WNW l llHHlWINlllllflllfltllllilWWW! __THS This is to certify that the thesis entitled Analysis of the Spectrum of the Single Integral Equation for MS. Scattering from Dielectric Objects presented by Jun Yuan has been accepted towards fulfillment of the requirements for the degree in Electrical and Computer Erflmeering (QM/A Major Professor’s Signature Jilxfifiafl Date I MS U is an Affirmative Action/Equal Opportunity Institution UBRARY Michigan State University .- ~.—.-.-.-o-c—-n-o--.- PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 p:ICIRC/DateDue.indd-p.1 Analysis of the Spectrum of the Single Integral Equation for Scattering from Dielectric Objects By Jun Yuan A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Electrical and Computer Engineering 2006 ABSTRACT Analysis of the Spectrum of the Single Integral Equation for Scattering from Dielectric Objects By Jun Yuan Surface integral equations are widely used to analyze electromagnetic scattering from a dielectric object residing in free space. Methods for analyzing scattering from dielectric bodies have largely relied on either the PMCHWT or the Muller formulations. It is well known that the PMCHWT formulation result in a first kind Fredholm integral equation, is not well-conditioned and leads to slow convergence. The Miiller formulation, on the other hand, results in a second-kind integral equation and is well-conditioned as the hyper-singular terms nearly cancel for low-contrast ratios. However, the Miiller formulation is not very accurate for very high-contrast materials. Alternatively, it has been shown that scattering from a dielectric body can be computed using a single unknown and a set of cascaded equations, viz., the single integral equation (SIE) [1]. Existing literature [2] has reported that the condition number of the impedance matrix is excellent. However, no work has been attempted to analyze the convergence and uniqueness of the SIE operator. In this thesis, a detailed analysis is carried out to reveal the underlying mathematical properties of the operator. It will be shown that this operator does not produce unique solutions at internal resonance frequencies. Nonetheless, we suggest a method to overcome spurious resonances and propose an integral equation that is accurate for arbitrary material contrast ratios, while still preserving its well-conditioned nature. To My Parents iii ACKNOWLEDGMENTS In finishing this research, I have profited intellectually from the support, comments and suggestions of my professors, my colleagues and my friends. First, I would like to highlight the role of Prof. Shanker Balasubramaniam for being my major advisor, who helped me with his technical knowledge, timely guidance and constant encouragement. I am grateful to Dr. Gregory Kobidze for his instrumental discussions with me on the regular basis. Thanks to Prof. Dennis Nyquist for his review of my presentation in the EM seminar series. I would also like to express my sincere thanks to Prof. Edward Rothwell for consolidating my background knowledge and theory in advanced EM research. Many valuable suggestions from Prof. Leo Kempel have greatly contributed to my research publications. I would also like to thank my friends and fellow graduate students Chuan Lu, Jun Gao, Jorge Villa, and Pedro Barba for their frequent help in the my study and research on this thesis. Without your comments, I wouldn’t have been able to grasp all the materials in such an efficient manner. Many thanks are also given to Brad Perry for providing me with the ETEX style files that made the writing of the thesis an enjoyable work. Most of all I would like to thank my parents for their support and education through all levels of my studies. Your recognition and pride in your son has made all this possible. iv TABLE OF CONTENTS LIST OF FIGURES ................................ vi KEY TO SYMBOLS AND ABBREVIATIONS ................. viii CHAPTER 1 Introduction ..................................... 1 CHAPTER 2 Integral Equations for Scattering from Dielectric bodies ............. 3 2.1 Definition of the dielectric scattering problem ............. 3 2.2 Surface Equivalence Theorem ...................... 3 2.3 Single Integral Equation ......................... 6 CHAPTER 3 Numerical Solution using the Method of Moments ................ 12 3.1 Method of Moments ........................... 12 3.2 Current Basis Functions ......................... 14 3.3 Projection and Inner Product ...................... 15 3.4 Construction of the Moment-Method Matrix .............. 17 3.5 Augmenting with the Fast Multipole Method .............. 19 3.6 Numerical Results ............................. 20 CHAPTER 4 Spectral Analysis of the SIE Operator ...................... 31 4.1 Definition of the Spectrum Analysis ................... 31 4.2 Spectrum Properties of the Integral Operators ............. 35 4.3 Analysis of the SIE Spectrum ...................... 37 4.4 Well-behaved CSIE Operator ...................... 39 4.5 An alternative integral equation scheme ................. 41 4.6 Implementation and solution of the augmented integral equations . . 47 CHAPTER 5 Conclusions ..................................... 58 APPENDIX A Definition of a second kind integral operator ................... 61 BIBLIOGRAPHY ................................. 63 Figure 2.1 Figure 2.2 Figure 2.3 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 LIST OF FIGURES Homogeneous dielectric scattering object (52, #2) embedded in the medium (51, p1) ............................ Equivalence principle models. .................... Equivalence principle models of SIE. Triangulated meshing of a dielectric sphere with r = 1.0. The number of triangles is 2904, and the number of edges is 4356. Triangle pair (Ti+, Ti') and parameters associated with the ith edge. Geometry parameters of the triangle pair Ti+ and Ti' . . . . . . . RCS pattern of an 52 = 250 sphere in the 33-31 plane at f0 = 160MHz. RCS pattern of an 52 = 450 sphere in the 55—31 plane at f0 = 160MHz. RCS pattern of an 52 = 450 sphere in the :r-y plane at f0 = 225MHz. RCS pattern of an 52 = 480 sphere in the :r-y plane at f0 = 300MHz. RCS pattern of an 82 = 450 cubic box in the x-y plane at f0 = 257MHz ................................. RCS pattern of a fat almond in the :r-y plane at f0 = 100MHz. Plot of three spectral functions of Mi. X for kla E [001,6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of M} U for kla E [001,6], and n E [1,3,5]. .............................. Plot of three spectral functions of XICIX for kla E [0.01, 6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of AR+,U for ha 6 [0.01, 6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of NICK for ha 6 [001,6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of X1211} for ha 6 [001,6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of A32 for kla E [001,6], and n 6 [1,3,5]. .............................. Plot of three spectral functions of ”Sim, X for kla E [0.01, 6], :—:_ = 4.0 and n E [2,3, 5] ........................... vi 10 10 11 22 23 24 25 26 27 28 29 30 49 50 51 52 53 54 55 56 . . E F lgure 4.9 Plot of three spectral functions of Agm, U for ha 6 [0.01, 6], 2% = 4.0 and n 6 [2,3,5] ........................... 57 vii KEY TO SYMBOLS AND ABBREVIATIONS CEM: Computational Electromagnetics MoM: Method of Moment EFIE: Electric Field Integral Equation MFIE: Magnetic Field Integral Equation CFIE: Combined Field Integral Equation PMCHWT: Poggio—Miller—Chang—Harrington-Wu-Tai SIE: Single Integral Equation FMM: Fast Multipole Method CSIE: Combined Source Integral Equation ASIE: Augmented Single Integral Equation viii CHAPTER 1 INTRODUCTION Analysis of electromagnetic scattering from arbitrary shaped three-dimensional (3- D) homogeneous or layered homogeneous dielectric bodies is of considerable interest and is largely motivated by possible application in the society of computational elec- tromagetics, due to the wide application of materials in a variety of radar targets. The early studies focused mainly on spherical or nearly spherical scatterers by using classical analysis. But in many applications the scatterer is an arbitrarily shaped 3—D object. For dielectric objects with a surface of arbitrary shape, one has to re- sort to some approximate numerical techniques based on either integral or differen- tial equations. Recent developments in the field of computational electromagnetics (CEM) have greatly expanded the palette of analysis tools to include problems in which the boundary conditions or the shape of the scatterer makes previous classical mathematical approaches intractable. Surface integral equations are often preferred for homogeneous or layered homogeneous objects, as they permit the use of surface equivalent currents. Compared with volume integral equations, the computational cost is greatly reduced from 0(N3) to 0(N2). However, the efficiency of this method depends heavily on the mathematical for- mulation. Given the fact that the CEM engineers need to solve practical problems of millions of unknowns, it is essential that the formulation yield a well-conditioned matrix system, so that the solution can converge to an accurate result rapidly. On the other hand, the formulation should yield a unique solution. In other words, the mathematical model should produce unique results at all frequencies. An overview of the mathematic models for analyzing the EM scattering from a dielectric body of arbitrary shape is given in Chapter 2, including a description of different integral equation approaches with an emphasis on the single integral equation (SIE) formulation. Chapter 3 covers the MOM implementation of the SIE scheme. Details of the method are elaborated upon, especially, the basis functions used, and the method to compute the inner product with the testing function. Also covered in Chapter 3 is the adaptation of the FMM algorithm for this specific scenario. Numerical results are illustrated at the end of this chapter, and these results are compared against the solutions obtained either using the classic approach or using an extant numerical code developed by Dr. Shanker Balasubramaniam at Michigan State University. The spectral analysis of integral equation operator is introduced in Chapter 4. The convergence properties of various equations are explained and illustrated in this chapter. The spectrum of the peculiar SIE operator is investigated from the mathe- matic perspective. As will be shown, the SIE operator does not yield unique solutions at resonance frequencies. Hence, we proceed to devise an augmented SIE operator and prove its validity thereafter. Chapter 5 serves to conclude this thesis. Contributions to this research are dis- cussed, along with future work necessary to expand the applicability of this new scheme in handling the EM scattering phenomena. CHAPTER 2 INTEGRAL EQUATIONS FOR SCATTERING FROM DIELECTRIC BODIES 2.1 Definition of the dielectric scattering problem Consider a homogeneous dielectric body of volume V, as shown in Figure 2.1, whose boundary is denoted by 9. Additionally, 9+ and 9‘ denote surfaces that are con- formal to and lie just outside and inside 9, and n denotes the outward pointing normal to {2. The regions inside (denoted as R1) and outside (denoted as R2) 9 are characterized by material parameters (6:1, pl) and (52, #2), respectively. Impressed sources residing within R1 produce incident electromagnetic fields electric and mag- netic fields {EinC,HinC}. The interaction of the incident fields with 9 gives rise to the scattered fields Eicaltr), Hicat (r). The total fields {BECK Him} in R1 comprise both the incident and scattered fields, viz., Eti°t(r) = Eincm + Eicatu) (2.1a) H]°t(r) —_— Hin°(r) + Hicaim (2.1b) Inside Q, the fields are expressed by Eacatfi), Hacatfi). 2.2 Surface Equivalence Theorem The equivalence principle [1] is applied to the scattering problem from a dielectric body, as illustrated by Figure 2.2. One can set up a problem equivalent to the original problem external to Q as follows. Let the original field exist external to $2, and the null field internal to (I, while the whole region is characterized by (51, #1). This is shown in Figure 2.2(a). To support the scattered field, there must exist surface currents J1, M1 on 9 according to the continuity conditions across the boundary. The currents therefore satisfy the constraints J1 = a x [Hinc + Hficat] (2.2a) M1 = [Einc + Rim] x a (2.2b) where fl is an outward pointing normal. Since the currents radiate in unbounded homogenous space, we can determine the scattered field using Eicmfi‘) = 721L1(J1) - K1(M1) (23a) area) ‘= K101) + UllLl(M1) (2.31») where L1{X} é —jk1/ [7+ —1-VV G1(r r') - X(r’) dS’ (2.4a) n k? ’ K1{X} i f V X [X(r’)G1(r,r')] (15' (2.41)) 9 I= is the idem factor, or unit dyad. Similarly, we can set up another equivalence for the field internal to Q as shown in Figure 2.2(b). Another set of equivalent currents J 2, M2, prescribed by J2 = -fi x Ham (2.5a) M2 = 433““ x a (2.5b) reside over Q in the homogeneous space of (52, p2), and produce the null field external to Q and the original field Egan“), chaWr) internal to 9, which is determined as Eicatm = 71214202} - K2{M2} (26a) Hacatir) = K202} + n§1L2{M2} (26b) Our specification of the null field internal / external to Q is overly restrictive in the preceding models. Any other field would serve equally well, given that the resulting equivalent currents satisfy the field continuity constraints. Yet, our proposed choices prescribe the complementary relation between the currents in Figure 2.2( a) and Figure 2.2(b), viz., J1 = a x [Hinc + Him] = (412) x chat = —J2 (2.7a) M1 = [Einc + Bicat] x a : E302“ x (432) = —M2 (2.7b) due to the fact that ii X [Hinc + Hicat] = fi x Hacat (2.83) a x [Einc + Eicat] = a x 1336“ (2.8b) and fig = —fi (2.9) is the normal vector inward to 9. Using Eqs. [2.3 - 2.6], the classical Poggio—Miller-Chang-Harrington-Wu-Tai (PM- CHWT) formulation [2] is derived, leading to a x Eincl9+ = —fi x Eficai|n+ + a x Egcatln- (2.103.) a x Hinclmt = is x Hficailfli, + a x chatln- (2.10b) An alternate approach suggested by Miiller [3] is to scale the interior and exterior field operators by the constitutive parameters and then subtract eqs. 2.5 from eqs. 2.2, representing the interior and exterior problems, respectively. The Miiller equations read as —fi x Einclfi = (1+ 0) M1 + a x (Hamil,2+ — aagcatlfl-) (2.1m) —fi x Hinclmt = —— (1 + ,3) J1 + a x (HiC‘iH,2+ + fichatln-) (2.11b) where a = 52/51, and fi = ,ug /,u1, as chosen by Miiller in his construction of integral equations. This formulation has some advantages over the PMCHWT formulation for low contrast materials (i.e., 5,. < 20). Specifically: 1) it behaves as a second-kind integral equation; 2) the static terms of the L-operator in Eqs. 2.4 cancel in the limit as [r — r’ | —+ 0, effectively canceling the hypersingular term [3, pg. 300]; and 3) the Miiller formulation has a lower condition number than the PMCHWT formulation for moderate to low contrast materials. 2.3 Single Integral Equation The coupled vector integral equations, PMCHWT and Miiller require one to solve for a set of unknown equivalent electric and magnetic currents. Marx [4] developed, in both the time and frequency domains, 3. single integral equation for scattering problems involving homogeneous dielectric bodies. Glisson [5] elaborated upon the technique for 3-D dielectric objects in the frequency domain using the techniques and terms that EM researchers are more familiar with. Numerical demonstration was recently reported by Yeung [6] and Tsang [7]. To determine the scattered field using SIE, we employ the equivalence principle to develop two different models. For the same original problem, two models equivalent in the external and internal regions, are illustrated in Figure 2.3. A model equivalent to the exterior region is shown in Figure 2.3(a). In contrast to the classical model in Figure 2.2(a), only a single current J eff resides in the homogeneous medium (51, #1) to produce the correct scattered field external to Q. The equivalent source J eff, however, is not unique unless the scattered field is specified internal to Q. In the approach usually followed, the auxiliary field is set to 0. And J eff is uniquely determined by Jeff = a x [Hinc + H‘i‘catoefl, 0)] _ (2.12) = fl x HInc + ft x K1{Jefir} Equivalence to the original problem inside (I (Figure 2.3(b)), the sources are con- structed by imposing (J2, M2). These sources radiate in a homogeneous medium (52, [12) and produce the correct scattered field inside S2 and a null field outside SI. Continuity of the true field is enforced to relate the equivalent currents Jeff and (J 2, M2) using the relations, a x a x Egcat = a x a x [EiInc + Eicat'] (2.13) a x H3O?“ = a x [Hinc + Hficat] (2.14) Thus, J2 = —fi x chat [9+ = —fi x [Hinc + Hfical] [9+ (215) = -fi X HmC -— fl X K1{Jeff} [9+ M2 = —Egcat' X fl [9+ = _(Einc + meat] x a (9+ (216) = _E1nc X f1 — 01L1{Jeff} X fl [9+ We may restate Eqs. 2.13 and 4.28 in terms of Jeff as n x fi >< Einc = fi X 13 X (772L2{J2} - K2{M2} - 771L1{JeH}) = ft x f1 x (n2L2{—fi x Hinc — f1 x K1{Jeff}} . (2.17) — K2{—EInc x I“! - n1L1{JeH} X a} — 771L1{Jeff}) A . A 1 n x HlnC = n X (EL2{M2} + K2{J2l — KliJeffl) 1 . = fl X —L2{—Emc X fl — 01L1{J } X fl} (n2 eff (2.18) + K2{—fi x Hinc - r“: x KliJeflll “KliJeffD Rearranging both sides of eqn. 2.17 yields the EF IE equation -772fiXfiXL2ifiXKliJeffl}—771fi><fiXK2{fiXL1{Jeff}}-fl1fixfiXLliJeff} = Erhs (2.19) where the term Erhs = f1 x f1 x Einc + 7721“] x f1 x L2{f1 x Hinc — 1‘1 x f1 x K2{Einc x fi}} (2.20) is the right-hand-side given by the incident field. Similarly, the MFIE equation can be derived, leading to . .. 77 . . .. n x K2{n x K1{Jefl—}} + in x L2{n x L1{Jeflr}} — n x K1983} = Hrhs (2.21). where H,hs = a x HlnC + a x K2{fi x Hmc} — -n—f1 x L2{fi x Emc} (2.22) 2 The unknown fields in Eqs. 4.28 and 2.13 can be expressed in terms of integrals over their respective sources. Then Eqs. 2.15 and 2.16 give J2 and M2 in terms of the single current Jeff, and Eqs. 2.19 or 2.21, or an appropriate linear combination of these equations represents an integral equation to be solved for J eff- In the next few chapters, we will investigate the desirable properties of the MF IE equation (in eqn. 2.21). It will be shown that the MF IE equation is a second kind integral operator (Appendix A). (€1,111) f1 tot tot E1 ,H1 / Einc \. Figure 2.1. Homogeneous dielectric scattering object (52, p2) embedded in the medium (61, #1). (61,111) Em, Hi“ (£2412) (a) (b) Figure 2.2. Equivalence principle models. 10 (6240) 0, 0 (6‘1, #1) Eli“, H‘iOt m_~ *1“ » ~ «2.112) /.. .3 3. ' ‘ 1 .\ : scat t :. [I (611 #1) "‘3‘ E! E2 ’Iigca - "\k l 1 ' E21112 ' .1 li fl \. ' w. ' r \ £3... , EM»! ‘2‘ ..;.;, ‘ .r‘zi \M“~*W’ Jeff (8) (b) Figure 2.3. Equivalence principle models of SIE. 11 CHAPTER 3 NUMERICAL SOLUTION USING THE METHOD OF MOMENTS 3.1 Method of Moments The method of moments is employed to convert the MFIE into a matrix equation, which can be algebraically solved to determine the unknown effective current [8] The general method of converting a linear integrodifferential equation £{f(x)} = 9(1) (3-1) into a matrix equation is discussed here. The first step in converting the integral equa- tion to a matrix equation is to expand the unknown in a finite number of subdomain basis functions N f(r) = Z aifBJ-(x) (3.2) 1:1 Inserting this into the general equation and using the linearity of the operator L result in n E acumen = 9(1‘) (3.3) 1:1 The next step is to test this equation with a testing function set. The testing pro- cedure is accomplished by taking the inner product of the equation with a testing function. The inner product of two real functions is defined as the integral of their product over their region of support, b (fawn) = / f(:v) he) dz. (3.4) 12 Using this definition, eqn. 3.3 can be converted into N linear equations by testing it with N different testing functions n 2 cm), aim-(ma,- '2 merge», 2‘: 1, N (3.5) i=1 The integral equation for the unknown continuous function f (x) has now been con- verted into a system of N equations in N unknowns. The unknowns now are the coefficients of the basis functions, i.e., aj. Converting into matrix notation gives i - a? = g (3.6) where E is an N X N matrix, and 517 and 57 are N -dimensional vectors with elements Lij = (3-7) and Qi = = 11/] 1,.r{rj} d1 (3.19) n . 1 With the given inner product, the intermediate expansion coefficients If and Iim 17 are computed from the coefficients of the effective current I jefi Ie ZWe 1h wh [193’] (3.20) Z . where [19], [1m] and [Ieff] are length-N column vectors of the respective coefficients, and [Z We] and [Z Wh] are N x N matrices, the entries of which are determined by the pre—defined integral operators in eqn. 2.4 and eqs. 3.17 and 3.18 2” ’i’j’=ai/27r6ij— 6ij)i1/:J/ [j],- )x VGl(r — r ’)]dS’ dl , Q .» ZVl/ifinz —]”1/1/jlij- (1")Gl(r— r') (15' dl (3.21) —]/l {251/ V’f r)[Gl (ira—r)— G1(r[)—r')] dS’ where ai is the angle between the planes of the triangles Ti+ and Ti measured in the exterior region, Tj = Tj+ + T]? and 5ij is the Kroneck delta function. Also, r? and r? are the two endpoints of the edge 2', such that the unit vector Ii points from rib to rid. G 1(r — r’) is the Green’s function for the exterior region G1(r — r') = e’jkl [r — r’]/47r|r — r'] (3.22) where k1 is the wavevector in the exterior region. Applying the same procedure to the MFIE equation 2.21, one can construct the moment-method matrix equation ZWe [ ZUe ZUm ] [Jeff] - [2W9] [Jeff] =[hol + [ZUellhol zwh (3.23) - [Z U ml [60] 18 In the above equation, hO and 60 are the length-N excitation vectors, the elements of which are obtained as ho 1= / HinC(r).1, dl 1 1° 1 . (3.24) 60’ i = /1’ Emc(r) - 1i d1 1 Z U 6 and Z U m are N x N matrices, with the entries ZUfi = — ai/27r6ij+(1- 6ij)1/li£ [I li'lf:1(rl)x VG2(I‘ — r')]dS" dl i j ZUinn =—stl/li/li [1‘ li-f3(r')G'2(r—r') dS'dl (3.25) J _9._ r..' ?_’_ b—' ' liflul TV 5(r)[02(r1 1') 01“! fl] d5 3.5 Augmenting with the Fast Multipole Method It is well known that the computational costs and the memory of classical MOM solvers that are augmented by FMM schemes scale as 0(Ns log NS) and 0(Ns), re- spectively. As such, the development of FMM based schemes has been a subject of intense study for a over a decade following the seminal paper by Rokhlin [10]. F MM employs a divide and conquer strategy to reduce the overall computational cost; this is achieved by embedding the body in a fictitious cubical box, and recursively dividing this into eight smaller boxes. A box that is subdivided into smaller boxes is termed the “parent” of the “child” boxes that result from the operation. This leads to a uni- form oct-tree structure. For an N + 1-level scheme, this subdivision proceeds N times. At the lowest level, the boxes are populated by basis functions or equivalently a set of point electric and magnetic dipoles. Fields due to these dipoles are computed at other locations by upward and downward traversal of the tree. In order to accomplish this in a hierarchical manner, the following dictum is used to create interaction lists: 19 a pair of boxes at any level are said to be in the far field of each other if the distance between their Centers is greater than a prescribed distance and if their parents are in the near field of each other. In practice, this distance, at any level, is chosen to be twice the linear dimension of the box at that level. Thus for any given box at a level greater than one, interactions with boxes in the near field have to be resolved at lower levels in the tree. Interaction with a box in its far field can be computed at a higher level in the tree provided that the respective parent boxes are in the far field of each other. The matrix equation 3.23 can be solved using the iterative methods, e.g., the transpose-free quasi-minimal residuals method (TFQMR) [11]. TFQMR is acceler- ated by the F MM method to achieve optimized computation time and memory cost [12]. 3.6 Numerical Results The aim of this section is to test the efficiency and accuracy of the single integral equations discussed in the last section. This is accomplished by applying SIE to the problems of electromagnetic scatttering of a plane wave by arbitrary shaped dielectric objects. In what follows, it is assumed that the dielectric scatterer is immersed in the free space (51 = 50, M = [10, c1 = 3 x 108 m/s), and the material is non-magnetic (p2 = [.10). These results are compared against the analytical data obtained by the Mie series [13] or the existing computer code based on the Muller formulation. As shown in Figure 3.4, the scattering from a dielectric sphere of 1 m radius and 62 = 480, centered at the origin and illuminated by an incident plane wave with Ex = :1“: polarization, it = ~23 incident direction, and f0 = 250MHz, is analyzed. There are totally 3600 unknowns involved in solving this problem. The data agree very well with those obtained using the Mie series. However, for this example, the SIE code converges almost twice as fast as that are based on the Miiller formulation. 20 In a second example, scattering from the same sphere as given in Figure 3.4, discretized in terms of 4020 spatial basis functions, is analyzed at f0 = 160 MHz. The sphere is excited by an electromagnetic wave that is Ex = :i: polarized and incident from the It = —:2 direction. The RCS pattern in the :r-y plane compare very well with the Mie series, as illustrated in Figure 3.5. The next example shows the scattering from a dielectric sphere of the same size but with 82 = 450. The incident wave is Ex = :i: polarized, and impinges on the scatterer from the If: = —:2 direction. The incidence frequency is f0 = 225MHz. There are totally 9414 unknown basis functions. Figure 3.6 demonstrates good agreement between the RC8 patterns obtained using the SIE solver and the Miiller code. Our final example of the sphere, given by the second example but analyzed at a higher frequency, is shown in Figure 3.7. The boundary is modeled by 30318 unknown. The incident plane wave at the frequency f0 = 300MHz is Ex = :f: polarized, and illuminates the sphere from the ft = —2 direction. Again, results obtained using both SIE equation and Mie series compare satisfactorily. It turns out that the SIE code can handle not only the smooth geometries, e.g., the sphere, but non-smooth boundaries as well. As shown in Figure 3.8, a cubic box of dimensions 1.0 mx1.0 mx1.0 m is excited by an electromagnetic wave that is polarized in the :2: direction and incident from the -2 direction. The total number of unknowns is 4077, and the incidence frequency is f0 = 257MHz. The agreement between the results obtained by SIE and Miiller codes is again very good. In the last example, scattering from a “fat” almond, discretized using 2208 basis functions is analyzed using SIE and results verified by the Miiller formulation, as demonstrated in Figure 3.9. The almond fits in a box of dimensions 1.5 mx1.0 mx0.8 In. The incident field travels in the I} = —:3 direction, is Ex = :i: polarized, and has a frequency of f0 = 100MHz. 21 EF;;‘;¢_— ; .‘- u‘V~ ‘« - ‘w’fin. fin” ”AVA? _| . ‘ 53; 5:1” ‘1 «flat; 3"” :5"ng F ‘T 1'." g 15’ i n . 1 4’4? “it?“ 5’. 5).: “0%“ mnuw in“ *i‘hv¢¢uvmvmum§t¢ : v , 3- {a $3”; ‘VAYAV‘YavaVAVfifi ”M Vigil" * v, AVAYAVAYafl Figure 3.1. Triangulated meshing of a dielectric sphere with r = 1.0. The number of triangles is 2904, and the number of edges is 4356. 22 Figure 3.2. Triangle pair (Ti+, Ti') and parameters associated with the ith edge. 23 H + N :L + ,...v... L 1' 1?: Figure 3.3. Geometry parameters of the triangle pair Ti+ and Ti" . 24 RCS (stm) 14 12 10 ,_\ —- Mie Series -— SIE fEm \ / I \ \ / / \ \ / \ I l I l L l l I l l l 20 40 60 80 100 120 140 160 180 200 ¢(DEG)-> Figure 3.4. RC8 pattern of an 52 = 250 sphere in the 13-3; plane at f0 = 160MHz. 25 —— Mie Series — Single htegral Equation fEx 10 ~ 5 _ E m m E $3 0 .. -5 _ _10 1 1 1 l 1 1 J J 0 50 100 150 200 250 300 350 400 (b (DEG) —) Figure 3.5. RC8 pattern of an 52 = 450 sphere in the 17-3; plane at f0 = 160MHz. 26 15F — Mie Series E — Single htegral Equation (B 10 — 5 .. A I E 8 3 o - co 0 Ir -5 _ -10 - _15 1 1 1 1 1 1 ' 1 1 0 50 100 150 200 250 300 350 400 «p (DEG) -—> Figure 3.6. RC8 pattern of an 52 = 450 sphere in the x-y plane at f0 = 225MHz. 27 Dielectric Sphere: er = 4.0, 0/1 = 4.0. NS = 30.318 25 r r f T l L 4 ' ' ' ' MieSeries - — - Single Integral Equation 20 ............................................................................................... .l 15 .............................................................................................. «a g I m : 31o ........... - (n : Q . “r 2 5- ........... .1 0 ................................................................................ ........... .. 1 f _5 J 1 1 1 1 1 4 50 100 150 200 250 300 350 400 (13—) Figure 3.7. RCS pattern of an 52 = 450 sphere in the x-y plane at f0 = 300MHz. 28 I — Muller ~— SIE E .. rn m B Q - .4 _30 1 1 L 1 l l l 50 100 150 200 250 300 350 400 (b (DEG) —) Figure 3.8. RC8 pattern of an 52 = 450 cubic box in the :r-y plane at f0 = 257MHz. 29 E J m in 3 g a _l 40 _ _ .45 - _ —— Muller —— Single Integral Equation _50 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 400 d) (DEG) —) Figure 3.9. RC8 pattern of a fat almond in the :r-y plane at f0 = 100MHz. 30 CHAPTER 4 SPECTRAL ANALYSIS OF THE SIE OPERATOR Method for scattering analysis from dielectric objects have been dominated by the PMCHWT and Miiller formulations have dominated the computational electromag- netic (CEM) society for decades. However, the constraints as discussed in section 2.2 have prompted the CEM researchers to search for desirable integral formulations. Recent progress in the construction of “fast” methods for the solution of the boundary integral equations, in both frequency [10] and time [14] domains, has vastly expanded the scope of tractable problems. Most of the integral formulations for analyzing scattering from dielectric bodies to date are in essence various combinations of L1{*} and K1{*} operators as given in 2.4. Understanding the performance and behavior of new boundary integral formu- lations requires a rigorous mathematical investigation of the basic integral operators. Throughout this chapter, we investigate in detail the spectral properties of different operators, and specifically analyze the SIE operator. 4.1 Definition of the Spectrum Analysis We illustrate spectral analysis of operators used for analyzing scattering from a per- fectly conducting (PEC) sphere of radius a. First, any surface current J on a sphere may be given in terms of the surface Helmholtz decomposition J = Vtcp + f x vii/J (4.1) 31 with 00 11 d1 2 90 = Z Zdn / mbanm (4.2a) 11:1 m2- 1] 111 = Z Zdl nm 2YCnm (42b) 11:1 m=- 'D where anfi‘) = Prllml(cos 6)e'jm¢, n 2 O, |m| g n (4.3) is the spherical harmonics [1], and _ (n. -— Iml)!(2n +1) dnm — (n + lml)!47rn(n + 1) (4'4) is the normalization constant to simplify the subsequent calculations. The coefficients bum and cum may be any complex numbers. Using the conjugate relation 7:1 = Yfim, it is a straightforward exercise to establish the orthogonality of the spherical harmonics: 1 2 1 2 A — I A I [9 dnfn dn'/m'Yrin(r)Y33 (r) d8 21r — —a 2/0 / d1/2dIII/Ii,P,'1ml(cos 6)Pr'1r,n l(cos 6) e‘iim'm >13 sin0 d6 dqs - 25nn'5mm' n(n + 1) 32 1/2 1/2 —m’ nl / dnmdnlmrvtynn:l(f) ‘ VtY (i) dig, S = / dfifidifi, [i- x vii/3(1)] . [f x v37$'(i)] dS’ S = [02" f0” difidfi/Ii, [digpgmkcosm %P},‘P"(cosa) (4-6) +ST:::;:PI|1ml(cos 6)PI|1I,nll(cos 6)] e'j(m'm,)¢’ sin0 d6 do = 5nn'5mm' and f8 vii/3(3). [f- x vii/"3113] dS’ = 0. (4.7) In eqs. 4.3 to 4.7, (6, (1)) are the spherical polar angles, and i‘ is unit vector normal to the spherical surface. From the equations above, a complete set of basis functions on the surface of a sphere of radius a is given by the vector spherical harmonics 36.1mm» = f x vii/mm) (4.8a) -j n(n + 1) mam = r x inmw, 45) (4.812) Next, the conventional electric field integral equation (EF IE) and magnetic field integral equation (MFIE) for electromagnetic scattering from PEC surfaces may be rewritten as fl X Einc = 171T1 O J (4.9) a x Hinc = (g + K1) 0 J (4.10) 33 where the integral operators T1 and K1 are defined [15] by T1 0.] =T(k1)OJ = —jk1 f1 X [S ds’{G(k1,r,r')J(r’) (4.11) +ki2V[VG(k1,rir') 400]} '1 (4.12) = _fi x/S ds’VG(k1,r,r') x J(r’) 1 1 Applying T1, K: E (K1 + 5) and Ki E (K1 — 2) to each basis function in 4.8 yields 76 —11 k a 1111 k. a {—5 T(kl)o A“ = “( 1 ) ”( 1 L“ (4.13) Unm Jh(k1a)Hh(kla)Xnm i’ - '1' k a H k a x’ K+(k1)0 _,nm = J n( l ) Il( 1 :nm (4.14) Unm jlln(kla)ll'll’n(kla)Unm x 411 k. a 1111' k a if K'(kl)o gum = J n( 1 l n( 1 an (4.15) Unm jjh(kla)lHln(kla)Unm where Jim and Mn are Riccati-Bessel and first-kind Riccati-Hankel functions of order n, and k1 is the wavenumber associated with the kernel of the each integral operator. The Riccati-Bessel and Riccati-Hankel functions are defined in terms of spherical Bessel jn(:r.) and Hankel hfll)(:r) functions by link?) = 1711(1) (4.16a) Il-lln(:r) = $IL£1)(1:) (4.16b) The spectrum of an integral operator is defined as the function preceding each 34 basis function on the right-hand-sides in eqs. 4.13, 4.15 and 4.14. There are gener- ally two spectrum factors associated with each basis function, respectively. For the convenience of what follows, Alli X denotes the spectrum function of order 71 due to —> ——> T operating on X, and Arrf U represents its U cousin. 4.2 Spectrum Properties of the Integral Operators Well-behaved integral operators are the sum of a constant operator and a compact operator (see Appendix). The operators lead to second-kind integral equations, which can be solved with fully controlled error. However, boundary integral operators in- volved in scattering analysis typically violate this requirement in one or more of three ways. 1. The operator may accumulate at zero. A typical example is the spectrum function Mi X' As shown in Figure 4.1, a plot of three spectrum curves as a function of ha is presented for orders 1, 3 and 5. It is worth noting that for a given kla, A? X eventually vanish, as indicated by the position of point A on the complex plane. 2. The operator may have an unbounded spectrum, such as a hypersingular oper- ator. The spectrum function /\% U falls into this category. A similar plot in Figure 4.2 demonstrate the singularity of A1} U when the order increases. For a fixed kla, A% U would eventually blow up to 00. 3. The operator may have trivial spectrum values associated with resonances, often nonphysical. These are often referred to as “spurious resonances”, which can be observed in the case of the operator T1. It is well-known that the MF IE operator (or K1+ ) is a second-kind integral oper— ator. As illustrated in Figure 4.3 and Figure 4.4, both A§+ and Ali‘l' converge ,X ,U 35 to —1/2 in the limit of large 71. Therefore, the operator K+ maintains a bounded spectrum. However, as is evident from eqn. 4.14, Figure 4.3 and Figure 4.4, the oper- ator K + yields null eigenvalues at the zeros of .lthcla) for Yum, and at the zeros of .lln(k1a) for finm, it is not regarded as a well-behaved operator. By the same token, the operator K ‘ seems spectrally bounded, but is not free of resonances. Similar plots are given in Figure 4.5 and Figure 4.6. Our analysis shows that the operator T1 does not have a bounded spectrum, either. Yet a cascaded operator T 2(k1) = T(k1) o T(k1) seems to possess better spectral properties. 32’ 33 T2(k1)° _,nm =r1112" Hum Unm Unm (4.17) , , xnm =-.lln(k1allHln(kla-l-Un(kla)Hn(kla) —> Unm It is worth noting that the basis functions inm and finm are eigenfunctions of the operator T2(k1), and its eigenvalues accumulate at -1/4, a result which follows from the asymptotic properties of jn(:r) and hg1)(a:) [16], and is illustrated in Figure 4.7. ' An interesting and yet useful derivation [15] 3'5 K+ o K’(k1)° sum Unm Y = ,i}. 0 K1 0 411m (4.18) um I I 3sum 2 —.Un(kla)ll‘lln(k1a).Un(k1a)IHIn(k1a) —§ Unm 36 reveals the identity T2(k) = K2(k) — — = K+(1~.) o K’(k) (4.19) 4.3 Analysis of the SIE Spectrum Following the notation of the operators T and K, we can rewrite the SIE operator on the left—hand—side of eqn. 2.21 as SmOJerr= K'oI<-oT oJeg (4.20) Our previous analysis reveals that both of the operators T o T and K + 0 K' have bounded eigenvalues that asymptotically converge, for large order n, to given values. Thus we “predict” that the linear combination of K :2 0 K i and T 2 0 T1 possesses a similar spectral behavior. It is a straightforward exercise now to derive the eigenvalues of the Sm operator. if —'11 k 111' leaf nm =K'(k2)o .7 n( la) n( 1 ) nm Unm )1;,(k1a)nn(k1a)finm —.ll kalHl kaU —:7—1—T(l.72)0 n(1) 11(an111 ’72 llh(kia)Hh(kia)Xnm ——> —_Iin(k.1a)lfll’n(k1a).lln(k2€l)Hh(k20) xnm —-> _J;,(k1a)111n(k1a).ll;,(k2a)Hn(k20)Unm 771 ’ln(k1alHn(k1a)llh(k2a)Hh(k2a)inm "2 —i;.in(kzammam’nm 37 Jn(k10)H'n(k2a) %Jh(k20lfln(klal — 111(k2allHleUc1a) Fine 4 21 lh(kia)Hn(k2a) %Jn(k2almh(kia) - llh(kzallHln(k1a) finm ( I ) Figure 4.8 and Figure 4.9 show the eigenvalue curves of a dielectric sphere with k2 = 21:1. As the order 71 increases, the eigenvalues accumulate, for the given material contrast ratio, at 1 / 2 for )‘ISlmX’ and at 5 / 4 for ”gmfl' However, the SIE operator Sm suffers from the internal resonances, due to the fact that the eigenvalues vanish at the zeros of .Iln(k1a) for the inm and at the zeros of 11;,(k1a) for the finm. As a result. the operator Sm is not regarded as a well-behaved second-kind integral operator, and therefore the SIE formulation, by itself, is not a suitable integral equation for analyzing closed dielectric objects. Yeung [6] claims that although the EF IE 2.19 and MF IE 2.21 are individually singular at the same resonant frequency, a linear combination of the EF IE and MFIE, namely the CFIE = [(1 — a)A]MFIE + amEFIE, where 0 < a < 1.0 and A is the average length of the triangular-patch model, is non-singular at all frequencies. The operator Se as in eqn. 2.19 can be expressed in terms of T and K as Se 0 Jeff = 7)] Z—2f X TM?) 0 K_(kl) + f‘ X K-(k2) O T051) 0 Jeff (4.22) 1 And the eigenvalues can be obtained 36 Sec Hum Unm —- Ill k a IHI’ k a i =771'77—2f‘XTUC2lo ]n(1) n(1)—)nm ”1 jig,(k1a)11n(k1a)Unm —.II I: a 1111 k a 1—1’ +fo_(k2)o 11(1) 11(1an Jh(Aila)Hh(klalxnm 38 jlln(kla)lHI’n(k1a).lln(k2a)lliln(k2a)finm _ 772. -01 —I‘X —> "1 1;.(k1amnwlawakeamewea)Xnm fix —j-1ln(klalmn(klal-Uh(k2a)Hn(k2alfinm ~11;(name(k1a>1nH;. (4.23) j-lln(k1a)Hn(kla)Jh(k2a)Hn(k2a)Xnm ’j-lli1(k1(l)Hh(klal~lln(k20)Hh(k2a)I—jnm j.lln(k10)Hn(k20) [.ll’n(k2a)lHIn(k1a) =n1 n1 —j-llii(kia)Hh(k2a) [Jn(k2a)Hh(k10) —%21—11wnT(k'1)+—T(’€2)°K (’91) 0 _, 722 -Xnm 11' k a 1111' k a if =01 K‘(k.2)o n( 1 l n( 1 )_,nm .Iln(kla)ll'lln(kla)Unm '1' k a 1111 k a U +n—1T(k2)o J n( 1 l n( 1 )finm 772 j-lln(kla)Hh(kla)Xnm 7) -J'llii(k10)H'n(kia)Jn(kza)Hh(k20)Xnm : 1 _’ flinwlalfln(klal3h(k2a)Hn(k20lUnm +n_1 'j-llii(klalmn(klal-Uh(k2a)Hh(k20)Xnm "2 —11n(k1a)H'n(klawneeamneeaffinm -J‘~li’n(k10)Hii(k2a) [ n(k2allHl'n(kial - %Jh(k2a)HD(kla):| Yum j.lln(k1a)lllln(k2a) [.llfikgafllilnwla) — -:;—;.lln(k2a)llll’n(k1a)] Unm (4.26) Apparently, they don’t share any zeros with the eigenvalues of operator S. Thus, we can write an expression for a well-behaved combined source single integral equation 40 (CSIE) operator SCS 0 Jeff : Sm O Jeff‘l' Sm? 0 (fl X Jefi) ——- [Iran 0 K111) — Z—:T(k2) 0 mm] 0 Jay (42?) — m [1:11:21 0 T1k1>+ Zine) o K'(k1)] o (a x Jee) The condition number of the CSIE, though larger than that of the SIE, is still domi- nated by the well-behaved property of S. 4.5 An alternative integral equation scheme It should be noted that the CSIE formulation is not the only approach for eliminating the spurious resonances. We spend the rest of this chapter analyzing an integral equation scheme from a different perspective. At interior resonant frequencies, the E field tangential to 9 calculated from the SIE surface currents is not continuous across the boundary. It follows that the H field normal to Q will not necessarily be continuous. It is suggested that the inclusion of the normal boundary conditions of the magnetic field would augment the SIE to yield the unique exterior solution at all frequencies [18],[19]. fi- [#vzHica’tl = fi- [#1(Hinc + H1635] (4.28) 41 It can be re-written, in analog to eqn. 2.17, in the following form. . inc _ . (1'2 11' [#111 l- n - (331%le + #2K2{J2} - MiKllJefil) _ - ”2L _Einc .. _ L. J .. -n' 752{ Xn n11iefrlxn} + Mszi-fi >< Him - 13 X K1{Jeir}} -u1K1{JeH}) (429) = 13' (Ciin—Einc X f! — 771L1{Je£f} X fl} 2 + #2K2i-fi >< Him - fl >< KliJefrll —#1K1{Jeff}) where c2 = 1/, #32112 is the velocity of light in the interior media. Rearrangement of the terms in the above equation gives the “augmented” single integral equation (ASIE) .. .. 77 .. ,. .. n ' #2K2{n >< KliJefrll + $11142“! X LiiJesll - n ' u1K1iJeH} = H5115 (430) where H318 = a . lem0 + a - 112K2{fi x HmC} — 51112911 x EmC} (4.31) Next, we prove that this ASIE shares no resonance with the existing SIE spectrum. By inspection, two differences can be observed between ASIE and eqn. 2.20. 0 Due to the normal product (fr), ASIE is a scalar equation; 0 The term —fi - p1K1{Jefl»} cannot be combined with f1 - 112K2{fi x K1{Jefi}}. 42 Using notation similar to eqn. (4.20), we define the ASIE operator as 33'. 0 Jeff = {-11sz - [K2 0 K(k1)l + ‘2‘?” [L2 0 T(/~71)l - mfi ° K1} 0 Jeff (43?) Utilizing the mapping of T <———> n x L and K +——+ -fi x K, and representing the unknown Jeff with inm and Unm, the right-hand-side of the ASIE equation can be rewritten as = -u2fi - K2 0 [Ken o xnml + Pin - L2 0 [Tim 0 Km] — 111a. [K1 0 36mm] _ ~ + . 1 —* ’71 ~ ~ T _ —112n-K2 e [K (1.1) — 5] o Xnm + 2241- L2 0 [Mil 0 Xnml = —u2fi ' K2 0 lK+(k1) 0 inml + Z—éfi - L2 0 [YT/~71) 0 Km] ,. —> 1 - _’ — #1“' [K1 0 xnml — EHZD' [K2 0 Xnm] = j11.2.ll;1(kla)Hn(kla) {fi- [K2 0 Xnml} - flfln(kla)H11(kla){fi'lL2 ° Unml} C2 .. —-> 1 - '2 — pln '[K10 Xnm] _ 5H2“ ' [K2 0 xnm] . , 1 . —* = #2 (Jinwlamnma) — 5) {n ~ [K2 0 Xnml} — 21—.lln(llc1(1)1H1n(klal{fi‘lL2 0 Umnl} C2 — #lfi‘ [K1O i*nml (4.33) 43 and Si 0 1—er1111 = {“Wfi ' [K2 0 K(kl)l + 13131112 0 T(klll — #lfi ' K1} 0 fjum : _,.2fi - K2 0 [K(k1) 0 1‘1an Z—éa - L2 0 [T(k1) o Unm] — #14 [K1 0 finml : _p,2fi . K2 0 [K+(k1) — é] o Unm + Z—éfi - L2 0 [T(k1) 0 finml — mfi- [K10 I7’an : —112fi-K2 o [K+(k1) o 17’an 761:4: - L2 olT(k1)o rim] — Mfr [K1 0 Unm] — ‘21‘#2fi' [K2 0 i3nml = —juein _ I 1 :r s: o xnm = #2 [11..(kla1111n124a) — 5] 14.3.32) — 27—31}ana.)lliln(kla)‘ptrllm(k2) _ ulqflfimwl) Sm 0 Xnm = Jn(kla)Hh(k2a) [1% h(k20)Hn(klal - ‘Unfk2athW10'l SEnm (4.40) -——+ . ’ 1 u Sit. o Unm = -#-2 [21114151111314 + 5] enmrke) 77 I I 1' u + éfln(k1a)lliln(kla)q’nm(k2) — minim/£1) sm 0 Unm = lunamnwea) [Z—é-Uflbalmflft’la) — 1114251114315] Unm (4.41) At interior resonant frequencies, i.e., nulls corresponds to .lln(kla) or .Ilh(kla), eqs. 4.40 and 4.45 would reduce to Sean 0 Xnm = #2 [NMMHWnUCial — i] (141111le - II-l‘l’iim(k1) Smo Xnm : (4.42) 46 3% 0 Unm = —%#2¢"rim(k2) + 321-121(1710)Hh(k10)‘1’3m(k2) — #lq)iim(k1) s... o Unm = arklamnwea) [Z—énrmmam — itHnl Unm (4.43) for .lln(k1a) = 0, respectively, and 3% ° xnm = —%~Un(kla)Hn(kla)‘I’iim(k2l — #I‘I’fim(kll Sm 0 76mm = Jniklalmmkzal [glhfizalflnflual - JnU‘zalflhUflal) 32mm (4.44) Si. 0 Unm = -M2 [jlln(k1a)lfll;1(k1a)+ i] ‘I’iimezl — H1¢iim(k1) (4 45) Sm ° I_jnm = 0 for .ll'n(kla) = 0. Now, the foregoing analysis should suffice to conclude that by solving SIE and ASIE, together, Sm 0 Jeff 2 Hrhs (4.463) s; o 18,,» = H315 (4.46b) spurious resonances from the exterior scattering or radiating problem can be elimi- nated. 4.6 Implementation and solution of the augmented integral equations From the prior analysis, one can say that ASIE is a viable remedy for the interior resonant issue of SIE. However, it proves no advantage unless several difficulties are addressed when implemented using MoM. o The augmented equations are overdetermined. Two equations need to be solved for only one unknown current J eff o Cascaded operators, e.g., K'(k2) o K'(k1) and T(k2) o T(k1), are encountered 47 in eqs. 4.46, and they can possess hyper-singular component, especially for T032) 0 T031). 0 Both vector and scalar equations show up in the augmented equations. There is a standard procedure for finding the least-squared solution to an overde- termined set of equations [18], [20]. Thus we are allowed to solve eqs. 4.46 by multiplying the equations by the Hermitian conjugate of its coefficient matrix, and then solving the resulting even-determined, Hermitian set of equations. Since we have proved that the overdetermined equations can yield a unique solution at all frequencies, the least-squared solution becomes identical to this unique solution. As for the second bullet, appropriate intermediate projection spaces need to be chosen discretely for the each of the inner products. Adams et al. [21] have proposed using the surface Helmholtz complement of the RWG subspace ({fi x fi}[:l) for the intermediate projection. With the appropriate projection subspaces, a correct discretization procedure for the T(k2) o T(k1) operator can be achieved. Since eqn. 4.46b is a scalar function, a set of scalar testing functions are required. Pisharody et al. [19] recently reported that spatial scalar testing functions can be constructed using the Silvester polynomials [22] in the normalized parametric coordi- nates. The most straight-forward category are the “hat” functions, which are unity at a given node, vanish at all neighboring nodes. 48 — 1st order — 3nd order 0.8 — — 51h order 0.6- 0.4- F~fi >1 5, 0.2 _ ’< w—v 3 o- E H 41.2 _ A: ka = 2 8: ka = 3 -0.4 ~ C: ka = 4 0: ka = 5 -0.6 — -1J.2-i T .ole ole ch d2 r) 0:2 34 Rea|{)\T,X} Figure 4.1. Plot of three spectral functions of /\% X for ha 6 [0.01, 6], and n 6 [1,3,5]. 49 12_ TA — 1st order ‘ -— 3MOMm I —- 5th order 0.8 - Imag{)\T,U} O I A: ka = 2.0 8: ka = 2.5 -O.4 —- C: ka = 3.0 D: ka = 4.0 -0.2 0 0.2 0.6 0.8 1 0.4 Real{)\T,U} Figure 4.2. Plot of three spectral functions of Hf U for kla 6 [001,6], and n 6 [1,3,5]. 50 3rd order ———- 5th order -— is! order \ -«\__-—-r___-__ \ 1 1 l ‘~4 l I l l I I I I .l I l I II \ I I 1 ""’""T"""‘"F """"‘-“l"""'"T"'-""I""" l I I f— I I I I" I I I I l "‘"'"7""-----1"""‘"T"‘""" I I I 'l"-’-""1"-'-"'T-""-" I 3 A I I I A._' / I —----4-------—?--—-----r----—-- ‘ i ____-___q__ __g 3 A:ka=2 B:ka= / I I l C //l T:- _ - s 1 1 :7? --_-_¢ __-..__+-__--_.._.._L \ \\ \ \ §Q§ _..-- ---._(_--_ “'V\ B l I I l l l \‘ I l 1 l 0.4 FQEHBLLAI(+.)(} 1 I I I 0.2 IIIIIIIIIIIIIIIIIIIIII OP—--—-----‘--—---fi----- c.3—-—-—--—- 344------" o.5~—------- 1.2 0.8 0.6 -0.2 for ha 6 [001,6], and K+,x Plot of three spectral functions of A" Figure 4.3. n E [1,3,5]. .. ....... - ....... _ 1111111 _ -1---)1.-1---11.-1-1111_ 1111111 .r a. . _ _ . _ "mm-m” " n u n u . . _ . . . _ .1 h. _ . _ 23. 5 _ .mwa. . . _ = 3.” = _ _ _1_ ...... L. ..... _ ---L_---.mn..m..ml-._ ....... n _ u _ u _ Renee. " _ _ _ _ _ . _ _ . _ . _ _ . . . . . _ . . _ . _ _ _ _ . . _ _ . _ _ _ . . . _ _ _ _ n ...... to ..... 1. ...... .u ...... 1. ..... o ..... ._ ....... n u n u u n u n u u a n u u n . _ film. _ _ _ _ . _ \\®\\I l/QUA . . _ . _ \D. C . . _ . -i.--.\-1_ ...... Mme ..... ._ ...... ._ ..... ._ ....... . _ x . . . . . . \ . A . _ _ n 1 n u u u n . _. _ . _ _ _ u i u u u u a u . 11111 G. 111111 . 1111111 . 111111 1. 111111 .— 111.114 111111 J _ a _ _ . _ _ u A: u u u n 4 n _ _ _ _ . 4 _ . a . . _ . _ u I u u u n \ u - ||||||| [I'll-4". lllllll — ll-‘lll—l"-lllu— "IIIII— """"" . _ . . . _ x u u / n n u ._ u . . I _ _ _ _ _ \ u u 1 n u . u H u u e._. x n _\ n u . 1111111 . 1111111 .1111]: l 111.111111.._11111..1_ 1111111 n u n .0 u u u . . _ _ _ . _ _ . _ _ . _ . . . . _ . _ _ _ . _ _ _ _ . _ . . _ _ _ . _ . _ _ _ _ . _ _ _ _ _ _ H 6. 4. 2 o 2 4. 6. 8 Eek-«vamp: 1.2 0.8 0.6 0.4 0.2 -0.2 for kla 6 [001,6], and K+,U Plot of three spectral functions of A" Figure 4.4. n 6 [1,3,5]. 52 lilifllrldIll4111411Il.11||..1||l_.llll_llil_111141114111] _ . _ . . _ . _ _ . More _ . . _ . . . _ _ . . . _ _ . . _ . _ . WW _ . _ . . _ . _ . . h . . . _ . _ . .23.45. . _ . _ . . . . _ . MS . _ _ _ D . . _= =... =. . _ . . 1... . . .ha.88_ _ _ -.---._--e._-\- .. ...... it _ . .1 . . _./. . _ . _ _ _ \.. _ . _ 1 .AB_CD. . _\ _ _ . . 7/ _ _ _ _ \ . _ . _ . I . _ .x . _ . _ . _ -_ . _ _ _ _ . _ _ . _ _ _ \ . _ . _ . _ /_/ _ _ liki111.....111_1111.1111_1111...111.1111.14 1.111w1114 _ _ _ _ _ . . _/ . \ _ _ _ . _ _ . . . _ _ . . _ . _ _ . . . _ e _ . . _ _ _ . _ / _ _ _ . _ . _ . _ . _ . . _ _ . _ _ . _ _ . _ . _ . _ _ . CWGA _ _ _ _ _ _ _ . _ _ _ 111.1111p:111.1111.1111_1111_1111_riil_11.. .1 liprirr . _ _ _ _ _ _ . fi. . _ . _ . _ _ _ _ \. . _ _ _ . _ _ _ . _ _ . . _ . _ . _ e. _ _ _ . _ . _ _ _ . _ . _ _ . . . . _ . _ k _ . _ \ _ _ _ _ . _ _ \\ _ . ---".---.T-L_---._--B. --.+--+o.- 7-4141} _ _ _ . _ _ . C. x. _ _ . . . _ _ AfiHMf.l\\L.W.\\. . _ _ . _ . _ ...II .E _ _ . . . . . _ . B _ _ _ . . . . . _ _ _ _ . . . . _ _ _ . _ . _ . _ _ . _ _ . . _ _ _ C a iiiirru e 111Flinn:11.7111.11-1.1111r111r111 e: 15-..: . 2 . _ . _ . _ . .nu _ 4 . _ . . _ . . . _ . _ . _ _ . _ . _ _ . . . . _ . _ . . _ _ . _ . . _ _ _ _ . _ . . _ _ . _ _ _ . _ . _ . _ _ _ . . . _ _ _ . _ . _ _ _ _ _ _ _ _ iilirnlirilipiiluiii .. 11111 r111r111e111h1111 _ . _ _ _ . . . . _ . . _ . . _ . . _ _ . . . _ . _ _ . _ _ _ _ _ _ . _ . _ _ _ _ _ _ . _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ . _ _ . _ _ . . . _ . _ . _ _ _ _ . . . . _ _ _ _ _ . _ _ _ _ _ 5 4. 3 2 1 0 1. 2. 3 4. 5. o o o o o o o a o o a 1.2 0.8 0.6 Real{AK_,X} 0.4 53 0.2 41.2 Figure 4.5. Plot of three spectral functions of AICX for kla 6 [001,6], and n 6 [1,3,5]. III .I III III lul 'I'IIIJIIIII' III'IIII I I l'l'l‘ - I- 4 J Ii . Ii — _ .-.|J.lk|K‘KtlI-"II1 . on on“ '1‘ Ad I -I' 14 III III IIIIIIII'II. I II I II III (I lI-"| 'I'II-d' III 'III “I. I- nd 5 Id 9 [A III 'IIIIIIIII III 'I lull II \ II _ \ I'I... I'll I'I lllvlell'"'- II-Illelv'IIIII-IA . III II- I- \ 'III' \ 'Iv \ e — t I— III» lull-III'I'l' II'IIII' I Il‘lll IL / III \ IIIIII \ II l'II I — _\l\'l|. ll llw'l'l'I-L .II|/ II II I lhl "I / Al ' .‘lIII o o u I “- Tank-$095 1.2 0.8 0.6 0.4 54 Real{).K_,U} 0.2 (1 001,6], an " for ha 6 [ tral functions of [\K’U hree spec Plot of t -0.2 Figure 4.6. n 6 [1,3, 5]. 03~-—--——-- 111.. 111111 .“1 1114111 . . 1111111 . . . 1111111 _ . . _ 111.41.... . . _ . . 6 .lll _ _ _ . . 14.5 .rr _ . . _ _ = =.=“ . e . . . . . .a 1111 "WWW" . . . _ lube-flak!“ — — III '| II it .- .t. h. . C— 11141111 4 ABHCD .aelvwflm. . ,,., 1/. . _ . . _ _ x n . _ _ u . _ 1111 . 11141.... _ 11111 . . 111111 . . 111 _ _ _ . 111m . .11 . . . . _ 11111 _ IIIJII . _ . 11111 . . .. _ _ _ . 1. _ .1 . . . . . _ ....... . _ 111. . . . _ . n _ --- :3 n .1 11111 _ . _ - . _ 1 _ _ 111111 . 111111. . 1 . . 1111111 . . 1111111 . . 111111 _ .1111/ _ . . . _ _ _ / 1. IIIIIII . . .._ 111111 . . . L 111111 . . . . 1. _ . _ . I . . _ . .. v . . _ _ 111111 _ ./DD. . _ . . . . _ _ . . _ . . . . . . . . . _ 4 _ . . . _ 3 . _ . . _ 2 . _ _ . 1. e. 4 . . p 1 0 0 2. 0. o 0.1 -O.3 10.2 -O.1 414 FQEBEIL{J\QHQC} -0.5 -0.6 O ‘ 12 I I p A 55 1.5” -—2ndonder _ _ _‘ — 31dorder 3;:2'; /-/‘ ‘\- —5thorder . = ’ \ 11- C2k8=4 ’I / ‘\ Dzka=5 D ’ ‘ / \ I \ 1 \ 0.5- \1 \ PM 1 >3 \ 5 . é” o— c _.. 1 w 9/: /A\ ' U) / g / , ’ m A ‘ 3.. /_ \’ I, g / / / \ 1 41.51 1 / I ¢ / K11] \ \ C 1 I, .B / \ J z ' 1 , ’ o \ \ / / , / -1~ ‘ \ @7\ ‘ x — r_, - - / \ / D / ” \ ~_. —/ I B . / \- \ l/f _15 L l - - l l J I -O.5 0 2 2.5 1 1.5 Real{Asm,X} Figure 4.8. Plot of three spectral functions of )‘ISlm, X for ha 6 [001,6], 3 = 4.0 and n 6 [2,3,5]. 56 2 F — 2nd onder -- 3rd order — 5th order 1.5 - 1 _ \ ~— \ D. a? f B 3 0.5 '- m I m / 1g 0 1- _0.5 _ A: k8 =1 8: ka = 3 C: ka = 4 D: ka = 5 -1 _ 4.5 1 1 1 1 1 -0.5 0.5 1 1.5 2 2.5 Real{ASm,U} Figure 4.9. Plot of three spectral functions of Agm, U for ha 6 [001,6], and n 6 [2,3,5]. 57 01er 1-* to II 11>- O CHAPTER 5 CONCLUSIONS This thesis presented an mathematical perspective of a new integral equation for analyzing the scattering from arbitrary dielectric bodies. The equivalence models, the numerical implementation using MOM, and the rigorous analysis of the spectrum properties of the SIE Operator was covered. A modified SIE scheme was devised and proven to be second-kind and resonance free for spheres. First, the standard models for analyzing the scattering from dielectric objects were introduced, followed by a in—depth discussion of their respective pros and cons. The SIE integral equation, with the subsets of MFIE and EFIE, was presented in detail. Second, the general method of moments was covered, with the emphasizing on the conventional steps converting a linear integro—differential equation into a matrix equation. The RWG triangular basis function was reviewed, and used to represent the effective unknown current in a discretized form. A two—stage projection technique was specified, i.e., the inner product was defined and two cascaded matrices were constructed to replace the SIE integral equation with a matrix equation. Then the accelerative algorithm, F MM was included to expedite the solving phase. Numerical results generated with various geometries were illustrated, and in comparison with the known data, the accuracy and the convergence of the SIE scheme were clearly demonstrated. Third, the mathematical foundations of the spectrum analysis were covered, start.- ing with the spherical basis functions. All fundamental integral operators were inves- tigated and their spectrum properties were discussion by showing the spectral curves in the complex plane. Then the same steps were applied to the SIE operator. And it is shown that the SIE operator behaves like a second-kind integral operator, but 58 suffers from spurious resonances. A supplemental magnetic source turned out. to be an appropriate remedy, and would yield a combined-source SIE, which were found to be free from resonances. The thesis extended to discuss an alternative approach to overcome the interior resonance. In this new approach, the normal fields are included to augmented the original SIE-MP IE equation. Rigorous proof shows that this set of augmented field equations possess a unique solution at all frequencies. Crucial difficulties in practical implementation are examined and necessary information is provided as to solve the whole system. 59 APPENDICES 60 APPENDIX A DEFINITION OF A SECOND KIND INTEGRAL OPERATOR The standard definition of a second kind integral operator is an operator for the form /\I + K (A.1) where A is a constant, I is the identity, and K is a compact operator. In scattering theory, one encounters operators of the form AIPI + A2P2 + K (A.2) where /\1 and A2 are constants and P1 and P2 are orthogonal projection operators such that P1 + P2 = I (A3) Operators of the form A.2 possess most of the desirable properties of second integral operators. Such expressions are referred as second kind integral operators throughout this paper. 61 BIBLIOGRAPHY 62 [1] l2] l3] [4] [5] [6] l7] [8] [9] [10] [11] BIBLIOGRAPHY R.F. Harrington, Time-harmonic Electromagnetic Fields, McGraw—Hill, 1961. J .R. Mautz and RF. Harrington, “Electromagnetic scattering from a. homoge- neous material body of revolution,” AE U, vol. 33, pp. 71C80, Feb. 1979. Miiller, C. Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, 1969 E. Marx, “Integral equation for scattering by a dielectric,” IEEE Trans. Antennas Propagat, vol. 32, pp. 166-172, 1984. A. W. 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