaned wth these functons are more accurate than the sub doman bass functons. Ths results n a faster and more accurate soluton, thus reducng the computatonal cost. One of the requrements of the entre doman bass functons s a pror knowledge of the dstrbuton of the unknown quantty for the knd of the structure under consderaton. The effectveness of a MoM numercal soluton depends on a judcous choce of bass functons. The optmal choce of the bass functon s one that provdes solutons wth the fewest number of expanson terms and n shortest computatonal tme. These functons should ncorporate as closely as possble the physcal condtons of the actual dstrbuton of the unknown quantty on the regon of nterest. In ths paper, entre doman polynomal bass functon s utlzed

2 255 whch results n the reducton of computatonal cost and an ncrease n the accuracy of the result. The other aspect of the MoM formulaton s the problem of the sngularty of the functon that s to be ntegrated to obtan the matrx elements, n both the approaches of the MoM formulaton. In the pont matchng MoM approach usng sub-doman bass functons, only a few matrx elements, whereas usng entre doman approach, all the matrx elements are obtaned by ntegraton of sngular functons. Varous analytcal and numercal technques have been adopted to deal wth such ntegrals. The Monte Carlo Integraton (MCI) technque [6,7] proposed n ths paper s not only capable of solvng the scatterng problem but also deals effcently wth the problem of sngularty. To demonstrate the capablty of the above mentoned technque, the problem s formulated n terms of an ntegral equaton to determne the current dstrbuton on square metallc plate. The MCI technque has been proposed to tackle scatterng problem from an nfntesmally thn square metallc plate structures n the MoM formulaton of the problem. The entre doman bass functons are utlzed n the MoM matrx soluton and hence reduce the computatonal cost to the great extent. Besdes, t s also capable of handlng the sngularty aspect of the Green s functon easly and more effcently. The typcal smulaton demonstrates the applcaton of the proposed technque and also valdates the result aganst the Benchmark soluton [8]. 2. Mathematcal Concept Monte Carlo methods are useful for obtanng solutons to problems nvolvng ntegraton whch are too complcated to be solved analytcally or by other numercal methods. Standard numercal ntegraton technques do not work very well on hgh-dmensonal domans, especally when the ntegrand s not smooth. Although the quadrature rules of ntegraton typcally work very well for one-dmensonal ntegrals, problems occur when extendng them to hgher dmensons. Monte Carlo methods have advantages over numercal methods n a space of many dmensons. Ther effcences relatve to other numercal methods ncrease when the dmenson of the problem ncreases e.g. Quadrature formula becomes very complex whle MCI technque remans almost unchanged n more than one dmenson. In addton to ths, the convergence of the MCI s ndependent of dmensonalty regardless of the smoothness of the ntegrand. Monte Carlo ntegraton s smple snce only two basc operatons are requred, namely samplng and pont evaluaton. It s also suted for large structures and hghly complex problems for whch defnte ntegral formulaton s not obvous and standard analytcal technques are neffectve. Samplng can be used even on domans that are not well-suted to numercal quadrature. The dea of Monte Carlo ntegraton s to evaluate the ntegral usng random samplng as I f( x) dx (1) where f s a functon of vector x, s doman of ntegraton. The Monte Carlo ntegraton s popular for complex f and/or. In ts basc form, ths s done by ndependently samplng ponts x 1,,x accordng to some convenent densty functon p, and then computng the estmate f ( x ) F (2) p( x ) 1 where p(x ) s the probablty densty functon or pdf. Here the notaton F s used rather than I to emphasze that the result s approxmate, and that ts propertes depend on how many sample ponts are chosen. If p(x ) s the unform probablty densty, then the ntegral s smply I f( x ) (3) 1 The MCI methods are better suted than quadrature methods for ntegrands wth sngulartes. It s partcularly helpful for ntegrand that have large values on a relatvely small part of the doman due to sngulartes. It can be appled to handle such ntegrands effectvely, even n stuatons where there s no analytc transformaton avalable to remove the sngularty. The smple way to handle the sngularty usng MCI s to gnore a regon around the sngularty and let ths regon become smaller as ncreases [9 13]. In ths approach the total regon s splt nto r r mn, where the numercal ntegraton s perform, and a small regon r r whch s apparently left mn out as long as t has a small or neglgble contrbuton for large values of. The event generators use therefore a cutoff rm n to avod ths regon of sngularty.e., the random ponts generated n MCI are restrcted to fall wthn ths excluded regon. Ths does not requre any extra effort to handle the sngularty problem as the requred condton can be embedded drectly n the MCI technque tself n a sngle statement of the MATLAB code employed for the purpose n smulaton. 3. umercal Example As an example, the electrc feld ntegral equaton s solved by method of moments for the unknown surface current densty on a square metallc plate. The plate s an nfntesmally thn λ x λ square n free space, wth lmts -1.0m to 1.0m along both the x and y axes. The scatterer s excted normally by an ncdent plane wave wth the electrc feld E havng a magntude 1.0 Vm -1 and polarzed along a scatterer edge, n ths case the y- axs. The

3 256 feld as: E s (1 / j 0 0 )[ ( A ) k 2 A ] where jk r r ' ' e dr ' A(r ) 0 J ( r ) r r' 4 conductngsurface Fgure 1. Geometry of the λ x λ square P. E. C. scatterer ncdent upon by a plane wave wth E polarzed along the yaxs, λ = 2m geometry s shown n Fgure 1. From Maxwell s equatons we can obtan vector expresson for the total electrc feld Et E E s (4) s where E s the ncdent feld and E s the scattered t (a) (5) (6) s the vector potental. Applyng the boundary condton n E t 0 on the scatterer, we get the electrc feld ntegral equaton (EFIE) for the unknown current densty J (r ' ) : E E s (1 / j 0 0 )[ ( A ) k 2 A ] (7) For method of moments (MoM) soluton of the ntegral equaton, takng some known bass functon f n (r ' ), the unknown current on the conductng surface can be expanded as: M J (r ' ) a n f n (r ' ) (8) n 1 (b) (c) (d) Fgure 2. (a) Soluton for the Jy-current component (real part) along the X axs of the scatterer obtaned for dfferent number of random ponts generatons; (b) Soluton for the Jy-current component (real part) along the Y axs of the scatterer obtaned for dfferent number of random ponts generatons; (c) Soluton for the Jy-current component (magnary part) along the X axs of the scatterer obtaned for dfferent number of random ponts generatons; (d) Soluton for the Jy-current component (magnary part) along the Y axs of the scatterer obtaned for dfferent number of random ponts generatons

4 257 where a n ; n = 1, 2,..., M are unknown ampltudes of the bass functons and are to be determned. Ths expanson s appled over the same number of feld ponts as the number of expanson terms, whch transforms the ntegral equaton nto a set of smultaneous algebrac equaton n the unknown coeffcents, whch can be wrtten n the matrx form as: or Z mn an En (9) an Z mn 1 En (10) The matrx elements of the matrx equaton are formed usng numercal ntegraton (such as MCI technque) of the sngular functon that results after dfferentaton of the Green s functon. The problem under nvestgaton s an entre doman MoM problem. The entre doman bass functons for the unknown current densty are the modfed polynomal functons, wth edge correcton and symmetry consderatons as stated n the benchmark soluton [8] and presented as follows: nxx y j J x ( x, y ) axj ( x x) 2 1 j 3 1 y (2) (2) (11) nyy x j J y ( x, y ) a yj ( y 1) 2 0 j 3 1 x ( 2) (2) (12) (a) n yx n xy where nxx = nyy = 8 and nxy = nyx = 7. The subscrpt (2) below the summaton sgn means that the ndces and j are to be ncreased wth a step sze 2. Thus the total number of expanson terms for x-current s 12 and for y-current s 16, thus makng the total number of unknown coeffcents = 28. Ths leads to the formaton of the Z-matrx of order 28 X 28, far less than the number of coeffcents requred n sub doman analyss, whch n case of such a large scatterer becomes extremely large for accurate analyss. Though, the Galerkn s method n MoM s a sutable choce for the problem under nvestgaton, the present formulaton employs the pont matchng technque n MoM specfcally to demonstrate the sngularty aspect of ntegrand. The pont matchng technque makes t essental that all the matrx elements that are evaluated, nvolve ntegraton of sngular ntegrands (sngular kernels of the ntegral equaton). Thus t s necessary to adopt means that can take care of the sngularty nsde the ntegral and gve a good approxmaton of the actual result. The technque adopted here s the Monte Carlo Integraton (MCI) technque that overcomes the sngularty, makng ntegraton much smpler and justfed n case of two dmensonal and three dmenson (b) (c) Fgure 3. (a) Soluton for the Jx current component (Real part) over the scatterer; (b) Soluton for the Jx current component (Imagnary part) over the Scatterer; (c) Soluton for the Jx current component (Magntude) over the Scatterer

5 258 al scatterng problems. The method that has dealt wth the sngularty here s named as the local correcton entfc and Industral Research (CSIR), Pusa, ew Delh, Inda technque. Snce MCI s based on random generaton of ponts nsde the entre doman of ntegraton, care must REFERECES taken as to prevent the ponts fall nsde a certan regon [1] R. F. Harrngton, Feld computaton by moment methods, ew York, Macmllan, r r, the regon of sngularty. mn The Fgures 2(a) and 2(b) show the real part of J y as [2] C. A. Balans, Antenna theory: Analyss and desgn, seen along the x-axs whle the Fgures 2(c) and 2(d) Harper & Row, ew York, pp show the real and magnary parts of J y as seen along the [3] C. M. Bulter and D. R. Wlton, Analyss of varous numercal technques appled to thn-wre scatterers, IEEE y-axs, for dfferent numbers of random ponts taken for the MCI. The fgures make evdent two thngs. Frst, the Transactons, Vol. AP 23, o. 4, pp , results for all the values of show a good agreement wth [4] B. M. otaros and B. D. Popovc, General entre-doman the results obtaned n the benchmark soluton [8] usng method for analyss of delectrc scatterers, IEE Proceedngs - Mcrowaves, Antennas and Propagaton, Vol ponts converge to that obtaned for = and 143, o. 6, pp , MoM for the soluton. Second, the soluton wth = = ponts, statng that = ponts are suffcent enough n the MCI for the problem under n vestga- [5] M. Djordjevc and B. M. otaros, Double hgher order method of moments for surface ntegral equaton modelng of metallc and delectrc antennas and scatterers, ton, thereby decreasng the computatonal burden to a great extent. Fgures 3(a), 3(b) and 3(c) show the results IEEE Transactons on Antennas and Propagaton, Vol. 52, for the 3-D current dstrbuton for the real part, magnary o. 8, pp , part and magntude of J x. Total number of random ponts [6] W. H. Press, S. A. Teukolsky, W. T. Vetterlng, and B. P. for the MCI to evaluate the elements n the mpedance Flannery, umercal recpes, second edton, Cambrdge Unversty Press, matrx for MoM soluton has been taken to be 10,000. It s found that the results show good agreement wth those [7] M.. O. Sadku, umercal technques n electromagnetcs, CRC Press, ew York. obtaned n benchmark soluton [8]. 4. Conclusons [8] B. M. Kolundžja, Accurate soluton of square scatterer as benchmark for valdaton of electromagnetc modelng The Monte Carlo ntegraton technque n the MoM soluton of the ntegral equatons has been proposed. The Propagaton, Vol. 46, o. 7, pp , of plate structures, IEEE Transactons on Antennas and technque employs the entre-doman polynomal bass [9] T. Pllards, Quas-Monte Carlo ntegraton over a smplex and the entre space, Ph. D. Thess, Katholeke Un- functons along wth Monte Carlo Integraton technque n MOM soluton of the problem under nvestgaton. In verstet Leuven, Belgum, ISB , the Monte Carlo Integraton technque, a local correcton [10] J. Hartnger, R. F. Kanhofer, and R. F. Tchy, Quastechnque s employed to deal wth the sngularty aspect Monte Carlo algorthms for unbounded, weghted ntegraton problems, Journal of Complexty, Vol. 5, o. 20, of the kernel very effcently and easly. For the demonstraton of the present method, the current dstrbuton s pp , 2004, nvestgated on a two dmensonal square plate scatterer. [11] A. B. Owen, Quas-Monte Carlo for ntegrands wth Where comparsons are avalable, t s found that ths pont sngulartes at unknown locatons, Monte Carlo technque yelds results whch compare very closely to and Quas-Monte Carlo Methods, pp , those of other methods. In addton to the rapd convergence of the Monte Carlo ntegral wth respect to total gral equatons n electromagnetc scatterng usng Monte [12] M. Mshra and. Gupta, Sngularty treatment for nte- number of random ponts, the small number of bass Carlo ntegraton technque, Mcrowave and Optcal functons (< 10) s needed to provde accurate results. Technology Letters, Vol. 50, o. 6, pp , June Further research s beng carred out to extend the method to three dmensonal structures [13] M. Mshra and. Gupta, Monte Carlo ntegraton technque for the analyss of electromagnetc scatterng from 5. Acknowledgements conductng surfaces, Progress In Electromagnetcs Research, Ths research work was supported by the Councl of Sc- PIER 79, pp , 2008.