XXIX CILAMCE November 4 th to 7 th, 2008 Maceió - Brazil

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1 XXIX CILAMCE Nvember 4 th t 7 th, 8 Maceó - Bral ELECTROMAGNETIC SCATTERING PROBLEM SOLVED BY BOTH NODAL AND GALERKIN FEM-BEM APPROACHES M. M. Afns M. O. Schreder T. A. S. Olvera marcmatas@des.cefetmg.br schreder@des.cefetmg.br tarcs@des.cefetmg.br Electrcal Engneerng Department - CEFET-MG. Av. Amanas, 7675 Nva Gamelera, 35. Bel Hrnte, MG - Bral J. A. Vascncels R. C. Mesquta ja@cpdee.ufmg.br renat@ufmg.br Electrcal Engneerng Department - UFMG Av. Antn Carls, BH - MG - Bral Abstract. In ths paper the electrmagnetc prblem s slved by bth ndal and Galerkn hybrds Fnte Element Methd (FEM) Bundary Element Methd (BEM). The hybrdatn prcess prduce a mre pwerful methd because t nhert the mst mprtant advantages f each ndvdual technque whle cast away ts man dsadvantages. Ths s especally true when nn-hmgeneus materals charactere tw r mre regns where ne f them s the free space. The ndvdual bundary element methd s well sutable t hld the free space because t ncrprates the Smerfeld radatn cndtn whle fnte element methd s the best t treat nhmgeneus nes. Ths wrk presents the ndal FEM BEM and the Galerkn FEM BEM frmulatns. The advantages, dsadvantage, and the matrx system characterstcs are dscursed. The effectveness f the hybrds methds are analyed thrugh the cmparsn f numercal and analytcal results btaned by arbtrary tw-dmensnal targets.

2 Keywrds: Electrmagnetc Scatterng, Hybrd Ndal FEM BEM, Hybrd Galerkn FEM BEM.. INTRODUCTION The prblem f electrmagnetc scatterng analyss frm an arbtrarly shaped materal s cnsdered, n ths paper. Ths prblem can be slved usng many technques lke fnte element methd (FEM) wth lcal absrbng bundary cndtns, mult-flament current methd, mment methd, cupled methds, lke mult-flament current methd cupled wth fnte element methd and thers (Gme, Cckrell, Levatan). The fnte element methd s a dfferental numercal technque apprprate t handle clsed arbtrary regns fllng by any materal type (Jn, Yuan). The FEM frmulatn s relatvely smple and rgnates sparse, banded and symmetrc algebrac system equatn (Vlaks). Als, the FEM characterstcs allw mre flexblty and faclty n cmputer cde mplementatn (Arvas, Jn). The Bundary Element Methd s a pwerful ntegral technque. Ths methd s mre apprprated t hld regns fllng by hmgenus, strpc and lnear materals. Furthermre, t autmatcally ncrprates Smmerfeld radatn cndtn thrugh the use f apprprate Green s functn. The dvergence therem s used t change dman ntegral nt bundary ntegral whch allws reductn by ne n the dmensn f the prblem. Ths methd s cnsdered exact hwever t gves rse t full ppulated and nn-symmetrc matrces (Afns, Yuan). In summary, bundary element methd s strngly ndcated t represent pen prblems n free space. The BEM frmulatn culd be perfrmed by bth ndal and Galerkn methds. Ndal frmulatn s easer and better knwn but the Galerkn frmulatn prvdes mre accurate results. The Fnte Element and the Bundary Element Methds can be cnsdered cmplementary t each ther n many cases. It s partcularly true t prblems where the dman cnsdered s frmed by tw (r mre) regns where ne f them s charactered by nn-hmgeneus materal whle the ther s free space. Ths knd f prblem s dffcult t be slved by any ndvdually methd descrbed abve. Therefre, a cmbnatn f tw prevus technques cmes up as a pwerful way t slve t (Afns).. PROBLEM FORMULATION Fgure shws a general tw-dmensnal target whse materal prpertes are unfrm alng ts nfnte axs ( axs). The scatterng prblem ccurs when the bject s llumnated by an electrmagnetc wave. T slve ths prblem by hybrd technque an artfcal surface must be chse. Fr ths case, the better chce s take bject surface as an artfcal surface. It allws dvde scatterng prblems n tw regns. The frst, Ω, s the free space wth permeablty μ

3 and permttvty ε. The secnd regn Ω may cnsst n general f nn-hmgeneus materal wth permeablty μ (x,y) and permttvty ε (x,y). Once the tw regns are knwn, the electrmagnetc felds n each ne can be frmulated. These frmulatns fr bth nterr and exterr felds can be then cupled at the bundary surface thrugh cntnuty cndtns (Afns, Balans, Jn). In ths wrk, the frmulatn fr bth TM and TE plaratn are jwt furnshed, the e tme cnventn s assumed and, the ncdent feld (x,y), gven by u u jk (x cssϕ + ysnϕ ) = e, () Fgure Delectrc cylnder llumnated by a. Fnte element frmulatn TM plane wave (Jn). The Helmhlt dfferental equatn descrbes the feld behavr n regn Ω. In general frm, the Helmhtlt equatn culd be wrtten as (Balans, Jn) ( α ) + k α u. () u = Where, k = w μ ε represent the wave number. Als, fr electrc feld plaratn α = / μr, α = εr and u = E whle fr magnetc feld plaratn α = / εr, α = μr. and u = M. The strng prblem frmulatn fr ths prblem can be n fllwng way. Gven α, α and k, fund u such that (Afns): ( α ) + k α u x Ω (3) u =

4 α u = ψ n c = φ u x Γg x Γ (4) (5) In abve equatns x = (x, y), Γ e and Γ g are exterr and Drchlet bundary, respectvely. Als, Ψ s the nrmal dervatve and φ s Drchlet ptental. The weak frmulatn shwn belw s very easy and fr cnvenence ts dervatn wll be mtted here (Jn). Ω ( α u ) dω kαw u dω u w wα dγ = (6) Ω Γ n In Eq. (6), u s an apprxmatn fr u and w s the weght functn. Applyng Galerkn s prcedure n Eq. (6) and dvdng the dman n a cllectn f trangular elements, t s pssble t determne the general set f equatns that can be wrtten n matrx frm as (Jn), [ K ]{} d [ C]{} k = where [ ] + (7) K s a n C s a n m rectangular matrx, wth m and n representng the ttal number f the ndes n the bundary and the nterr dman, respectvely. Als, {} d s a clumn vectr fr electrc feld arguments and { k } s a clumn vectr fr the nrmal dervatve feld arguments. n square matrx and [ ]. Bundary element frmulatn In free space, Ω, the feld als are frmulated by Helmhlt equatn. The general frmulatn fr bth electrc and magnetc felds are u (r) + k u (r) = f (r) r Ω (8) In Eq. (8), fr electrc plaratn u = E, f (r) = jk ZJ M and fr magnetc plaratn u = H, f (r) = j(k Z )M J. Where, Z s the ntrnsc mpedance f the free space, J represent all the current that flws alng -axs and M s the mpressed magnetc surce. T frmulate the felds n Ω, n ntrduce the free space Green s functn G, whch satsfes the dfferental equatn G ( r,r ) + k G ( r,r ) = δ( r,r) r Ω (9)

5 and the Smmerfeld radatn cndtn (Jn). In Eq. (9), ( r r ) The slutn fr ths equatn s (Jn). ( ) r,r = H k r δ represent Drac delta functn. G r. () 4j The bundary felds are derved frm prduct f Eq. (8) by G, the applcatn f secnd scalar Green s therem and Drac prpertes (Afns, Jn). G (r,r ) n u(r ) n.5 u(r ) = u (r) + u(r ) dγ G (r,r ) dγ Γ Γ () In Eq. (), the prmed sgn ndcate peratn under ntegratn pnt, the factr.5 s the sld angle saw by bserver n the bundary and u ( r ) s the ncdent feld gven by the ntegral u (r ) = Ω s G ( r,r ) f ( r ) dωs () where, Ω s dente the surce regn fr the free space. In ths methd, Eq. () must be wrtten fr all ndes n the bundary. S, f there are m ndes n the bundary s, a m algebrac equatns s btaned. These algebrac equatns culd be arranged n system f equatn (Afns) {} u {} b + [ H]{ d} + [ Q]{} k =, (3) where, [ H ] and [ ] Q are m m ncdent feld clumn vectr. m matrces btaned frm dscretatn f Eq. () and { } b s a.3 Galerkn bundary element frmulatn The Eq. () can als be evaluated alternatvely by applyng Galergn methd. It cnsst n premultplyng Eq. () wth a weghtng functn selected t be the same as thse used fr the expansn f the apprxmate slutn and ntegratng ver the bundary leads t G u uwd Γ = u r wdγ + u dγ wdγ G d wdγ n Γ n.5 ( ) Γ Γ Γ Γ Γ Γ (4) where fr brevty the crdnates dependence have be mtted. Ths equatn usually leads t the mst accurate slutn and als can agan be wrtten n matrx frm as shwn n Eq. (3).

6 3. Cupled FEM BEM methd The system resultng frm FEM BEM frmulatns can be cast n the n fllw frm: K H C d = Q k b (5) Ths equatn system nherts all characterstcs f bth fnte element and bundary element equatn system. Fr that reasn, ths system s partally sparse and symmetrc and partally full and nn symmetrc (Afns, Jn, Vlaks). There are varus frms t wrte FEM BEM equatn system hwever, the frm present here s the easest ne (Jn). The equatn (5) s cmpsed by ( m + n) ( m + n) algebrac equatns and slve bth fnte element and bundary element equatn systems smultaneusly. Althugh Eq. (5) be the best frm t understand the cupled FEM BEM methd t s nt the better way t slve t because the symmetry f FEM subsystem are nt explted (Jn). T slve ths system mre effcently see Jn and Lu. Especal attentn s requred n [ H ] and [ Q ] evaluatng. In ths prcess, there are sngularty when bth ntegratn and bservatn pnt cncdes. Ths subject s well explan by Afns et al n ( Afns). 4. Results The results btaned frm FEM BEM Ndal and Galerkn frmulatns fr TM wave are cmpared. A cmputatnal cde was mplemented based n these methds and frm ths sftware s pssble t btan ttal electrc feld and the scattered felds. T shw the cupled methd effectveness and t valdate the frmulatns, n cmpare the numercal FEM BEM Ndal and Galerkn results wth the analytcal slutn. The errr s cmputed by the abslute value f the dfference f exact electrc feld and apprxmated numercal electrc feld as fllwng: r r = E e E a, (6) where E r e s the exact electrc feld and E r a stand fr apprxmated numercal electrc feld. Althugh the FEM ablty f slve cmplex dman, the gemetry cnsdered here s a smple cylnder wth crcular crss-sectn whse mesh s shwn n Fg.. It cnsst n a delectrc crcular cylnder wth.3λ dameter, ε r = 3 and wave length λ =. Tw dscretatn s analyed. The frst mesh has 7 pnts and 3 secnd rder trangles elements. The secnd ne has 939 pnts and 94 secnd rder trangles elements. The cylnder s llumnated by a TM plane wave wht the ncdent angle ϕ = 8.

7 Fgure Delectrc cylnder trangles dscretatn. The electrc feld dstrbutn ver the cylnder bundary s shwn n Fg. fr the frst mesh descrbed prevusly. Fgure Slutn fr delectrc cylnder fr the frst mesh.

8 Althugh the use f a pr mesh the numercal results are very clse t the analytcal ne. It shws the accuracy f the hybrd numercal technques. Hwever, the FEM-BEM dscretatn va Galerkn methd gves mre accurate results than Ndal frmulatn f the FEM-BEM methd. The abslute errr f ndal and Galerkn frmulatn s shwn n Fg. 3. Frm ths fgure n can see that Galerkn frmulatn results n mre stable and accurate slutn than ndal ne. Fgure 3 Abslute FEM-BEM Ndal and Galerkn errr fr the frst mesh. In Fgs. 4 and 5 the feld and the abslute errr s shwn fr the secnd mesh. In ths case the numercal results s better than the prevus ne because the better dscretatn. And als the Galerkn frmulatn gves better slutn. The Fg. 5 shws a tny and nn scllatry errr fr FEM-BEM wth Galerkn dscretatn.

9 Fgure 4 Slutn fr delectrc cylnder fr the secnd mesh. Fgure 5 Abslute FEM-BEM Ndal and Galerkn errr

10 5. Cnclusns A hybrd methd that cmbnes the Fnte Element and the Bundary Element methds has been develped t treat D scatterng prblems due t delectrc cylnders. Bth ndal and Galerkn frmulatns are used. Frm the results, ne can see that t s pssble t cmpute the electrc feld accurately by hybrd methd. The numercal results agree well wth the analytcal nes. But Galerkn frmulatn prduces better and nn scllatry slutn. 6. Acknwledgements Ths wrk has had fnancal supprt f CEFET MG, CNPq and CAPES COFECUB. REFERENCES Afns, M. M. & Vascncels, J. A.,, An accurate evaluatn f sngular ntegrals arsng n bundary element methd fr D electrmagnetc prblems, Cngress brasler de eletrmagnetsm - CBMAG, vl., pp Afns, M. M., J. A. Vascncels, Tw dmensnal scatterng prblems slved by fnte element bundary element methd In: CILAMCE,, Campnas. Iberan Latn- Amercan Cngress n Cmputatnal Methds n Engneerng.,. Arvas, E. & Sarkar, K. T., 989, RCS f tw dmensnal structures cnsstng f bth delectrcs and cnductrs f arbtrary crss sectn, IEEE Transactns n antennas and prpagatn, vl. 37, n. 5, pp Gme, A. N., AFONSO, M. M., J. A. Vascncels, R. C. Mesquta, C. Vllare, L. Nclas, Hybrd FEM-BEM and FEM-MFCM technques n electrmagnetc scatterng applcatn In: CILAMCE, 3. Jn, J., 993, The fnte element methd n electrmagnetcs, Wley, New Yrk. Balans, A. C., 989, Advanced engneerng electrmagnetcs, Wley, New Yrk. Cckrell, C. R. & Reddy, C. J., 996, Electrmagnetc scatterng analyss f arbtrarly shaped materal cylnder by FEM BEM methd, Nasa techncal paper Levatan, Y. & Bag, A., 987, Analyss f electrmagnetc scatterng frm delectrc cylnders usng a multflament current mdel, IEEE Transactns n antennas and prpagatn, vl. AP 35, n., pp Lu, N. & Jn, J. M., 996, Applcatn f fast multple methd t fnte element bundary ntegral slutn f scatterng prblems, IEEE Transactn n antennas and prpagatn, vl. 44, n. 6, pp Vascncels, J. A. & Lmônac, A. G. & Saldanha, R. R. & Ramíre, J. A.,, FEM ABC and MFCM technques appled t the slutn f D scatterng prblems, IEEE Transactn n Magnetcs, vl. 36, n. 4, pp Vlaks, J. L. & Chatterjee, 995, A selectve revew f the fnte element ABC and the fnte element bundary ntegral methds fr electrmagnetc scatterng, Ann. Télécmmun., vl. 5, n. 5 6, pp Yuan, X. & Lynch, D. R., 99, Cuplng f fnte element and Mment Methds fr electrmagnetc scatterng frm nhmgeneus bjects, IEEE Transactn n antennas and prpagatn, vl. 38, n. 3, pp