Method Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems

Transcription

1 Internatonal Workhop on MehFree Method Method Of Fundamental Soluton For Modelng lectromagnetc Wave Scatterng Problem Der-Lang Young (1) and Jhh-We Ruan (1) Abtract: In th paper we attempt to contruct the electromagnetc wave catterng feld by a gven ncdent wave. For two-dmenonal problem the normal ncdent plane wave catterng by a conductng crcular cylnder wth nfnte dmenon n the -drecton are dcued. For three-dmenonal problem we focu on the electromagnetc catterng wave by a conductng phere. The method of fundamental oluton (MFS) for the vector Helmholt equaton n the frequency doman are employed to mulate the electromagnetc wave problem. Both the D and 3D homogeneou electromagnetc wave catterng are compared wth the analytcal a well a other numercal method, uch a the fnte element method (FM) or the boundary element method (BM). The MFS ha demontrated to render very effcent and accurate reult a comparng wth the analytcal and other numercal oluton. 1. Introducton The catterng of electromagnetc wave by object an mportant problem both n academc reearche a well a ndutral applcaton. Hao et al. [1] appled the boundary element method to mulate three dmenonal electromagnetc wave catterng equaton. Morgan et al. [] [3] employed the fnte element tme doman method to olve the three dmenonal catterng problem. Ledger [4] ued the edge fnte element method to mulate the two-dmenonal homogenou and delectrc catterng phenomena. And n the mean tme the analytcal oluton of catterng problem can be found n other reearch area [5]. Dfferent numercal method have been ued to nvetgate the two and three dmenonal catterng problem of electromagnetc wave n our reearch group. For example, Chen [6] ued the conventonal BM to mulate D and 3D catterng wave problem. Chu [7] ued the non-ngular BM to tudy the catterng of electromagnetc wave over a conductng crcular cylnder and a conductng phere. In th tudy, we wll try to ue the method of fundamental oluton (MFS) a a mehle numercal method to olve the homogenou vector Helmholt equaton n the frequency doman, whch repreent the governng equaton of the electromagnetc wave catterng of a perfect electrc conductor. The preent method wll be ued to compare wth exact oluton and other numercal reult to ae the accuracy and effcency of the mulaton. (1) Department of Cvl ngneerng & Hydrotech Reearch Inttute, Natonal Tawan Unverty, Tape, TAIWAN (Correpondng to DL YOUNG, Tel/Fax: , mal:dlyoung@hy.ntu.edu.tw)

2 Der-Lang Young, Jhh-We Ruan. Mathematcal formulaton.1 Governng equaton Maxwell equaton whch governng the propagaton of electromagnetc wave are reduced to the homogenou vector Helmholt equaton n the frequency doman by aumng that the electrc and magnetc feld are both tme-harmonc. The homogenou vector Helmholt equaton are condered n the followng form. 0 + k 0 = 0 (1) H0+ k H0 = 0 () where 0 and H 0 denote the tme-harmonc electrc and magnetc feld ntenty vector n thee equaton repectvely, and k repreent the wave number, determned by the followng formula k = ω µε (3) where µ and ε denote the permeablty and permttvty repectvely, whlt ω the angular frequency of the materal. The total electrc and magnetc feld, regarded a beng combned wth ncdent and cattered component, were repreented n thee form. t = + (4) t H = H + H (5) The upercrpt and ndcate the ncdent feld and the cattered feld repectvely. Due to the lnearty, cattered feld and H wll alo atfy quaton (1) and quaton (). In other word, the governng equaton of cattered feld could be wrtten a 0 + k 0 = 0 (6) H + k H = 0 (7) 0 0. Boundary condton The regon outde the perfect electrc conductor the man concerned doman. Applyng the ntegral form of Maxwell equaton to a mall regon, we obtan the tangental component of total electrc feld and the normal component of the total magnetc feld. Both vanh on the urface of a perfect electrcal conductor,.e. n = n (8) n H = n H (9) where n denote the unt normal vector to the conductor urface.

3 Internatonal Workhop on MehFree Method Numercal algorthm 3.1 Method of fundamental oluton(mfs) Snce quaton (6) and (7) are the homogenou vector Helmholt equaton, we may aume that the homogeneou oluton a lnear combnaton of the fundamental oluton of the Helmholt operator [8],.e. M Φ (x ) = β g( r ) (10) h j j = 1 where 1 1 Y0( kr) J0( kr) for two dmenon 4 4 gr () = (11) co( kr) n( kr) for three dmenon 4πr 4πr are the fundamental oluton of the Helmholt operator, rj = ξ x the dtance j between a feld pont x j and the ource pont ξ, and M the number of ource node. Where the dervatve of the homogeneou oluton obtaned through M Φ h(x j) g( rj) = β (1) xk = 1 xk After β have been olved through the method of the collocaton of the boundary and ource pont, we can fnd the homogeneou oluton through the equaton (10). 3. Computaton of the radar cro ecton(rcs) The radar cro ecton(rcs), obtaned by employng the tranformaton of a near feld to the far feld, defned a [9] where RCS = 10log ( σ ) σ σ TM 10 = lm πr n two dmenon (13) r H 3D = lm 4πr lm 4πr r r = n three dmenon (14) H 4. Numercal xample 4.1 Scatterng by a conductng crcular cylnder A perfect electrc crcular cylndrcal conductor, one of the mot wdely ued to repreent practcal catterer, ha been computed to demontrate the method of fundamental oluton of the propoed procedure. The frt example we aume that a tranvere magnetc ( TM ) plane wave normally ncdent upon the perfect

4 4 Der-Lang Young, Jhh-We Ruan conductng crcular cylnder wth 1 unt radu. The ncdent wave ha a wavelength of π unt and propagate n drecton. To acheve th catterng problem, we make the pace cont of 40 node and take permttvty and permeablty equal to 1. Fg.1 how the contour of the real part of cattered electrc feld,. To compare the exact oluton [5] and the numercal mulaton, Fg. how the dtrbuton of exact and computed magnary part of along radu equal to unt. And the radar cro ecton (RCS) of the catterer alo hown n Fg.3. Smlar to th procedure, we can alo get the catterng magnetc feld depcted by T wave through equaton (1) Computed xact 0.40 catterng component Fg.1 The contour dtrbuton of Re [.] Degree Fg. Comparon of Im[ ] between the exact oluton (old lne) and the computed oluton (trangle) at the radu unt. 4. Scatterng by a conductng phere A perfect electrc phercal conductor a very clac problem for the catterng of electromagnetc wave. We aume that a tranvere electrc feld of a unform plane wave polared n the x drecton travelng along the ax, ncdent on a conductng phere wth radu 0.5 unt. After the tranformaton [10] the catterng Intenty of electrc feld can be expreed a x = nθ coφr + coθ coφθ nφφ (15) where r, θ and φ can be obtaned ung the equaton (8) and equaton (9), therefore we can compute the catterng electrc feld ntenty by ettng permttvty and permeablty contant equal to 1. Fg.4 how the contour of the real part of the catterng electrc component n drecton on x=0 plane and urface of the phere. Fnally to compare the accuracy and effcency of the propoed MFS, the radar cro ecton (RCS) of exact oluton [5] and other numercal oluton ncludng of the MFS we propoed, the conventonal BM of Chen [6], the non-ngular BM of Chu [7] and the FM of Morgan [3] are all llutrated n Fg.5. It worthy whle to oberve that all the oluton gve the ame accurate reult a compared wth the exact oluton. However, only 800 nodal pont are ued n the MFS, 300 nodal pont are employed n both the conventonal BM and non-ngular BM, whle 589,505 element,706,999 edge and 99,991 node are ued n the FM formulaton.

5 Internatonal Workhop on MehFree Method Computed xact 1.00 RCS degree Fg.3. Comparon of the RCS between computed oluton (trangle) and exact oluton (old lne) Fg.4. The contour of the Re[ ] on x=0 plane and urface of the phere XACT BM NSBM FM MFS RCS degree Fg.5. Comparon of the RCS between MFS(quare), BM(cro), NSBM(crcle), FM(trangle) and exact oluton(old lne) at φ = 0 5. Concluon The nnovatve numercal cheme of applcaton of the method of fundamental oluton (MFS) to olve the electromagnetc catterng problem ha revealed that the MFS better off than the tradtonal numercal methodologe Both a perfect electrc D cylndrcal conductor and a 3D perfect electrc phercal conductor are performed n th tudy. The MFS ha already atfed the governng equaton and only the boundary node need to be collocated o that one dmenon can be reduced and alo the mehfree mert acheved. We have demontrated that for the mulaton of an electromagnetc wave catterng of a 3D conductng phere, the uage of 800 boundary node wll render the ame reoluton a thoe obtaned by the conventonal BM, the no-ngular BM a well a the FM cheme. All the numercal mulaton poe the ame accuracy a comparng to the exact oluton. However the MFS ha hown the mple,

6 6 Der-Lang Young, Jhh-We Ruan effcent, and powerful apect a far a the numercal algorthm concerned. The numercal mulaton of the catterng of electromagnetc wave by the method of fundamental oluton (MFS) ha opened a new fronter for the computatonal electromagnetc wave reearch. Acknowledgment Th tudy upported by the Natonal Scence Councl, Tawan, t greatly apprecated. Reference [1] G. C. Hao, R.. Klenman, D.-Q. Wang, Applcaton of boundary ntegral equaton method n 3D electromagnetc catterng, Journal of Computatonal and Appled Mathematc Vol. 104, pp89-110, 1999 [] K. Morgan, O. Haan, J. Perare, A tme doman untructured grd approach to the mulaton of electromagnetc catterng n pecewe homogeneou meda, Computer Method n Appled Mechanc and ngneerng, Vol. 134, pp17-36, [3] K. Morgan, P. J. Brooke, O. Haan, N. P. Weatherll, Parallel proceng for the mulaton of problem nvolvng catterng of electromagnetc wave, Computer Method n Appled Mechanc and ngneerng, Vol. 15, pp , [4] P. D. Ledger, dge lement for Wavegude and Scatterng Problem, MS the, Unverty of Wale, Swanea, [5] C. A. Balan, Advanced ngneerng lectromagnetc, John Wley & Son, Inc., [6] G. Q. Chen, Applcaton of Helmholt equaton by boundary element method to the wavegude and catterng propagaton problem, MS the, Natonal Tawan Unverty, Tawan, 00. [7] C. L. Chu, Non-ngular Boundary Integral quaton for the Analy of lectromagnetc Problem, MS the, Natonal Tawan Unverty, Tawan, 00. [8] C. C. Ta, Mehle Numercal Method and ther ngneerng Applcaton, Ph.D. The. Natonal Tawan Unverty, Tawan, 00. [9] C. A. Balan, Antenna Theory, John Wley & Son, Inc., 198. [10] R. F. Harrngton, Tme-Harmonc electromagnetc Feld, New York, NY: McGraw-Hll Book Company, Inc., 1961.