Electromagnetic Analysis of Arbitrary Antennas on Large Composite Platforms Using PO Driven MoM

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1 December, 2013 Microwve Review Electromgnetic Anlysis of Arbitrry Antenns on Lrge Composite Pltforms Using PO Driven MoM Miodrg S. Tsic 1, Brno M. Kolundzi 2 Abstrct Electromgnetic nlysis of electriclly lrge problems using numericlly exct method of moments (MoM) is enormous burden for personl computer. So clled fst techniques cut down CPU time, but storge problem remins. Here we present novel itertive method which converges towrd MoM solution, but does not del with lrge mtrices. The method is especly suitble for isolted ntenns on lrge obects. Keywords Electromgnetic nlysis, Method of moments, Antenns, Physicl optics. I. ITRODUCTIO Surfce integrl equtions (SIEs) of electromgnetic field in frequency domin cn be solved using method of moments (MoM) [1]. MoM trnsforms SIEs into system of liner equtions, which unnowns re weighting coefficients of dopted bsis functions (BFs). MoM solution is expressed s finite series (liner combintion of BFs) so, essently, it is pproximte. However, by proper choice of BFs, the solution converges towrd exct solution when number of BFs increses, i.e. it is numericlly exct. The min drwbc of MoM is poor sclbity - the number of BFs per wvelength squred is fixed, hence totl number of BFs () rising fst by incresing frequency. Furthermore, memory occupncy is O( 2 ), nd CPU solution time is O( 3 ). So, there is cler limit on electricl size of problems tht cn be solved using certin computer. There re number of techniques for ccelerting MoM: hybridiztion with symptotic methods [2,3], fst multipole method [4,5], compression of MoM mtrix [6]. For ll of them, t some point, the entire MoM mtrix should be vble. So, whe these methods cut down CPU time, the storge remins the bottlenec. Another pproch is use of chrcteristic bsis functions, suitble for the problem prticulr geometry [7,8]. In [9] we proposed technique which hs some simrities with such pproch. It is itertive method which converges towrd MoM solution by employing correctionl currents creted in physicl optics (PO) mnner. (Tht is why the method ws clled PO driven MoM PDM). These currents re then used to crete mcro bsis functions, pproprite for tht prticulr structure. PDM does not need to store entire MoM mtrix t once, or to del with it s whole. 1 Miodrg S. Tsic is with the School of Electricl Engineering, University of Belgrde, Bulevr rl Alesndr 73, Belgrde, Serbi, E-m: tsic@etf.rs 2 Brno M. Kolundzi is with the School of Electricl Engineering, University of Belgrde, Bulevr rl Alesndr 73, Belgrde, Serbi, E-m: ol@etf.rs This wy, PDM significntly decreses memory occupncy nd CPU time, whe preserving sufficient ccurcy. In [9], PDM is formulted s method for nlyzing perfectly conducting (PEC) closed sctterers (nd prticulrly well suited for electriclly lrge problems). In [10,11] we introduced some modifictions in the PDM method, considering dptive grouping nd nlysing composite sctteres, mde of metl nd dielectric. In [12] we presented some results for PDM pplied to ntenn problems, but without dets of the theory of opertion. Complete theory of PDM pplied to ntenn problems ws first described in [13]. Here we extend tht wor. The essence of the method is the sme s for the sctterers we dd new mcro bsis functions (MBFs) in ech itertion nd dust their weighting coefficients to minimize the difference between PDM nd MoM solution. However, only perfectly conducting prts tht belong to closed surfces cn be treted in modified PO wy [9], wheres MBFs in the rest of the structure re obtined using new pproch, discussed here. Theory of PDM pplied to ntenn problems is given in section II, numericl exmples in section III, nd conclusion in section IV. II. PDM APPLIED TO ATEA PROBLEMS We strt from n existing MoM solution of surfce integrl equtions. Unnown vector function f is pproximted by liner combintion ( f ) of nown vector BFs ( f ) multiplied by unnown complex sclr coefficients ( ) Coefficients equtions f 1 f. (1) re obtined from the MoM system of 1 z v, 1,..., where z nd v re nown complex sclrs. In PDM, coefficients re obtined in itertive mnner. In ith itertion we determine one set of vlues, (for ll coefficients, or some of them), nd, using them, bud i M MBFs, i F l (2) 21

2 Mirotlsn revi Decembr l 1 i i F b f, l 1,..., M. (3) l l ( i ) Coefficient b ( 0 b l 1) controls membership of th BF to lth MBF in ith itertion. Init solution is referenced s 0th itertion. Coefficients wl be clled init coefficients for i 0 nd correctionl coefficients for i 0. PDM solution in nth itertion is dopted s n M i f c F, n 0, (4) i0 l1 l where c re unnown coefficients tht should be determined. By substituting expression for F i l from (3) to (4), fter some rerrngements, (4) cn be expressed s n n f A f. (5) 1 Expression (5) is formlly the sme s (1), except tht coefficients A do not stisfy MoM system (2). Insted, for ech of equtions in (2) we cn define residul error n R v z 1 A nd men vlue for ll residul errors - Residuum, (6) i ( ) 1 n M n n R R v c Z l, (7) 1 where we introduced uxiry mtrix i Z l z 1 l b i0 l1. (8) Coefficients c re determined in wy to minimize Residuum (7). By imposing condition n R 0, c m 0,.., n we obtin PDM system of equtions m 1,.., M i, (9) i n M n i c * Z l Z * m v Z m 0 l (10) i 0,..., n, m 1,..., M i As solution of the system (10) we obtin coefficients c. ote tht, for efficient clcultion of elements of PDM mtrix (in (10) given by expression in the brcet), we do not need to now entire MoM mtrix t once, but only single row t once. Hence, if the MoM mtrix is lrge, insted of clculting it one time nd store it, we cn clculte it in ech itertion, one row t time, thus decresing memory demnds from O( 2 ) to O(). In ll prcticl situtions order of the PDM mtrix is by few orders smller thn order of the MoM mtrix. BFs re seprted in two res. BFs defined over PEC surfces tht re prt of closed surfce cn belong to PO re. All other BFs must be in MoM re. These two res differ in wy tht MBFs re creted. For simplicity, we wl ssume tht single MoM re exists nd tht first MoM BFs belong to it, wheres the rest of BFs belong to the PO re. In PO re we determine correctionl currents [2] in ith itertion ust below the surfce S of the structure (i.e. t S - ) i i-1 J s 2n H tot r, r S, (11) nd pproximte these currents using BFs from PO re [2], i.e. we obtin correctionl coefficients i i J f. (12) s MoM 1 Correctionl currents determined in this wy loclly cncel mgnetic field tht exists within PEC prt of the structure (wheres it should be zero). This pproch hs no mening outside closed PEC region, i.e. in the MoM re. By forml extrction of MoM re from MoM system of equtions we obtin prt system of equtions. Correctionl coefficients in MoM re re obtined using residul errors from previous itertion s excittion for prt system of equtions, i.e. MoM i i 1 z R, 1,.., MoM. (13) 1 Init coefficients of equtions MoM 1 (0) re obtined by solving prt system 0 z v, 1,.., MoM. (14) If the cse of multiple MoM res, we solve prt systems of equtions (13) nd (14) for ech of them. Such sitution cn occur if we hve multiple ntenns mounted on the pltform, fr from ech other, or we hve dielectric prts on otherwise metl pltform (composite pltform). One MoM zone tht includes ll these prts cn be too lrge nd thus ineffective, so it is the best to hve one MoM re for ech isolted prt tht cnnot be in the PO Are. 22

3 December, 2013 III. UMERICAL EXAMPLES The first exmple is microstrip ptch ntenn (MPA) mounted on perfectly conducting, 59 wvelengths long, helicopter. The helicopter s nose is directed long x-xis of Descrtes coordinte system, wheres z-xis is orthogonl to helicopter s floor nd directed towrd its propeller. The structure is modeled using surfce biner ptches. MoM settings from [14,15] re dopted, i.e. we use polynom higher order vector BFs. For this model, second order BFs re predominntly used, resulting in totlly BFs. BFs cn be defined over single ptch or over more ptches. MoM re includes MPA nd certin number of neighboring ptches. Ptches in PO re re subdivided into M groups. The grouping is performed uniformly throughout volume of the structure, with M proportionl to the root of number of BFs (s rule of thumb). In ech itertion, except the first, we crete one MBF using BFs tht belong entirely to the MoM re. In ech itertion, except the zeroth, for ech group of ptches, we crete MBF using BFs tht belong to tht group (in (3) we use b l 1 if BF entirely belongs to the ( group, b i ) L if BF belongs to L different groups, nd l 1 b l 0 otherwise). Prticulrly, ech BF tht belongs to both MoM re nd PO re ( bridge BF) is introduced s single MBF in the first itertion (note tht this is only ppernce for these BFs). Microwve Review As mesure of convergence we use normlized Residuum R norm R 1 1 v 2, (15) which is 0 for MoM solution nd is 1 if ll coefficients in (1) re zero. As we see in Fig. 2, Residuum decreses with itertions for ll models. Convergence, though, is not uniform it is the best t the beginning of the itertive procedure. Fig. 2. Residuum s function of umber of itertions Chnging size of MoM re to 2 nd to 8, whe eeping number of groups close to 698 (711 groups for 2 model, nd 666 groups for 8 model) ust slightly shifts this curve up or down. This is shown in Fig. 3. Fig. 1. Model of MPA on the helicopter, with ptches, BFs, 4 MoM re, nd 698 groups of ptches in PO re The model of the helicopter with MPA mounted on its floor, shown in Fig. 1 (enlrged MPA is shown on the inset picture), hs MoM re (gry ptches) ccommodted in sphere (centered t MPA) which rdius is four wvelength ( 4 ), nd hs 698 groups of ptches (neighboring groups re differently colored) in PO re. Groups re distributed uniformly throughout the volume of the model. The groups do not contin shrp edges. We lso creted models with 342, 1345, nd 2694 groups (ll of them loo simr, but with different size of the groups). Fig. 3. Residuum s function of size of MoM re We cn lso see tht models with higher number of groups hve lower Residuum fter the sme number of itertions. However, if we loo Residuum s function of number of MBFs in Fig. 4, we see tht efficiency (number of MBFs for certin Residuum) of different models depends on the wnted Residuum. Residuum s function of CPU time is shown in Fig. 5. ote tht CPU time for full MoM nlysis on the sme 23

4 Mirotlsn revi destop PC (processor Intel Qud Core 2,67GHz, RAM 4 GB, grphics nvidi GTX 560) is bout 5,5 hours. Since we clculte MoM mtrix in ech itertion, rw by rw, the memory occupncy depends solely of number of MBFs. Looing t Fig. 4, we cn see tht we wor with number of MBFs tht is t lest order of mgnitude lower thn number of MoM BFs. Although we describe convergence by Residuum, it is n integrl mesure, wheres one is mostly interested in opertive vlues of his ntenn, such s gin nd rdition pttern. Decembr nd the side lobes hve lrger devition from MoM solution, but the levels remin prcticlly the sme. The simr thing, only in opposite direction, hppens by expnsion of MoM re to 8, s shown in Fig. 11. Fig. 6. Gin, Mom vs PDM solution of 4 model, itertion 0 Fig. 4. Residuum s function of umber of MBFs Fig. 7. Gin, Mom vs PDM solution of 4 model, itertion 1 Fig. 5. Residuum s function of CPU time In Figs. 6-9 re shown gins obtined using PDM solution of model in Fig. 1 fter 0, 1, 2 nd 7 itertions, compred to gins obtined using MoM solution. The gins re clculted in plne 0 ( is clculted from +x-xis, nd is clculted from xy-plne). Itertion 0 is init solution essently, it is MoM solution for extrcted MoM re. Itertion 1 includes both bridge BFs nd PO re BFs, wheres strting from itertion 2 we constntly dd one MBF from MoM re nd M MBFs from PO re. We see tht the first itertion significntly improves the min lobe, wheres other itertions improve the side lobes. After 7 itertions (Residuum bout 0.02) we hve very good greement with MoM solution. The effect of reducing the MoM re to 2 on init solution is shown in Fig. 10 both the min lobe 24 Fig. 8. Gin, Mom vs PDM solution of 4 model, itertion 2

5 December, 2013 Microwve Review wheres z-xis is orthogonl to irplne s floor nd directed from it. The irplne is modeled using surfce biner ptches. Second order BFs re predominntly used, resulting in totlly BFs. The model of the irplne with dipole ntenn, shown in Fig. 12 (enlrged dipole is shown on the inset picture), hs MoM re (gry ptches) ccommodted in sphere (centered t the dipole) which rdius is four wvelength ( 4 ), nd hs 316 groups of ptches (colored) in PO re. We lso creted models with 665, 1320, nd 2605 groups (ll of them loo simr, but with different size of the groups). Fig. 9. Gin, Mom vs PDM solution of 4 model, itertion 7 Fig. 12. Model of the dipole on the irplne, with ptches, BFs, 4 MoM re, nd 316 groups of ptches in PO re Fig. 10. Gin, Mom vs PDM solution of 2 model, itertion 0 Residu for these models, s function of number of itertions, re shown in Fig. 13. As we cn see, these residu re from two to three orders of mgnitude smller (btter) thn those for helicopter models. In order to determine reson for such discrepncy, let s first loo t the residu for three models with simr number of groups (round 316) nd different size of MoM re defined by spheres of 2 (319 groups), 4 (model in Fig. 12) nd 8 (306 groups), shown in Fig. 14. We cn see tht init solution (itertion 0) for these models differs significntly 0,034 for 2 model, 0,0075 for 4 model, nd 0,0017 for 8 model. Fig. 11. Gin, Mom vs PDM solution of 8 model, itertion 0 The second exmple is hlf-wvelength dipole ntenn positioned qurter wvelength from the floor of the perfectly conducting irplne, 61 wvelengths long. The irplne s nose is directed long x-xis of Descrtes coordinte system, Fig. 13. Residuum s function of umber of itertions 25

6 Mirotlsn revi The models with lrger MoM res then preserves this dvntge trough the itertions. However, s seen in Fig. 14, the convergence is better for models with smller MoM re. The models with lrger MoM re hve very good init solution, nd it s hrd to improve something tht is lredy good. In Figs re shown gins obtined using PDM solution of models 2, 4, nd 8 in 0th itertion, compred to gins obtined using MoM solution. The gins re clculted in plne 0 (the irplne symmetric plne). Obviously, the chnge in the MoM re size hve much more influence here thn with MPA on the helicopter. Enlrging MoM re (from 2 to 8 ) in certin degree improves the shpe of the min lobe, but, in much lrger extent, improves side lobes, both in shpe nd level. So, it seems tht ll importnt things in the nlysis of the irplne models come in the vicinity of the dipole. MPA distributes its influence in wider spce, so it is hrd to ctch ll these interctions in smll number of itertions. For dipole ntenn, these interctions re cught fst. The gins fter 1st itertion for 2 nd 4 models re shown in Figs. 18 nd 19. Gin for 2 is now much better, for 4 excellent, nd for 8 (not shown here) is lmost perfect mtch with MoM. Decembr It is cler tht number of MBFs here is extremely smll (prcticlly, we found perfect solution in single itertion) bout two orders of mgnitude smller thn number of MoM BFs. Finlly, let s loo how CPU time depends on number of itertions, s shown in Fig. 20. For smller number of groups nd smller number of itertions, the dependence is prcticlly liner. For lrge number of groups second, third nd higher order of dependence rise, s predicted in [9]. Fig. 16. Gin, Mom vs PDM solution of 4 model, itertion 0 Fig. 14. Residuum s function of MoM re size Fig. 17. Gin, Mom vs PDM solution of 8 model, itertion 0 Fig. 15. Gin, Mom vs PDM solution of 2 model, itertion 0 Fig. 18. Gin, Mom vs PDM solution of 2 model, itertion 1 26

7 December, 2013 Microwve Review Stellite, Cble nd Brodcsting Services - TELSIKS 2013, held in October 2013 in iš, Serbi. REFERECES Fig. 19. Gin, Mom vs PDM solution of 8 model, itertion 1 Fig. 20. CPU Time s function of umber of itertions IV. COCLUSIO In this pper we presented PDM method (Physicl optics Driven Method of moments) pplied to ntenn problems. We found tht pure metl structures converge fster thn composite metllic nd dielectric structures. umericl exmple shows good greement with MoM solution, with reduction in memory occupncy from one to two orders of mgnitude, nd moderte reduction in CPU time. In this wy PDM enbles by two orders electriclly lrger problems to be solved on the sme destop PC, compred to clssicl MoM nlysis. ACKOWLEDGEMET This wor ws supported by the Serbin Ministry of Eduction, Science, nd Technologicl Development under grnt TR This is n extended version of the pper "Electromgnetic Anlysis of Antenns on Lrge Pltforms Using Physicl Optics Driven Method of Moments," presented t the 11th Interntionl Conference on Telecommunictions in Modern [1] R. F. Hrrington, Field computtions by moment methods. ew Yor: McMln, [2] T. J. Kim nd G. A. Thiele, A hybrid diffrction technique- Generl theory nd pplictions, IEEE Trns. Antenns nd Propgt., vol. AP-30, pp , Sept [3] U. Jobus nd F. M. Lndstorfer, Improved PO-MM hybrid formultion for scttering from three-dimensionl perfectly conducting bodies of rbitrry shpe, IEEE Trns. Antenns Propg., vol. 43, no. 2, pp , Feb [4] R. Coifmn, V. Rohlin, nd S. Wndzur, The fst multipole method for the wve eqution: A pedestrin prescription, IEEE Antenns Propgt. Mg., vol. 35, no. 3, pp. 7 12, June [5] J. M. Song, C. C. Lu, nd W. C. Chew, Multevel fst multipole lgorithm for electromgnetic scttering by lrge complex obects, IEEE Trns. Antenns Propgt., vol. 45, pp , Oct [6] J. Sheffer, Direct Solve of Electriclly Lrge Integrl Equtions for Problem Sizes to 1 M Unnowns, Antenns nd Propgtion, IEEE Trnsctions on, vol.56, no.8, pp , Aug [7] V. V. S. Prsh nd R. Mittr, "Chrcteristic bsis function method: A new technique for efficient solution of Method of Moments mtrix equtions," Microwve nd Opticl Technology Letters, Vol. 36, o. 2, pp , Jn [8] L. Mteovits, V. A. Lz, nd G. Vecchi, "Anlysis of lrge complex structures with the synthetic-functions pproch," IEEE Trnsctions on Antenns nd Propgtion, Vol. 55, o 9, pp , Sep [9] M. Tsic nd B. Kolundzi, Efficient Anlysis of Lrge Sctterers by Physicl Optics Driven Method of Moments, Antenns nd Propgtion, IEEE Trnsctions on, vol.59, no.8, pp , Aug [10] M. Tsic nd B. Kolundzi, Physicl optics driven method of moments bsed on dptive grouping technique, Microwve Review, vol. 18, no. 2, pp. 2-7, Belgrde, December [11] M. Tsić, nd B. Kolundži, Eletromgnets nliz eletriči veliih metlno-dieletričnih rseč orišćenem metode moment vođene fizičom optiom, Zborni rdov 57. ETRA Konferencie, 3-6. un 2013, Zltibor, Srbi, pp. AP [12] B. Kolundzi, M. Tsic, D. Olcn, D. Zoric, nd S. Stevnetic, Advnced techniques for efficient modeling of electriclly lrge structures on destop PCs, Applied Computtionl Electromgnetics Society Journl, Spec Issue on Computtionl Electromgnetics Worshop, CEM 11, vol. 27, no. 2, Februry 2012, pp [13] M. Tsić, nd B. Kolundži, "Electromgnetic Anlysis of Antenns on Lrge Pltforms Using Physicl Optics Driven Method of Moments," TELSIKS 2013, Proceedings of Ppers, Volume 2, pp , is, Serbi, [14] B. Kolundzi nd A. Dordevic, Electromgnetic modeling of composite metllic nd dielectric structures, orwood, USA: Artech House, [15] WIPL-D Pro v10.0, 3D EM solver,