Physical Optics Driven Method of Moments Based on Adaptive Grouping Technique

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1 December, 0 Mcrowve evew Phycl Optc Drven Method of Moment Bed on Adptve Groupng Technque Modrg S. Tć, Brno M. Kolundž Abtrct ecently we ntroduced new tertve method for nlyng lrge perfectly conductng ctterer, clled Phycl Optc Drven Method of Moment (PDM). PDM perform groupng of orgnl b functon nd crete mcro b functon ung thee group. In th pper we preent PDM reult ung vrble number of group per terton. Keyword Method of moment, B functon, Phycl optc, Perfectly conductng ctterer, Itertve method. I. ITODUCTIO Surfce ntegrl equton (SIE of electromgnetc feld n frequency domn cn be olved ung method of moment (MoM) []. MoM trnform SIE nto ytem of lner equton, whch unnown re weghtng coeffcent of dopted b functon (BF. MoM oluton expreed fnte ere (lner combnton of BF o, eently, t pproxmte. However, by proper choce of BF, the oluton converge towrd exct oluton when number of BF ncree (complete et of BF,.e. t numerclly exct. The mn drwbc of MoM poor clbty - the number of BF per wvelength qured fxed, hence totl number of BF () rng ft by ncreng frequency. Furthermore, memory occupncy O( ), nd CPU oluton tme O( 3 ). Dfferent trtege for overcomng th problem re propoed: hybrdzton wth ymptotc technque [,3], peedng up mtrx vector product n tertve oluton of MoM ytem of equton [4], compreng MoM mtrx [5], nd, mong ll, ung pecfc BF [6,7]. The de behnd pecfc BF to contruct BF whch cover lrger urfce (thn typcl BF, hvng n mnd prtculr geometry nd exctton of the problem. ecently we propoed method [8] whch, n wy, belong to pecfc BF ctegory. The method tertve nd t converge towrd MoM oluton by employng correctonl current creted n phycl optc (PO) mnner. Tht why the method w clled PO drven MoM (PDM). PDM formulted method for nlyzng perfectly conductng cloed ctterer. Prtculrly, PDM well uted for electrclly lrge problem. PDM trt wth phycl optc (PO) oluton nd then tre to mprove t. To do o, PDM etmte correctonl vlue for MoM unnown (weghtng fctor for BF, then group BF whch mght Modrg S. Tć nd Brno M. Kolunž re wth School of Electrcl Engneerng, Unverty of Belgrde, 0 Belgrde, Serb., E-m: tc@etf.r, ol@etf.r hve the mr level of etmton qulty, crete mcro b functon (MBF ung BF grou nd fnlly determne weghtng coeffcent for MBF n wy to mnmze dfference wth repect to MoM. PDM need le memory nd le CPU tme, pd wth poorer ccurcy thn MoM. However, from engneerng pont of vew, PDM cn provde uffcent ccurcy n CPU tme unrechble to MoM. In [8] we notced tht convergence rte of PDM decree trough the terton. In th pper we wl preent mple modfcton tht cn mprove convergence of PDM to ome extent. II. PO DIVE MOM (PDM) A. Formulton of the Problem A cloed body mde of perfect electrc conductor (PEC), plced n vcuum nd excted by tme hrmonc ncdent electromgnetc feld of frequency f nd electrc nd mgnetc feld vector E nc nd H nc. A reult, electrc current of denty J re nduced over the body urfce, o tht the ncdent feld nde the body nnhted. Once thee current re determned, ll other quntte of nteret cn be ey evluted. Current J cn be determned by olvng electrc feld ntegrl equton (EFIE) [9], whch belong to the lner opertor equton n generl form L f g () where g nown vector functon (exctton), L lner opertor, nd f unnown vector functon to be determned (repone). B. MoM MoM generl method for oluton of lner opertor equton gven by (). Unnown functon f pproxmted by lner combnton of nown vector BF f multpled by unnown coeffcent f f. () Th pproxmton reult n error of oluton, E f f, nd error n tfyng equton (), clled reduum, L f g. Wht MoM doe choong BF tht cn repreent unnown

2 Mrotln rev Decembr 0. functon f nd then to dut coeffcent n order to mnmze error of oluton n ome ene. For tht purpoe reduum forced to be orthogonl to the pce of o clled tet functon w ung nner product w, 0,,...,. Thu we obtn the ytem of lner equton, whch oluton gve u coeffcent,.e. pproxmte oluton (), where z C. Theory of PDM z v,,..., w, Lf nd v w, g. PDM te over MoM geometrcl modellng, BF (f ), element of MoM mtrx (z ), nd rght hnd de term (v ). The dfference tht PDM doe not olve the ytem of equton (3) whch the mot cotly MoM operton but determne n tertve procedure. Suppoe we hve PDM oluton for J n terton -, ( ) f. Snce the oluton pproxmte, totl mgnetc feld ut below the urfce of the (cloed) PEC body, H tot, wl ext (where t hould be zero for exct oluton). ote tht th totl feld um of mgnetc feld from ource outde the body (.e. nown ncdent mgnetc feld H nc ), nd mgnetc feld due to urfce current (equvlent urfce ( ) ource ( ) H f. f, We ntroduce correctonl urfce current - (3) J n Htot r, rs (4) where r poton vector, nd n unt vector norml to the urfce S of the body nd drected outwrd. Eq. (4) tte tht correctonl current t ome pont of the urfce clculted ung mgnetc feld ut below tht pont. Loclly, thee current hould cncel extng mgnetc feld below the urfce (t pt pont). Globlly, the cncelton wl not hppen becue ll other urfce current (other thn tht prtculr t pt pont) wl chnge the feld t pt pont. everthele, current gven by (4) re ueful led n wht drecton the correcton hould go. In order to ue them, we need to expre them lner combnton of b functon, n form gven by () J f. (5) Coeffcent re determned n wy to mnmze reduum gven by J f ds. (6) J S Once we determne thee coeffcent, we cn ue them to bud mcro b functon (MBF F l b l f, l,..., M. (7) Let u loo cloer t (7). In th terton we bud M MBF. At frt glnce t eem tht ech MBF lner combnton () of ll BF f. But, by ettng weghtng coeffcent b l to zero, th BF omtted from lth MBF n th terton. ( ) Generlly, 0 b l, o we cn nclude th BF to one or more MBF. PDM pproxmte oluton n nth terton (n>0) expreed lner combnton of ll extng MBF n M f c F, n 0 (8) 0 l l where c re unnown coeffcent tht hould be determned. By ubttutng expreon for MBF from (7) n (8), nd fter ome rerrngement, (8) cn be wrtten where A n f A f (9) n M 0 l c b l. (0) PDM oluton gven by (9) h the me form MoM oluton gven by (). However, coeffcent A generlly won t tfy ytem of equton gven by (3), nd nted wl generte reduum for ech equton n n z A v,,...,. () The men qure reduum fter nth terton clculted wrtten n. After ome rerrngement, t cn be n M 0 l c Z l v () 3

3 December, 0 Mcrowve evew where Z l z bl. (3) By mpong condton tht reduum () hould be mnmzed (wth proper choce of coeffcent c ) we obtn (PDM) ytem of equton n M n c * Z l Z * m v Z m 0 l (4) 0,..., n, m,..., M from whch we determne coeffcent c,.e. pproxmte PDM oluton n nth terton, gven by (7). In ytem of equton (3), =0 refer to nt oluton (0th terton). A nt oluton we ue phycl optc (PO) current, obtned PO 0, r hdow regon of S J r. (5) n Hncr, r lt regon of S We tret thee PO current mr to correctonl current (4) we expre them n form () nd then ue coeffcent PO 0 J f (6) 0 to crete MBF n 0th terton. Though thee MBF cn be ued to crete oluton of the form (8), we ue (6) nt PDM oluton. The ey pont of ung PDM to vod oluton of MoM ytem of equton (3) for lrge, t become hghly neffcent. Inted, n ech terton we olve PDM ytem of equton (4). The order of PDM ytem (.e. the number of MBF (for lrge ) much lower thn, enblng greter effcency thn MoM. D. elzton of PDM For the MoM prt of PDM we ue WIPL-D ernel [0]. EFIE trnformed nto the ytem of lner equton ung Glern tetng (BF re ued tetng functon nd the ytem olved ung LU decompoton. Fg.. Setch of bner urfce Geometrcl modelng performed ung curved qudrterl (plte, hown n Fg., decrbed by prmetrc equton r( 4 r [ ( ) p][ ( ) ]. (7) The current dtrbuted over plte re decompoed nto two component n locl po coordnte ytem. In prtculr, the -component expnded n p n ( P( 0 0 J (8) ( / 0 P ( p ( / (9) p where n p nd n re order of pproxmton long p nd coordnte, re unnown coeffcent, P (, = 0,, re edge BF, P (, >, re ptch BF, r( p, nd r(. The p-component expnon obtned by nterchngng the coordnte p nd n expreon (8) nd (8). Edge BF re common for two plte nd re clled doublet, where ptch BF re defned on ngle plte nd re clled nglet. Correctonl current (4) re clculted n dcreet pont on n n. the urfce of the body uffcent number Condton (6) mpoed for ech ptch eprtely. Hence doublet wl be clculted twce nd men vlue wl be dopted correpondng coeffcent n (5). Mgnetc feld hould be clculted n the body, ut below thee pont. In ech clculton pont th feld cn be decompoed nto the prt due to current n the pont ut bove tht pont nd the prt due to ll other current. Snce we cn clculte the frt prt ung boundry condton, nd the econd prt prctclly the me on the urfce nd ut p p 4

4 Mrotln rev Decembr 0. below the urfce, we cn ue pont on the urfce for mgnetc feld clculton lo. In cretng MBF ccordng to (7) ntly (n 0th terton) we ue ndrect pproch. Frt we crete group of phyclly connected plte (ee [8] for det. For ech group of plte we crete one group of BF, ncludng ll BF defned over the plte of the group. ow we cn crete MBF ung (0) group of BF. Coeffcent b l wl be f BF f belong only to lth group of BF, wl be / f BF f belong to the lth group nd one other group of BF, nd wl be 0 otherwe. In ubequent terton uch cheme (groupng) cn be preerved or cn be chnged. ote tht, n 0th terton, MBF creted ung group of BF whch re completely n the hdow re (where PO current re zero) re zero, o we omt them. III. UMEICAL ESULTS Conder PEC rplne, bout 40λ long, plced long z-x n Crten coordnte ytem, wth wng pnned n xoz plne, hown n Fg.. The rplne excted by crculrly polrzed plne wve ncomng from drecton gven by ngle 45 nd 45 (θ meured from xoy plne to z-x. The rplne modeled by 7445 plte. The nd order pproxmton ued for lmot ll plte, o tht totl number of MoM b functon = 590. We ue M = 643 group of plte (386 re n lt zone), hown n Fg. b, for creton of nt MBF. BF,.., % 00, Then we clculte reduum for ech group of BF. (0) G l bl BF, l,.., M, () nd fnlly we clculte cumultve reduum CUM n 00 l Gl, 0,.., M. () When clcultng cumultve reduum ung () we uppoe tht Gl Gl,.e. we orted redu of group of BF n non-growng order. Hence, cumultve reduum y whch percent of totl reduum wl preerve fter we remove BF group wth hghet redu. Cumultve reduum for number of terton n from to 7 hown n Fg. 3. We cn ee tht e.g. fter even terton bout 00 group of BF re crryng bout 60% of totl reduum. Fg. 3. ormlzed cumultve reduum for dfferent number of terton Cn we mprove convergence f we plt thee bd group? In order to exmne th, we wl modfy groupng cheme lttle bt. After ech terton n (n>0) we wl clculte cumultve reduum () for extng group of BF. Then we wl fnd mnml c uch tht Fg.. ) PEC rplne, excted by crculrly polrzed wve, modeled wth 7445 plte, nd b) 643 group of plte Frt we performed PDM nly eepng nt groupng cheme throughout the PDM nly. It men tht we hve 643 fxed group of BF (ddtonl 643 MBF n ech terton). In ech terton we clculte reduum () for ech BF, nd then we qure nd normlze t module CUM n cutoff c () CUM cutoff where CUM cumultve reduum cut off vlue for grou whch we wl dopt. Then we wl plt ech group of BF from to c n two (f c = 0 there no plttng). Doe t me dfference how we wl plt the group of BF? To chec th, we wl clculte normlzed cumultve reduum for ech group nd et cut off vlue for BF n the 5

5 December, 0 me wy for group. Then we wl plt BF from ngle group n two group ung cut of vlue threhold. Preented technque for plttng group wl be referred to dptve groupng technque (AGT), nd the method PDM bed AGT. Integrl meure of PDM convergence eduum norm v. (3) eduum, defned n (3), cn te vlue between 0 (for MoM oluton) nd (zero oluton). We expect vlue bout 0.0 for uffcently ccurte oluton, nd bout 0.00 for excellent greement wth MoM reult. The rte of eduum decree the peed of PDM convergence. Mcrowve evew decreng cut off vlue for grou number of group tht wl be plt ncree nd, hopefully, convergence too. The plttng of the group w performed ccordng to cumultve reduum cut off vlue for BF. In Fg. 4 th vlue 30%, where n Fg. 4b t 70%. Obvouly reult re very mr, nd the me refer to vlue n between, whch re not preented here. So t eem tht 50% good prctcl choce for BF cut off. ow, loong t ny of the Fg. 4-b, we ee tht eduum of 0.00 reched fter 5 terton wth 0% group cut off vlue, but n 7 terton wthout plttng (00% group cut off). Snce CPU tme per terton mr for both procedure (number of MBF become gnfcnt only for lrge number of terton, procedure wth 0% group cut off wl rech the me eduum n horter CPU tme. Fg. 5 how how reduum chnge functon of number of MBF (.e. effcency of the method). ) ) b) Fg. 4. PDM-AGT ppled to rplne, eduum v Iterton, for cut off vlue for BF of () 30%, nd (b) 70% Fg. 4 how convergence trough PDM terton, for dfferent cumultve reduum cut off vlue for group (00%, 70%, 50%, 30% nd 0%). Vlue 00% men tht there no plttng t ll, vlue 70% men tht group of BF (wth hghet reduum) mng 30% of totl reduum wl be plt (ech n two new grou nd o on. By b) Fg. 5. PDM-AGT ppled to rplne, eduum v MBF for BF cut off vlue of () 30%, nd (b) 70% Obvouly, PDM-AGT for group cut off 0% the let effcent (gven eduum reched wth the lrget number of MBF, but ll other vlue gve mr reult. It men tht we cn obtn the me eduum wth mr number of MBF. 6

6 Mrotln rev Decembr 0. Fg. 6. PEC helcopter wth 7445 plte nd h 40 group Second exmple PEC helcopter, bout 65λ long, plced long x-x, hown n Fg. 6. The helcopter excted by crculrly polrzed plne wve ncomng from drecton gven by ngle 35 nd 45. The helcopter modelled by 377 plte. The nd order pproxmton ued for lmot ll plte, o tht totl number of BF = We ue M = 40 group of plte (34 re n lt zone). eduum trough the terton of PDM-AGT hown n Fg. 7. BF cut off 50%. We ee tht group cut off 50% curve drop lmot lnerly, nd tht reche eduum of 0.0 n 5 terton, more thn terton erler thn 00% curve (no plttng). eduum veru number of MBF hown n Fg. 8 (terton re mred for ech curve) lghtly better for 00% curve, but nothng drmtc. CS (dr Cro Secton) obtned by PDM-AGT oluton (group cut off 50%) n 5th terton compred to MoM reult n Fg. 9. very good greement. Fg. 8. PDM-AGT nly of helcopter, eduum v MBF Fg. 9. CS for helcopter MoM v PDM-AGT (cut off 50%) IV. COCLUSIO In th pper we ntroduced mple technque for mprovng convergence rte of PDM method (PDM-AGT). In ech terton group of BF wth hghet reduum (error of oluton) were dvded (ech n two new grou thu llowng better correcton n the followng terton. Snce there no certn rule how to chooe nt number of group of BF for ech prtculr model, th technque enble dptve correcton once the PDM trted. umercl reult how tht the technque ndeed mprove peed of convergence. ACKOWLEDGEMET Fg. 7. PDM-AGT nly of helcopter, eduum v Iterton Th wor w upported by the Serbn Mntry of Scence nd Technologcl Development under Grnt T

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