COMPUTATION OF RADAR CROSS SECTIONS OF COMPLEX TARGETS BY SHOOTING AND BOUNCING RAY METHOD

COMPUTATION OF RADAR CROSS SECTIONS OF COMPLEX TARGETS BY SHOOTING AND BOUNCING RAY METHOD A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY SALİM ÖZGÜN IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONICS ENGINEERING AUGUST 2009

Apprval f the thess: COMPUTATION OF RADAR CROSS SECTIONS OF COMPLEX TARGETS BY SHOOTING AND BOUNCING RAY METHOD Submtted by SALİM ÖZGÜN n partal fulfllment f the requrements fr the degree f Master f Scence n Electrcal and Electrncs Engneerng, Mddle East Techncal Unversty by, Prf. Dr. Canan Özgen Dean, Graduate Schl f Natural and Appled Scences Prf. Dr. İsmet Erkmen Head f Department, Electrcal and Electrncs Engneerng Prf. Dr. Mustafa Kuzuğlu Supervsr, Electrcal and Electrncs Engneerng, METU Examnng Cmmttee Members: Prf. Dr. Gönül Turhan Sayan Electrcal and Electrncs Engneerng, METU Prf. Dr. Mustafa Kuzuğlu Electrcal and Electrncs Engneerng, METU Prf. Dr. Gülbn Dural Electrcal and Electrncs Engneerng, METU Assst. Prf. Dr. Lale Alatan Electrcal and Electrncs Engneerng, METU Assst. Prf. Dr. Egemen Yılmaz Electrncs Engneerng, Ankara Unversty Date:

I hereby declare that all nfrmatn n ths dcument has been btaned and presented n accrdance wth academc rules and ethcal cnduct. I als declare that, as requred by these rules and cnduct, I have fully cted and referenced all materal and results that are nt rgnal t ths wrk. Name, Last name: Salm Özgün Sgnature :

ABSTRACT COMPUTATION OF RADAR CROSS SECTIONS OF COMPLEX TARGETS BY SHOOTING AND BOUNCING RAY METHOD Özgün, Salm M.Sc., Department f Electrcal and Electrncs Engneerng Supervsr: Prf. Dr. Mustafa Kuzuğlu August 2009, 66 Pages In ths study, a MATLAB cde based n the Shtng and Buncng Ray (SBR) algrthm s develped t cmpute the Radar Crss Sectn (RCS) f cmplex targets. SBR s based n ray tracng and cmbne Gemetrc Optcs (GO) and Physcal Optcs (PO) appraches t cmpute the RCS f arbtrary scatterers. The presented algrthm s examned n tw parts; the frst part addresses a new aperture selectn strategy named as cnfrmal aperture, whch s prpsed and frmulated t ncrease the perfrmance f the cde utsde the specular regns, and the secnd part s devted t testng the multple scatterng and shadwng perfrmance f the cde. The cnfrmal aperture apprach cnssts f a cnfguratn that gathers all rays buncng back frm the target, and calculates ther cntrbutn t RCS. Multple scatterng capablty f the algrthm s verfed and tested ver smple shapes. Ray tracng part f the cde s als used as v

a shadwng algrthm. In the frst nstance, smple shapes lke sphere, plate, cylnder and plyhedrn are used t mdel smple targets. Wth prmtve shapes, cmplex targets can be mdeled up t sme degree. Later, patch representatn s used t mdel cmplex targets accurately. In rder t test the whle cde ver cmplex targets, a Cmputer Aded Desgn (CAD) frmat knwn as Stere Lthgraphy (STL) mesh s used. Targets that are cmpsed n CAD tls are mprted n STL mesh frmat and handled n the cde. Dfferent sweep gemetres are defned t cmpute the RCS f targets wth respect t aspect angles. Cmplex targets are selected accrdng t ther RCS characterstcs t test the cde further. In addtn t these, results are cmpared wth PO, Methd f Mments (MM) and Multlevel Fast Multple Methd (MLFMM) results btaned frm the FEKO sftware. These cmparsns enabled us t mprve the cde as pssble as t s. Keywrds: Shtng and Buncng Ray (SBR) Methd, Radar Crss Sectn (RCS), Cnfrmal Aperture, FEKO v

ÖZ KARMAŞIK HEDEFLERİN RADAR KESİT ALANININ SEKEN IŞIN YÖNTEMİYLE HESAPLANMASI Özgün, Salm Yüksek Lsans, Elektrk-Elektrnk Mühendslğ Bölümü Tez Yönetcs: Prf. Dr. Mustafa Kuzuğlu Ağusts 2009, 66 sayfa Bu çalışmada csmlern Radar Kest Alanının (RKA) hesaplanması çn Seken Işın Yöntem (SIY) algrtması kullanılarak MATLAB tabanlı br kd gelştrlmştr. SIY rastgele seçlmş hedeflern RKA değern hesaplamak çn Gemetrk Optk (GO) ve Fzksel Optk (FO) yöntemlern beraber kullanan ışın takbne dayalı br algrtmadır. Sunulan algrtma k bölümde ncelenmştr; lk bölümde dğrudan yansıma dışında kalan bölgelerde kdun perfrmansını artırmak çn yen br yöntem lan uyumlu açıklık önerlmş ve frmüle edlmş, knc bölümde se algrtmanın çklu yansımaları hesaplama ve gölgeleme perfrmansı test edlmştr. Uyumlu açıklık yöntem le hedeften ger seken v

tüm ışınlar tplanıp RKA ya katkıları hesaplanmıştır. Algrtmanın çklu yansıma kablyet sadece bast şekller üzernde test edlmştr. Kdun ışın takb bölümü aynı zamanda gölgeleme algrtması larak kullanılmıştır. İlk aşamada bast hedefler mdellemek çn küre, plaka, slndr ve plhedrn gb bast şekller kullanılmıştır. Ancak bast şekller le karmaşık hedefler br dereceye kadar mdelleneblmektedr. Daha snrak aşamalarda karmaşık şekllern daha y mdelleneblmes çn üçgen parçalar kullanılmıştır. Kdu karmaşık hedefler üzernde test etmek çn Blgsayar Destekl Tasarım (BDT) frmatı lan STL ağ yapısı kullanılmıştır. BDT rtamında mdellenen hedefler STL ağ frmatında kda taşınmış ve şlem yapılmıştır. Csmlern RKA değerlern bakış açısına göre hesaplayablmek çn farklı döndürme gemetrler kullanılmıştır. Seçlen karmaşık hedeflerde RKA karakterstklernn farklı lması göz önünde bulundurulmuştur. Bu çalışmalara ek larak kddan elde edlen snuçlar, FEKO yazılımından elde edlen FO, MM ve MLFMM snuçları le karşılaştırılmıştır. Bu karşılaştırma, kdu mümkün lduğunca gelştrme mkânını bze sunmuştur. Anahtar Sözcükler: Seken Işın Yöntem (SIY), Radar Kest Alanı (RKA), Uyumlu Açıklık, FEKO v

T my Famly v

ACKNOWLEDGEMENTS I wuld lke t thank my thess supervsr, Prf. Dr. Mustafa Kuzuğlu fr hs gudance, understandng and suggestns. Hs brad vsn and mtvatn has been a great prvlege. I wuld als thank my leader Harun Slmaz and age f empres team; Mehmet Karakaş, Selm Kcabay, Mehmetcan Apaydın, Seçkn Arıbal, Aykut Changr, Musa Cvl, Erdem Kazaklı, Baran Kızıltan and Şükrü Bülent Tker fr ther great frendshp and enjyable age f empres the cnquerrs tmes durng lunch. Als, I wuld lke t express my thanks t my famly fr ther understandng and patence. Lastly, thanks t TAI fr supplyng the FEKO sftware and supprtng me fr ths wrk. x

TABLE OF CONTENTS ABSTRACT... v ÖZ... v ACKNOWLEDGEMENTS... x TABLE OF CONTENTS... x LIST OF TABLES... x LIST OF FIGURES... x LIST OF FIGURES... x CHAPTER 1 INTRODUCTION... 1 1.1 Lterature Revew... 3 1.2 Overvew f the Thess... 4 1.3 Outlne f the Thess... 5 2 SHOOTING AND BOUNCING RAY METHOD... 7 2.1 Frmulatn f the Shtng and Buncng Ray Methd... 8 2.2 Numercal Results fr Smple Targets... 19 2.3 SBR Fnal Integratn Aperture... 39 2.4 SBR Multple Scatterng Capablty... 44 2.5 Shadwng... 46 3 APPLICATION OF SBR TO COMPLEX TARGETS... 48 x

3.1 Target Mdelng... 49 3.2 Target Rtatn... 52 3.3 Numercal Results... 53 4 CONCLUSIONS... 61 4.1 Summary f the Thess... 61 4.2 Advantages and Dsadvantages f SBR... 62 4.3 Future Wrk... 63 REFERENCES... 65 x

LIST OF TABLES TABLES Table 3-1 Cmputatn tmes fr tank mdel smulatn... 54 Table 3-2 Cmputatn tmes fr shp mdel smulatn... 56 Table 3-3 Cmputatn tmes fr F-117 mdel smulatn... 58 x

LIST OF FIGURES FIGURES Fgure 2.1 SBR, ncdent rays and ext aperture... 9 Fgure 2.2 Trangle defned wth rght hand rule... 11 Fgure 2.3 A trangle cnstructed frm three half spaces... 12 Fgure 2.4 Reflectn f a ray... 13 Fgure 2.5 Ray tube... 16 Fgure 2.6 Shape f ext ray tube... 18 Fgure 2.7 Smulatn setup fr mnstatc RCS f PEC plate... 21 Fgure 2.8 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC plate wth respect t frequency... 22 Fgure 2.9 Smulatn setup fr bstatc RCS f PEC plate... 23 Fgure 2.10 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate wth respect t frequency... 23 Fgure 2.11 Smulatn setup fr mnstatc RCS f PEC plate... 24 Fgure 2.12 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC plate... 25 Fgure 2.13 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC plate... 26 Fgure 2.14 Smulatn setup fr bstatc RCS f PEC plate... 27 Fgure 2.15 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate... 28 Fgure 2.16 Smulatn setup fr bstatc RCS f PEC plate... 29 Fgure 2.17 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate wth respect t frequency... 30 Fgure 2.18 Smulatn setup fr mnstatc RCS f PEC flare-lke shape... 31 x

Fgure 2.19 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC flare-lke shape fr planar aperture case... 32 Fgure 2.20 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC flare-lke shape fr cnfrmal aperture case... 33 Fgure 2.21 Smulatn setup fr mnstatc RCS f PEC arbtrary plyhedrn. 34 Fgure 2.22 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn... 34 Fgure 2.23 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn after crrectn n shadwng algrthm... 35 Fgure 2.24 Shadwng effect... 36 Fgure 2.25 Cmparsn f PO, MM and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn... 37 Fgure 2.26 Smulatn setup fr mnstatc RCS f PEC arbtrary shape... 38 Fgure 2.27 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary shape... 38 Fgure 2.28 Planar ext aperture apprach... 40 Fgure 2.29 Mnstatc RCS f PEC cube wth planar ext aperture apprach... 41 Fgure 2.30 Cnfrmal ext aperture apprach... 43 Fgure 2.31 Mnstatc RCS results wth cnfrmal ext aperture apprach... 44 Fgure 2.32 Smulatn setup fr mnstatc RCS f PEC dhedral... 45 Fgure 2.33 Cmparsn f PO, MM and SBR results fr θθ plarzed mnstatc RCS f PEC dhedral... 46 Fgure 2.34 Shadwng prblem... 47 Fgure 3.1 Target and Ray Wndw Gemetry... 49 Fgure 3.2 STL mesh structure... 50 Fgure 3.3 Smulatn setup fr mnstatc RCS f PEC trangular patch... 51 Fgure 3.4 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC trangle... 51 Fgure 3.5 Fxed rtatn arund Y axs... 53 Fgure 3.6 Tank mdel cnstructed frm 132 trangular patches... 54 xv

Fgure 3.7 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC tank mdel... 55 Fgure 3.8 Mnstatc RCS f shp mdel... 56 Fgure 3.9 Cmparsn f PO, MLFMM and SBR results fr θθ plarzed mnstatc RCS f PEC shp mdel... 57 Fgure 3.10 F-117 mdel... 58 Fgure 3.11 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC F-117 mdel... 59 Fgure 3.12 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC F-117 mdel... 59 xv

CHAPTER 1 INTRODUCTION Durng the secnd half f th 19 century, James Clerk Maxwell cmpleted the classcal thery f electrmagnetsm, syntheszng all the prevus knwledge develped by prmnent physcsts and mathematcans such as Culmb, Ampere, Oersted, Faraday and Hertz. Hs famus equatns, knwn as Maxwell Equatns, have been wdely used untl the present tme by physcsts and engneers n numerus applcatns. A majr applcatn area f Maxwell s equatns s scatterng f electrmagnetc waves by bstacles. Ths tpc ganed a lt f mprtance after the develpment f radar systems durng Wrld War II. In radar applcatns, the electrmagnetc pwer ntercepted by a target s mdeled va a hypthetcal area knwn as Radar Crss Sectn (RCS). The pwer ntercepted by the bject s calculated as the prduct f RCS and the pwer densty at the target lcatn. As a result, calculatn f the RCS f targets s ne f the mprtant aspects f radar engneerng. The RCS f an bject s clsely related f the cnsttutve parameters (permttvty, permeablty and cnductvty) and the gemetry (r shape). In desgnng targets almst nvsble t radars (.e. stealth targets), t s the parameter that shuld be mnmzed. In early 1970s, stealth technlgy has becme ppular, wth a lt f emphass placed n reductn f the RCS f targets. Even thugh the mdern vew f stealth desgn s t acheve an ptmum balance between RCS 1

reductn, ncrpratn f Electrmagnetc Warfare (EW) and peratnal perfrmance; RCS reductn s stll the mst mprtant aspect f stealth technlgy. Mdern radars are nt nly capable f detectng lw RCS targets, but they can examne and classfy targets accrdng t ther scatterng characterstcs. The man task f RCS desgn engneer s t ntrduce stealth targets effectvely nvsble t these pwerful mdern radars. At ths pnt, develpers f stealth targets use extensvely the tls f Cmputatnal Electrmagnetcs (CEM) t smulate the scatterng f electrmagnetc waves by targets wth cmplex gemetry and/r electrcal parameters. The methds f CEM are based n the numercal apprxmatn f Maxwell s Equatns fr mdelng the nteractn f electrmagnetc felds wth bjects. Durng the last few decades ths area has becme crucal fr stealth engneerng. The same methds are als used extensvely n antenna desgn and placement, radme desgn, electrmagnetc cmpatblty (EMC) analyss, RF cmpnent desgn and b-electrmagnetcs prblems. Many cmputatnal electrmagnetcs methds have been develped and mplemented t slve dfferent knds f prblems. Bascally, these methds can be classfed nt tw grups, namely as full-wave and apprxmate slutn technques. Full-wave methds have been develped fr the numercal dscretzatn and apprxmatn f equatns deduced frm Maxwell s equatns. The Methd f Mments (MM), Fnte Dfference Tme Dman (FDTD) methd and Fnte Element Methd (FEM) are the mst cmmnly-used full-wave methds. These methds are smetmes named as lw frequency methds because f the requrement f hgh cmputer resurces at hgh frequences. In recent years, sme new full-wave based methds have been ntrduced n rder t slve electrcally large prblems. Fr example, the Multlevel Fast Mult Ple Methd (MLFMM) s a methd that uses the frmulatn f MM. Its man dfference frm the MM s that t grups the bass functns and calculates the nteractn between these grups. Thrugh ths mdfcatn, ths apprach can handle electrcally large 2

prblems such as delectrc structures wth memry and run-tme requrements frmdable even fr tday s super cmputers. Apprxmate slutns are mplemented bascally t handle electrcally large prblems. Snce these methds are effectve at hgh frequences, they are als knwn as hgh frequency slutn technques. Basc hgh frequency technques are Physcal Optcs (PO), Gemetrc Optcs (GO), Physcal Thery f Dffractn (PTD), Gemetrc Thery f Dffractn (GTD) and Unfrm Thery f Dffractn (UTD). PO and GO can gve gd slutns n specular regns but fr cmplex structures ther results are nt very relable. Utlzatn f GTD and PTD can mprve the results f PO and GO n regns where dffracted rays dmnate. In stealth desgns, snce specular reflectns are avded, calculatn f the RCS must take nt cnsderatn the effect f dffractn and multple reflectns. Cntrbutn f dffractn can be calculated wth PTD. In recent years a new methd, namely the Shtng and Buncng Ray (SBR) methd s mplemented t slve the effects f multple scatterng. SBR s a methd that uses a cmbnatn f the GO and PO technques. It s a ray tracng technque sutable t use n stealth desgn applcatns. 1.1 Lterature Revew Multple Scatterng plays an mprtant rle n RCS calculatn. Cavty structures (such as jet-engne nlets) are the regns where multple scatterng phenmenn ccurs. Snce such structures cntrbute extensvely t RCS, the analyss f ths prblem has been an mprtant tpc n RCS lterature. Tradtnally, RCS f cavty structures were calculated by mdel analyss r full wave methds. Fr cmplex structures, mdel analyss was nt adequate, and full wave methds were nt capable f slvng electrcally large prblems. In 1987, SBR was mplemented fr the frst tme by Chu et al. [1] n a prject wrk supprted by NASA. In ths methd, rays are launched twards the cavty and traced nsde wth GO rules. Materal prpertes f the cavty walls are taken nt accunt and 3

rays are captured n the cavty penng n rder t calculate the far feld values. Later, Lng et al. presented ths wrk as an artcle n [2]. SBR has been develped rgnally fr cavty structures, but Baldauf et al. mdfed ths technque fr pen scatterers [3]. In cavty structures, rays that leave the cavty are captured n the cavty penng. Hwever fr pen scatterers the fnal ntegratn aperture s nt easy t chse. In [3], an aperture that cncdes wth the scatterer tself s suggested. Wth ths aperture chce, the slutn s dentcal t the cnventnal PO slutn, except that multple scatterng effects f the rays are als mdeled. In ths wrk, frmulatn s based n Huygens Prncple. In 2005, t extend the capablty f SBR fr the smulatn f scatterng frm dffusve and gratng structures, Dffusve Ray Algrthm s ntrduced by Gallway et al. [4]. Ths algrthm reduces t the basc SBR frmulatn fr targets cnstructed as a superpstn f flat plates, but fr curved structures a sngle ray tube that reflects frm a curved part wll generate a famly f ther ray tubes [4]. 1.2 Overvew f the Thess In ths thess, a basc SBR cde s develped n MATLAB. Numercal results taken frm the cde are cmpared wth the results btaned frm FEKO sftware. FEKO s an electrmagnetc (EM) analyss sftware whch ncludes bth fullwave and apprxmate slutn methds and whch has the capablty t slve a wde range f electrmagnetc prblems [5]. Fr smple targets results are btaned frm FEKO s MM and PO slvers. Fr electrcally large prblems results are btaned frm MLFMM slver n a redhat based super cmputer wth 18GB RAM. 4

SBR cde s tested ver smple targets lke plate, dhedral and cube fr mnstatc and bstatc RCS calculatns, fr a range f frequences. T test the SBR cde fr target n whch multple scatterng phenmena s effectve, a dhedral was used. Clearly, scatterng frm targets lke dhedral and trhedral s dmnated by multple bunces and therefre cmpses a gd test envrnment fr SBR [3]. The cde s mplemented fr pen scatterers where the aperture chsen fr fnal aperture ntegratn becmes crucal. If the electrmagnetc feld values n the ext aperture were knwn exactly, regardless f the chsen ext aperture, exact far feld result can be btaned [3]. Plane aperture and cnfrmal aperture (cncdng wth the scatterer tself) are mplemented fr fnal aperture ntegratn. Results are crtcally evaluated t chse the best ext aperture t acheve accurate results. Bth fr smple and cmplex targets, gemetres are specfed n STL mesh frmat. In STL mesh frmat, targets are cnstructed frm trangular patches. Crner pnts and nrmal vectrs f trangular patches are specfed n ths frmat. RCS s typcally presented n plts f RCS values versus aspect angle. Aspect angle s the angle f target wth respect t radar pstn. Dfferent sweep cnfguratns are mplemented n the SBR cde. Fr sme cases, the target and recever are fxed and the transmtter s rtated, fr sme cases the transmtter and recever are fxed, and the target s rtated. 1.3 Outlne f the Thess The thess s rganzed n three man chapters. In Chapter 2 frmulatn f basc SBR algrthm s dscussed. The cde s tested ver smple test targets lke plate, cube, plyhedrn and dhedral. Fnal aperture ntegratn part f the cde s dscussed n detal and a new apprach s develped n Chapter 2. Als, shadwng and multple scatterng are studed n ths chapter. 5

In Chapter 3, applcatn f SBR t cmplex targets s dscussed. Methds f cnstructng a CAD mdel are studed. The chapter cntnues wth the sweep defntn. RCS f cmplex targets lke shp, arcraft and tank are smulated n ths part as well. Results are cmpared wth FEKO results. The thess s summarzed n Chapter 4. Advantages and dsadvantages f the methd are dscussed. Cncludng remarks are made n numercal results. The Chapter termnates wth a summary f pssble future wrk where pssble mdfcatns/enhancements f the algrthm are dscussed. 6

CHAPTER 2 SHOOTING AND BOUNCING RAY METHOD In ths chapter, the Shtng and Buncng Ray (SBR) methd s dscussed. It shuld be mentned that the presented algrthm s a ray-based technque whch can cmpute the RCS f electrcally large, arbtrarly-shaped Perfect Electrc Cnductr (PEC) targets. Bascally, the algrthm cnssts f ray tracng, ampltude trackng and fnal aperture ntegratn parts. The man blcks f the algrthm are dscussed n Sectn 2.1. In Sectn 2.3, fnal aperture ntegratn part f the algrthm s studed n detals. Essentally, the algrthm s used t calculate the RCS f pen scatterers. S, t s nt cnvenent t perfrm the ray-tube ntegratn ver a cmmn ext plane aperture [6]. SBR algrthm s pwerful t calculate multple scatterng effects. In Sectn 2.4 ths attrbute f the algrthm s tested ver prblems dmnated by multple bunce effects. In Sectn 2.2 numercal results f SBR cde are cmpared wth the results f FEKO sftware. The last sectn s devted t the shadwng effect. 7

2.1 Frmulatn f the Shtng and Buncng Ray Methd Shtng and Buncng Rays (SBR) methd s based n ray tracng. The prblem s t determne the scattered feld frm an pen scatterer. Bascally, rays that smulate the ncdent plane wave are sht nt the target, traced n the target and fnally gathered n an ext aperture. Electrc feld s traced wthn the rays wth Gemetrc Optc (GO) rules. The algrthm wll be dscussed n three man parts: 1) ray tracng 2) ampltude trackng and 3) physcal ptcs apprxmatn n ext aperture. SBR uses bth GO and PO rules. Referrng t Fgure 2.1 cnsder an arbtrary - shaped PEC bject and an ncdent plane wave. 8

Fgure 2.1 SBR, ncdent rays and ext aperture The ncdent plane wave s represented by a dense grd f rays. A ray s defned by an rgn, P = ( x, y, z ) and a drectn vectr D ( s, s, s ) fr the ray s; p p p =. The equatn x y z r I( t) = P + td, t 0 (2.1) where, 9

s x s y s z = snθ csφ (2.2) = snθ snφ (2.3) = csθ (2.4) θ and φ are angles that dente the drectn f ncdence. The defned rays are sht nt the PEC bject. Let s z = f ( x, y) be the equatn f the surface f the scatterer. The ntersectn pnts f rays and the surface are fund by slvng equatns f the surface and rays smultaneusly. Fr example, f the bject s a plate (.e. the surface s a plane) we can fnd the ntersectn pnt as fllws; a plane can be defned by a nrmal vectr N and a pnt n the plane Q. A pnt P s n the plate f: N r ( P Q) = 0 (2.5) By cmbnng the equatns f plane and ray tgether we end up wth the fllwng equatn: r r N Q ( P + td ) = 0 (2.6) where t = r N( Q P) r r N D (2.7) Fr smple shapes lke plate, sphere r cylnder a sngle mplct equatn s enugh t defne the shape f the surface. Fr cmplex targets, a superpstn f trangular patches can be used t apprxmate the gemetry. In ths case ntersectng a ray wth a trangle s mre cmplcated than smple shapes. Bascally, ray trangle ntersectn can be accmplshed n tw steps; 10

Intersectng the ray and the plane f the trangle, Checkng whether the ntersectn pnt s nsde r utsde the trangle. Ray-plane ntersectn s already dscussed n equatns (2.5), (2.6) and (2.7). After fndng the ntersectn pnt, we must decde whether the pnt s nsde r utsde the trangle. A trangle s defned wth three vertces p0, p1 and p2. The nrmal vectr f the trangle (n) s defned va the rght hand rule. p2 n p2-p0 p1 p0 p1-p0 Fgure 2.2 Trangle defned wth rght hand rule nˆ = ( p1 p0) ( p2 p0) (2.8) A trangle s the ntersectn f three half spaces as n Fgure 2.3. Each edge f the trangle les n a lne and a pnt s nsde the trangle f t s n the crrect sde f each ne f the three lnes [9]. 11

Fgure 2.3 A trangle cnstructed frm three half spaces Usng ths defntn, a pnt s nsde the trangle f t s n the left sde f each edge. Va the crss prduct, t can be decded whether a pnt x s n the left r rght f the edges wth the fllwng equatns; (( p 1 p0) ( x p0)) nˆ 0 (2.9) (( p 2 p1) ( x p1)) nˆ 0 (2.10) (( p 0 p2) ( x p2)) nˆ 0 (2.11) Next, the reflected ray s determned by usng the rules f Snell s law [2]; 1) Reflected ray must le n the plane f ncdence 2) The angle f reflectn must be equal t the angle f ncdence 12

ray r ray N î î yˆl xˆl Lcal Crdnate System zˆl ĝz ĝy ĝx Glbal Crdnate System Fgure 2.4 Reflectn f a ray Referrng t Fgure 2.4 the reflected ray can be determned accrdng t Snell s law. A unt vectr s defned as r mˆ = ( ray N) / snθ (2.12) where ray s the ncdent ray and N s the reflectn nrmal. Nte that m s perpendcular t the plane f ncdence. Referrng t Fgure 2.4, defne the lcal crdnates y l z l x l = mˆ (2.13) r = N (2.14) r = ( mˆ N) (2.15) In sphercal crdnates rayr can be determned as fllws 13

ray r = ( r, θ, φ) (2.16) r = 1, θ = π / 2 ˆ θ ), φ = 0 (2.17) ( Next, x, y, z ) crdnates f rayr n the lcal crdnate system can be ( l l l calculated by sphercal t Cartesan crdnate transfrmatn. As a fnal step lcal t glbal crdnate transfrmatn s acheved as belw; 1 0 0 A = 0 1 0 0 0 1 (2.18) xl (1) xl (2) xl (3) C = yl (1) yl (2) yl (3) z (1) (2) (3) l zl zl (2.19) 1 B = A C (2.20) ( xg, y g, z g ) = B ( xl, yl, zl ) (2.21) where, B s the transfrmatn matrx. By usng the reflected ray as a new ncdent ray ths prcedure s appled untl the ray ends t bunce n the gemetry. In ray paths the feld ampltude s als traced. In Gemetrcal Optcs, the ampltude, phase and plarzatn f electrc feld can be updated wth the fllwng equatn. r E ( x+ 1, y+ 1, z+ 1) ( DF). Γ r =. E( x, y, z ). e jφ (2.22) Where φ and DF s the dvergence 2 2 2 1/ 2 = k0 [( x+ 1 x ) + ( y+ 1 y ) + ( z+ 1 z ) ] factr whch calculates the spreadng f ray tubes. DF s applcable fr curved surfaces and fr planar surfaces takes the value f 1. Γ s the planar reflectn 14

ceffcent. Fr PECs, planar reflectn ceffcent can be appled wth the equatn (2.23). r E r = ( E ) + (2( nˆ E ) nˆ) r (2.23) The scattered far feld can be cmputed by applyng the basc physcal ptcs apprxmatn n the aperture where ext rays are gathered. The magnetc current sheet M r ver the aperture s; s M r = 2 E r ( x, y, z) ˆ s n a (2.24) where, n r a s aperture nrmal. Frm ths magnetc current sheet, the scattered feld can be calculated. The scattered far feld s the sum f cntrbutns frm ndvdual ray tubes. The cntrbutn f a ray tube s calculated as; r E s = e jk0r r [ ˆ θ A θ + ˆ φ A φ ] (2.25) and, r M s = [ E ˆ 0 θθ + E0φ ˆ] φ 2zˆ (2.26) where, ) ) ) ) φ = csθ csφx + csθ snφy snθz (2.27) ) ) ) θ = sn φx + csφy (2.28) r = E [sn( φ) yˆ + cs( φ) xˆ] + 2E [ cs( θ )cs( φ) yˆ cs( θ )sn( φ) ˆ] (2.29) M s 2 0θ 0φ + x 15

A = jk0 4π M s e jk. r dxdy (2.30) jk 0 jk0 ( ux+ uy) Aθ = dxdye [ E x csφ + E 2π A y snφ ] (2.31) jk 0 jk0 ( ux+ uy) Aφ = dxdye [( E x sn φ + E 2π A y csφ ) csθ ] (2.32) u = snθ csφ (2.33) v = snθ snφ (2.34) Fr the bstatc case, θ and φ values n equatns (2.31), (2.32), (2.33) and (2.34) are replaced wth bservatn angles. Snce the utgng rays are nt unfrm, the ntegratns cannt be evaluated easly [2]. At ths pnt, shtng and buncng ray methd has a way ut. A small ray tube s sht nt the gemetry. The ray tube bunces n the gemetry and cmes t the aperture fnally. Then the scattered feld s calculated frm ths ray tube. Enugh ray tubes are sht nt the gemetry t mdel the ncdent plane wave. The ttal scattered feld s the sum f all cntrbutns frm the ray tubes. Aperture Ray tube ẑ ŷ ˆx Fgure 2.5 Ray tube 16

Assume that the ncdent ray tube has an area f Δ x Δ ). The central ray wth drectn vectr ( s s, s ) ( s s x y z ( 0 y0, hts the aperture. The ray tube wll have an area f Δ x Δy ) n the aperture and the feld wthn the exstng ray tube can be apprxmated as ( x x, y) E x ( x, y ) jk0[ sx ( x x ) + s y ( y y )] E E y ( x, y) = E y e ( x, y ) (2.35) Ths means that the ampltude f the feld s same n the ray tube and acrss the ray tube there s a lnear phase varatn [2]. If the sze f exstng ray tube s t large, the apprxmatn wll nt be vald. Wth equatns belw; A A θ φ = = jk 2π jk 2π 0 jk ( ux+ vy) jk0[ sx ( x x ) + s y ( y y )] 0 ray tube dxdye e [ E ( x, y )csφ + E ( x, y )snφ ] (2.36) x 0 jk ( ux+ vy) jk0[ sx ( x x ) + s y ( y y )] 0 ray tube dxdye e y [( E ( x, y )snφ + E ( x, y )csφ )csθ ] (2.37) x y Snce E x, y ) and E x, y ) are ndependent f ntegral varables they can be x ( taken ut f the ntegral; y ( jk0 A θ = [ E x ( x y )csφ + E y ( x, y )snφ ] 2π e jk0 ( sxx + s y y ) ( Δx Δy ) I (2.38) jk0 A φ = [( E x ( x y )snφ + E y ( x, y )csφ )csθ ] 2π 17

e jk0 ( sxx + s y y ) ( Δx Δy ) I (2.39) where I = 1 ( Δx Δy ) ray tube dxdye jk0 [( u sx ) x+ ( v s y ) y] (2.40) The ntegral n equatn (2.40) s the phase factr n standard physcal ptcs thery [3]. Smetmes t s called shape functn and t s the Furer Transfrm f the ray tube shape. In rder t take advantage f ths methd fr each ray tube, fur rays arund a central ray are sht nt the gemetry. Fgure 2.6 Shape f ext ray tube The pstn vectrs f rays are dented by r γ = x xˆ y yˆ, n = 1,2,3, 4. Furer nˆ n + transfrm can be evaluated as descrbed n [2], [7]. Smpler prf f ths methd based n Stkes therem can als be cnsdered n [10]. n 18

I = S( p, q) / S(0,0) (2.41) where, S( p, q) = 4 e ( x x x )( y y ) p + ( y y ) ( y ) q][( x y )( x x x ) p + ( y jw γ n n+ 1 n n n 1 n+ 1 n n n 1 (2.42) [( x n= 1 n n 1 n n 1 n+ 1 n n+ 1 S 0,0) = ( x y x y ) + ( x y x y ) + ( x y x y ) + ( x y x ) / 2 (2.43) ) y n ) q] ( 1 2 2 1 2 3 3 2 3 4 4 3 4 1 1y4 p = k ( u s ) (2.44) 0 x q = k ( v s ) (2.45) 0 y r w = pxˆ + qyˆ (2.46) As a fnal step, RCS can be calculated as n equatns (2.47) and (2.48). RCS RCS 2 4π Aθ = (Vertcal plarzatn) (2.47) 2 4π Aφ = (Hrzntal plarzatn) (2.48) 2.2 Numercal Results fr Smple Targets In ths sectn, RCS values f sme smple targets are cmpared wth Methd f Mments (MM) and Physcal Optcs (PO) results. FEKO, a sftware prduct fr the smulatn f electrmagnetc felds, s used t btan MM and PO results. Dfferent slutn technques mplemented wthn FEKO make t applcable fr a wde range f prblems [5]. The targets f nterest are a plate, a cube and a flarelke shape. Generally, mnstatc RCS f targets were calculated fr vertcal plarzatn. Targets are swept n θ angle. FEKO sftware runs n cmputers; Cmputer 1; Intel (R) Cre 2 Du CPU, E4500@2.20GHz 3.25GB RAM Cmputer 2; (Dual) AMD Optern Prcessr 248 14.926 GB RAM 19

Cmputer 2 s used fr electrcally large prblems. Even wth cmputer 2 sme prblems cannt be slved wth MM. Multlevel Fast Multple Methd (MLFMM) whch s a fast alternatve frmulatn f the MM s applcable t much larger structures than the MM [5]. MLFMM makes t pssble t btan full-wave slutns fr electrcally large structures [5]. Wth ths methd and cmputer 2, a mltary arcraft at 2.5GHz r a shp at 400MHz can be handled. In rder t slve the same prblems wth MM, we need 30TByte memry apprxmately. Dfferent sweep gemetres are used t calculate the RCS values f smple PEC bstacles. In the plate gemetry, plane wave (prvded by the transmtter) s swept arund the target n θ angle and plate s kept fxed n rgn. Fr cube and flare lke shape, transmtter and ext aperture (n the drectn f the recever) are kept fxed and the target s rtated n y axs. Bth appraches can be used t calculate the mnstatc RCS. The plate gemetry s frmulated n the prblem drectly by usng ts equatn. The cube and flare lke gemetry are created frm trangle patches that are bg enugh t permt t cnstruct these shapes. The number f the trangular patches s kept mnmal t decrease the runtme snce the SBR cde generated n MATLAB des nt nclude a specal algrthm that speeds up the ray-trangle ntersectn test. Ray-trangle ntersectn s tested wth brute frce methds, whch mples that runtme ncreases dramatcally wth the number f patches. The frst smulatn was cnducted n a perfect electrc cnductr (PEC) flat plate as llustrated n Fgure 2.7. In ths smulatn bth mnstatc and bstatc RCS f a square PEC plate wth sde length f 0.25 meters were cmputed and cmpared wth MM and PO results taken frm FEKO. 20

The plate was llumnated frm φ = 0 and θ = 0 and ts mnstatc RCS was cmputed wth respect t frequency frm 2GHz t 6GHz as shwn n Fgure 2.7. The result f the SBR s cmpared wth the MM and gven n Fgure 2.8. Fgure 2.7 Smulatn setup fr mnstatc RCS f PEC plate Rays are sht twards the plate wth λ/10 densty, where λ s wave length. Fr each ray, fur rays neghbrng the central ray are sht t buld up a ray tube. Snce specular reflectns are dmnant, the results f SBR slutn and MM are perfectly matched. 21

14 Mnstatc RCS f a PEC plate wth respect t frequency 12 RCS [dbm 2 ] 10 8 6 4 MOM FEKO,MM Standard SBR SBR 2 2 2.5 3 3.5 4 4.5 5 5.5 6 frequency [GHz] Fgure 2.8 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC plate wth respect t frequency Fr bstatc case, the plate s llumnated frm φ = 0 and θ = 10 and ts bstatc RCS s cmputed at φ = 0 and θ =10 wth respect t frequency frm 2GHz t 6GHz as shwn n Fgure 2.9. The result f the SBR was cmpared wth the MM and gven n Fgure 2.10. 22

Fgure 2.9 Smulatn setup fr bstatc RCS f PEC plate As a result fr flat surfaces n specular regns the results are matchng wth MM results perfectly fr bth mnstatc and bstatc cases. 14 Bstatc RCS f a PEC plate wth respect t frequency 12 RCS [dbm 2 ] 10 8 6 4 MOM FEKO,MM Standard SBR SBR 2 2 2.5 3 3.5 4 4.5 5 5.5 6 frequency [GHz] Fgure 2.10 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate wth respect t frequency 23

In the fllwng example, the plate s llumnated at φ = 0 frm θ = 60 t θ = 60 and ts mnstatc RCS s cmputed Fgure 2.11. The result f the SBR was cmpared wth the MM n Fgure 2.12 and wth PO n Fgure 2.13. Fgure 2.11 Smulatn setup fr mnstatc RCS f PEC plate 24

15 10 Mnstatc RCS f PEC Plate SBR FEKO, MM 5 0 RCS [dbm 2 ] -5-10 -15-20 -25-30 -60-40 -20 0 20 40 60 Theta [degrees] Fgure 2.12 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC plate The results are perfectly matchng wth MM near specular regn. Results are nt agreeng at angles where dffractn effects are dmnant. SBR cde s unable t mdel dffractn. When we cmpare the results wth PO cde n FEKO we can see that the results are n gd agreement, snce PO s als unable t take dffractn nt accunt. 25

20 10 Mnstatc RCS f PEC plate FEKO, PO SBR 0 RCS [dbm 2 ] -10-20 -30-40 -50-60 -40-20 0 20 40 60 Theta [degrees] Fgure 2.13 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC plate In the ther case the plate s llumnated frm φ = 0 and θ = 0 and ts bstatc RCS s cmputed at φ = 0 frm θ = 90 t θ = 90 as n Fgure 2.14. The result f the SBR was cmpared wth the MM and gven n Fgure 2.15. 26

Fgure 2.14 Smulatn setup fr bstatc RCS f PEC plate The results are harmnus wth MM near specular regns. In rear sdes where dffractn cntrbutn s mprtant, the results are nt matchng well wth the MM results. 27

5 0 Bstatc RCS f PEC plate SBR FEKO,MM -5-10 RCS[dBm 2 ] -15-20 -25-30 -35-40 2 2.5 3 3.5 4 4.5 5 5.5 6 frequency[ghz] Fgure 2.15 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate Fnally, fr the plate case, bstatc RCS s cmputed wth respect t frequency. The plate s llumnated frm φ = 0 and θ = 0, RCS values are cmputed at φ = 0 and θ = 15 wth respect t frequency frm 2GHz t 6GHz as shwn n Fgure 2.16. The result f the SBR was cmpared wth the MM and gven n Fgure 2.17. 28

Fgure 2.16 Smulatn setup fr bstatc RCS f PEC plate Out f the specular regn where dffractn mechansm s effectve, agan the results are nt n agreement wth MM results. Hence, t can be cncluded that by crrectly mdelng the dffractn effects, mre accurate results can be btaned ut f specular regns. 29

5 0 Bstatc RCS f PEC plate SBR FEKO, MM -5-10 RCS [dbm 2 ] -15-20 -25-30 -35-40 2 2.5 3 3.5 4 4.5 5 5.5 6 frequency [GHz] Fgure 2.17 Cmparsn f MM and SBR results fr θθ plarzed bstatc RCS f PEC plate wth respect t frequency The secnd example was cnducted n a flare lke bject shwn n Fgure 2.18. The bject s represented by 20 trangle patches. Maxmum length f the bject s 0.5 meter. The mnstatc RCS s calculated at φ = 0 frm θ = 0 t θ =180 fr vertcal plarzatn at 6GHz. Fr each mnstatc RCS value 72000 rays are traced wth λ/10 densty. Results are cmpared wth the PO slutn n Fgure 2.19. 30

Fgure 2.18 Smulatn setup fr mnstatc RCS f PEC flare-lke shape Fr the angle ranges 15 < 50 0 0 < θ, 80 < 90 0 0 < θ and 120 <160 0 0 < θ, the RCS values are nt matchng wth PO results. Ths dscrepancy between the results s due t the shape f the aperture ver whch the rays are cllected. At these angles, sme f the rays buncng frm the gemetry cannt be gathered at the planar ext aperture. After a detaled analyss, t s seen that the aperture shape must be mdfed n rder t btan crrect results. Alternatve slutn suggestns and frmulatns related t ths pnt are cnsdered n Sectn 2.3. In ths sectn, t wll be shwn that cnfrmal apertures that encmpass the scatterer are effectve n cllectng rays n a prper fashn. Althugh ths pnt wll be fully clarfed n the next sectn, crrected results btaned by a cnfrmal aperture are represented n Fgure 2.20. Wth ths new frmulatn that wll be presented n Sectn 2.3, results agree very well wth the PO slutn. 31

20 Mnstatc RCS f PEC flare-lke shape SBR FEKO,PO 10 0 RCS [dbm 2 ] -10-20 -30-40 0 20 40 60 80 100 120 140 160 180 Theta [degrees] Fgure 2.19 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC flare-lke shape fr planar aperture case Results n Fgure 2.20 shw that SBR algrthm gves the same results as PO slutn. Addtnally, ray tracng capablty f the cde handles the multple scatterng and shadwng prblems. 32

20 Mnstatc RCS f PEC flare-lke shape SBR FEKO,PO 10 0 RCS [dbm 2 ] -10-20 -30-40 0 20 40 60 80 100 120 140 160 180 Theta [degrees] Fgure 2.20 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC flare-lke shape fr cnfrmal aperture case The thrd example was cnducted n an arbtrary prsm shwn n Fgure 2.21. The shape s cmpsed f 14 trangular patches. The maxmum length f the shape s 0.35 meters. Mnstatc RCS s calculated at φ = 0 frm θ = 0 t θ = 360 fr vertcal plarzatns at 5GHz. The results are cmpared wth PO and MM results. 33

Fgure 2.21 Smulatn setup fr mnstatc RCS f PEC arbtrary plyhedrn 20 Mnstatc RCS f arbtrary plyhedrn FEKO,PO SBR 10 0 RCS [dbm 2 ] -10-20 -30-40 0 50 100 150 200 250 300 350 Theta [degrees] Fgure 2.22 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn 34

In Fgure 2.22, t s seen that the results are clse t the PO slutn except fr angles frm θ = 60 up tθ = 130. The prblem at these angles can be revealed when the gemetry s nvestgated mre clsely n Fgure 2.24. At prblematc aspect angles, rays that are sht twards the gemetry are ntersectng wth fur pnts lcated n the surface. The nave ray tracng algrthm f the cde s unable t handle such stuatns, where the bject s cncave and a ray may ntersect wth several pnts n the surface. In sectn 2.5, t s explaned hw the algrthm s vervewed t vercme ths prblem. SBR s a ray tracng methd and naturally ray tracng algrthms may crrectly handle the shadwng prblem. Althugh the mdfcatns ntrduced n the cde are explaned n Sectn 2.5, the crrect results btaned after these amendments are shwn n Fgure 2.23. 20 Mnstatc RCS f arbtrary plyhedrn SBR FEKO,PO 10 0 RCS [dbm 2 ] -10-20 -30-40 0 50 100 150 200 250 300 350 Theta [degrees] Fgure 2.23 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn after crrectn n shadwng algrthm After these mdfcatns n the cde, the results agreed wth the PO slutn except at angles190 0 < θ < 230. At these angles, multple scatterng s dmnant. 35

Fr ths example, multple scatterng s nt mplemented n the cde. The results are als cmpared wth the MM results. In Fgure 2.25, multple scatterng effects can be seen n MM results n an bvus fashn. Multple scatterng effects wll be mplemented n Sectn 2.4. Fgure 2.24 Shadwng effect 36

20 10 Mnstatc RCS f arbtrary plyhedrn FEKO,MM FEKO,PO SBR 0 RCS [dbm 2 ] -10-20 -30-40 0 50 100 150 200 250 300 350 Theta [degrees] Fgure 2.25 Cmparsn f PO, MM and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn The furth example was cnducted n an arbtrary plyhedrn shwn n Fgure 2.26. The shape s cmpsed f 8 trangular patches. The maxmum length f the shape s 0.3 meters. The mnstatc RCS s calculated at φ = 0 frm θ = 0 t θ = 180 fr vertcal plarzatns at 5GHz. The results are cmpared wth PO and MM results. 37

Fgure 2.26 Smulatn setup fr mnstatc RCS f PEC arbtrary plyhedrn 10 0 Mnstatc RCS f arbrary shape SBR FEKO,PO -10 RCS [dbm 2 ] -20-30 -40-50 -60 0 20 40 60 80 100 120 140 160 180 Theta [degrees] Fgure 2.27 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC arbtrary plyhedrn 38

The results agree very well wth the PO slutn fr ths arbtrary plyhedrn. The man reasn fr ths cnclusn s the cnvex shape f the bject that avds all the dffcultes encuntered n the prevus example. 2.3 SBR Fnal Integratn Aperture The Shtng and Buncng Ray methd has been frst appled n cavty structures lke a jet nlet r an arcraft cabn [2], where pen muth f the cavty has been chsen as the fnal ntegratn aperture. Hwever, fr pen scatterers there are several ptns fr the chce f an aperture chse. If the ext aperture s chsen as planar, ths chce gves gd results n specular regns. A planar aperture cannt be adequate fr gatherng the ext rays fr sme prblems. Bascally, the prblem can be nvestgated by usng the cube gemetry shwn n Fgure 2.28. Wth a planar ext aperture, t s nt pssble t cllect utgng rays at angles clse t 45 degrees. By fxng the transmtter (plane wave) and ext aperture (recever) and rtatng the gemetry, the mnstatc RCS f the cube can be calculated as shwn n Fgure 2.29. 39

Fgure 2.28 Planar ext aperture apprach 0 0 At angles 0 < θ < 30, 60 <120 0 0 < θ and 150 <180 0 0 < θ the results are n gd agreement wth bth MM and PO results. On the ther hand, at angles between 30 < 60 0 0 < θ and 120 <150 0 0 < θ the feld values are well belw thse btaned by MM and PO. At 45º SBR cde yelds value. If the MM and PO results are evaluated tgether, the dfference between them n these prblematc angles s nt much. S, t can be stated that the prblem s nt due t the absence f dffractn effect n the cde. At angles near 45º the rays buncng frm cube cannt be gathered at the planar aperture. Ths leads us t state that cntrbutn f rays at these angles are greater than dffractn effects, and ther cntrbutn t the RCS value must be taken nt accunt. T vercme ths prblem, a new ext aperture s recmmended and frmulated. 40

10 0 Mnstatc RCS f PEC cube SBR,Planar Aperture FEKO,MOM FEKO,PO -10 RCS [dbm 2 ] -20-30 -40-50 0 20 40 60 80 100 120 140 160 180 Thetas [degrees] Fgure 2.29 Mnstatc RCS f PEC cube wth planar ext aperture apprach A cnfrmal ext aperture as shwn n Fgure 2.30 can be the best chce t gather the ext rays. In cnfrmal ext aperture, all utgng rays are gathered and ther cntrbutns t verall RCS value can be calculated. Bascally, the cnfrmal ext aperture s a replca f the gemetry f the surface tself. Ths guarantes that all rays buncng frm the gemetry can be evaluated fr ther cntrbutn t RCS. Frmulatn f a cnfrmal ext aperture s mre cmplcated than a planar ext aperture frmulatn [8]. In rder t smplfy the resultng equatns, fr each element (planar part) f the cnfrmal ext aperture, a lcal crdnate system ( l l l x, y, z ) shall be defned as shwn n Fgure 2.30. The lcal crdnates are defned usng the fllwng equatns; z l = N (2.49) 41

N s the nrmal f the trangular patch wth whch the rays ntersect. T defne a lcal crdnate system, lcal z crdnate s prjected nt glbal x, y plane accrdng t the fllwng equatn; u = z ( z a )) a (2.50) l l z z Then, t fnd the glbal-t-lcal transfrmatn matrx, rtatn angles φ and θ are calculated by means f equatns (2.51) and (2.52). φ = cs 1 ( a x u] / u ) (2.51) θ = cs 1 ( a z N] / N ) (2.52) The glbal-t-lcal transfrmatn matrx can be defned thrugh the multplcatn f rtatn matrces. cs( φ) sn( φ) 0 cs( θ ) 0 sn( θ ) T = sn( φ) cs( φ) 0 0 1 0 (2.53) 0 0 1 sn( θ ) 0 cs( θ ) After defnng the lcal crdnate system values n equatn (2.42) and (2.43), are transfrmed nt lcal crdnate system. Then, the ncdent feld gven n the glbal crdnate system s transfrmed nt the lcal crdnate system as fllws: E ( x, y, z ) = E ( x, y, z ) T (2.54) l l l g g g Fnally, φ and θ parameters n equatn (2.51) and (2.52) are used n equatns (2.31) and (2.32). 42

yl zl z l y l x l x l z y x x l x l y l z l z l y l Fgure 2.30 Cnfrmal ext aperture apprach Wth aperture mdfcatn, at angles between 30 0 < θ < 60 and 120 0 < θ <150 the calculated values agree well wth MM and PO results as shwn n Fgure 2.31. 43

10 0 Mnstatc RCS f PEC cube SBR, Cnfrmal Aperture FEKO,PO FEKO,MM -10 RCS [dbm 2 ] -20-30 -40-50 0 20 40 60 80 100 120 140 160 180 Theta [degrees] Fgure 2.31 Mnstatc RCS results wth cnfrmal ext aperture apprach 2.4 SBR Multple Scatterng Capablty In the SBR cde, rays must be allwed t bunce wthn the gemetry untl they ext. The structure f the algrthm allws us t cmpute multple scatterng effects wthut a lmt unless we have extremely lng runtmes. Wth nn-cnvex targets that have cncave surfaces such as cavtes and crners, numercal results f PO devate frm ur smulatn results. Targets lke dhedral and trhedral are apprprate t test the cde fr multple scatterng capabltes. A PEC dhedral shwn n Fgure 2.32 wll be a gd test bject fr SBR cde. Clearly, scatterng frm a dhedral s dmnated wth multple scatterng effects. As shwn n Fgure 2.32 a dhedral s llumnated frm ts mnstatc RCS was cmputed wth respect t θ at 6GHz. θ = 60 t θ = 60 and 44

Fgure 2.32 Smulatn setup fr mnstatc RCS f PEC dhedral In Fgure 2.33 results are btaned wth and wthut cnsderng the multple scatterng effects. Clearly, the RCS values fr a prblem where multple scatterng s dmnant are nt matchng wth MM results when multple scatterng s nt mdeled n the cde. The SBR RCS values are n gd agreement wth the MM results when multple scatterng mechansm s mplemented n the cde. 45

10 0 Mnstatc RCS f PEC dhedral SBR FEKO, MM FEKO, PO SBR, w/ multple scatterng capablty -10 RCS [dbm 2 ] -20-30 -40-50 -60-40 -20 0 20 40 60 Theta [degrees] Fgure 2.33 Cmparsn f PO, MM and SBR results fr θθ plarzed mnstatc RCS f PEC dhedral 2.5 Shadwng Fr a specfc aspect angle, the ncdent electrmagnetc wave llumnates sme parts f the target and the rest f the target wll stay n dark. In addtn, sme parts f the target can be shadwed by ther parts f the target tself. Especally, cncave and separated parts f targets cause shadwng. Fr example, wngs f arplanes nduce shadwng frm certan aspect angles ver the bdy f the arcrafts. Many algrthms are develped t slve ths prblem fr the PO apprach. Naturally, SBR s usng the ray tracng methd, whch handles the shadwng effect n a natural way. Bascally, f a ray ntersects wth a part f the target fr 46

the frst tme, ths part wll be the llumnated part and the rest wll stay n the shadw regn. Fgure 2.34 Shadwng prblem In Fgure 2.34, trangle 9 s shadwed by trangle 15. The ray ntersects frst trangle 15 whch s taken as the llumnated part. As a summary, ray tracng s tself a natural shadwng algrthm. Ray tracng s mplemented t fnd the raytrangle ntersectn pnts, and the prmary ntersectn part s taken as the llumnated part, and the rest are assumed t le n shadw. 47

CHAPTER 3 APPLICATION OF SBR TO COMPLEX TARGETS In the prevus chapter, the SBR cde s tested by usng smple PEC bjects. Hwever, realstc targets pssess much mre cmplex gemetres wth several scatterng mechansms. The nteractns between parts f cmplex gemetres wll affect dramatcally the RCS values. At ths pnt the SBR cde, wth ts multple scatterng capablty, wll be superr t ther ray based RCS predctn tls. The RCS predctn cde develped n the prevus chapter wll be appled t cmplex targets n ths chapter, n rder t evaluate the perfrmance f the methd fr realstc cases. In ur smulatns, targets are mdeled usng large trangular patches. In rder t calculate the mnstatc RCS f a target frm dfferent aspect angles, tw dfferent methds are used. In the frst methd, the target kept fxed and the ray wndw n Fgure 3.1 s rtated. Ths methd s appled t smple shapes. 48

Fgure 3.1 Target and Ray Wndw Gemetry In the secnd methd, whch s used fr cmplex targets, the target s rtated. 3.1 Target Mdelng By usng a superpstn f smple prmtve bjects lke prsms, spheres r cylnders, cmplex targets can be mdeled. Hwever, such mdels may suffer frm precsn. Surface patch representatns wll yeld accurate target mdels. Trangular patches are very flexble fr mdelng targets effectvely. Hwever, f ur gemetrc mdel has a large number f patches, the ray-trangle ntersectn algrthm wll be the man cntrbutr t runtme. By usng brute frce search technques t s nt pssble t slve hgh frequency prblems n reasnable tmes. In ur examples, we cnstruct gemetres by usng large trangular patches shwn n Fgure 3.3. The gemetres are cnstructed n a Cmputer Aded Desgn (CAD) 49

mdelng tl. The gemetry s mprted n STL mesh frmat shwn n Fgure 3.2 and used n SBR MATLAB cde. Fgure 3.2 STL mesh structure Althugh cmplex targets are cnsdered n ths sectn, we start wth a smple case t remember the essental pnts n RCS calculatns. Fr a sngle trangular patch shwn n Fgure 3.3, the mnstatc RCS value s calculated. In the specular regn SBR results agree very well wth MM results. Outsde the specular regn, where dffractn s effectve, SBR results are devatng frm MM results. Snce ur cmplex targets are all cmpsed f trangular patches, t can be clamed that thse patches cntrbutng t specular reflectn wll dmnate. In the absence f such patches, SBR and MM wll nt match, due t the dmnant dffractn effects. In the fllwng examples, three dfferent cmplex targets are 50

smulated; frst a tank mdel whch s mdeled by 132 trangular patches, secnd a smple shp mdel cnstructed frm 28 trangular patches and last a stealth arcraft desgn cnstructed frm 20 trangular patches, whch s mre lke an F- 117 arcraft. Fgure 3.3 Smulatn setup fr mnstatc RCS f PEC trangular patch 15 10 Mnstatc RCS f PEC trangle SBR FEKO,MM 5 0-5 RCS[dBm 2 ] -10-15 -20-25 -30-35 -40-60 -40-20 0 20 40 60 Theta[degrees] Fgure 3.4 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC trangle 51

3.2 Target Rtatn Fr the vsualzatn f RCS varatn, we btan plts f RCS versus aspect angles. Aspect angle s the rentatn f target relatve t the surce pstn whch s a ray wndw. In rder t calculate the RCS as a functn f aspect angle, the target s rtated. In rder t rtate the target n three dmensns, three basc rtatn matrces are used; 1 0 0 R x ( α) = 0 cs( α) sn( α) 0 sn( α) cs( α) (3.1) cs( α) 0 sn( α) R = y ( α) 0 1 0 sn( α) 0 cs( α) (3.2) cs( α) sn( α) 0 R = z ( α) sn( α) cs( α) 0 (3.3) 0 0 1 Each f these matrces rtates a target n cunterclckwse drectn arund a fxed crdnate axs, by an angle fα. Rtatn drectn s determned by the rght-hand rules. Other rtatn matrces are derved frm these three basc matrces. In ur smulatns, fr rtatng a target that s n STL mesh frmat, all vertces f trangular patches are rtated abut the rthgnal axes f the glbal crdnate system. Smple rtatn gemetry s llustrated n Fgure 3.5. 52

Fgure 3.5 Fxed rtatn arund Y axs 3.3 Numercal Results The frst smulatn s cnducted n a smple tank mdel whch has a length f 1 m and a heght f 0.5 m. Mnstatc RCS values are calculated at angles frm θ = 0 t θ = 180 at 6GHz. The results btaned frm SBR MATLAB cde are cmpared wth thse btaned frm the FEKO sftware. In these smulatns, tw cmputers are used; Cmputer 1; Intel (R) Cre 2 Du CPU, E4500@2.20GHz 3.25GB RAM Cmputer 2; (Dual) AMD Optern Prcessr 248 14.926 GB RAM The tank s mdeled n CADFEKO, whch s the mdelng tl f FEKO sftware, and exprted n STL mesh frmat. The mdel cnssts f 132 trangular patches. 53

Fgure 3.6 Tank mdel cnstructed frm 132 trangular patches Table 3-1 Cmputatn tmes fr tank mdel smulatn Tme/Sample Ttal Tme RAM Sample Secnds Hurs MB Cmputer SBR, MATLAB 1181 181 59.4 --------- 1 PO, FEKO 0.19 181 0.01 13 1 54

20 10 Mnstatc RCS f tank mdel FEKO,PO SBR 0-10 RCS[dBm 2 ] -20-30 -40-50 -60-80 -60-40 -20 0 20 40 60 80 Theta [degrees] Fgure 3.7 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC tank mdel The results f FEKO (PO) and MATLAB (SBR) match clsely as shwn n Fgure 3.7. Althugh the mdel cntans nly a mderate number f patches, the runtme s very lng due t the lack f a fast ray-trangle ntersectn algrthm. Wth brute-frce ray-trangle ntersectn test, t takes 20 mnutes t calculate the mnstatc fr a sngle aspect angle. The secnd example s cnducted n a smple shp mdel wth length 1 m and heght 0.5 m. Mnstatc RCS values are calculated at angles frm θ = 0 t θ = 180 at 6GHz. The shp s mdeled n CADFEKO, mdelng tl f FEKO sftware, and exprted n STL mesh frmat. The mdel cnssts f 28 trangular patches. In ths example, the slutn s btaned by usng a planar ext aperture. Althugh we have mentned that the cnfrmal aperture apprach yelds ptmum results, ths example shws that the planar ext aperture chce can gve gd results fr targets whse scatterng characterstcs are dmnated wth specular regns. 55

Fgure 3.8 Mnstatc RCS f shp mdel Table 3-2 Cmputatn tmes fr shp mdel smulatn Tme/Sample Ttal Tme RAM Sample Secnds Hurs MB Cmputer SBR, Matlab 551 151 23.2 --------- 1 PO, FEKO 0.25 181 0.012 13 1 MLFMM, FEKO 2880 31 24.9 875 2 56

25 20 15 Mnstatc RCS f shp mdel SBR FEKO, PO FEKO, MLFMM 10 5 RCS [dbm 2 ] 0-5 -10-15 -20-25 -30 0 20 40 60 80 100 120 140 160 180 Theta [degrees] Fgure 3.9 Cmparsn f PO, MLFMM and SBR results fr θθ plarzed mnstatc RCS f PEC shp mdel The results btaned frm FEKO (PO and MLFMM) and SBR agree well n specular regns. Due t the fact that specular scatterng s dmnant n ths mdel the results match clsely wth thse btaned by FEKO (MLFMM) n the wde spectrum frm θ = 0 tθ = 180. The thrd smulatn s cnducted n a stealth arcraft desgn [11], whch s 1 m lng and has a 1 m wng span. The mnstatc RCS values are calculated at angles frm θ = 30 t θ = 150 at 6GHz. The results btaned frm the SBR MATLAB cde are cmpared wth the results f the FEKO sftware. 57

Fgure 3.10 F-117 mdel Table 3-3 Cmputatn tmes fr F-117 mdel smulatn Tme/Sample Ttal Tme RAM Sample Secnds Hurs MB Cmputer SBR, MATLAB 276 121 9.27 --------- 1 PO, FEKO 0.78 121 0.026 20 1 MLFMM, FEKO 2484 121 83.49 875 2 58

10 0 Mnstatc RCS f F-117 mdel arcraft SBR FEKO,PO -10 RCS [dbm 2 ] -20-30 -40-50 -60 40 60 80 100 120 140 Theta [degrees] Fgure 3.11 Cmparsn f PO and SBR results fr θθ plarzed mnstatc RCS f PEC F-117 mdel The results f PO and SBR agreed fr the angles between 0 0 < θ <120 as shwn n Fgure 3.11. Fr the angles120 0 < θ <150, the results are nt cnsstent. 10 Mnstatc RCS f F-117 mdel arcraft SBR FEKO,MLFMM 0-10 RCS [dbm 2 ] -20-30 -40-50 40 60 80 100 120 140 Theta [degrees] Fgure 3.12 Cmparsn f MM and SBR results fr θθ plarzed mnstatc RCS f PEC F-117 mdel 59

When the results are cmpared wth thse btaned by the MLFMM cde, t s clear that the results d nt match clsely. The man reasn f ths ncnsstency s the dffractn weghted structure f the gemetry. Snce the gemetry s a stealth arcraft, specular reflectns are avded fr all angles. Als, the SBR cde des nt nclude a multple scatterng capablty fr cmplex targets. Ths s the secnd reasn fr the ncnsstency between the results f SBR and MLFMM. 60

CHAPTER 4 CONCLUSIONS 4.1 Summary f the Thess A MATLAB cde based n SBR s mplemented t cmpute the RCS f cmplex targets. Results are cmpared wth bth full-wave and hgh frequency methds f FEKO sftware. Bascally, SBR s a hgh frequency methd that cmbnes the advantages f Gemetrc Optcs (GO) and Physcal Optcs (PO) appraches. In SBR, rays that smulate the ncdent plane wave are sht twards the target, bunce back and frth between the facets mdelng the PEC bject surface and fnally are captured n an ext aperture. The phase and ampltude f the electrcal feld s prpagated alng the rays wth the GO rules. In an ext aperture utgng rays are captured and the feld can be determned. Wth the PO apprxmatn a current sheet s determned n the aperture. Integratn n ths aperture s nt easy t carry ut. Instead f ths, a small ray tube s sht twards the gemetry and the scattered feld s cmputed by takng nt accunt ts sze and shape. Wth a suffcent number f ray tubes that mdels the ncdent feld sht nt the gemetry, the RCS s calculated. SBR was mplemented fr the frst tme t slve cavty prblems, n whch the ext aperture s the muth f the cavty, whch guarantees that all the utgng rays are captured there. In ths thess, the methd s mplemented fr pen 61

scatterers where the ext aperture selectn s crtcal. A new ext aperture cnfguratn named as cnfrmal aperture s prpsed and the results f ths cnfguratn are cmpared wth thse btaned va the cmmn aperture selectns. Wth ths aperture cnfguratn, the results matched the PO results. The results btaned frm the MATLAB cde match clsely wth PO and MM results near specular regns. Out f the specular regns, the results d nt match due t the fact that dffractn effects are nt mdeled n the cde. Bascally, SBR s a multple scatterng algrthm. Ths capablty f the cde s nvestgated and verfed. Als, t s a ray tracng apprach whch handles the shadwng prblem easly. Sme cmplex target mdels are generated n CADFEKO tl and mprted t SBR cde n STL mesh frmat. Fr the ray-trangle ntersectn test, brute-frce methds are used whch turned ut t be the man reasn fr the excessvely lng run-tmes. Ths dffculty led us t cnsder targets mdeled wth a mderate number f patches. The cde s develped n MATLAB 2007b versn. There s n need a specal MATLAB tlbx t mplement ths algrthm. 4.2 Advantages and Dsadvantages f SBR RCS predctn tls are generally used t desgn stealth structures. In stealth desgns usually specular reflectns, that cntrbute a lt t the RCS, are tred t be avded. Therefre, fr such applcatns t becmes crucal t predct multple scatterng and/r dffractn effects. SBR s a cde that handles multple scatterng up t any arbtrary rder. Wth the help f dffractn cdes lke Physcal Thery f Dffractn (PTD) r Gemetrc Thery f Dffractn (GTD), ths algrthm can be useful fr smulatns that are utlzed n stealth desgn. Als, SBR can 62

handle shadwng prblem autmatcally wth the help f ts ray tracng capablty. Snce SBR s a ray tracng based apprach, t s nt effcent unless a fast raytrangle ntersectn algrthm s used. Wth brute frce methds n ray-trangle ntersectn, t s nt feasble t handle realstc prblems mdeled by a large number f facets. 4.3 Future Wrk In ths wrk, a cmputatnal electrmagnetcs cde s created by usng the SBR algrthm n MATLAB. It s stated that fr pen scatterers the ext aperture selectn s crtcal. A cnfrmal ext aperture algrthm s develped and appled ver cmplex targets. The cde s tested fr ts multple scatterng capabltes. Fr future wrk, the cde can be mdfed and expanded t handle delectrc structures and catngs. Als, wth brute-frce ray-trangle ntersectn test, the cde s nt effcent wth regard t runtme. Wth a fast ray-trangle ntersectn algrthm, lke malbx r kd-tree, the cde can be effcently used fr mre realstc targets. The SBR algrthm cannt handle dffractn effects. S, by ncludng a rutne that handles dffractn effects (.e. a cde based n Physcal Thery f Dffractn (PTD) r Gemetrc Thery f Dffractn (GTD)), the cde can be used n stealth desgn applcatns. Anther applcatn may be the generatn f Inverse Synthetc Aperture Radar (ISAR) mages wth the help f SBR cde. A cmputatnal electrmagnetc tl that nly calculates the RCS f realstc targets s nt sufcent t acheve stealth desgn. By vsualzatn tls lke ISAR magng r range prflng, scatterng centers n a target can be detected. It s pssble t create the ISAR mage f a target frm mnstatc r bstatc RCS data f the SBR cde. Ths wll be the pst prcessng stage f the SBR cde and 63

may help us t fnd and/r annhlate scatterng centers f bjects n stealth applcatns. 64

REFERENCES [1] Chu R., Lng H., Lee S.W., Reductn f the Radar Crss Sectn f Arbtrarly Shaped Cavty Structures, Techncal Reprt, Unversty f Illns August 1987, N. 87-6. [2] Lng H., Chu R., Lee S.W., Shtng and Buncng Rays: Calculatng the RCS f an Arbtrarly Shaped Cavty, IEEE Transactns f Antennas and Prpagatn, Vl. 37 N.2, February 1989, pp. 194-205. [3] Baldauf J., Lee S.W., Ln L., Jeng S., Scarbrugh S., Yu C.L., Hgh Frequency Scatterng frm Trhedral Crner Reflectrs and Other Benchmark Targets: SBR Versus Experment, IEEE Transactns f Antennas and Prpagatn, Vl. 39 N. 9, September 1991, pp. 1345-1351. [4] Gallway E.R., Welsh T.D., Extensn t the Shtng Buncng Ray Algrthm fr Scatterng frm Dffusve and Gratng Structures, Internatnal Radar Sympsum, Inda, 2005. [5] Cmprehensve Electrmagnetc Slutns FEKO, http://www.fek.nf/, last vsted n August 2009. [6] Lee S.W., Chu R., Lng H., Ray-Tube Integratn n Shtng and Buncng Ray Methd, Mcrwave and Optcal Technlgy Letters, Vl.1 N.8, Octber 1988, pp. 286-289. [7] Lee S.W., Mttra R., Furer Transfrm f a Plygn Shape Functn and Its Applcatn n Electrmagnetcs, IEEE Transactns n Antennas and Prpagatn, Vl. AP-31,N. 1, January 1993, pp. 99-103. [8] Albayrak N.A., RCS Cmputatns wth PO/PTD fr Cnductng and Impedance Objects Mdeled as large Flat Plates, M.Sc. Thess, Blkent Unversty, Ankara, July 2005. 65

[9] Marschner S., CS465 Ntes: Smple Ray-Trangle Intersectn, Curse Ntes, Crnell Unversty, Octber 2003 [10] Chu F., Huang C., On the Calculatn f the Furer Transfrm f a Plygnal Shape Functn, Jurnal f Physcs A: Mathematcal and General, Vl. 22 N. 14, Department f Electrcal Engneerng f Tatung Insttute f Technlgy, Aprl 1989, pp. 671-672. [11] RCS Calculatns Usng the Physcal Optcs Cdes, http://arcraftdesgn.nuaa.edu.cn/l/sftware/pcdes.pdf, last vsted n August 2009. 66