FOR ELECTROMAGNETIC SCATTERING FROM PENETRABLE OBJECTS DISSERTATION. Nilufer A. Ozdemir, M.S. * * * * * The Ohio State University. Prof.

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1 METHOD OF MOMENTS SOLUTION OF A NONCONFORMAL VOLUME INTEGRAL EQUATION VIA THE IE-FFT ALGORITHM FOR ELECTROMAGNETIC SCATTERING FROM PENETRABLE OBJECTS DISSERTATION Presented in Partial Fulfillent f the Requireents fr the Degree Dctr f Philsphy in the Graduate Schl f The Ohi State University By Nilufer A. Ozdeir, M.S. * * * * * The Ohi State University 007 Dissertatin Cittee: Prf. Jin-Fa Lee, Adviser Prf. Edward Newan Dr. Rbert Burkhlder Graduate Prgra in Electrical and Cputer Engineering Prf. Sandip Mazuder

2 ABSTRACT This thesis is aied at develping a vlue integral equatin which 1) is well-psed, i.e., when successively finer eshes are utilized t prduce re accurate results by the ethd f ents, the cnditin nuber f the cefficient atrix reains bunded and the eigenvalue distributin preserves itself; ) leads t a syetric cefficient atrix when slved by the ethd f ents prvided that the ediu that fills the bject f interest is reciprcal; 3) is applicable t a nncnfral esh where the ndes d nt atch alng an interface. Previus studies and develpents in the ethd f ents slutin f vlue integral equatins require a cnfral esh t discretize a penetrable bject due t the atheatical cnstraints ipsed n the basis and/r testing functins. In rder t avid any atheatical cnstraints n the basis and/r testing functins, we apply the gradientgradient peratr n the integral equatin kernel, which is the free space Green s functin, instead f splitting it t perate n the basis and/r testing functins. This gives us the flexibility t chse piecewise cnstant functins fr expansin and testing. Hence the esh t discretize the bject f interest need nt be cnfral. Mrever, the cefficient atrix is syetric fr a reciprcal ediu due t the syetry f the ii

3 vlue integral peratrs in the frulatin. Hwever, applying the gradient-gradient peratr n the Green s functin leads t a hypersingular integral which needs t be regularized. In this study, we adpt a regularizatin schee which adds and subtracts a functin, which can be integrated analytically n the surface f a vlue eleent, t the integral equatin kernel. A atheatical prf is prvided t shw that the nncnfral vlue integral peratr is cercive; in ther wrds, the cnditin nuber f the cefficient atrix is bunded. The nuerical study supprts the well-psedness f the frulatin. The analysis f penetrable structures has traditinally been carried ut using partial differential equatin ethds due t the large cputatin tie and ery requireents f integral equatin ethds. T alleviate this prble, this thesis extends the fast algrith tered IE-FFT t the ethd f ents slutin f nncnfral vlue integral equatins. Previus studies shw that the IE-FFT algrith reduces the cputatinal tie and ery requireent t O (N 1.5 lgn) and O (N 1.5 ), respectively, fr surface integral equatins t slve electragnetic scattering prbles where N is the nuber f unknwns. When the IE-FFT algrith is applied t nncnfral vlue integral equatins, the cputatinal cplexity and ery requireent reduce t O (NlgN) and O (N), respectively. iii

4 VITA April 3, Brn Diyarbakir Turkey B.S. Electrical and Electrnics Engineering, Middle East Technical University, Ankara, Turkey 000 M.S. Electrical and Electrnics Engineering, Middle East Technical University, Ankara, Turkey Graduate Teaching and Research Assciate, The Ohi State University, Clubus Ohi FIELDS OF STUDY Majr Field: Electrical and Cputer Engineering iv

5 TABLE OF CONTENTS ABSTRACT ii VITA. iv LIST OF FIGURES.vii LIST OF TABLES...x CHAPTER 1. INTRODUCTION Overview Suary Outline CHAPTER. A NONCONFORMAL VOLUME INTEGRAL EQUATION FOR ELECTROMAGNETIC SCATTERING FROM PENETRABLE STRUCTURES.1.1. Intrductin Thery Prble Stateent Integral Representatin Discretizatin Testing Prcedure Matrix Slutin Singular Integral Treatent Nuerical Results Validatin...8 Cnvergence Study...40 v

6 CHAPTER 3. IE-FFT ALGORITHM FOR VOLUME INTEGRAL EQUATIONS Overview Interplatin f the Green s Functin By Lagrangian Plynials Lagrange Interplatin Teplitz Matrix G Representatin f Π Matrices Crrectin f Matrix Entries fr Near Eleents Fast Matrix Vectr Prduct Nuerical Results Validatin...58 Cplexity Study...64 CHAPTER 4 CONCLUSIONS.. 68 APPENDIX A INTEGRAL EQUATION REPRESENTATION OF Er ( ) Hr ( ) AND APPENDIX B REGULARIZATION OF THE HYPERSINGULAR INTEGRAL IN VOLUME INTEGRAL EQUATIONS APPENDIX C THE COERCIVITY OF THE VOLUME INTEGRAL EQUATION OPERATOR C.1. Integral Equatin Representatin C.. Nr f the Vlue Integral Operatr and its Bund C.3. Cercivity f the Vlue Integral Operatr APPENDIX D DUFFY TRANSFORM FOR WEAKLY SINGULAR INTEGRALS OVER A TETRAHEDRON,. 104 LIST OF REFERENCES vi

7 LIST OF FIGURES Figure Page.1 An bject with bth ε( r ) ( r ) and μ (with bth electric and agnetic prperties) in free space can be replaced with a ediu with distributed plarizatin current J( r) M( r) and that radiate in free space Cparisn f bistatic RCS f a dielectric sphere in E and H planes with the Mie inc -jkz series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ Cparisn f bistatic RCS f a agnetic sphere in E and H planes with the Mie inc -jkz series slutin The sphere is illuinated at θ = 0 with E = xe ˆ Cparisn f bistatic RCS f a dielectric sphere f r = 0.5λ in E and H planes with the Mie series slutin The sphere is illuinated at θ= 0 with inc -jkz E = xe ˆ Cparisn f bistatic RCS f a agnetdielectric sphere in E and H planes with Mie series slutin. The sphere is illuinated at θ = 0 inc -jkz with E = xe ˆ Cparisn f bistatic RCS f a dielectric shell in E and H planes with the VIE slutin by Schaubert et al. 65H [1].The shell is illuinated at θ = 0 inc -jkz with E = xe ˆ Crss sectin f the nncnfral esh fr the layered sphere exaple alng the center is illustrated. The inner and uter spheres have radius f 0.18λ and 0.λ, respectively Cparisn f bistatic RCS f a tw-layer sphere in E and H planes with the Mie inc -jkz series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ vii

8 .9 Crss sectin f the nncnfral esh fr the layered shell exaple alng the center is illustrated Cparisn f bistatic RCS f a 3-layer dielectric shell in E and H planes with the Mie series slutin. The shell is illuinated at θ= 0 with inc -jkz E = xe ˆ Relative errr versus the nuber f unknwns fr the cnvergence study Nuber f iteratins versus the nuber f unknwns fr the cnvergence study A -D pictrial representatin f the IE-FFT algrith (a) Side view f a sphere discretized int 180 tetrahedra, with spatial decpsitin int a 3 x 3 x 3 array f cells. (b) Superipsed grid pints crrespnding t the cell decpsitin f (a), with p = One Cartesian eleent when p= One diensinal FFT in ne directin f a Cartesian eleent Blck Teplitz structure fr the 3-D FFT at a Cartesian eleent Cparisn f bistatic RCS f a dielectric sphere ( ε =. r ) in E and H planes with the Mie series slutin. The shell is illuinated at θ= 0 with inc -jkz E = xe ˆ Cparisn f bistatic RCS f a lssy dielectric sphere ( ε = 1.5 j0.5 r ) in E and H planes with the Mie series slutin. The shell is illuinated at θ= 0 with inc -jkz E = xe ˆ Cparisn f nuerical and analytical results fr electragnetic scattering fr dielectric shell cated dielectric sphere Cparisn f nuerical and analytical results fr electragnetic scattering fr a three-layer dielectric sphere Mery requireent t stre Z crr and Π atrixes versus nuber f unknwns viii

9 crr 3.11 CPU tie required t asseble Z Z,, Π atrices and CPU tie required fr a single iteratin in the iterative slver B.1 Gaussian quadrature is perfred fr Iii R k l r = r we take the integral inside a sphere f vlue. Instead f the undefined value fr k w...76 B. Geetric definitins fr the evaluatin f the singular integral when P is in the surce triangle..79 B.3 Geetric definitins fr the evaluatin f singular integral when P is utside the plane f surce face.. 83 C.1 V is an arbitrary vlue in Ω cntaining the bservatin pint r...95 D.1 The ain tetrahedrn is brken int fur sub-tetrahedra which share vertex P D. Affine transfr aps an arbitrary tetrahedrn t the aster tetrahedrn D.3 The Duffy transfratin aps the aster tetrahedrn t a triangular pris ix

10 LIST OF TABLES Table 1. Perfrance f the IE-FFT Algrith fr Nncnfral VIE.. 66 Table. Gaussian Quadrature Rule n the Base f the Triangular Pris fr Quadratic Variatin..111 Table 3. Gaussian Quadrature Rule Alng the Height f the Triangular Pris fr Quadratic Variatin.. 11 Table 4. Gaussian Quadrature Rule ver the Triangular Pris fr Quadratic Variatin in ( x,y,z ) Crdinates Table 5. Gaussian Quadrature Rule ver the Master Tetrahedrn fr Quadratic Variatin in ( x,y,z ) Crdinates Table 6. Gaussian Quadrature Rule ver an Arbitrary Tetrahedrn fr Quadratic Variatin in ( ς, ς, ς, ς 1 3) Crdinates x

11 CHAPTER 1 INTRODUCTION 1.1. Overview Electragnetic scattering and radiatin prbles invlving penetrable structures find a wide range f applicatins. Se f these applicatins are 1) radi wave prpagatin thrugh buildings and frests in rain and snw [1], ) radar crss sectin cputatin f aterial cated PEC targets [1] 3) analysis f printed circuit and icrstrip antenna n a finite dielectric substrate [3], 4) indr radi wave prpagatin [4], 5) analysis f scattering by airbrne particulates [5], [6] 6) rade design [7], 7) interactin f electragnetic fields with bilgical edia [8], 8) nnlinear eddy current analysis [9], 9) surface tgraphy, 10) analysis f scattering by anistrpic and high cntrast cplex edia such as newly intrduced agnetic phtnic crystals [10]-[13]. The frulatins gverning these applicatins ay be in the fr f either a differential r an integral equatin. Ang the differential equatin ethds, finite eleent ethd (FEM) [14] and finite-difference tie dain (FDTD) ethd [15] are st widely applied. Bth ethds 1

12 require the incrpratin f esh truncatin schees t iic the electragnetic waves prpagating in unbunded dain [16]. On the ther hand, bth the vlue and surface integral equatins satisfy the radiatin cnditin inherently because the Green s functin exists in the integral equatin kernel. Mrever, nntrivial backgrunds like ultilayer structures can easily be included in the Green s functin [17]. Integral equatin frulatin fr electragnetic scattering by penetrable bjects has been the tpic f extensive research since the initial study by Richnd [18]-[19], wh eplyed ethd f ents (MM) fr the prble f tw-diensinal (-D) scattering fr hgeneus dielectric cylinders f arbitrary crss sectin. In this study, the scatterer crss sectin has been deled by square cells, piecesewise cnstant basis functins are used t apprxiate the unknwn electric field within each cell and pint atching at the cell centers is applied fr testing. This study is fllwed by Livesay and Chen [8] wh extended it t three-diensinal (3-D) hgeneus bjects deled by cubical eleents. Since these initial studies research n integral equatin frulatins t slve electragnetic scattering by penetrable bjects has fcused n tw ain tpics: 1) geetric deling f the bject f interest and apprpriate basis functins t apprxiate the unknwn vectrs within the scatterer [0]-[9]; ) efficient ethds fr the ethd f ents slutin f integral equatins. [30]-[38]. The purpse f this study is t develp a vlue integral equatin (VIE) which 1) is well-psed, i.e., when successively finer eshes are utilized t prduce re accurate results by the ethd f ents, the cnditin nuber f the cefficient atrix reains bunded and the eigenvalue distributin preserves itself; ) leads t a syetric

13 cefficient atrix when slved by the ethd f ents prvided that the ediu that fills the bject f interest is reciprcal; 3) is applicable t a nncnfral esh where the ndes d nt atch alng an interface as seen in Figure 1.1. Furtherre, the efficiency and accuracy f the IE-FFT algrith [39] is investigated fr the MM slutin f the nncnfral VIE. Figure 1.1. Crss sectin f the nncnfral esh fr the layered sphere exaple alng the center is illustrated. The inner and uter spheres have radius f 0.18λ and 0.λ, respectively. 3

14 1.. Suary Previus develpents and applicatins f VIEs require a cnfral esh in discretizing penetrable structures due t the atheatical cnstraints n the basis and/r testing functins. Basis functins that expand the unknwn vectrs are required t fllw the atheatical cnstraints inherited fr the integral equatin frulatin. While divergence ( ) peratr ipsed n the unknwn vectr enfrces the basis functin t be divergence cnfring, curl ( ) peratr enfrces it t be curl cnfring. Divergence and curl cnfring basis functins ensure nral and tangential cntinuity f the unknwn vectrs acrss the faces f vlue eleents, respectively. Mst f the earlier papers n VIEs have used piecewise cnstant representatin f the electric plarizatin current J r electric field intensity E in square and cubical eleents fr -D and 3-D prbles, respectively [18]-[0]. Hwever, a piecewise cnstant representatin fr the electric field r the plarizatin current intrduces fictitius charge layers at every cell bundary, with resulting errr in the fields. The errr tends t grw as the relative perittivity increases and the cell size decreases. The divergence cnfring vectr basis functins intrduced by Schaubert et al. [1] which ensure the nral cntinuity f the electric flux density D acrss the cn faces f adjacent tetrahedra are re rbust and therefre re frequently used. In the ipleentatin f the divergence cnfring vectr basis functins, ne intrduces surface charge prprtinal t the difference in the perittivities f the adjacent tetrahedra n their cn face t satisfy the cntinuity cnditin fr J. 4

15 The nuber f unknwns fr the face-based divergence cnfring vectr basis functins is equal t the nuber f faces f the esh. Fr a tetrahedral esh the nuber f faces is cnsiderably greater than the nuber f edges. This is a disadvantage cpared t the edge-based FEM basis functins where the nuber f independent unknwns fr the syste atrix is even saller than the nuber f edges. Thus Carvalh and Mendes [3] have eplyed basis functins that still acquire the cnditins f the cntinuus nral D -cpnent, but include all the tetrahedra that share the edge which is siilar t the edge-based divergence free basis functins. Hwever, ne needs t perfr a preliinary peratin which eliinates the null space f the basis set. T btain a nncnfral VIE frulatin ne can avid ipsing the divergence r curl peratrs n the unknwn vectrs by applying instead the gradient-gradient peratr n the Green s functin. This prvides the flexibility t chse piecewise cnstant functins fr expansin and testing. Hence the esh that discretizes the bject f interest need nt be cnfral. In ther wrds, the axiu edge length in discretizing an inhgeneus structure can be deterined independently fr each inhgeneity, resulting in fewer vlue eleents, and therefre fewer unknwns. Mrever, the resulting cefficient atrix is syetric fr a reciprcal ediu due t the syetry f the vlue integral peratrs in the frulatin. Hwever, applying the gradient peratr n the Green s functin leads t a 3 ter 1R in the integral equatin kernel, where R is the distance between a surce and an bservatin pint. When the surce and bservatin pints verlap r lie in the sae eleent, the crrespnding integrals diverge withut a regularizatin schee [13], 5

16 [40]. In general, the regularizatin schee requires subtracting fr the integrand a functin with the sae singular behavir, but which can be integrated analytically. Hence the riginal integral can be separated int regular and singular parts. The kernel f the regular integral exhibits reduced degree f singularity; therefre, it can be evaluated nuerically. There are tw appraches t cpute the singular part f the riginal singular integral. The first ne is t take the Cauchy principal value by intrducing an exclusin vlue arund the singularity, which is well suited fr siple eleents such as sphere r cube, but ay be difficult t btain fr a general eleent such as tetrahedrn [41]. Thus ne ay prefer t use the secnd apprach that reduces the riginal vlue integral t a surface integral n the eleent bundary [13], [40]. In this study, we adpt the regularizatin schee in [13] which has been applied in -D prbles and denstrate that this schee can be extended t 3-D prbles t prvide reliable results. In rder t btain a cnvergent nuerical slutin, ne needs t regularize the integrals that represent interactins between neighbr eleents using the sae schee as well. After regularizing the self and near-interactin integrals, the nuber f iteratins fr the syste even withut a precnditiner stays alst cnstant when finer discretizatins are utilized fr re accurate nuerical slutins. This iplies that successively finer discretizatins f the VIE prduce cefficient atrices with bunded cnditin nuber and apprxiately the sae eigenvalue distributin and thus the sae cnvergence prperties. In [4], Rahla et al. have even estiated the cnvergence speed f the iterative slver based n the eigenvalue distributin f the cefficient atrix. Nevertheless, the develped estiate is nt practical because it requires cputing the eigenvalues f the cefficient atrix, which ay bece tedius fr large prbles. 6

17 Due t their high cputatinal cst, VIEs are largely superseded by the differential equatin ethds. When VIEs are slved by MM, slutin f the atrix equatin by a direct atrix slver, such as Gaussian eliinatin, t btain the unknwn vectr 3 distributin within the scatterer requires O( N ) ery and O( N ) slutin tie, where N is the ttal nuber f unknwns needed t characterize the vluetric scatterer. Even fr an electrically sall vluetric scatterer, this cputatinal cst can exhaust the available resurces. Hence fast slvers bece essential tls t slve cplex real life prbles. T reduce the cputatinal cplexity f the direct slutin f MM via a atrix equatin with Gaussian eliinatin Chew et al. [30]- [33] have develped a series f recursive T-atrix algriths: recursive aggregate T-atrix algrith (RATMA) [30], recursive aggregate interactin atrix algrith (RAIMA) [31] and nested equivalence principle algrith (NEPAL) [3], [33]. The basic idea f these appraches is t nest ne algrith within anther s that a saller prble is slved befre a larger ne. RATMA 73 has O( N ) and ( ) O N cputatinal cplexity fr -D and 3-D scattering prbles, respectively. RAIMA is a dificatin f RATMA and has the sae rder f cputatinal cplexity as RATMA. NEPAL slves the VIEs with cputatinal cplexity f 1.5 N in -D and principle. N in 3-D by expliting the Huygen s equivalence Using iterative slvers, such as the cnjugate gradient (CG) ethd r the generalized iniu residual (GMRES) ethd instead f direct slvers invlves atrix-vectr 7

18 prduct that has O( N ) cputatinal cplexity. This results in ( ) O pn final cputatinal cplexity, where p is the nuber f iteratins fr the iterative slver t cnverge t the desired slutin within predeterined tlerance. Althugh the iterative slvers are re efficient than the direct slvers, the cputatinal deands ay still be unbearable fr realistic prbles. In an attept t expedite the atrix-vectr prduct in iterative slvers, several ethds/algriths have been prpsed and successfully applied. The st widely used apprach t slve VIEs is the cnjugate-gradient (CG) fast Furier transfr (FFT) ethd [34]. This ethd requires the vlue f the bject t be discretized int unifr hexahedral cells in rder t use the Teplitz prperty f the cefficient atrix and hence t cpute the atrix vectr prduct via FFT. Therefre the ethd requires O( N ) ery and O( NlgN ) slutin tie. Hwever, the staircasing errr intrduced by the unifr grid t achieve the Teplitz syste atrix can be a ajr liitatin fr bjects with curved surfaces and thin cating. In rder t avid this liitatin se irregular esh based algriths such as the ultilevel fast ultiple algrith (MLFMA) [35], [36], which is the ultilevel ipleentatin f fast ultiple ethd (FMM) [43], the precrrected FFT ethd [37], which has widely been applied in lw frequency regie [44], and the adaptive integral ethd (AIM) [38], which has initially been prpsed fr surface integral equatins (SIEs) [45], have been extended t speed up the slutin f VIEs. All f these ethds achieve their efficiency by apprxiating the far zne interactins. 8

19 The FMM represents the Green s functin by ultiple spherical expansin fr far zne interactins. The ultilevel ipleentatin f the ethd recycles infratin between cnsecutive levels which helps t reduce the cputatinal cplexity. It has been shwn thrugh nuerical study that MLFMA requires O( N ) ery and O( NlgN ) slutin tie t slve VIEs [35], [36]. Bth the precrrected FFT ethd [37] and AIM [38] use fictitius unifr grid t prject the actual surces in the nnunifr esh, i.e., the grid pints represent the equivalent surces that prduce the sae effect as the actual surces at a predefined testing pint. Hence the interactin between the equivalent surces n the unifr grid can be cputed via FFT. The cputed ptentials are then interplated in the nnunifr esh. The near interactins are cputed directly. By the virtue f prjectin f the actual surces nt the unifr grid and the ipleentatin f atrix-vectr prduct via FFT, bth ethds reduce the ery requireent t O( N) and slutin tie t O( NlgN ), respectively. There exists anther class f slvers that are nt as efficient as the afreentined ethds, yet relatively less dependent n the integral equatin kernel. This class includes IES 3 [46], adaptive crss-apprxiatin (ACA) ethd [47], [48], single level IE-QR algrith [49], which cpress the glbal cefficient atrix thrugh lw rank apprxiatins f lcal atrices that represent interactins between physically wellseparated grups f eleents. This lw rank apprxiatin can either be achieved by QR factrizatin as in the case f IES 3 and single-level IE-QR algrith r clun pivted Gaussian eliinatin as in the case f ACA ethd. These slvers suit the integral equatins that invlve nn-scillatry kernels which ccur in electrstatic applicatins 9

20 such as parasitic paraeter extractin. Hwever, the cputer cdes that ipleent these ethds can still accelerate the slutin f electragnetic scattering and radiatin prbles in a wide range f applicatins with a few r n dificatins [49]. We prpse t extend the IE-FFT algrith [39], which has been develped t slve SIEs that arise in electragnetic scattering fr PEC bjects, t the MM slutin f VIEs. The IE-FFT algrith is cnsidered in the class f irregular esh based grid ethds. The algrith relies n the sapling f the integral kernel (and its gradient) n a Cartesian tensr grid and its interplatin by Lagrangian plynials. The actin f this kernel representatin n a vectr can be cputed efficiently by eans f FFT. Fr the near interactins, a crrectin ter is required, since the accuracy f the sapled representatin wuld be insufficient. One shuld nte that the chice f Lagrangian plynials fr interplatin is due t its siplicity; nly inr dificatins are needed t extend the interplatin t ther standard plynials such as Newtnian r trignetric plynials. The IE-FFT algrith is ipleented in fur ain steps: 1) interplatin f the Green s functin by Lagrangian plynials; ) prjectin f the basis functins nt the Cartesian grid; 3) crrectin f the near interactin ters; 4) atrix-vectr prduct by FFT. By the virtue f the prjectin f the basis functins nt the Cartesian tensr grid and atrix-vectr prduct using FFT, the algrith is expected t require O( N ) ery and O( NlgN ) slutin tie in the slutin f VIEs Outline The thesis can be brken int tw ain bdies. Chapter is devted t the develpent f a nncnfral VIE with its atheatical fundatins. Nuerical 10

21 results are prvided t verify the frulatin. Appendices A, B C, and D supprt Chapter. Appendix A derives the integral representatin; Appendix B describes the regularizatin schee t treat the hypersingular integral in detail; Appendix C prvides the atheatical prf f the cercivity f the nncnfral vlue integral peratr, i.e., well-psedness f the nncnfral VIE; Appendix D suarizes the Duffy transfratin t treat weakly singular vlue integral equatins. Chapter 3 extends the IE-FFT algrith t the MM slutin f the nncnfral VIE. The nuerical results n cannical exaples prve the accuracy and efficiency f the algrith. Finally; Chapter 4 suarizes the cnclusins f the study. 11

22 CHAPTER A NONCONFORMAL VOLUME INTEGRAL EQUATION FOR ELECTROMAGNETIC SCATTERING FROM PENETRABLE STRUCTURES.1. Intrductin In this chapter, we revisit the VIE frulatin fr electragnetic scattering fr inhgeneus anistrpic edia with nntrivial cplex perittivity and pereability tensrs. We btain a cupled integral equatin that represents bth E( r) and H( r) t include bth electric and agnetic prperties f the scatterer. T btain a nncnfral VIE frulatin we avid ipsing the divergence r curl peratrs n the unknwn vectrs by applying instead the gradient-gradient peratr n the Green s functin. Applying the gradient-gradient peratr n the Green s functin instead f splitting it t perate n the basis and/r testing functin gives us the flexibility t chse piecewise cnstant functins t expand the electric and agnetic field intensities, E( r) and H( r) and t test the electric and agnetic plarizatin current 1

23 densities, J( r) and M( r), respectively. Hence the tetrahedral esh, which prvides an accurate and flexible geetric del f the scatterer, need nt be cnfral, i.e. the ndes at the interfaces f tw tetrahedra d nt need t atch. The rganizatin f this chapter is as fllws. In Sectin. we state the prble fr electragnetic scattering by inhgeneus, anistrpic 3-D bjects with nntrivial cplex perittivity and pereability tensrs ( r ; r ) μ I ε I and present the VIE frulatin t represent E and H. The variatinal stateent t cnvert these representatin frulae int a atrix equatin is als given in Sectin. alng with the details n the cputatin f the atrix entries. The nuerical results are presented in Sectin.3... Thery..1. Prble Stateent We assue that a ediu filling a dain Ω bunded by surface S is characterized by the perittivity,ε and the pereability,μ tensrs whse cpnents can be functins f crdinates. Outside the dain Ω, the paraeters f the ediu are cnstant and the ediu is istrpic, i.e., ε=ε = cnstant andμ =μ = cnstant. We seek t deterine inc the electragnetic field excited in the ediu by an incident plane wave E ( r) r inc extraneus currents J ( r) inc and M ( r) with tie dependence j t e ω. One can present this prble by the fllwing atheatical frulatin: 13

24 Seek vectr functins E( r) and H( r) that satisfy the Maxwell s equatins inc H( r) = jω εe( r) + J ( r), (.1) E r = jω H r M r inc ( ) μ ( ) ( ) everywhere, and the Silver-Muller radiatin cnditin at infinity where k =ω μ ε is the free space wave nuber and ω = π f is the angular frequency f peratin. One can als express (.1) in equivalent fr as = ωε + E r = jωμ H r M r (.) H( r) j E( r) J( r) ( ) ( ) ( ) where inc p inc J( r) = J ( r) + J ( r) = J ( r) + jω( ε εi) E( r). (.3) M r M r M r M r j H r inc p inc ( ) = ( ) + ( ) = ( ) + ω( μ μ I) ( ) Here, p J r j E r ( ) = ω( ε ε I) ( ) (.4) is the electric plarizatin current density and p M r j H r ( ) = ω( μ μ I) ( ) (.5) is the agnetic plarizatin current density, which is nnzer in dain Ω [50]. Hence ne can view (.) as the Maxwell s equatins in a hgeneus ediu; i.e. ne ay assue that the sught-fr electragnetic fields E( r) and H( r) are prduced 14

25 p by currents J ( r) p replaced by J ( r) p and M ( r) p and M ( r) in free space. Thus the riginal dain Ω can be radiating in free space. One can see the equivalent prble in Figure.1. E H inc J inc inc ( r) ( r) ( r) Ω ( μ,ε) inc Ωe J ( r) Ωe ( μ, ε ) ( μ, ε ) inc E ( r) inc S H ( r) p J ( r) p M ( r) ( μ, ε ) Figure.1. An bject with bth ε( r ) and ( r ) μ (with bth electric and agnetic prperties) in free space can be replaced with a ediu with distributed plarizatin J r M r that radiate in free space. current ( ) and ( )... Integral Representatin Starting with the Maxwell s equatins given in ne can btain the fllwing cupled integral equatins 15

26 E r E r k A E r G E r jk M H r ( ) ( ) ( ) ( ) ( ( )) ( ( )) inc = ε ε μ η η H r =η H r k A η H r G η H r + jk M E r ( ) ( ) ( ) ( ) ( ( )) ( ( )) inc μ μ ε, (.6), (.7) where εr I A ε ( υ ( r) ) = g( r;r ) υ( r ) dω ( μ) Ω μr I (.8) εr I G ε ( υ ( r) ) = g( r;r ) υ( r ) dω ( μ), (.9) Ω μr I εr I M ε ( υ ( r) ) = g( r;r ) υ( r ) dω ( μ) Ω μr I (.10) In (.6) and (.7), the unknwn vectr H( r) is replaced with η H ( r ) fr prper scaling. ε r and μ r dente relative perittivity and pereability tensrs at surce pint r. g( r;r ) is the usual kernel t the scalar Helhltz equatin, naely jk r r e g( r;r ) = fr bservatin and surce pints r and r, respectively. The 4π r r derivatin f (.6) and (.7) can be fund in APPENDIX A. One shuld nte that the chice f E( r) and η H ( r ), as the unknwn vectrs instead f the electric and J agnetic plarizatin current densities ( r) and η ( ) 16 M r, respectively, has the advantage f leading t a syetric cefficient atrix when (.6) and (.7) are slved by the ethd f ents (MM).

27 The nuerical slutin f (.6) and (.7) by MM cnsists f three basic steps: 1) Expansin f the unknwn vectrs by a set f basis functins and discretizatin f the scatterer; ) Testing the integral equatin frulatin t btain an algebraic atrix equatin; 3) Slving the atrix equatin btained in step. The fllwing sectins briefly g ver these steps...3. Discretizatin In discretizing (.6) and (.7), the first step is t represent the unknwn vectrs, E( r) and H( r) by a set f N basis functins: E r E e r ; H r H h r, (.11) N N j j j j j= 0 j= 0 ( ) = ( ) ( ) = ( ) where e j and h j are the j th basis functin with the expansin cefficients E j and H j, respectively. Substituting (.11) int (.6) results in N N inc E ( r) = Ejej k Ej g( r;r ) εr I ej( r ) dω j= 0 j= 0 Ω N Ej g( r;r ) εr I ej( r ) dω j= 0 Ω N jkη H j g ( r; r ) μr I h j( r ) dω j= 0 Ω. (.1) The secnd step is t apprxiate the dain Ω by a unin f siple eleents. In ur case, we chse tetrahedrn, which is a siplex in 3-D, as ur vlue eleent. In ther 17

28 wrds, we assue that Ω is divided int M tetrahedral hgeneus eleents, h M h i.e. Ω Ω = T and the aterial prperties are assued cnstant in T h. The ntatin and = 0 h Ω dentes the apprxiatin f Ω with tetrahedra with typical edge length h ; h T is the th tetrahedrn in the esh whse edge length is f rder h. After geetric discretizatin (.1) can be revised as N M inc E ( r) E( r) k E =,j g( r;r ) r ej( r ) dω ε I h j= 0= 0 T N M E,j g( r;r ) r ej ( r ) dω ε I h j= 0= 0 T N M jk H η,j g ( r; r ) r h j( r ) dω μ I h j= 0= 0 T. (.13) Applying the gradient-gradient peratr n the Green s functin under the integral instead f splitting it t perate n the basis functin avids any cnstraints n the basis functins in transitin fr (.1) t (.13). Hence the basis functins t expand E( r) and H( r) can siply be chsen t be piecewise cnstant, i.e.( ) ( ) ( ) ( ( )) e,h H Ω H Ω 3 3 where H ( ) Ω dentes the space f square integrable functins: e = x; ˆ e1 = y; ˆ e = zˆ E r xe ye ze P x,y,z ( ) = ( ˆ + ˆ + ˆ ) ( ) 1 (.14) 18

29 where ( ) P x,y,z h ( ) 1, if x, y, z T = 0, therwise. (.15) and ( E,E 1,E ) are the expansin cefficients f E( r) in h T. The discretizatin f (.7) is evident fr the discretizatin f (.6): N M inc H ( r) H( r) k H =,j g( r;r ) r hj( r ) dω μ I h j= 0= 0 T N M H,j g( r;r ) r hj ( r ) dω + μ I h j= 0= 0 T N M jk E + η,j g( r; r ) r ej( r ) dω ε I h j= 0= 0 T (.16) Althugh piecewise cnstant basis functins are used t expand E( r) H η r in and ( ) (.13) and (.16), due t the cntinuus nature f the integral peratrs ne shuld expect bth fields t be cntinuus acrss the interfaces as successively finer eshes are utilized...4. Testing Prcedure In this sectin, we shall g ver the variatinal stateent t cnvert the integral equatin frulatin in (.6) and (.7) int an algebraic atrix equatin t slve fr 19

30 nuerical apprxiate slutins. The vectrs t test are J( r) = ( εr I) E( r) and η M r =η ( ) ( μ I) H( r) r which represent the electric and agnetic plarizatin current densities, respectively. By ultiplying ( εr I ) and ( r ) respectively, befre testing, we arrive at the fllwing Galerkin stateent: μ I t (.6) and (.7), ( ) ( ) Seek ( ) ( ) ( ) ( ( )) E r,h r H Ω H Ω 3 3 such that inc ( εr I) ( ) = ( ε ) ( ) ( ) ( ( )) r I εr I ε e, ( εr I) Gε( E( r) ) jk e, ( εr I) Mμ ηh( r) e, E r e, E r k e, A E r ( ),(.17) inc ( μr I) ( ) ( μr I) ( ) ( μr I) μ ( ( )) h, ( μr I) Gμ( η H( r) ) + jk h, ( μr I) Mε E( r) h, η H r = h, η H r k h, A η H r (.18) ( ) ( ) ( ) ( ( )) e,h H Ω 3 H Ω 3.Here a,b Ω a bd Ω { } and H ( ): f L ( ) ( ) represents the usual cplex dual pairing Ω = Ω is the space f square integrable functins...5. Matrix Slutin Equatins (.17) and (.18) can be represented in atrix fr as inc E E [ Z] = inc H H (.19) where inc E and inc H are 3Nd 1 and 3N 1 vectrs with entries 0

31 E H inc i inc i inc = ei εr I Ei d T h ()Ω r inc = h i r I Hi d T h ()Ω r, (.0) μ, (.1) e i and h i are testing functins fr E and H, respectively, in T h, E is a 3Nd 1 vectr which cntains the expansin cefficients f E( r), H is a cntains the expansin cefficients f η H ( r ), Z is a ( ) 3N 1 vectr which ( 3 N + N ) ( 3 ( N + N )) d d cefficient atrix which represents the interactins between tetrahedral eleents, N d and N dente the nuber f tetrahedra with nntrivial perittivity and pereability tensrs, respectively. The syste atrix Z can be decpsed int fur atrix blcks based n the physical nature f the interactins between tetrahedral eleents: Z [ ZEE ] [ ZEH ] [ Z ] [ Z ] N N N N d d d = HE N HH Nd N N. (.) Z EE and ZHH represent the interactins between bservatin and surce eleents that are dielectric r agnetic, respectively. On the ther hand, ZEH and Z HE represent the interactins between bservatin and surce eleents that are bth dielectric and agnetic. Finally, ne can btain the entries fr each atrix blck as in the fllwing: 1

32 Z = e ε I e dω k e ε I g r;r ε I e dω dω n EE ij i r j i r r j T T T ( ) ( ) ( ) ( ) T n ( r ) ( ) ei ε I g r;r εr I ejdω dω T ( ) ( ) ( ) ( ) ( r ) ( ) n Z = h μ I η h dω k h μ I g r;r μ I η h dω dω n HH ij i r j i r r j T T T T n h μ I g r;r μ I η h dω dω i r j T n n EH ij i r j T T ( ) ( r ) ( ) n, (.3),(.4) Z = jk e ε I g r; r μ I η h dω dω, (.5) Z = jk h μ I g r; r ε I e dω dω. (.6) n HE ij i r r j T T ( ) ( ) ( ) n We shuld nte that ( k I + ) g( r;r ) and g( r;r ) peratrs are syetric and skew-syetric, respectively. Hence if the ediu is reciprcal, i.e. the perittivity and pereability tensrs are syetric, Z Z Z EE,ij HH,ij EH,ij = Z = Z = Z EE, ji HH, ji HE, ji (.7) Hence ne can reduce the ery requireent as well as the CPU tie fr atrix cputatin and atrix-vectr prduct by a factr f. If the ediu is istrpic, further reductin can be achieved if ne separates ( ) n EE Z and ( Z ) n HH int tw parts

33 ( Z ) ij ( εr I) ( ) ( ) ( r ) ( ) EE i j T n EE ij i r j T T T (I) = e e dω ( r ) ( ) T n Z II = k e ε I g r; r ε I e dω dω, (.8) ei ε I g r;r εr I ejdω dω n ( Z ) ij ( μr I) ( ) ( ) ( ) ( ) HH i j T n HH ij i r r j T T T ( r ) ( ) T n n (I) = h h dω Z II = k h μ I g r; r μ I h dω dω, (.9) hi μ I g r;r μr I hjdω dω n ) n HH ij which enables ne t express the entries in ( Z in ters f the entries in ( Z ) n EE as ij n HH ij ( Z ) ( I r ) ( Z ) ( I) n HH ij ( Z ) ( II) = r n EE ij ( μr 1.0)( μ r 1.0) ( )( ) ( ) n ZEE ( II ) ε 1.0 ε 1.0 ij r μ 1.0 = ε 1.0 r. (.30) Hence when the ediu is istrpic, ne nly needs t stre the upper r lwer triangular part f factr f 4. Z EE and Z EH atrix blcks, which eans a ery reductin by a..6. Singular Integral Treatent Applying under the integral in (.8) and (.9) invlves differentiatin f the 3 Green s functin twice which leads t a ter 1R in the kernel f the integral 3

34 equatin, where R is the distance between a surce and bservatin pint. The singularity f the dyadic Green s functin ay be relaxed by a regularizatin schee that adds and subtracts a functin with the sae singular behavir, but which can be evaluated analytically. There are tw cases t study in regularizing the singular integral: 1) T and T n, which dente the tetrahedra in which the bservatin and surce pints reside, are the sae eleents; ) T and T n are neighbr eleents, i.e. T and T n have at least ne cn vertex. In the fllwing we assue a dielectric scatterer. The treatent shuld be the sae in the case f a agnetic scatterer. One shuld nte that the singular integral frulatin belw is alst identical t the ne given in [13] which has been validated n -D prbles. Case 1: Surce and Observatin Tetrahedra are Identical In this case, the singular integral is divided int tw parts as regular and singular parts: R S ii ii ii I = I + I (.31) where where g ( r;r ) ( εr I) ( ) ( ) ( εr I) I = k e G r;r g r;r e dω dω, (.3) R s ii i i T T k = 4π r r s 1 I = e g r;r ε I e dω dω, (.33) ( )( r ) S s ii i i T T is the static Green s functin, G( r;r ) = I + g( r;r ) k 4

35 is the Green s tensr, g( r;r ) jk r r e = is the dynaic Green s functin and e 4π r r i is piecewise cnstant basis and testing functin in T. It can be shwn that the kernel f the regular integral in (.3) exhibits t O1R ( ) singularity. Hence the nuerical evaluatin f the integral in (.3) invlves Gaussian quadrature with a special treatent when the surce and bservatin quadrature pints verlap. When the surce and bservatin pints verlap, the interactin between the surce and bservatin pints can be apprxiated as the interactin between the bservatin pint and the pints within an infinitesially sall sphere arund it. Frtunately, the integral due t this interactin can be analytically cputed. Hence the regular integral can be apprxiated as ( εr I) ( ) ( ) ( εr I) K K R k l k l s k l ii i i k= 1l= 1 k l k I V V k w w e G r,r g r,r e + ( εr I) ( ) ( ) ( εr I) K k k s k i i k= 1 S k + V w e G r,r g r,r e ds k.(.34) where V is the vlue f T, K is the ttal nuber f Gaussian quadrature pints, ( ) kl r and kl w ( ) are the k() l th Gaussian quadrature pint and its crrespnding weight in T, k S dentes the sphere centered arund k r with vlue w k. One shuld nte that the radius f this sphere deterines the accuracy f this apprxiatin. Fr pint k r ne can epirically deterine the radius as 5

36 R k 3 k = w 4π 13. (.35) Analytical evaluatin f the vlue integral in (.34) finalizes the regular integral frulatin: ( εr I) ( ) ( ) ( εr I) K K R k l k l s k l ii i i k= 1l= 1 k k l I = V V k w w e G r,r g r,r e + k jkr 1 k ( εr I) I ( εr I) K k i i k= 1 3k k 3k + V w e e + jr e. (.36) The singular integral in (.33) can be apprxiated as ( εr I) ( )( εr I) K S k s k ii i i k= 1 V. (.37) I V w e g r ;r edω One can refrulate the integral in (.37) as V 1 ( )( r ) ( ) ˆ ˆ ε I ( εr I) g r ;r e dω = g r,r nr eds. (.38) s k s k i i R V Furtherre (.38) can be revised as in the fllwing if piecewise cnstant basis functins are used t expand the unknwn vectrs in a hgeneus ediu: V s g r ;r e dω = ds e k 1 nr ˆ ˆ ( )( εr I) ( εr I) i i 4π V R (.39) 6

37 Substituting (.39) in (.33) results in I w e ds e nr ˆ ˆ ( εr I) ( εr I) K S V ii i i 4π k= 1 V R k. (.40) The surface integral in (.40) can be analytically evaluated. Case : Surce and Observatin Tetrahedra are Neighbrs The eleents that have at least ne cn vertex are defined as neighbr eleents. The integral in this case is als divided int regular and singular parts. Fr the basis functins i and j, which reside in bservatin and surce tetrahedra T and T n, respectively, the regular integral Here, R I ij and the singular integral I S ij are given as ( εr I) ( ) ( ) ( εr I) I k e G r;r g r;r e d d R s ij = i j Ω Ω T T k n, (.41) K Kn k l k l s k l VVn wwnei ( εr I) G( r,rn) g ( r,rn) εr I ej k= 1l= 1 k V and S s Iij = ei εr I g ( r;r ) εr I ejdω dω T T n. (.4) K V ˆ k nr ˆ wei( εr I) ds ε r I ej 4π k= 1 V R n V n are the vlues f the bservatin and surce tetrahedrns, k r and l r n are th k and th l quadrature pints with the weights k w and l w n in T and T n, respectively The regular integral is evaluated by apprpriate Gaussian quadrature rules bth in T and T n because there are n singularity pints within T n anyre. Siilar t 7

38 the singular integral fr identical eleents the surface integral in (.4) can be analytically evaluated. The details f the regularizatin f the hyper singular integral are given in APPENDIX B..3. Nuerical Results In this sectin, we intend t denstrate that the nncnfral VIE frulatin prduces accurate results fr five cannical prbles: 1) dielectric and agnetic sphere; ) agnetdielectric sphere; 3) dielectric shell; 4) -layer dielectric sphere, 5) 3-layer dielectric shell. We cpare the bistatic RCS fr nncnfral VIE slutin with the bistatic RCS btained fr the Mie series slutin and the VIE slutin presented by Schaubert et al. in [1]. The cnvergence study n dielectric sphere shws that the relative errr curve ntnically decreases and that the nuber f iteratins reains cnstant as we increase the nuber f unknwns Validatin 1. Bistatic RCS f a Dielectric (Magnetic) Sphere T denstrate the accuracy f the nncnfral VIE frulatin given by (.6) and (.7) we study an exaple fr which we can btain analytical slutin t cpare the bistatic radar crss-sectin (RCS). Hence we study electragnetic scattering by a hgeneus sphere in which E( r) and/r H( r) can be expanded by Mie series. First, we verify that the nncnfral VIE prduces reliable results fr the bistatic RCS f a dielectric sphere in bth E and H planes, i.e. at φ= 0 and φ= 90, respectively. The relative perittivity f the ediu filling the sphere is.. The sphere 8

39 is discretized int 658 tetrahedra. Fr piecewise cnstant basis functins this results in unknwns because there are three unknwn cpnents f E( r) in each tetrahedrn. The frequency f peratin is 600 MHz at which the sphere radius is λ 5, where λ is the free space wavelength at the frequency f peratin. The axiu edge length in the sphere esh is defined t be ( ) illuinated at ( ) = ˆ inc jk z E r xe Δ =λ 10 ε 0.067λ. The sphere is ax r θ= 0 by an incident plane wave plarized in ˆx directin, i.e.. It takes 7 iteratins fr GMRES withut a restart t cnverge t the desired slutin within tlerance. Figure. cpares the bistatic RCS btained fr the nuerical and analytical slutins in E and H planes, respectively. One can easily ntice the excellent agreeent between these tw slutins. We repeat the sae nuerical study fr a agnetic sphere with relative pereability.. Since the frulatin fr the agnetic scatterer and dielectric scatterer are atheatically dual, we expect the sae results except fr the interchange between the bistatic RCS curves in E and H planes. Our bservatins validate ur expectatins: It takes 7 iteratins fr GMRES t cnverge and Figure.3 agrees well with Figure.. Next we study an electrically larger size f dielectric sphere. The sphere has λ radius and is filled with dielectric aterial with relative perittivity f.. The sphere is discretized int 916 tetrahedra which leads t 7648 unknwns. It takes 5 iteratins fr GMRES t reach the desired slutin within tlerance. The cparisn f the bistatic RCS curves btained fr the nncnfral VIE and Mie series expansin shws excellent agreeent in Figure.4. 9

40 Figure.. Cparisn f bistatic RCS f a dielectric sphere in E and H planes with the inc -jkz Mie series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ. Figure.3. Cparisn f bistatic RCS f a agnetic sphere in E and H planes with the inc -jkz Mie series slutin The sphere is illuinated at θ = 0 with E = xe ˆ. 30

41 Figure.4. Cparisn f bistatic RCS f a dielectric sphere f planes with the Mie series slutin The sphere is illuinated at inc -jkz E = xe ˆ r = 0.5λ in E and H θ= 0 with. Bistatic RCS f a Magnetdielectric Sphere We denstrate that the nncnfral VIE prduces reliable results fr the bistatic RCS f a agnetdielectric sphere in bth E and H planes, i.e. at φ = 0 and φ= 90, respectively. The relative perittivity and pereability f the ediu filling the sphere is. and 1.37, respectively. The sphere is discretized int 669 tetrahedra. Fr piecewise cnstant basis functins this results in unknwns because there are 3 unknwn cpnents f bth E( r) and H( r) in each tetrahedrn. The frequency f peratin is 600 MHz at which the sphere radius is λ 5, where λ is the free space 31

42 wavelength at the frequency f peratin. The axiu edge length in the sphere esh is defined t be ( ) Δ =λ 10 ε μ 0.058λ by an incident plane wave plarized in ˆx directin, i.e. ( ) ax r r inc jk z E r xe ˆ =. It takes 8 iteratins fr GMRES t cnverge t the desired slutin within tlerance. Figure.5 cpares the bistatic RCS btained fr the nuerical and analytical slutins in E and H planes, respectively. One can easily ntice the excellent agreeent between these tw slutins. Figure.5. Cparisn f bistatic RCS f a agnetdielectric sphere in E and H planes inc -jkz with Mie series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ 3

43 3. Bistatic RCS f an Anistrpic Uniaxial Sphere We denstrate that the nncnfral VIE prduces reliable results fr the bistatic RCS f a uniaxial anistrpic dielectric sphere in bth E and H planes, i.e. at φ= 0 and φ= 90, respectively. The electrical size f the sphere is ka= π and the perittivity tensr is characterized by ε t = ε and ε z = 3ε. The sphere is discretized int 6791 tetrahedra. Fr piecewise cnstant basis functins this results in 0373 unknwns because there are 3 unknwn cpnents f bth E( r) in each tetrahedrn. The sphere inc jk z E r xe ˆ is illuinated by an incident plane wave plarized in ˆx directin, i.e. ( ) It takes 7 iteratins fr GMRES t cnverge t the desired slutin within tlerance. Figure.6 depicts the bistatic RCS btained fr the nncnfral VIE in E and H planes. This slutin agrees well with the slutin given in [5]. =. 33

44 Figure.6. Cparisn f bistatic RCS f a agnetdielectric sphere in E and H planes inc -jkz with Mie series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ 4. Bistatic RCS f a Dielectric Shell The secnd exaple that we shall study t validate accuracy f the nncnfral VIE frulatin is a lssy dielectric shell. The shell has inner radius f radius f λ and uter 1.6λ. The aterial that fills the shell has relative perittivity f 1.5-j0.5. It is discretized int 8904 tetrahedral eleents and illuinated at inc jk z E r xe ˆ wave plarized in ˆx directin, i.e. ( ) = θ = 0 by an incident plane. It takes 9 iteratins fr GMRES t cnverge t the desired slutin within tlerance when nncnfral VIE is slved. In this study we cpare tw different nuerical slutins f the sae prble, naely the slutin f the nncnfral VIE and the slutin f the VIE presented by Schaubert et al. in [1] which eplys divergence cnfring basis functins in the

45 ixed ptential integral equatin frulatin. Figure.7 depicts the bistatic RCS btained by these ethds. The cparisn f the curves shws that the slutin f the nncnfral VIE agrees with the slutin f the VIE frulatin presented in [1]. Figure.7. Cparisn f bistatic RCS f a dielectric shell in E and H planes with the VIE slutin by Schaubert et al. [1].The shell is illuinated at θ= 0 inc -jkz with E = xe ˆ 4. Bistatic RCS f a Tw-Layer Dielectric Sphere T cnfir that the VIE frulatin presented is applicable t nncnfral esh t prduce accurate results we study a layered sphere exaple. The inner sphere has 0.18λ radius and uter sphere has 0.λ radius. The inner sphere is filled with aterial f 35

46 relative perittivity 1.5 and the uter sphere is filled with aterial f relative perittivity 10. Hence the cntrast rati is 8.0. The inner sphere is discretized int 6686 tetrahedra and the uter sphere is discretized int 6144 tetrahedra. The crss-sectin f the nncnfral esh alng the center plane is illustrated in Figure.8 The layered sphere is illuinated at ( ) = ˆ inc jk z E r xe θ= 0 by an incident plane wave plarized in ˆx directin, i.e.. It takes 0 iteratins fr GMRES t cnverge t the desired slutin within tlerance. The cparisn f the bistatic RCS curves btained fr the nncnfral VIE slutin and Mie series in Figure.9 shws excellent agreeent. 36

47 Figure.8. Crss sectin f the nncnfral esh fr the layered sphere exaple alng the center is illustrated. The inner and uter spheres have radius f 0.18λ and 0.λ, respectively. 37

48 Figure.9. Cparisn f bistatic RCS f a tw-layer sphere in E and H planes with the inc -jkz Mie series slutin. The sphere is illuinated at θ = 0 with E = xe ˆ. 5. Bistatic RCS f a Three-Layer Dielectric Shell T cnfir that the VIE frulatin presented is applicable t nncnfral esh t prduce accurate results we study a three-layer dielectric shell. The radii f the three layers are ( 1.0 λ,1.1λ ), ( 1.,1.3 ) λ λ, respectively. The relative perittivities f the layers are 1., 1.6 and.0, respectively. The layers are discretized int 5548, 1773 and 0778 tetrahedra, respectively.. The crss sectin f the esh alng the center plane is illustrated in Figure.10. The layered sphere is illuinated at inc jk z E r xe ˆ plane wave plarized in ˆx directin, i.e. ( ) = θ= 0 by an incident. It takes 4 iteratins fr 38

49 GMRES t cnverge t the desired slutin within tlerance. The cparisn f the bistatic RCS curves btained fr the nncnfral VIE slutin and Mie series expansin in Figure.11 shws gd agreeent. Figure.10. Crss sectin f the nncnfral esh fr the layered shell exaple alng the center is illustrated 39

50 Figure.11. Cparisn f bistatic RCS f a 3-layer dielectric shell in E and H planes inc -jkz with the Mie series slutin. The shell is illuinated at θ = 0 with E = xe ˆ..3.. Cnvergence Study In this sectin, we study the effect f regularizing the integrals that represent interactins between adjacent eleents n the nuerical cnvergence f the iterative slutin. Hence we discretize a sphere esh successively and bserve the trend in the relative errr and the nuber f iteratins. The sphere is filled with a ediu which has relative perittivity f We define the relative errr as 40

51 rel. errr = π 0 Mie ( ) VIE ( ) σ θφ=, 0 σ θφ=, 0 dθ π 0 Mie ( ) σ θ, φ= 0 dθ, (.43) where σ Mie is the bistatic RCS btained by Mie series slutin, σ VIE is the bistatic RCS btained by VIE frulatin fr elevatin angle θ and aziuth angle the relative errr in GMRES is φ= 0 when T shw the cnvergence f the fralis Figure.1 gives this relative errr as a functin f nuber f unknwns used t discretize the geetry. As the nuber f unknwns increases, the relative errr in bistatic RCS decreases ntnically. We als nitr the nuber f iteratins versus the nuber f unknwns in Figure.13. One can easily ntice that the nuber f iteratins stays bunded as the nuber f unknwns increases. This nuerically validates that the cnditin nuber f the cefficient atrix is bunded and the eigenvalue distributin preserves itself as the nuber f unknwns increases. The atheatical prf f the cercivity f the vlue integral peratr is given in APPENDIX C. 41

52 Figure.1. Relative errr versus the nuber f unknwns fr the cnvergence study Figure.13. Nuber f iteratins versus the nuber f unknwns fr the cnvergence studyequatin Chapter 3 Sectin 1 4

53 CHAPTER 3 IE-FFT ALGORITHM FOR VOLUME INTEGRAL EQUATIONS 3.1. Overview The starting pint f the IE-FFT algrith is t interplate the Green s functin by Lagrange plynials n a unifr Cartesian grid. Interplatin f the Green s functin enables us t decuple the surce and bservatin variables; in ther wrds, it allws us t separate the integrals in the surce and bservatin dains. The Lagrange cefficients f the Green s functins n the unifr Cartesian grid prvide the link between the surce and bservatin integrals. In fact, st f the fast integral equatin slvers ake use f the sae idea such as FMM [4] and panel clustering ethd [51], which apprxiate the Green s functin by the spherical ultiple expansin and the Taylr series expansin, respectively. The Green s functin can be interplated as jk r r N g 1Ng 1 e p p g( r;r ) = βl ( r) gll βl ( r ) r r (3.1) l= 0 l = 0 where p is the rder f Lagrange plynial, p N is the nuber f grid pints, β ( r ) g l 43

54 p and ( r ) β are the l th p rder Lagrange plynials n bservatin and surce grids, r = ( x,y,z) and r = ( x,y,z ), respectively, g l,l are the Lagrange cefficients f the Green s functin, l and l One can interplate g( r,r ) and g( r,r ) are diensinal indices f grids in r and r, respectively. jk r r jk r r in (.3) and (.5) as p p ( ) ( ) ( ) N 1N 1 g g l= 0 l = 0 p l p ( r) g ( r ) ll l N 1N 1 g g e e g r;r = = βl r gll βl r r r r r β β jk r r l= 0 l = 0 N g 1Ng 1 Ng 1Ng 1 l ll l l ll l l= 0 l = 0 l= 0 l = 0, (3.) e p p p p g( r;r ) = β ( r) g β ( r ) β ( r) g β ( r ) r r. (3.3) Substituting(3.1), (3.) and (3.3) fr g( r;r ), ( g r,r ) and g( r,r ) (.3) and (.5) can be revised t separate the integrals ver the surce and bservatin dains: Z Z ε 1.0 e e dω ( ) EE,ij r i j T k ε 1.0 ε 1.0 g e β r d Ω. e β r dω +, (3.4) N 1N 1 r r ll i j g g l l l 0 l = = 0 T Tn p p ( )( ) ( ) ( ) N 1N 1 + ε 1.0 ε 1.0 g e β r dω e β r dω g g l l l 0 l = = 0 T Tn p p ( )( ) ( ) ( ) r r ll i j N 1N 1 jk g e r d η ε μ β Ω β r e d Ω.(3.5) g g l l l= 0 l = 0 T Tn p p ( )( ) ( ) ( ) EH,ij r r ll i j The prduct frs f cefficient atrix ters in (3.4) and (3.5) are valid fr all interactins except the nes between the tetrahedra that reside in the sae and adjacent 44

55 cells. At these lcatins, g l,l' = when l= l, the Green s functin values need t be apprpriately crrected. One can suarize the IE-FFT algrith in fur ain steps: 1) Representatin f the Green s functin in siple Lagrange plynials ) Prjectin f the basis functins nt the Cartesian grid 3) Crrectin f the atrix entries fr near-interactin eleents 4) Acceleratin f the atrix-vectr prduct using FFT Figure 3.1 gives a pictrial descriptin f the IE-FFT algrith in -D. The grey regin cnsists f cells that include the surce tetrahedrn and its neighbr cells. Adjacent cells are defined t be the neighbr cells. The near interactin zne f the surce tetrahedrn falls int the sphere f radius αλ, where λ is the free space wavelength at the frequency f peratin and α is a paraeter that deterines the accuracy. The center f this sphere cincides with the center f the surce tetrahedrn. All tetrahedra with centers in the defined sphere are cnsidered near interactin tetrahedra. The interactins between the surce tetrahedrn and the near interactin tetrahedra are cputed directly and this step is dented by (4) in Figure 3.1. T cpute the interactins between the surce tetrahedrn and the distant tetrahedra that fall utside the near interactin zne the first step is t prject the basis functins nt the unifr grid as dented by (1) in Figure 3.1. The interactin is then cputed by ultiplying the prjected basis functins with the Lagrange cefficients f the Green s functin assciated with the ndes f the cells in which the surce and bservatin tetrahedra reside. This ultiplicatin can be efficiently cputed via FFT by the virtue f unifr sapling as dented by () in Figure

56 αλ Figure 3.1. A -D pictrial representatin f the IE-FFT algrith Interactins with near tetrahedra are cputed directly, interactins between distant tetrahedra are cputed using the grid. 3.. Interplatin f the Green s Functin By Lagrangian Plynials First we cnstruct a rectangular bx that enclses the esh discretizing the scatterer. Let us dente the side lengths f the bunding bx as L, x L y and L z in x, y and z directins, respectively. Secnd we cnstruct a unifr 3-D grid within this bx. The nuber f grid pints in x, y and z directins are Nx = Lx d, Ny = Ly d, Nz = Lz d 46

57 where d is the sapling distance. The diensinal indices fr the surce and bservatin grids can be expressed as l= ( i,j,k) and l ( i,j,k ) where =, 0 i,i Nx, 0 j,j Ny, 0 k,k Nz, l = i + j+ k and l = i + j + k. We shuld ephasize that the grid and the irregular tetrahedral esh discretizing the scatterer are independent f each ther. Therefre the IE-FFT algrith is cnsidered in the class f irregular esh based grid algriths. The th p rder interplatin basis functins ne-diensinal Lagrange plynials n a Cartesian grid: p β l in 3-D are cnsidered as the prduct f β =β β β ( r) ( x) ( y) ( z) p p p p l i j k (3.6) Eplying (3.6) the integral equatin kernels in (.3) and (.5) can be written in atrix fr as g r;r = r r ( ) β( ) ( ) ( ) ( ) ( ) T ( ) G β( ) g r;r = r r T ( β ) G β( ) g r;r = r r T ( β ) G β( ), (3.7) where β p p p β0( x) β0( y) β0( z) = β0 β =, (3.8) Ng 1 p p p β N ( x) β ( y) β ( z) x 1 Ny 1 Nz 1 p p ( r) ( r) ( r) T 47

58 β p p p β0( x) β0( y) β0( z), (3.9) p p ( r) ( r) ( r) = β0 β = Ng 1 p p p β N 1( x) β N 1( y) β N 1( z ) x y z T g0,0 g0,1 g0,ng 1 g1,0 g1,1 g1,ng 1 G =, (3.10) gng 1,0 gng 1,1 g Ng 1,Ng 1 Ng = Nx Ny Nz. The nuerical values f g ll fr l = l are siply set t zer and are apprpriately crrected during the crrectin f the near interactin eleents. Since β ( r) in (3.9) is a vectr-valued vectr, ne can represent this vectr in an alternative fr as x y z x y z ( x) ( y) ( z) ( x) ( y) ( z) p p p N 1 N 1 N 1 x y z T p p p p p p ( ) ( ) ( ) ( ) ( ) ( ) 0 x 0 y 0 z β0 x β0 y β0 z β β β y x β( r) = xˆ + yˆ + p p p p p p β β N 1 N 1( y) β N 1( y) β N 1( x) β N 1( y) β N 1( z) x y p p p β β β z β β β z T ẑ T (3.11) Fr a general Green s functin ne needs t stre the entire G which results in ( g ) O N ery requireent and cputatinal cplexity. Since the free-space Green s functin in VIEs is f a difference-fr, i.e. g( r;r ) = g( r r ), G is a Teplitz atrix. Hence ne nly needs t cpute and stre nly ne rw and clun f G, 48

59 which cntain 8 N eleents. Therefre, ery requireent and cputatinal g cplexity f G is O( N g ). Since st f tie the unifr grid is uch carser than the irregular esh that discretizes the bject, the cputatinal cplexity and ery requireent f this step is alst negligible. (a) (b) Figure 3.. (a) Side view f a sphere discretized int 180 tetrahedra, with spatial decpsitin int a 3 x 3 x 3 array f cells. (b) Superipsed grid pints crrespnding t the cell decpsitin f (a), with p = 3. Figure 3. shws the grid ipsed n the 3x3x3 cell structure. This eans in each cell, a 3 x 3 x 3 array f grid pints is used t interplate the Green s functin at the quadrature pints in each tetrahedrn within the cell. Se f the grid pints are shared 49