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DATE DUE DATE DUE DATE DUE it’Eb 9 i Mi 6/01 cJCIFiC/Datoouepss-p. 15 MODELING THE RADIATION FROM CAVITY-BACKED ANTENNAS ON PROLATE SPHEROIDS USING A HYBRID FINITE ELEMENT-BOUNDARY INTEGRAL METHOD By Charles Alphonso Macon A DISSERTATION Submitted to Michigan State University in partial fillfilhnent of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical and Computer Engineering 2001 ABSTRACT MODELING THE RADIATION FROM CAVITY-BACKED ANTENNAS ON PROLATE SPHEROIDS USING A HYBRID FINITE ELEMENT-BOUNDARY INTEGRAL METHOD By Charles Alphonso Macon Conformal antennas are increasingly being deployed on the surfaces of air and land vehicles. Quite Often, the mounting surfaces are doubly curved. A characteristic property of these antennas is the curvature dependence of their input impedance, resonant fiequency, and radiation pattern. In light of this, it is vital that conformal antenna models include surface curvature so that the effects of local surface geometry on their resonant behavior and radiation pattern can be predicted more precisely. This is especially important for a highly resonant antenna, such as the micostrip patch, due to its narrow bandwidth. In addition, advanced material antenna loadings are increasingly being used in practice. These factors motivate the development of a new approach to modeling the radiation from conformal antennas on convex, doubly curved platforms utilizing the hybrid finite element-boundary integral (FE-BI) method. The hybrid F E-BI method, which combines the finite element method with the method of moments, is extended to model convex, doubly curved platforms by means of a specially formulated asymptotic dyadic Green’s function. This asymptotic Green’s function, formulated within the context of the uniform theory of diffraction (UTD), incorporates the physics of interactions on the surface of an electrically large, perfect electrically conducting prolate spheroid and is highly amenable to numerical applications. The prolate spheroid is a canonical shape that is sufficiently general to model the curvature of a convex, doubly curved mounting platform. The F E-BI method is used to investigate the effect of curvature variation on the resonant input impedance of a cavity-backed slot and a cavity-backed patch antenna recessed in the surfaces of prolate spheroids of varying dimensions. The effect of curvature variation on the far field radiation pattern of a cavity-backed patch antenna recessed in the surfaces of prolate spheroids of varying dimensions is also investigated using this method. Measured input impedance data for a patch antenna mounted on a planar and a doubly curved surface also is presented. Covyright by CHARLES A. MACON 2001 In memory Of my father Charles A. Macon Sr. deg enc m that pers lab ACKNOWLEDGMENTS Many people have assisted my on this long journey towards the completion of this degree. I am especially grateful to my advisor Dr. Leo Kempel for his constant encouragement, support, and guidance throughout the years. He has been an outstanding mentor and fi'iend. I would also like to thank my committee members. I would like to thank Dr. Edward Rothwell for providing me with useful advice on the experimental aspects of this research and Dr. Dennis Nyquist for Offering his valuable theoretical insight. I would also like to thank Dr. Byron Drachman for providing his mathematical perspective. I am very grateful to Dr. Stephen Schneider of the Air Force Research Laboratory (AF RL/SNRP) for his sponsorship and enthusiastic support of this research. I also offer thanks to my office colleagues and friends: Chi-Wei Wu, Jeong Lee, and Jong Oh and to my lab colleagues and friends: Christopher Coleman, Jeff Meese, Ben Wilmhoff, and Dr. Mike Havrilla for all of their friendship and assistance through these years. I would also like to thank Dr. Tim Hogan and Dr. Sangeeta Lal for their generosity and assistance during the construction phase of my research. The support of my family has been equally important throughout this journey. I am forever grateful to my father for teaching me how to persevere in the midst of life’s challenges and to always put my trust in God. I am forever grateful to my mother for her love and support. I also would like to thank Samuel Parks and his family and Willy and Etta Tate for all of their strong support through very difficult times. Finally, but certainly not last, I am especially grateful to my loving wife Donyale for all of her love, support, encouragement, and patience with me during my years of school. vi TABLE OF CONTENTS LIST OF TABLES ................................... ix LIST OF FIGURES .................................... x CHAPTER 1 Introduction ........................................ 1 CHAPTER 2 Uniform Theory of Diffraction .............................. 7 2.1 Introduction ................................... 7 2.2 Curved Surface Diffraction .......................... 13 2.2.1 Uniform Asymptotic Evaluation of the Dyadic Green’s Function for an Electrically Large Infinite Circular Cylinder ......... 17 2.2.1.1 On—Surface ......................... 17 2.2.1.2 Far Zone ........................... 31 2.2.1.3 Axial Singularities ..................... 40 2.2.2 Generalization to Doubly Curved Surfaces .............. 43 CHAPTER 3 Finite Element-Boundary Integral Method ....................... 47 3.1 Introduction .................................. 47 3.2 FE-BI Formulation .............................. 50 3.3 Finite Element Matrix Elements ....................... 53 3.4 Boundary Integral Matrix Elements ...................... 57 3.4.1 Selfcell Evaluation of the Boundary Surface Integral ........ 57 3.4.2 Asymptotic Dyadic Green’s Function Formulation .......... 67 3.4.2.1 Prolate Spheroid Coordinate System ............ 68 3.4.2.2 Surface Geometry ...................... 72 3.4.2.3 Calculating the Geodesic Path ............... 74 3.4.2.4 UTD Surface Ray Parameters ................ 85 3.4.3 Validation of the Prolate Spheroid Dyadic Green’s Function . . . . 87 3.4.3.1 Analytical .......................... 87 3.4.3.2 Numerical .......................... 90 3.5 Solving the FE-BI System .......................... 92 3.6 Radiation ................................... 94 3.6.1 Input Impedance ............................ 94 3.6.2 Near-to-Far Field Transformation ................... 95 CHAPTER 4 Numerical Results .................................... 103 4.1 Introduction .................................. 103 4.2 Input Impedance Studies ........................... 103 vii 4.2.1 Cavity-Backed Slot Antenna ..................... 103 4.2.2 Cavity-Backed Conformal Patch Antenna .............. 114 4.2.2.1 2.5 cm x 2.5 cm Patch .................... 114 4.2.2.2 3.0 cm x 3.0 cm Patch .................... 124 4.3 F ar-field Radiation Pattern .......................... 136 CHAPTER 5 Experimental Results .................................. 142 5.1 Introduction .................................. 142 5.2 Antenna Fabrication .............................. 142 5.3 Experimental Setup and Measurements .................... 147 5.3.1 Ground Plane ............................. 147 5.3.2 Prolate Spheroid ............................ 155 CHAPTER 6 Conclusion ....................................... 161 6.1 Summary ................................... 161 6.2 Future Studies ................................. 164 APPENDIX A Evaluation of Potential Surface Integrals over Triangular Regions .......... 166 A.1 l-, 4-, and 7-Point Approximation Weights .................. 166 A2 Analytical Formulas .............................. 167 APPENDIX B Parameterization Of the Prolate Spheroid Unit Vectors in Terms of Spherical Coordinates ................................. 170 APPENDIX C Derivation of the Exact Eigenfunction Series for the Circular Cylinder Dyadic Green’s Function ...................... 175 APPENDIX D Fock Functions ..................................... 180 APPENDIX E Biconjugate Gradient Pseudocode ........................... 186 BIBLIOGRAPHY .................................... 188 viii lal la‘ la la Table 3.1 Table A. 1.1 Table D. 1 Table D2 Table D3 LIST OF TABLES Comparison of approximate and exact geodesic path lengths between two points located on the midsections of two prolate spheroids ..... 84 Approximation weights for numerical integration over triangular regions ............................. 166 Zeros of the Fock-type airy function of the second-kind w2(r) and its derivative w2 '(r). ..................... 184 Constants for g‘°’(¢f) and g(”(§) ..................... 184 Constants for f (”(4‘) ........................... 185 ix Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 LIST OF FIGURES The field regions adjacent to a magnetic dipole situated on a perfectly conducting, convex surface ................... The mechanism of diffraction from a convex curved surface. ...... Spread of a surface diffracted ray strip due to energy conservation. . . . Fixed ray-based vector coordinate system ................. The Watson transform integration contour ................. The integration contour for the Watson transform split into two segments. ............................... Deformation of the integration contour around the complex poles of the integrand arising from the zeros of the Hankel function ....... Lowest order short and long creeping wave paths on a circular cylinder ............................... The integration path around the zeros of the F ock-type Airy fimction w2(r) and its derivative w2 '(r) in the complex 1 -plane. ............................. Figure 2.10 Deformation of the Ca contour into the steepest descent contour CSDP ................................ Figure 2.11 Position of source and Observation points on the surface Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 of a cylinder with respect to the origin. .................. Topological transformation of a prolate spheroid into a plane and a circular cylinder in the limits of zero azimuthal and axial curvatures and zero axial and finite azimuthal curvatures, respectively ................................. Subdivision of the two types of prisms into tetrahedra ........... RWG basis functions supported within the triangular regions 7: and 7: sharing a common edge 11 ........................ Local area coordinate system defined within a triangular region .................................... 10 15 16 20 20 21 23 25 28 3O 54 56 66 66 (IQ FigL Figure 3.5 Prolate spheroidal geometry ......................... 69 Figure 3.6 A geodesic on a prolate spheroid surface traced via numerical integration ............................ 82 Figure 3.7 The geodesic angle. ............................ 82 Figure 3.8 Limiting cases for the geodesic path length: (a) quasi-cylindrical, (b) circular arc, and (c) elliptical arc ......... 83 Figure 3.9 The geodesic trajectory between two points located at (a, = 79.o°.¢, = 0.0°) and (a, = 80.0°,¢f =160.0°) on a 40.02x4.02 prolate spheroid for which 6, =15.80 ............ 98 Figure 3.10 Comparison of the relative magnitudes of the prolate spheroidal asymptotic dyadic Green’s function components along the geodesic trajectory depicted in Figure 3.9 and the components of the cylindrical asymptotic dyadic Green’s function along the helical geodesic for which 6 =15.8° on a circular cylinder with an equivalent azimuthal radius . . . 98 Figure 3.11 The geodesic trajectory between two points located at (a, = 9o.o°. where ,08 is the radius of curvature along a geodesic, ds is the incremental distance along the geodesic, Q is the position of an observation point on the surface, and Q' is the position of the source point. Fock surmised that since the current distribution in the transition region depends only on the local geometry of the surface at the point of incidence and the magnitude Of the incident field at this point, Fock functions could represent the current distribution in the shadow transition region Of any convex surface with the same principle radii of curvature at the point of incidence. This assertion, known as the principle Of locality of the diffracted field in the penumbra region [23], forms the basis of UTD. By means of reciprocity, the same principle also can be applied to the field excited by an aperture on a convex surface. In the next section, expressions for the diffracted field excited by a magnetic dipole source on a perfectly conducting circular cylinder are derived via an asymptotic evaluation of the exact dyadic Green’s function for the circular cylinder. The canonical asymptotic solutions are expressed in terms of Fock functions that are convergent in the transition region and uniform across the illuminated and deep shadow regions. Next, the asymptotic solution for a magnetic dipole source on a perfectly conducting sphere is given. The procedure for generalizing these canonical solutions, within the context of UTD, to treat the problem of a magnetic dipole source on a perfectly conducting general convex surface will then be discussed. The expression for the dyadic Green’s function of 12 the surfs generali; LZCun An expl directly path. th: geodesic points 0 intinites surface giyen b Where The 5} 53316111 Come: rat’s t rEgior U3] 5C mCret the surface field excited by a magnetic dipole on a general convex surface based on the generalization procedure is given at the end of this chapter. 2.2 Curved Surface Diffraction An explanation of the phenomenon of diffraction by convex curved surfaces follows directly from postulate one of the previous section. In propagating along the least-time path, the portion of the diffracted ray path lying along the convex surface must follow a geodesic path. A geodesic is by definition the path Of minimal arc length joining two points on a surface or more precisely, the curve whose length is stationary with respect to infinitesimal pertubations in the path. Consider an aperture M situated in a convex curved surface S. The source in the aperture is represented by an equivalent magnetic dipole given by dM(r') = E(r') x r'idA (2.4) where E is the electric field in the aperture and dA is an element of area in the aperture. The symbols r' and r are position vectors directed from the origin of the coordinate system to source and Observation points on S , respectively. A magnetic dipole on a convex curved surface excites creeping waves that propagate along as surface diffracted rays that are directed away from the source in all directions to points in the shadow region. The surface diffracted rays shed energy along forward tangents to their trajectories. This phenomenon is depicted in Figure 2.2. The general form of the incremental surface field dF(r|r') excited by a magnetic dipole is given by [26] -16 e! dF(r|r') = ngQ') -T(r|r')D (2.5) S where s is the geodesic distance between source and observation points on the surface 13 and sun Int poi: of a I.) (J) ma, Gr: uh bin 10 fm and D is the surface ray divergence factor which quantifies the change in width of a surface diffracted ray tube due to energy conservation and is given by lipcdw In (2.6) dV/o is the angle between adjacent surface rays at the source point, dry is the angle between the backward tangents to a pair of adjacent surface rays at the observation point, and pc is the radius of curvature of the geodesic circle centered at r. The spread of a surface diffracted ray as it propagates along a curved surface is depicted in Figure 2.3. The parameter T(r|r') is a dyadic transfer function for the surface field excited by a magnetic dipole on a convex surface. It is proportional to the second-kind electric dyadic Green’s function through the relationship T(r|r')=—jkYEe2(r|r'). The parameter T(r|r') describes the launching of the surface ray field at r', the variation of the surface ray field between rand r', and the attachment of the ray field at r. This dyadic parameter is given by T(rlr? = Tit» T3134 T,r3i'+ nfifi' (2.7) A where t is a ray-fixed unit vector tangent to the direction of propagation, b is the binorrnal unit vector defined as b = txfi , with ii being the outward unit normal vector to the surface, and T are coefficients that are deduced from the uniform asymptotic solutions to canonical problems to be described in the next section. An illustration of the fixed ray-based coordinate system is provided in Figure 2.4. In order to satisfy the 14 Surface Diffracted Ray é .. Shed Ray Figure 2.2 The mechanism of diffraction from a convex curved surface. Surface Diffracted Ray Figure 2.3 Spread of a surface diffi'acted ray strip due to energy conservation. 15 v Figure 2.4 Fixed ray-based vector coordinate system. 16 requiren region. 1 paramet shadou series t asymptt the dee functio: cont ert Shadou cleyiati. Surface requirement for rapidly convergent solutions that are continuous across the transition region, the elements are expressed uniformly in terms of Fock functions with the distance parameter ,3 as their argument. In the deep shadow region where ,6 >> 0 , with the shadow boundary ,6 = 0 taken as reference, the Fock functions revert to creeping wave series by means of Cauchy’s residue theorem [22]. Appendix D presents details on the asymptotic behavior of Fock functions. The creeping wave series is rapidly convergent in the deep shadow region. Moreover, in the deep lit region where ,6 <> 1) . In order to Obtain a rapidly convergent expression for the dyadic Green’s function Of a large radius cylinder that is amenable to numerical evaluation, the Watson transformation [18] is employed. As explained previously, the Watson transformation effectively transforms a poorly convergent infinite eigenfimction series into a rapidly convergent series Of pole residues, also known as the creeping wave series. The value of the pole residue series asymptotically approaches that of the original eigenfunction series as the argument k pa increases. The poles residues are physically interpreted as creeping waves launched at the geometrical Optics shadow boundary and propagating along the cylinder surface into the Shadow zone. Hence, the number of terms in the series that are needed for a reasonably accurate representation of the diffracted field decreases with increasing radius. The Watson transform is given by [27] Sin wr n—co i e’"’f(n) = Eli eiVWWQdV (2.9) and from (2.8) the 2 component Of the surface field attributed to a i directed magnetic dipole is given by 18 In light of where v integrand maybe 5; 6:: = with sepe 2.6. Note The subs contour i illerefor The new in Figure ,, -1 .. — °° k Hf,”(y) G; =—— e "t "1" ’ dk, (2.10) 2 2 Z Ie akgH’EZ) '(7) In light of (2.9), (2.10) may be rewritten as Gzz (k [3).]. 0° e‘jk,d z dkzge e-jV(x-;)H(2)(7) (2 11) G—eZ fi’rkz (2) r . (217:) 0a 2 sin vrrH (y) where v is the complex order and C is the closed contour enclosing the poles of the integrand in (2.11), as depicted in Figure 2.5. The integral around the closed contour C may be split into two integrals w - wot-3) (2) ervor- Z) (2) 65,: 1,91. [e'jk’zdh [9, H; (7)dv+_[e H; mdv (2.12) (27r)2 k0 a 2 ‘ C. srn wer ”(7) sin er‘ ”(7) with seperate integration paths denoted by C1 and C2 , respectively, as shown in Figure 2.6. Note that the integration path is perturbed from the real axis by a tiny amount 0'. The substitution v :> -—v is made to reverse the direction of integration path C2. The two contour integrals are subsequently merged via analytic continuation [27] H5230) = e"’”H§2’(7) . (2.13) H 53’ '(7) = e"”’H§2’ '(7) Therefore, (2.12) becomes k , a, _ (e-J'V(II-3) +eJ’V(fi-;))H(2)(},) 05;: ;— i Ie-jk’zdkz [ , 2 " dv (2.14) (2—71r—)2 k0 a 2 _Q C! srnvrer, ”(7) The new integration path enclosing the complex zeros of the Hankel function is depicted jtrv in Figure 2.7. Factoring out e 19 Im(v) ..,..-- 1 1 ‘ a... WWW R60) ...... t y; .o" C 1m(V) jO' C2 / 4 F / WWW» Re(v) / P . r ,1 ”10' Cl Figure 2.6 The integration contour for the Watson transform Split into two segments. 20 Im(v) .: 1 Q C , Q 3 ® G C .0 u .- .- o . o ’’’’’ ------ ..... ......... .. ,. """""""" """""""""" .............. -------------------- > Re(v) Figure 2.7 Deformation of the integration contour around the complex poles Of the integrand arising from the zeros of the Hankel firnction. 21 ex; 0rd intc The den“ “her and; m wan-Z) 1v; (2) e (e +e )H, (y) z: p] °° -jkE e ’ dk dv 2.15 92: (27) kga 2 1 1! sin mm?) '(7) ( ) and noting that Im v < 0 from Figure 2.7, the expansion em —2 'ie'm’" (216) Sin wr 11:0 ' is utilized through a technique known as the Poisson sum formulation [18] where 1 represents the number of complete encirclements made by a creeping wave in either clockwise or counterclockwise directions. Since the magnitude of a creeping wave exponentially decays as it propagates along the surface, the contribution from the higher order (e. g. multiple encirclements for which I > 0) terms is negligible. Substituting (2.16) into (2.15) and retaining only the lowest order short and long path terms (refer to Figure 2.8) results in z, —1 kp °° .- (e""(2”';’+e”‘)H§2)(7) G, =—— e’jk’ dk, dv (2.17) 2 (27:)2 k561i. Ci H5”'(7) The leading terms in the uniform asymptotic expansion of the Hankel function and its derivative, for large 7 , in terms of Fock-type Airy functions are given by [28] H(2) ~jW2(T) . (7) 7‘5 H52) 1(7) ~ —jW2 '(T) mZJZ where w2(r) is the Fock-type Airy function (see Appendix D), m is as defined in (2.1), (2.18) and r is defined in [27] 22 Long Path Short Figure 2.8 Lowest order short and long creeping wave paths on a circular cylinder. 23 \Vit Omk TESL uh in I r—;(v—y) (2.19) Without a loss of generality, only the short path term e"; is considering from this point onward. Substituting (2.18) into (2.17) and employing the change of variable dv = mdr results in co , _ 2k _ _ G: ~ ——1—-2- I e”"" m2 " IWZFT) e’Mdr kz (2.20) (277) —>1 via saddle-point integration. To employ this method, (2.22) is recast into an appropriate standard form by means of the following polar coordinate transformation [29] k2 = ko sina kp = k() cosa a3=scos6 z=ssin6 ,B = m5 (2.23) where s is the geodesic distance between the source and Observation points on the cylinder surface and 6 is the angle subtended by the geodesic curve from the azimuthal 24 '. h. u,_ H. u" ..... ..‘ .. ....... ....... .................. Figure 2.9 The integration path around the zeros of the Fock-type Airy function w2(r) and its derivative w2 '(r) in the complex 2' -plane. 25 plane of the cylinder. The mapping of the steepest descent path (SDP) from the complex kz-plane onto the complex a-plane is accomplished by substituting (2.23) into (2.22). Hence (2.22) becomes 2 2 65;: 1 2 [ (WW—6"" “’8 0‘ [W251)e“’fl'dr a (2.24) (27;) C. 27m rlw2 (r) In order to determine the SDP contour, complex a is decomposed into real and —jcos(a—6) imaginary parts (a = a'+ ja") and the phase term e is re-evaluated for complex a resulting in e—jCOS(Cl -6)cosha +srn(a -5)Slflha (2.25) In order for (2.24) to converge, the constraint sin(a'— 6) Sinha" < 0 must be satisfied. Furthermore, in order to eliminate the oscillations Of the integrand along the SDP contour, the imaginary part of (2.25) must remain constant and equal to its value at the saddle point. Thus, the constraint cos(a '— 6) cosha " =1 determines the shape of the SDP contour in the complex a-plane. Expressing (2.24) in terms Of the hard surface Fock function given by (see Appendix C for details on the Fock functions) v(fl) = ([2 [M e‘jfl’dr (2.26) 471' 1.! W2 '(7) yields .. 1 mzcosza 47: _- _ c}; =___ __ _v e 1‘05“)“ 5) a 227) 2 (midi 2w \/ 1'13 m i” ‘ where ,8, defined in (2.3) Specializes to ,8 =£ for a circular cylinder, and pg is the Pg radius Of curvature along a geodesic given by 26 Deforming figure 3. ll point integ dyadic (it) Note that In the pr t surface ‘ Cun‘atur radiating Cypress. Warren result (5. the 6X] di’adic exPres VECIOr, eNines Empit a = 2.28 cos2 6 ( ) Pg Deforrning the Ca integration contour in (2.27) into the SDP contour, as depicted in Figure 2.10, and asymptotically evaluating (2.27) for large kos via the method of saddle- point integration, yields the asymptotic expression for the 22 component of the electric dyadic Green’s function for a magnetic dipole radiating on a circular cylinder —'ks e10 GS ~ v(,B) (2.29) 27:3 Note that (2.29) is identical to the dyadic Green’s firnction of a magnetic dipole radiating in the presence of a PEC ground plane, derived via image theory, modulated by the hard surface Fock function v(,B). The physical interpretation of this result is that as the curvature vanishes, (2.29) reverts to the dyadic Green’s function for a magnetic dipole radiating in the presence of a PEC infinite ground-plane. This result, however, is not expressed within the framework of UTD. In order to recast this result in terms of the invariant ray-based unit vectors (t ,b) of UTD, which can readily be compared with the result derived by Pathak [29], further manipulation is required. From physical reasoning, the expression for the cylindrical dyadic Green’s function should recover the planar dyadic Green’s fimction in the limit Of zero curvature. Based on this assumption, an expression for the cylindrical dyadic Green’s fimction in terms of the ray-based unit vectors may be heuristically developed by substituting (2.29) into the following expression for the planar dyadic Green’s function G = I —1 VV 8.),“ B 230 e + ' 2 kg 27rs v( ) ( ) Employing the identity from [30] which is given below 27 1111(0) A *— A MIN Saddle E CW Point at E a = 6 E h Re(a) . > Figure 2.10 Deformation of the Ca contour into the steepest descent contour CSDP. 28 VVV[ (fl) 8:]: {RR[%+(jko+%)z]—(I-RR)(jk Vii —ez}v(/3)S 9% (2.31) to evaluate (2.30) yields ___ ___ . . A A . A A . e_jk°s 0.2: I t—J— ——l- +RR 2 +—ZL— +RRJ— 1—— 129(5) kos kos (kos)2 kos kos kos 27:3 ___ . A = A .. A . e-jkos ={(I-RR)—-(I—RR)q(1—q)+RR(2q—2q2)}v (,6);r = A . 2 7‘05 ={1.[1_q(1_q)]+RR(2q-2q )}v(:r)-2—fl; (2.32) where qzk—j—, R=(r-r')/|r—r', and Iszf—RR. Referring to Figure 2.11, it is 03 apparent that R is tangential to the direction of propagation for a creeping wave between a source and Observation point on the cylinder surface. Therefore, setting R =t and I. = [313' allows (2.32) to be expressed in terms of the ray-based unit vectors of UTD. Therefore, (2.32) can be rewritten as e-jkos G.2={bb[1— q(1— q)]+tt'(2q— 2112e2)}tz(,3)7r (2.33) To facilitate the numerical computation of 5.2 , the explicit expressions for the ray-based unit vectors in terms of the geodesic angle 6 that are given below t: zsin6+tpcos6 (2.34) I) =(iicos6—zsin6 are substituted into (2.33). The subsequent evaluation of (2.33) yields the following expressions for all four of the components of the asymptotic dyadic Green’s function for a circular cylinder: 29 xA 1e Figure 2.11 Position of source and observation points on the surface of a cylinder with respect to the origin. 30 e-jkos 0:;(a,?¢i,2)~ [sin 6+q(1— q)(2- -3sin 6)]v(,6)ezfl (2.35) Gf’ 2=Gf§(a,¢,z)~—sin6cos6[l— 3q(l— q)]v(fl)e2” e-Jkos (2.36) Gfg(a, ¢, z)~ [cos 6+q(1-— q)(2— 3cos 6)]v(,6)e2 (W (2.37) Note that the (M) -component in [29] contains a mixed term comprising both the hard, v(,B), and soft, u()6), surface Fock functions. The soft Fock function u()8) arises from the asymptotic evaluation of the first term enclosed within the brackets of the ([16) - component Of the exact dyadic Green’s function (2.8) by the procedure outlined above. With the inclusion of the mixed term (2.37) may be rewritten as [29] 0301,03,?) ~2[COS 5+q(l ‘1)(2— 3cos2 6)]V(fl)ezfl eflkos +q [sec2 ,B(u(fl ,6)—v(,6))] 2.2.1.2 Far Zone (2.38) In this case, we begin with the expression for the exact electric dyadic Green’s function Of the second kind for an infinite, PEC circular cylinder given by = r 1 jn -_/kz —jnH(2)(x) In Hri2)'(x) "t‘r Ge2(p,¢,zla,¢az)= flZZe ‘jdk ii——— W21.) + zit-3) 11ml” n-.. 7 k k H0" (2)' (2) . . k k HO) . "'j z p 712 (x) [32’4" 2Hn2(x) _ nkz2 Hnafx) (Pq)r+ n 2 P (722 (x) (Pit 7k. HS "(7) 7H}. "(7) 16.7 xH. '(7) 7x xkoH. "(7) kk H") k ’ (Z) + "2,2,, (3).“) 26—1 (4] Hint“) 22' (2-39) 7 koH. (7) 7 k0 H. (7) where x = k p p. This time, however, we asymptotically evaluate the exact Green’s firnction for the case of a source point lying on the cylinder surface p = a , while an off- 31 surface observation point is allowed to recede to infinity. Since the Hankel function requires n >> kp p for convergence, the exact Green’s function becomes poorly convergent for large kpp. In order to alleviate this problem, the method of steepest descent must be applied to derive an asymptotic approximation of the exact Green’s function that is valid in the far zone. Substituting the following approximations, valid in the far field, into (2.39) lim Hf,” '(x)~ —jH,(,2)(x) (2.40) (2) lim iii) ~ lim . = 2.41 p—wo x p—pao elpp‘l—E ( ) and evaluating, results in = v v jn ~jk,z k: 2 HO) A: Gez(p,¢,z|a,¢,z)= (27:1,,=_,,)2 2e * jam {[7", [IT] H32),((:))]p¢ _kzkariZ) '(X) A A 1_ jHISZ) '(X) " " v+ nkszH’EZ) (x) " " ' z— —— z 7kg2 H ‘2’ '(7) 7H§”(7) 72165115” '(7) _1 k, H.900 (thank) Before proceeding with the application of the method of steepest descent to (2.42), it is clearly evident that k p goes through 0 along the interval of integration of kz. As a result, special consideration must be given to the asymptotic evaluation of this integral because the arguments x and 7 also pass through 0, thereby, challenging any asymptotic approximations that may be used. Since it is evident that (2.42) can be written in the form of a steepest descent integral via the substitution of an asymptotic expression for H 90:) into (2.42), it is known a priori that the major contribution to this integral comes from the 32 region near the saddle-point where k p at 0. Therefore, the use of the asymptotic form of the Hankel function for a large argument x=kpp >>1 is justified provided that the observation point is not in the vicinity of the axis of the cylinder where 6 = 0 or it radians. (Note: The near axis behavior will be discussed in the next section.) In light of this, the second-kind Hankel function is factored out of the numerator and replacing by its large argument form for x >>1 H:2)(X) ~ lie-jxejmr/Zejx/‘i (243) under the constraint that 6 at O or 7! radians. Grouping all is and 2 terms together with the same unit source vectors, (2.42) becomes = ~ 2 143%)” 11,5 nk, k . k_,, . 0.2 (2”)22e Joe {m’kH,E”'(7)ik—; p- k i']¢ k - fir/4 -J(k,,p+k.2) +—1——[k= fi—ii]i'———J—¢¢'}e e dk, (2.44) k J27zkpp k Making the substitution [71:21“)- ” z]=0 and decomposing (2. 42) into dyadic components yields 5.2 ~ (2:),M 2 fig"2 “[6 éq) +G,,éz mm“ (2.45) where no jk,'z jx/4 k e-JUcp p+k z)d G“: e e " 2.46 ‘3’ l _M(k a) 2,,kH;2>'(k a) e—de’ ( ) 0° ejk z'ejzr/4e‘1(kpp+kzz)d (2.47) Gefizz_ — Ie 2 _mka aH}, “(k a) e.——/2nkp 33 . ' ’ . -'k k, no em: eMM(—j)e J( pp+ 2) of; = I k <2) k k z .2 paH, ( pa)\/27r pp (2.48) Each of the integrals in (2.46)-(2.48) is amenable to evaluation by the method of steepest descent under the constraint that 6 1: 0 or 7: radians. The canonical steepest descent integral given by G = I F(kz)e"g"‘"dkz (2.49) SDP has first order solutions of the form (2.50) where, K denotes the large parameter, (,7 is the angle at which the SDP contour intersects the saddle-point, and k: denotes the saddle-point. Comparing (2.49) and (2.46) g(kz)=—j((/k02—kz2 sin6+kzcos6) (2.51) = 0. Consequently, the is obtained and the saddle-point can be found from (1% g(kz) z k, =k’ 1 saddle-point is given by k: = kcos6 , where 6 is the angle subtended from the z axis to an observation point in the far-zone. It follows that S - fl 5 j k, =—k d k, = 2.52 g(.) Joan g(.) [(08ng ( ) where the double prime denotes second-order differentiation with respect to the argument k2. Making the following substitutions, to express the parameters in (2.46) in terms of spherical coordinates in the far-zone: kp=ksin6, p=Rsin6,and z=Rcos6 (2.53) 34 and evaluating (2.46) at the saddle-point, the expression for F94, (k5) becomes ne’msoz'k cos 6 F k: = (2.54) 9" ( ) (ka sin a)’ 1:11,?) '(ka sin 0) sin 194/2sz Setting the parameters K = R and 1;! =3; and substituting (2.52) and (2.54) into (2.50) to obtain the asymptotic form of G3 which is valid in the far-zone of the cylinder . - 7r jkcos6z’ w nejni¢+2j -JkoR - a, e 12k0 cos6e (2.55) 6432 ~ 2 ° 2 (2)1 ' koR (27:) (ka sm6) "3..., H" (ka sm 6) The other components of the asymptotic dyadic Green’s function are obtained through the same procedure and are given by the following: 6: e-jkoR jzejkcos6z’ no ejn[;+%) Ge2 ~ 2 (2). , (2.56) koR (27:) a ,,,__,, H" (ka sm 6) —ij - )1: cos6z' to ”($125) ¢¢ e 2k0e e (2.57) GeZ " 2 - (2). - koR (27:) kasm6 "M, H" (ka sm 6) To facilitate numerical computation, the eigenfunction representations in (2.55)-(2.57) can be further simplified. Decomposing the infinite summation into two separate sums, (2.55) can be rewritten as 9,4 e""°R j2k0 cos6e jkcos6z' 0 on jn; on on jn; .22 ~ 2 "1 e +2 "1 e (2.58) M (27r)z(kasin6) ) ".0 "M, Hf,” '(ka sin6 H52) '(ka sin 6) Changing the interval of the first summation via the substitution n :> —n yields 6¢ -jkoR - jkcos6z' co _ --n -jn; co m 1n; e2 ~ e 12kO cos6e 2 (2)71] e . +2 (2)11] e . (2.59) koR (27:)2(kasin6) "=0 H_,, '(kasrn6) "=0 H" '(kasm6) 35 Anahtic applied 1 Euler' 5 Apply Whicl Whei proc Ins 8m; Analytic continuation of the Hankel functions, Hf? '(7) = e"’"’H,(,2’ '(y), is subsequently applied to merge the two summations -- . 4 . m J";_ "I"; , 9,, e "‘“R 12k0cos6e1k°°’9’ °° "1 (e e )2} .. ~ . _. 2.60 2 koR (27:)2(kasin6)2 "=0 H§Z)'(kas1n6) 2] ( ) Euler’s identity is invoked to simplify (2.60) which leads to the final form 66¢ ~ _e’fkoR 4k0 COS Oejkcos6z' co njn Sin(n3) (2 61) ’2 koR (27;)2 (kasin 6)2 "=6 Hf,” '(kasinB) ' Applying the same procedure to (2.56) leads to ’JkoR ~2ejkcos6z' co j" (e171; + e-j";) G$~e J 2 n . 3 a6» koR (27:) a "H, H ,2 '(ka sm 6) 2 which upon simplification, results in 6: e-jkok jzejkcosoz' co gnj" cos(n¢) (2.63) G .. ‘2 koR (27:)2a ,2... H32) '(kasin 6) where an is Neumann’s constant (8,, =1, n = 0 and an = 2 , n 1: O ). Following the same procedure, (2.57) becomes M e—JkoR ejkcost' co gnj" cos(n¢) G ~ 2.64 ’2 koR (27:)2kasin6,,=0 H§2)(kasin6) ( ) In summary, the asymptotic expressions for the far-zone dyadic Green’s function for an axially infinite, PEC circular cylinder in the shadow region are given by Ga .. -e""°” 4koc089e”‘°°”" °° "fsmlngl (2 65) "’ koR (2n)2(kasin6)2 "=1Hi”'(kasin6) ' GO: ~e-jkoR jzejkcos6z' co gnj" cos(n3) ‘2 koR (27:)2a M H,‘,Z"(kasin6) (2.66) 36 The asyn useful 01 diameter. \Vax’eleng consequ: argumer. approxir the far-; these as field rac B: R‘MTifle Follou Where 1 Figure the Sub. «5 e’fl‘oR ejkcosaz' «o a j "cos(n¢) G.~ 2.67 ‘2 koR (2n)2kasin6;H,fz)( (kasin6l) ( ) The asymptotic approximations to the dyadic Green’s function given in (2.65)-(2.67) are useful only for an electrically small cylinder on the order of a few wavelengths in diameter. For a cylinder with a large radius of curvature with respect to the operating wavelength, these expressions are slowly convergent. As discussed previously, this is a consequence of the poor convergence property of the Hankel function with a large argument. Consequently, the Watson transformation will be used to develop asymptotic approximations to the expressions in (2.65)-(2.67) that are valid in the shadow region of the far-zone for an electrically large circular cylinder. Just as in the previous section, these asymptotic expressions are physically intepretated as components of the diffracted field radiated by creeping waves propagating along the cylinder surface. By means of the Watson transformation, the axial component (2.56) can be rewritten as - I: _k Sin Hejkcosflz' jv(‘——2-) e d 471,2 ' (2) v V 7 C sm ver, (y) of; ~ (2.68) Following the same procedure as in the previous section, (2.68) is decomposed into two contour integrals dv (2.69) 0: Ge2 ~ —k sin 0e!" ”‘9" e437) I 843%] 4 2 ° Hm' dv+ - Ha). 7:7 Clsmwr , (y) Czsmvn' , (7) Where the integration paths of the two integrals in the complex v -plane are depicted in Figure 2.6. Factoring out e’"r as before, merging the two integration paths, and making the substitution given by (2.18), (2.69) can be rewritten as 37 where. l asymptc closed 2 arising radius. derivati uthed compui and de reuriti. Where Makin cOmpl. jlz cos 02' co . [:e-j'v(¢—zr/2+2zl) + e‘JV[(37‘/2‘;)+2’d]:l -jk srn0e ~dv (2.70) 0:” ~ 2 2.2. , El Ham where, for a large radius cylinder, 1: O is sufficient. In order to develop a uniform asymptotic representation for (2.70) in terms of a Fock function, the contour Cl must be closed at infinity in the lower half-plane to enclose the complex poles of the integrand arising from the zeros of the Hankel function. Since 7 is large for a cylinder of large radius, then in light of (2.21) v ~ y. Hence, the uniform asymptotic expansion of the derivative of the Hankel function in terms of the Fock-type Airy function in (2.18) will be utilized. These functions are tabulated (see Appendix D) and are amenable for computation. Making the necessary substitutions, as was done for the on-surface case, and deforming the integration contour into 1‘] according to Figure 2.9, (2.70) may be rewritten as - jkcosez' e-jvd), 4.3-j”? of; ~ k 5‘“ 9e [ 141] (2.71) 471' [W rl WI '(T) where (I)l =izi—3 and (D2 =5—g. Upon the substitution of (2.21), (2.71) is rewritten as k sin erkwsoz -Jm°zr 92~ “.001 jmolt 6,, 4” [e JL [w (1)dz‘+e‘”°’ J7; [8 W'(T) d7] (2.72) Making the substitution mCI)L2 = ,B from (2.23), (2.72) is expressed in terms of the Complex conjugate of the far-zone hard Fock function g‘“) (,6). which is given by [28] g“) "emrd (2 73 tiff. - > 38 where u ( terms of (j. Followin compone' In this c; soft Focl Noting ‘ the far-2 The 35\ function Cases an At [his into (2.7 where u denotes the order. Noting that w1'(r)= w2 '(r)' allows (2.72) to be expressed in terms of (2.73) as kSingejkcosaz' 2 -'70P 0 o 4” 2e ’ g( )(mCDp) (2.74) p=l l9: GeZ ~ Following the same procedure, the Watson transformation is applied to the azimuthal component (2.57), which yields . 2 ‘kcosflz' -jm0 r -jm Noting that WI (1) = W2 (1). , the asymptotic approximation of the azimuthal component of the far-zone dyadic Green’s function is given by -Ian2ejkcosoz' 2 - ' . of; ~ 1 2727 2e ”"°Pf<°>(m<1>p) (2.77) p=l The asymptotic evaluation of the cross-polarized dyadic component of the Green’s function in (2.55) via the Watson transformation is analogous to the axial and azimuthal Cases and leads to the following expression , . . -jv0 -jv2r —jm cos 0e (no, +k cos 6e (1,4,, I8 — I e (17' 47:7 77? w. W 4n 5., w. '(r) 49¢ GeZ ~ + jkm COS gejkoosaz' e-jfl), e-jmsblrd _k COS Hejkcosflz' e—jflb, -jm0,r 47!? \f— IL W2 'd(T) 47! 7: Iwz '(T) dr (2.79) Employing the complex conjugate of the far-zone hard Fock function defined in (2.73), (2.79) is rewritten as - jkcosBz' jkcosflz' GE; ~ jkm 0045 He e’ Wig“) (m (DI). _ kcos :le e"°'g(°) (m (D1). W 72' jkcosflz' _ - jkcosaz' . . jkm cos 9e e“ 1mg“) (m (1),) + kcos 9e {’0’ gm) (mCDZ) (2.80) 4717/ 471’ Simplifying (2.80) in the same manner as before leads to jkcosez' 2 631°”: Ze”""’(-I)”[g ‘°’)——7 “’(mcb )] (2.81) p=l In Summary, (2.74), (2.77), and (2.81) are rapidly convergent asymptotic representations of the electric dyadic Green’s function that are valid in the far-zone of the shadow region for a canonical PEC circular cylinder. As discussed earlier, the analytical representation in terms of Fock functions ensures the convergence of these expressions in the shadow boundary transition region consistent with a UTD formulation. Moreover, since the hard and soft far-zone F ock functions are tabulated, these expressions are amenable to numerical computation. 2.2.1.3 Axial Singularities The far-zone asymptotic dyadic Green’s function for a PEC circular cylinder becomes infinite when evaluated at the vertical axis. This anomalous behavior is due to the presence of a singularity in the dyadic Green’s function that is manifested only when the 40 observation angle 6 , subtended by an observation point in the far-zone from the z axis, approaches 0 or 7: radians. In this section, the component of the dyadic Green’s function that exhibits this singularity is isolated by means of a small argument approximation. The argument 7 of the Hankel function, as defined previously, is y = kasin6. As the observation angle 6 approaches the vertical cylinder axis at 6 = O radians, y —> 0". In light of this, the following small argument approximations of the second-kind Hankel function given by [31] . (2) , 2 11m H0 (y) ~ -_]—ll'l}’ (2.82) 7-20’ 71' and . (2) - 1 2 v 11m H, (7) ~ 1 —F(v) — , Re(v) > O (2.83) 7-20’ ft 7 where F(v) is the gamma function, in conjunction with the recurrence relationship [31] 2H5” '(7) = 1153(7) - H “”(7) (2.84) v+l are applied to evaluate the small argument approximations of the dyadic Green’s function components given by (2.65)-(2.67). Invoking the recurrence formula (2.84) to expand the derivative of the Hankel function in (2.65)-(2.67), in the following manner Himtr)=§[Hé”-H§”(r)] H90) = %[le’(7)-H§2’(7)j H§2"(7) = -:-[H2‘2’(7) - H900] (2.85) 41 and evaluating each term of (2.85) via the small argument approximations in (2.82) and (2.83), the small argument approximations of the Green’s dyadic components are obtained. Hence, (2.65)-(2.67) now are given by lim G“ ~ —e""°R 4nke”"'cos6j" —27:sin(3) +717 sin(23)+ ”7233163) 740+ .2 koR (27:)2 4j+7’(21n7-Ir) j(y’-8) 2j(7’—24) 3sin 4— +:—}f—(-2—(£g-+...] bounded (2.87) (22(2)) Pill-fill 77’ 7’ 77’ 7’ . w e—jkoR ke—Jkoz'jn 7r neosfi) nycos(23) fly’cos(3$) lrr(r)1+G,,2 ~ k , , + . + . + . 7-> OR 27: 7(77—121ny) 2] 4] 16] 3cos 4- +1Fifl+mi=w :>unbounded (233) J The 623’ component of the dyadic Green’s function becomes infinite as the observation point approaches the vertical axis giving rise to an infinite field at the vertical axis. The existence of singularities at the vertical axis is intrinsic to this type of problem and cannot be eliminated analytically. 42 Zlgit Thea: conse fiequ IEVCY spee‘ expr then 2.2.3 Generalization to Doubly Curved Surfaces The asymptotic form of canonical solutions are generalized to treat the case of the general convex curved surface by means of the principle of locality for the propagation of high frequency radiation discussed in the previous section. The generalized solution should revert to the canonical solutions for the circular cylinder, sphere, and plane when specialized to those cases. Once the circular cylinder and sphere canonical solutions are expressed in terms of the ray-based unit vectors i, b , and ii , the differences between them become apparent. Expressing the canonical circular cylinder solution in terms of the ray-based unit vectors following the methodology of [29] results in mm ~ M.[fiv5[(1-é)v(m+ DZ (21;) mg)”: -é{u(fl)-v(fl)}] +i'i[D’év(fl)+%u(,6)—2(é) 17(6)] +[t'b+fi'i]roé{u(fl)—v(fl)})DG(ks) (2.89) where the parameter 2'0 which uniquely specifies a helical geodesic path has been introduced. This parameter is defined as [29] Z K r, = (2.90) where T is the torsion of a surface diffracted ray and 7c is the surface curvature along a geodesic. The surface curvature is defined as K =1/pg . The expression for the surface field excited by a magnetic dipole on a perfectly conducting sphere may be found in a manner analogous to that of the circular cylinder. The ray-based expression for the 43 -uw--. surf: giVe ssh: C35: reac lQIS by l‘m pre EH‘ surface magnetic field excited by a magnetic dipole on the surface of a PEC sphere is given by [29] de ~M.[b’b[(l-%Jv(fl)+D’(—é) u(,6)]+t't[D’7i—v(fl) 2' _21; +4 “Jumjmum (2.91) . For the spherical where 6 specializes to ,6 = E for a sphere, s = a6 , and D = a sm6 case, the geodesic path of a creeping wave is a great circle. From the definition of D , it is readily apparent that the points 6 = 0 or 71' , are caustics of the surface diffracted rays. Thus, all surface diffracted rays converge at the two poles of the sphere. By setting the torsion factor To = 0 , the circular cylinder solution reverts to the spherical solution given by (2.91) except for the presence of the terms q’v(,6)G(ks) for the cylinder and q’u(,6)G(ks) for the sphere. Therefore, it is apparent that the differences between these two solutions are due to the effects of torsion on the surface diffi'acted rays and the presence of either a hard v(,6) or a soft u( ,6) Fock function in the limiting expressions. In view of this, the canonical cylinder and sphere solutions may be generalized by employing differential geometry to develop an expression for 7}, that is appropriate for a general convex surface and by introducing the dimensionless factors 7, and 7c to interpolate between the canonical cylinder and sphere solutions. Specifically, the terms q’v(,6)G(ks) and q’u(,6)G(ks) are properly weighted via 7, and r, such that the correct term is present once the generalized solution has been specialized to either the circular 44 cyhnder c difierentie uith K = the princ dimensit‘ \seightir Where ( prepen In add the all from . Surfac Note gCDEr cylinder or sphere case. The generalized torsion factor is given by (2.90) where differential geometry is employed to generalize T as _ sin26 2 T (x, —x,) (2.92) with K = K, cos’ 6 + K2 sin2 6 . The parameters K, and K2 are the surface curvatures along the principle surface directions (to be discussed in detail in Chapter 3). Furthermore, the dimensionless factors must satisfy the following constraint in order to provide the proper weighting 7. + 7. =1 (2.93) where ( 7: =1 , 7‘. = 0) for a sphere, and ( 7, =1, 7, = 0) for a cylinder. In view of these properties, the dimensionless interpolating factors are given by [29] K1 7.=—- and 7.=1-7. (2.94) “2 In addition, the generalized form of the distance parameter given by (2.3) is employed in the arguments of the Fock functions to treat the general convex surface. Consequently, from (2.90), (2.92), (2.94) via (2.89), and (2.91), the dyadic Green’s function for the surface field excited by a magnetic dipole on a general convex surface is given by [29] Ea(rlr')~(l3'l3{[1-q]v(fl)+D’q’[7.u(fl)+7.V(fl)]+‘r§q[u(fl)-v(fl)]} +3'3{D’qV(fl) + quw) — 242 l7.u(fl) + 7.V(.B)]}+(3'13 + 6'?) {4. qluw) (2.95) -v(.6)]})DG(kos) Note that this solution satisfies the criteria for an appropriate asymptotic solution for a general convex surface in that it reduces to the canonical cylinder solution when ( 75 = 0 , 7’.- =1) and to the canonical sphere solution when (7, =1, 7, =0) and 70 =0. The 45 generalizec geometry. ' This deri\ generalized Green’s function in (2.95) may be specialized to a prolate spheroidal geometry. The prolate spheroidal dyadic Green’s function components are given by 0&5, :6,(p|6’,(o') = {(C082 6—q[(D’ + 2) cos’ 5 —(D’ +l)])v(,6) +42 ([(122 + 2)]cos2 6 - 2)[7.u(/3) + mm] Y +(ro cosé' + sin 5)2 q[u(,6) — v(,6)]} D%’—qe"k°’ GE,” = 637(60 :6,(o|6',¢') = {-sin6c0s6(v(,6)—(D’ + 2)qv(,6) +(D’ + 2)q’ [y,u(6)+ 7cv(,6)])+[(2cos’ 6—1)ro — (T: —1)sin6cos 5]q[u(fl) _v(fl)]}k20_:,qe_jkos 0:,"(50 :6,¢ 03¢) = {(st 6--q[(D2 +2)sin’ 6 —(D’ +1)])v(,6) +42 ([022 + 2)]st 5-2)[7.u(fl) + mm] . _ 2 _ fl “1’95 +(ro smé’ c056) q[u(,3) V(fl)]}D 2” qe This derivation of (2.96)-(2.98) will be discussed in detail in Chapter 3. 46 (2.96) (2.97) (2.98) CHAPTER 3 FINITE ELEMENT-BOUNDARY INTEGRAL METHOD 3.1 Introduction The finite element-boundary integral (FE-BI) method is a hybrid computational technique for solving general electromagnetic radiation and scattering problems. This technique has been used with much success in the past for modeling cavity-backed aperture antennas recessed in both flat and curved substrates. The FE-BI technique was first successfully used to model the radiation by a cavity-backed, rectangular aperture recessed in a planar ground plane by Jin and Volakis [l] at the University of Michigan. In this implementation, the cavity region was tessellated into rectangular brick elements. The use of rectangular brick elements results in a uniform mesh giving rise to a block Toeplitz boundary integral matrix. Consequently, iterative solutions of the matrix can be accelerated through the use of a Fast Fourier Transform (FFT) [1]. The utility of rectangular bricks, however, is strictly limited to rectangular geometries. Gong, et al. [32] at the University of Michigan further refined the technique by utilizing tetrahedral finite elements to model arbitrarily shaped apertures. Tetrahedral elements are advantageous in that they are the simplest shape capable of modeling arbitrarily shaped volumes and may be generated automatically by commercial meshers. Kempel at the University of Michigan first extended the F E-BI technique to accommodate cavity-backed apertures and rnicrostrip patch antennas on curved substrates by utilizing specially formulated circular cylinder shell elements [2,3]. These shell elements are singly curved and capable of uniformly discretizing a volume bounded by a singly curved surface with a constant radius of curvature (e.g. the surface of an infinite circular cylinders). As the radius of 47 curvature approaches infinity (the flat case), a shell element becomes functionally equivalent to a rectangular brick. Analogous to the rectangular case, the uniform discretization of singly curved regions with shell elements results in a boundary integral matrix that is block Toeplitz and, therefore, amenable to a fast iterative solution employing FF T. The motivation for the use of bricks and shells was the need to minimize the computational burden associated with the boundary integral due to limitations in computer memory and processing speed at the time. The FFT-based iterative solver efficiently utilizes memory (0(Ns)) while minimizing compute time (0(N, log2 N,)), where N3 is the number of surface unknowns. Traditional vector matrix multiply routines require 0(Ns’) of memory and 0( N s3 ) of compute time. However, with the recent advent of high performance computers and the availability of large blocks of random access memory, the limitations on the complexity of conformal antennas that can be modeled has been relaxed. A major limitation of the brick and shell element approach is that they can only be used to accurately represent volumes delimited by canonical surfaces and, therefore, they are not applicable to arbitrary geometries. Consequently, in order to extend the range of applicability of the FE-BI technique to the most generally shaped structures while preserving its computational efficiency, right triangular prism elements were developed by Ozdemir et. al. at the University of Michigan [33]. Prism elements are advantageous in that they are capable of modeling arbitrary geometries while yielding fewer unknowns than tetrahedral elements [33], and they are not as geometry constrained as bricks and shells. However, there is a drawback in that distorted prisms are not functionally capable of accurately representing electric fields because they lack tangential continuity across their faces. This defect is the result 48 of their vertically oriented edges not being perpendicular to the planes of their triangular faces [30] resulting in a nonuniform cross-sectional surface area. In modeling cavity- backed apertures of arbitrary shape, the best features of prisms and tetrahedra are combined by the following procedure. The aperture is discretized into a mesh of triangular elements, which are then extruded by means of distorted prisms into the cavity. Each prism is subsequently decomposed into three tetrahedral elements. In this manner, extrusion can be used to form the volumetric mesh with elements that correctly represent the unknown electric field. This method was recently used by Macon et al. [34] for arbitrary apertures recessed in a circular cylinder. In this chapter, the FE-BI method will be extended to model cavity-backed, arbitrarily shaped apertures recessed in electrically large, doubly curved surfaces. A domain decomposition approach is inherent in the FE-BI formulation for modeling cavity-backed apertures in that the computational domain is broken into an interior and an exterior region. The finite element method is used to model the volumetric fields in the interior region. A boundary integral is employed to enforce the requisite conditions (6. g. tangential magnetic field continuity across the aperture) for mesh truncation at the doubly curved aperture surface via a specially formulated asymptotic electric dyadic Green’s function. The doubly curved surface is modeled as an electrically large, perfect electrically conducting (PEC) prolate spheroid. As illustrated in Figure 3.1, by allowing the axial and azimuthal radii of curvature, in turn, to approach infinity, the circular cylinder and plane may be recovered as limiting cases. The formulation of the asymptotic dyadic Green’s function within the context of UTD and its analytical and numerical 49 s'alidal regiur 3.2 l The Begi in T6 validation will be covered. Finally, the formulation of the far-zone fields in the exterior region by means of the surface equivalence principle will be covered. 3.2 FE-BI Formulation The F E-BI equation is derived from the weak form of the vector wave equation. Beginning with the time-harmonic form of Maxwell’s equations Vx 12"" = — 12020:, -H"" (3.1) Vx 11““ = 17,133. -'E“‘ +J (3.2) where Eim is the unknown interior electric field, Hint is the unknown interior magnetic Pi 80 field, k0 = 277/710 is the free-space wave number, Z, = is the free—space wave impedance, 2" , and Z, are the relative anisotropic pennittivity and permeability, respectively, given by = 8,“ 6x, 6,, a, = 6‘” a” 3),, (3.3) zr zy 22 = luxx luxy 6.: yr : ”y: ”W #yz (3.4) #3 lazy #2: Note that the e’“ time convention is assumed and suppressed throughout this dissertation. Substituting (3.2) into the curl of (3.1) we get the vector wave equation =—l . = . . V x [pr -V x E"" ] — k: 6', -E"“ = —jk0Z0J""” (3.5) where J ""P is the impressed current due to the excitation source. The method of weighted residuals is employed whereby the inner product of (3.5) and an edge-based, vector 50 testin the \\ whet \sei der. \\' testing function, W, , is taken over the computational volume V . This procedure yields the weak form of the vector wave equation which is given by =4 . = . . I{W, -Vx[pr ~VxE""]-k:VV, -8r ~E’m}dV = f,"" (3.6) V. where the interior excitation function used to model probe feeds is given by f,‘“‘ = —jk0Zo jw, ~J”"”dV (3.7) V, The weak form of the vector wave equation approximates the vector wave equation in a weighted sense over the computational domain V. Note that (3.6) contains second-order derivatives of the unknown electric field. Since constant tangential/linear normal (CT/LN) vector basis functions are used, it is necessary that the order of (3.6) be reduced through the application of the first vector Green’s theorem. The application of the theorem transfers a curl operator from the unknown electric field and onto the testing function, after which (3.6) becomes =-l . = ‘ A _ ‘ [[vxw, .p, -VxE”" +k§w,. .5, -E‘”’]dV—jkoZ0 L w, -§xH"“dS = f‘” (3.8) V. l where a is the outward-directed unit normal vector in the prolate spheroidal coordinate system. Equation (3.8) is underdeterrnined in that it contains unknown electric and magnetic fields; however, the testing function represents the unknown electric field only. The interior magnetic field Hint cannot be found easily; however, an expression for the total magnetic field, just exterior to the aperture, may be found from Hw=HW+HM+HW 09) 51 where H” is the incident magnetic field, H”f is the reflected magnetic field, and H“” , determined via the surface equivalence theorem, is the magnetic field attributed to the aperture fields which is given by Hap = jkOYO j a. .39 E’"‘dS' (3.10) 3 up An electric dyadic Green’s function of the second kind [10] is used to convert the tangential electric field in the aperture to an exterior magnetic field. The natural boundary condition, ex H” = Ex H’” , is enforced across the aperture surface by substituting the expression for the total magnetic field just exterior to the aperture into (3.8). Upon evaluating (3.8), we obtain the coupled F E-BI equation given by =_] = _ A = A . I]:wa -,ur -VxE"" -k:W ~49, oE'"’]dV+k: jI(§XW,)-Ge2 -(§'xE"")dS'dS y 35- (3-11) =ftnt+fext where f w“ is the exterior source excitation function given by ff“ = —jk,Z, jw, .Ex (H’m + H')dS (3.12) St Note that the surface integral in (3.11) has support only over the nonmetallic portions of the aperture. The FE-BI equation in (3.11) is not yet in a form that is suitable for numerical implementation. The unknown interior electric field must be expanded throughout the computational volume in terms of subdomain, edge-based vector expansion (e.g. basis) functions W]. E=fi8wj (3.13) 52 In this formulation, Galerkin’s testing procedure is utilized whereby the vector basis functions, W1. , are CT/LN functions and identical to the testing functions W,. Note that the expansion functions become identical to the testing functions on the aperture surface (4‘ = .50 ), thereby, enforcing the essential boundary condition éx E’” = Ex E‘” across the aperture surface. The unknown complex coefficient associated with each free edge of the volumetric finite element mesh is given by E j. A free edge is any edge that is not tangential to a PEC surface since a total electric field formulation is being used in this work. Hence, any edge that is tangential to a PEC surface has an expansion coefficient equal to zero. Substituting (3.13) into (3.11) gives the final discretized FE-BI equation that is amenable to computation r =4 = 7 N [{VXW‘flr -Vij—k02VV,'£r-WJ dV . 2E. " . = . =f.'“‘+f.”" (3.14) 14 +1.3 jj(§ij)oG.2-(§'ij)dS'dS 5,8, _ 3.3 Finite Element Matrix Elements In this formulation, the volumetric unknown electric field is expanded within a tetrahedral element in terms of CT/LN vector basis functions. CT/LN basis functions provide a constant tangential component along one edge, while the tangential component along the other edges equals zero. In addition, these basis functions provide a linearly varying normal component along each edge. Tetrahedral elements are formed from prism elements by first generating a planar surface mesh of triangular elements, mapping the surface mesh onto the prolate spheroid surface, and extruding each surface element into 53 a—>00 b is fixed circular cylinder Figure 3.1 Topological transformation of a prolate spheroid into a plane and a circular cylinder in the limits of zero azimuthal and axial curvatures and zero axial and finite azimuthal curvatures, respectively. 54 the cavity volume by means of prism elements. The prisms are, subsequently, divided into tetrahedra. The process of extrusion essentially amounts to growing the mesh along a direction that has been defined as normal to the surface in a particular orthogonal coordinate system (in this case, the a direction). This process entails generating finite elements for each layer of the mesh by duplicating the aperture node distribution in all of the lower layers. Thus, in order to form the layer, the aperture nodes are generated at the interface of the first and second layer. Elements for the current layer are generated from those nodes and the bottom nodes of the previous layer. Edges are subsequently formed based on the chosen finite element. The scheme that is used in subdividing prism elements into tetrahedral elements is illustrated in Figure 3.2 [35]. Two types of prism elements are used in order to prevent the diagonal edges of adjacent prisms from crossing, thereby, ensuring tangential field continuity across each face. Once the tetrahedral elements have been generated, the unknown electric field is expanded in terms of the vector basis function given by W]. = (leVsz — LJZVLflflJ (3.15) In (3.15), the subscripts denote the two local node numbers defining the edge directed h from jl to j 2 , Ij is the length of the j' edge, and the nodal basis functions are given by L = J 6V6 (a: +bfx+cjy+djz) (3.16) where the coefficients aj, bf, cj and d; are found fiom the coordinates of the four local nodes that define the tetrahedral element and V" is the volume of the tetrahedral element given by 55 Prism Type 2 Figure 3.2 Subdivision of the two types of prisms into tetrahedra. 56 (x1 ’x4)[(}’2 ‘Y4)(23 '24)‘(y3 -y4)(22 '24)]+ ,. 1 V = '6' (yl ‘J’4)[(zz —z4)(x3 'x4)—(Z3 -z4)(x2 —x4)]+ (3'17) (21‘24)[(x2 —x4)(y3 ”Y4)"(x3 —x4)(y2 'Y4)] The key benefit of using this type of element in the FE-BI formulation is that the vector basis firnction and its curl are easily expressed in terms of Cartesian unit vectors. In light of this, the curl of W1 is given by 21. Vx Wj = (6V: )2 [fie/Id], —cjzdfl )+j3(dj,bj2 —dj.2bjl )+ 2"(bflcj2 —b12cj.,)] (3.18) 3.4 Boundary Integral Matrix Elements 3.4.1 Selfcell Evaluation of the Boundary Surface Integral The selfcell evaluation is the local planar approximation due to the small cell dimensions relative to a wavelength. This is in regards to the surface integral term in (3.14). As discussed previously, the FE-BI method is a hybrid method combining the finite element method with the method of moments. The boundary integral is formulated as an integrodifferential equation that can be solved by the method of moments. The tangential electric field in the aperture is expanded in terms of a set of divergence free, vector basis functions having support over two triangular patch regions sharing an edge. These basis functions were first introduced by Rao, Wilton, and Glisson [36] and will, henceforth, be referred to as RWG basis functions. The expansion of the tangential electric field in the aperture in terms of RWG basis functions begins with the formulation of the magnetic field just exterior to the aperture in terms of an electric vector potential and a magnetic scalar potential given by H” = —ja)F—Vmg,fm>] = (fix warm) I": — =, ijF.fmdS— JVscDmg-fmdS= jaxn'" -r,,,ds (3.29) S S S where the inner product is denoted by (AB) = IA-BdS . Applying the surface vector S identity in [3 7] to the second term on the left-hand side of (3.29) ja) jF-fmdS— jV,.(<1>mgrm)dS+ jomgvs ~fmdS = fixH'm -fmdS (3.30) S S S S The second term on the left-hand side involves integration over the surface S of a closed three-dimensional body. This surface integral may be evaluated by splitting the closed surface S into two surfaces S1 and S2 bounded by the contours C1 and C2 , respectively, 60 directed in opposite directions and applying a two-dimensional version of the divergence theorem mag 77: jij-fmdS-[IVs-((Dmgfm)dS+IVs-( jij-rmdS-(qc 80>“;deij fi-(D r dz) S l 2 mag in + jomgvs -r,,,ds = [a x H’"‘ .rmds (3.31) S S Hence, (3.31) now becomes jar [F -r,,ds + jcbmgv, -fmdS = [fix H'” .rmds (3.32) S S S Employing the method of moments to solve this system, we expand the magnetic surface current in the aperture in the set of RWG basis functions N K,(r') 5 21,1, (r') (3.33) n=l where 1,, is the unknown weighting coefficient and N is the number of non-boundary edges. Note that K,(r') = E’” xfr and W” =r’rxfn (r'). Employing Galerkin’s method whereby the set of vector testing functions, denoted by fm (r) , is set equal to the RWG basis functions f" (r) we obtain a e‘ij l (”R ja) 0 [r',—{18' -f,,,dS- IVs-f" dS' ,-r,,,ds S 2775. R 3 277.1105. R = a x H... (3.34) Substituting (3.26) and (3.28) into (3.34), we obtain the boundary integral impedance matrix 61 nm Z =jk°”’7:"l —{jAn— —j’;,9.(r3 9405:],— S'S kOSS'AII where and {+1, reT; 6’ - . reT‘ m In light of (3.35), the electric vector potential given by (3.20) may be rewritten as =A 3A. 1 ] p.01 pm)??— '45 m n Ti T01 and the magnetic scalar potential given by (3.21) may be rewritten as e...ij (I) ”’ =,:1AA:,.,.R y+ds'ds}(3 35) (3.36) (3.37) (3.38) (3.39) The potential integrals given by (3.3 8) and (3.39) may be evaluated over the source and observation triangle regions Tmi and If most efficiently by expressing them in terms of normalized local area coordinates (g1,g2,g3) [38]. The local area coordinates are defined within a triangular region in the following manner A A2 A3 91: Aq -—=,g2 Aq —=,§3 7., 62 (3.40) where A, A, and A3 are the areas of the sub—triangles and A" is the area of the entire triangular patch. The local area coordinate system within a triangle is depicted in Figure 3.4. The normalized area coordinates satisfy the following constraint g+g+n=1 BAD As a result, only two coordinates are independent. The local area coordinates may be converted to Cartesian coordinates via the following vector transformation r=gq+gq+gg GAE where r, is a position vector from the origin to the ith vertex of a triangle. Surface integration over a triangular region T q effectively transforms the kernels of the integrals given in (3.3 8) and (3.39) from a function of position defined in Cartesian coordinates to a function of position defined in local area coordinates as given by l 1‘42 IK(r)dS =1 jKlgiri +9212 +(1-9'1 ‘92)r31d9'1d9'2 (3-43) After the transformation, the potential integrals in (3.38) and (3.39) are re-expressed in terms of local area coordinates. Hence, (3.3 8) and (3.39) may be rewritten as F—J—j *(r) —’—J‘ in)?!” dS' S (344) AP Tppm Aq Tan R ' and .1“; ¢m=i —’—j" dS' s. (3.45) Apfl.flfl R Before the integrals in (3.44) and (3 .45) can be numerically evaluated, the singularity in 63 each of their kernels must first be isolated if the test and source points coalesce. This is done inside of the bracketed expressions by adding and substracting out the singularity. Evaluating (3.44) in this manner yields the expression -ij- . 2”" . 1 e p'- p . FrlpnO'deS =FTJP:(I'V)R IdS'f-Alq {TIT-61S (3.46) +(p- p.) jldS ' T" R The first term on the right-hand side of (3.46) has a bounded kernel, while the bracketed term on the right-hand side contains the singularity. Having isolated the singularity, (3.46) may now be expressed in terms of the local area coordinates, resulting in the expression 1 1'61 1 =2] jp.(r)——ld§.'d§2'+ A, —-{ jig-ed? 7w 3117] p.160" (3.47) filo-72.)]! 11593} The first term on the right-hand side of (3.47) is now bounded and expressed in terms of local area coordinates. Therefore, it may be evaluated by numerical integration over each triangular patch on the surface [39]. However, the second term on the right-hand side is singular and must be evaluated analytically. Appendix A provides details on the evaluation of surface integrals over triangular regions. Evaluating the bracketed expression in (3.45) in a similar manner yields the following expression 1 rr-;, ‘3-ij 1 F —dS'= 2OH—dg1dg,+— j—ds' (3.48) 64 Analogous to (3.47), the first term is bounded and well-suited for evaluation by numerical integration over the triangular patch, while the second term containing the singularity must be evaluated analytically [39]. 65 Figure 3.3 RWG basis functions supported within the triangular regions 7;: and 7: sharing a common edge n. Figure 3.4 Local area coordinate system defined within a triangular region. 66 3.4.2 Asymptotic Dyadic Green’s Function Formulation The exact condition for finite element mesh truncation is provided by the boundary integral by means of a dyadic Green’s function. The electric dyadic Green’s function of the second kind [10] couples the tangential electric and magnetic fields in the aperture and enforces the boundary condition on the tangential electric field over the PEC prolate spheroid surface. This dyadic Green’s function is used in the hybrid FE-BI formulation (3.14) and is denoted by 5... Due to the poor convergence and high computational expense of an exact form of the dyadic Green’s function for electrically large bodies, an asymptotic form for an electrically large, PEC prolate spheroid will be derived. The asymptotic Green’s function physically represents surface diffracted rays (e.g. creeping waves) that are excited by a magnetic dipole (e.g. aperture) on the PEC prolate spheroid surface. The formulation begins with the UTD expression for the surface magnetic field excited by a unit infinitesimal magnetic dipole on an arbitrary convex curved surface developed by Pathak [29] which is given by (=3e2(r|r') ~(B'B{[1—q]v(/3)+D’qz [7.u(fl)+7.V(fl)]+rfq[u(fl)-V(fl)]} +i'%{quv(fl)+quw)—2qz[y.u 0 and G >0 for all elevation angles. Hence, the expressions inside the radicals of (3.91) and (3.92) are always positive in this work. Substituting (3.70), (3.72), (3.91) and (3.92) into (3.90) and solving for 77 CDT 9601 for . “he; Subs glVer “‘6 ol 6 yields the geodesic angle 6 (6,c,) in terms of the elevation angle 6 and integration constant c, 6(6,c,) = sin" [fig-5) (3.94) The equation for the geodesic angle corroborates physical intuition in that the geodesic angle of a point on a geodesic curve would depend not only on the location of a point on the surface but also on the particular geodesic upon which it lies. Closely associated with the geodesic path length is the generalized Fock parameter ,6. As discussed in Chapter 2, this dimensionless parameter expresses the ratio of the distance of a point from the geometrical optics shadow boundary to the width of the transition region. The expression for ,6 given in Chapter 2 is repeated here for convenience P. p s where as before m is given by [C 1/3 m =[ is) (3.96) Substituting (3.80) and (3.81) into Euler’s equation from differential geometry which is given by K(6) = K, cos’ 6(6) + K, sin’ 6 (6) (3.97) we obtain the expression for the geodesic curvature, which is given by ab4 + (a3 - ab’ )c,’ b” (a’ sin’ 6 + b’ cos’ 6)”’ K(6) = (3.98) Note that the geodesic curvature is angularly dependent as will be the case for most of the surface ray parameters. As mentioned previously, this is a consequence of the fact that 78 CE [0 be the geodesic path exhibits variable torsion. The geodesic radius of curvature pg (6) is just the reciprocal of (3.98) and is given by b3 (a’ sin’ 6 +b’ cos’ 6)”2 ab4 + (a3 - ab’)c,’ p869) = (3.99) Substituting (3.96), (3.99) and the reciprocal of (3.92) into (3.95) and evaluating leads to the expression for ,6 given by 7: )1/3 6,, asin6(b" +[a2 —b’]c,2 )2/3 d6 (3 100) fl(6)=[ 1/ 716 a, b[(a2 sin’ 6 +1;2 cos’ 6)(b’ sin’ 6 — c,’ )] 2 The numerical integrations involved in tracing the geodesic path and in the calculation of the associated geodesic parameters can be quite time consuming. In order to expedite the calculation of these parameters, the following limiting cases depicted in Figure 3.8 may be handled seperately: (1) The geodesic endpoints lie close together and are situated on a quasi-cylindrical midsection of the spheroid ( Figure 3.8a). (2) The geodesic endpoints share the same azimuthal angle, lying on a circular arc (Figure 3.8b). (3) The geodesic endpoints share the same elevation angle and, thus, lie on an elliptical arc (Figure 3.8c). For the first case, the following heuristic approximation to the geodesic path length has been found to be reasonably accurate: -— 2 supp eflpmgw) +E(6,e)’ (3.101) 79 w} \\'l SU 1h I'E where pm, is the average of the azimuthal radii of curvature of the two endpoints, (6 = (6 — (6' , and E (6,6) is Legendre’s elliptic integral of the second kind [42] given by 9: E(6,e) = 0 [J1 —e2 cos’ 6d6 (3.102) 9| where e is the eccentricity, defined previously, 6,, and 6, are the elevation angles subtended by the endpoints of the geodesic from the z-axis. From Table 3.1, it is apparent that the geodesic path length computed from (3.93) compares quite favorably with the approximation to the geodesic path length given by (3.98) over the quasi-cylindrical region. The approximation to the F ock distance parameter for the first case is given by 2/3 cos’ 6 z ks, —— (3.103) fl PP [fikpavg and the geodesic angle is approximated by E 6, (seem ii) (3.104) Pavgto For cases (2) and (3), (3.96) reverts to the circular arc length and the elliptic arc length formulas, respectively. For the second case, ,6 is given by 92 fl=j J1— ”cos 666 (3.105) 9: 31 Pg While for the third case, ,6 is given by ms p, (3.106) E || | 80 Note that 6 = O and 6 =% radians for the second and third cases, respectively. 81 Figure 3.6 A geodesic on a prolate spheroid surface traced via numerical integration. Figure 3.7 The geodesic angle. 82 (a) (b) (c) Figure 3.8 Limiting cases for the the geodesic path length: (a) quasi-cylindrical, (b) circular arc, and (c) elliptical arc. 83 Table 3.1 Comparison of approximate and exact geodesic path lengths between two points located on the midsections of two prolate spheroids. The approximate geodesic path length is denoted by s, and the exact geodesic path length is denoted by s geo ' PP Angular Position Major and Minor Major and Minor of Geodesic Axes of Prolate Axes of Prolate Endpoints Spheroid Spheroid as, 0 ’ ¢s’ (0 ( f) ( f) a=400cm a=50cm b = 40 cm b = 40 cm (90°, 33°),(0°, 10°) sgeo= 15.6017 cm Sgeo = 7.1870 cm so”, = 15.6072 cm supp = 7.1941 cm %error = 0.03535 %error = 0.09870 (90°, 33°),(0°,15°) SW = 17.5099 cm Sgeo = 10.7394 cm supp = 17.4491 cm SW = 10.6133 cm %error = 0.3473 %error = 1.1744 (90°, 33°), (0°, 20°) Sgeo = 19.9119 cm s,” = 14.3168 cm SW = 19.7411 cm SW = 14.0671 cm %error = 0.8574 %error = 1.7441 (90°, 33°),(00, 25°) Sgeo = 22.6185 cm s,“ = 17.8799 cm SW = 22.3452 cm sap, = 17.5350 cm %error = 1.2082 %error = 1.9289 (90°, 33°), (0°, 30°) sgw = 25.5564 cm sgeo = 21.5222 cm SW = 25.1646 cm %error = 1.5331 sap, =21.0102 cm %error = 2.3788 84 3.4 No ma the $3 as 51. Ill bi 3.4.2.4 UTD Surface Ray Parameters Now that the geodesic parameters have been determined, the UTD surface ray parameters may now be calculated. The torsion 7(6) is obtained by substituting (3.80) and (3.81) into the following expression: sin 26 1(9) = 2 (K2 _K1) 3.107 _ c,(b’ sin’ 6—c,’)”’a(b’ - a’) ( ) b’(a’ sin’ 6 +b’ cos’ 6)”2 The torsion factor To may now be calculated via 4(6) = L” K(6) (,2 (3.108) c,a(b’ -a’)[(b’ sin’ 6 —c,’ )(a’ sin’ 6 +b’ cos’ 6)] ab" +a’c,’ —ab’c,’ The ray divergence factor D , which quantifies the amount by which a surface diffracted ray spreads within a tube, is analytically determined by evaluating the angle between tangent vectors to adjacent geodesic paths. The adjacent geodesic paths, traced from the same source point, are angularly seperated by approximately 1.00. However, for this application, the attenuation in the magnitude of the Green’s function attributable to the surface divergence factor was found to be negligible. Hence, in order to expedite the numerical determination of this factor without imposing an unnecessary computational burden associated with the numerical computation of two geodesics for every source and observation point, a heuristic expression for D was derived based on the known values of D for a circular cylinder and a sphere D=[1.0—K‘(6’]+K‘(6) ‘9 “2(6) K2(9) Sin6 85 For ‘ sphe ICSU and ex] dir de Tl ar 6 = 1— —— 3.109 ( 7.)+7. Sing ( ) For the case of a sphere where K, = K, , D = 45 , which is the well-known result for a srn sphere; for the case of circular cylinder where K, = 0, D =1 , which is the well-known result for a circular cylinder. The interpolating factors 7, and y, are given by _ “1(6) 7.(6') — —K2 (6,) (3.110) _ a’ sin’6+b’ cos’6 and (9l-K (9) 7.(6>= "2 ‘ “2(9) (3.111) _ (a’ —b’)sin’ 6 a’ sin’ 6 + b’ cos’ 6 Now that explicit formulas for the UTD surface ray parameters have been derived, expressions for the ray-fixed unit vectors tand b with respect to the principle surface directions i1 and ([1 must be derived. Note that f]: —0. See Appendix B for the derivation of this result. The unit tangent vector t is given by t=ficos6+¢sin6 (3.112) and the unit binormal vector b is given by xr’i 5:3 =(ficos6—fisin6 (3.113) 86 where 6 is the geodesic angle, defined previously, and the unit normal vector to the surface is fr =8 Substituting (3.112) and (3.113) along with the UTD parameters into (3.49) and after considerable algebraic manipulation, the components of the asymptotic dyadic Green’s function for the PEC, electrically large prolate spheroid are obtained and are given by 0mg, :6,¢|6',¢') = {(COSZ 6-q[(D2 +2)cos’ 6-(D’ +1)])v(6) +q2 ([(D2 + 2)] cos’ 6—2)[y,u(6)+ y,v(,6)] (3.114) +(ro cos6 + sin 6)2 q [u(,6) — v(,6)]) D%qe"k°’ 6:; = 03%;, :6,¢|6',¢') = {—sin6cos6(v(,6)—(D’ +2)qv(6) +(D2 + 2)q2 [7,u(6)+ 7,v(,6)])+[(2008’ 6 —1)e, — (3.115) (I: ‘1)Sin6C056]q[u(fl)—v(fl)]}%§_qe-jkos GZ,"(§0 :6,¢|6',¢) = {(sin’ 6 —q[(1)2 + 2) sin’ 6 — (D’ +1)])v(,6) +q’ ([(D2 + 2)]sin2 6—2)[y,u(,6)+y,v(6)] (3.116) 2 +(ro srn6—cos6) q[u(§)—v(§)]}D-k£—fl°qe J . 3.4.3 Validation of the Prolate Spheroid Dyadic Green’s Function In this section, the validity of the asymptotic dyadic Green’s function for the prolate spheroid given in (3.114)-(3.116) is established analytically and numerically. 3.4.3.1 Analytical Beginning with the expressions for the Green’s ftmction given in (3.114)-(3.116), we proceed by allowing the radius of curvature along the axial direction to approach infinity, while maintaining a fixed azimuthal radius of curvature. In this case, the prolate spheroid topologically approximates an infinite circular cylinder. Consequently, the magnitude of 87 the asymptotic prolate spheroidal dyadic Green’s function should approach the magnitude of the asymptotic dyadic Green’s function for a PEC, infinite circular cylinder. The values of surface curvatures along the axial and azimuthal directions become 7:, =lim ab =0 (3.117) , 3/2 “"°° (a’ srn’ 6 +b’ cos’ 6) and x, =lim a =-’- (3.118) “7‘” b(a’ sin’ 6 + b’ cos’ 6)“2 b From (3.117) and (3.118), the geodesic curvature K now becomes K = K, cos’ 6+K, sin’ 6 sin’ 6 b (3.119) Ill The torsion factor 7,, becomes sin6cos6 .=-—K——( rm) ~ “’5" =cot6 , (3.120) srn 6 . . K K — K The 1nterpolat1ng factors now become 7,. = -—‘- = 0 and 7c = 2 I K, K, =1 For a circular cylinderD=1. Substituting (3.117), (3.118), (3.119), (3.120), and the interpolating factors into the dyadic components given in (3.1 14)-(3.116) we have for the G3” component 88 gig; 033(66 6',(6') = {(cos’ 6 —q[(D2 + 2) cos’ 6 —(D’ +1):|)v(§) +q’ ([(1)2 + 2)] cos’ 6 - 2)[y,u(g) + y,v(§)] . 2 kiYo —7k.s +(ro cos6 +srn 6) q[u(f)—v(§)])D—2—fl—qe = {(cos’ 6 -q[3 cos’ 6 - 2])V(4’) +q’ (3cos’ 6-2)V(5) 0036 2 k’Y . + - 6+ ' 6 - —° 0 ""°’ (Sim, cos sm ) q[u(6) 1)(€)]} 27, 96 = {cos’ 612(5) + q(1— q)(2 — 3 cos’ 6mg) 2 +qcsc’6[u(e)—v(e)]}5-,que”"°’ (3.121) 7: similarly, for the Gf,” = 62,“ components it}; 023” (19.62 63¢) = {—sin6eos6(v(;) —(02 + 2)qv(4=) +(D’ + 2)q2 [y,u(g) + y,v(§)]) + [(2 cos’ 6 —1)r, _(rg —1)Sin5C086]q[u(é)—V(§)]}£2(%qe—Jkos = {—sin 6 cos6(v(;) — 3qv(1,=) 6 +3 2 + 2 26—1cos — q v(£j) i:( COS )sin6 (’8’: f: —1)sin6cos6]q[u(§) —v(§)])-]E-"Z‘iqe”k°’ fl' srn 2 . kozyo -jk.s - {sm6cos6[1—3q(1—q)]v(2,‘))gqe (3.122) and finally for the G2,” component )1_1§1°c;,;',"(6,¢|63¢)={(sin2 6—q[3 sin’ 6 —2])v(§) +q’ (3 sin2 6 — 2)v(§) 6 . ’ k’Y _.,,, +(Ccs6 .sma—cos6) q[u(§)—v(§)])—§fliqe 1. sm 89 Gre mat that with the re along helical the m2 Green’, a113, fl indicati 2 = { sin’ 61(5) + q(1— q)(2 — 3 sin’ 6)v(.f)) 52—: qe-m (3,2,) As seen from the limits of the components in (3.121)-(3.123), the asymptotic prolate spheroid Green’s fitnction reverts to the asymptotic circular cylinder Green’s function in the limit of an infinite radius of curvature along the axial direction 77 [2]. 3.4.3.2 Numerical To further validate the prolate spheroid asymptotic dyadic Green’s function, the relative magnitudes of its components are compared with those of the circular cylinder asymptotic dyadic Green’s function as a function of the electrical geodesic path length s/ 2,. The electrical geodesic path length is expressed in terms of wavelengths. Based upon the analysis of the previous section, it is expected that the prolate spheroid asymptotic dyadic Green’s function will reduce to the circular cylinder asymptotic dyadic Green’s function in the limit of an infinite axial radius of curvature. A comparison can be made by first tracing the geodesic path between a set of source and observation points that are confined to the quasi-cylindrical midsection of a 40.071 x 4.0}. prolate spheroid with an initial geodesic angle 6, =15.8° as shown in Figure 3.9. A comparison between the relative magnitudes of the asymptotic prolate spheroid Green’s function components along this geodesic with those of the asymptotic cylindrical Green’s function along a helical geodesic for which 6 =15.80 is given in Figure 3.10. There is a rapid increase in the magnitude of the Green’s function near the origin due to the singularity of the Green’s function at the source point. As the creeping wave propagates a few wavelengths away from the source, the magnitude exhibits a constant rate of attenuation which is indicative of the characteristic exponential decay of a creeping wave. Along the spheroid 90 surface there is greater curvature along the (a direction than along the 7) direction, hence, the attenuation of the Gf’,’ component is greatest, while the attenuation of the 0:,” component is least. As expected, the relative magnitude and attenuation of the G3,” component lies in between values of G2," and G3,“ over the extent of the geodesic. Figure 3.11 depicts a geodesic path between a set of source and observation points that are oriented such that the initial geodesic angle 6, = 26.20 is larger than in the previous case. From Figure 3.12, it is evident that the prolate spheroid asymptotic Green’s function magnitudes along this geodesic are almost identical to the magnitudes of the cylindrical asymptotic Green’s function along a helical geodesic for which 6 = 26.20. The attenuation of each component is less because the geodesic path spans the portion of the spheroid surface which exhibits less curvature than in the previous case. In Figure 3.13, a geodesic path on a 40.071 x 4.071 prolate spheriod with an initial geodesic angle given by 6, = 30.10 is depicted. As seen in Figure 3.14, the relative magnitudes of the prolate spheroid Green’s function begins to deviate from the cylindrical Green’s function along the helical geodesic for which 6 = 30.10. This is due to the fact that the prolate spheroid surface exhibits curvature along both the axial and azimuthal directions along the geodesic trajectory depicted in Figure 3.13, while the circular cylinder exhibits curvature only in the azimuthal direction along the helical geodesic. The effect of moving the source and observation points closer to the tip of a prolate spheroid is examined next. For the geodesic trajectory depicted in Figure 3.15 and its associated dyadic component magnitudes shown in Figure 3.16, the attenuation of G2,” is greatest within four wavelengths of the source, tapering off to a steady decay rate 91 afterwards. This is most likely due to the fact that the geodesic does not follow a straight path along the 7] direction. Instead, it follows the variably curved surface contour. On the other hand, the G,”,” and GZ’,” components exhibit a constant rate of attenuation after approximately two wavelengths from the source. This is due to the constant rate of curvature along the (0 direction. As expected, the magnitude of the G3,” component lies in between the magnitudes of the other two components. Placing the source and observation points even closer to the tip, as shown in Figure 3.17, primarily effects the magnitude of G2," , as gleaned from an examination of Figure 3.18. In this figure, G2,” exhibits a rapid decay rate, followed by a slight plateau and culminating in a steady decay rate. This phenomenon is a consequence of the twisting of the geodesic curve, along the variably curved 77 direction. The behavior of the dyadic Green’s function components for the geodesic trajectory orientations considered in the previous cases appears to be consistent with the physical behavior that one would expect for creeping wave propagation along a variably curved surface [43, 44]. With the derivation and validation of an appropriate electric dyadic Green’s function for the electrically large, PEC prolate spheroid, the boundary integral is completely specified. 3.5 Solving the FE-BI System The coupled finite element and boundary integral equation given in (3.14) generates a large sparse matrix and a fully populately matrix, respectively. This type of system is amenable to solution by an iterative technique. An iterative approach for large sparse matrices is preferable to a direct approach due to the phenomenon of fill-in associated with direct methods, that utilize matrix factorization schemes such as LU decomposition. Specifically, the upper or lower triangular matrices, into which a large sparse matrix 92 would be factored, may not represent the sparsity pattern of the original sparse matrix. However, iterative solutions methods do not employ fill-in, which allows them to maintain the sparsity of the system. In order to employ an iterative technique, the FE-BI equation in (3.14) may be rewritten in matrix form as [2] A,,+G A, E” A, A0,. E“” G 0 E“? 0 . = . + . = m, (3.124) A... An- Em! A... An E W 0 0 E "u f where [A] is the finite element matrix, [G] is the boundary integral matrix, E ”" is the unknown electric field in the cavity, E‘” is the unknown electric field in the aperture, f '"‘ denotes the interior excitation due to a probe feed. Note that ff” = O in (3.14) for the case of interior excitation while fl” = 0 for the case of exterior excitation. The decomposition of the FE-BI matrix in this manner allows the matrix-vector product, which is the most computationally expensive task in the iterative approach, in each partition to be optimized for solution by an iterative solver. As an example, since the finite element matrix is sparse, the matrix can be stored in an efficient compressed sparse row (CSR) fashion [45] and the matrix multiply scheme can be optimized for a sparse matrix. Furthermore, although the boundary integral matrix is fully populated, it is symmetric. Hence, only the upper (or lower) triangle needs to be stored. Thus, the boundary integral matrix-vector product can be Optimized for a symmetric matrix. For this problem, the biconjugate gradient (BiCG) iterative scheme is chosen rather than the conjugate gradient scheme (CG). The BiCG scheme is a variation of the CG method and is applicable to asymmetric as well as symmetric systems of linear equations. The main advantage of using BiCG is that for symmetric matrices, Jacob’s algorithm 93 employs only one matrix-vector product, as opposed to the CG scheme which employs two matrix-vector products [46]. Moreover, the BiCG scheme converges faster than the CG scheme. The trade-off, however, is that the convergence of BiCG is more erratic than that of CG [30] (See Appendix E for a listing of BiCG pseudocode). 3.6 Radiation 3.61 Input Impedance Once the electric fields in the cavity E’"‘ and aperture Ea” have been determined by solving (3.124) with a suitable iterative solver such as the BiCG scheme, the input impedance can be found. The input impedance is calculated from the ratio of the voltage at the input port to the current flowing into the port. The simplest type of feed is a Hertzian dipole feed where the source is a filament of current. For a normally directed probe feed (e.g. directed along the é-direction) that is positioned at (775,625) on the surface of a prolate spheroid, (3.7) is evaluated as 73'" = —jk.Z.IIW. (77.5%) =_jkOZOII (3.125) For this case, the input impedance can be computed using Gauss’ Law II") _1 6 Z,,,=7—ZE,,,,,)“1.W d] (3.126) in me] where n is the number of edges, l denotes the orientation of the probe-feed, E100 are the coefficients of the electric field determined by solving the FE-BI system, and as defined previously W J are the vector basis functions. The total electric field at the feed (n) location is determined by summing over all the edges of the element, which would be the six edges of the tetrahedral containing the probe-feed in this case, and integrating over 94 the length of the probe. Since this approach relies upon an accurate field calculation in the vicinity of the feed, it is important to finely sample the computational volume in the vicinity of the probe-feed. 3.6.2 Near-to-Far Field Transformation Once the tangential electric field in the aperture has been determined, the field radiated by the aperture can be determined from the surface equivalence principle. In applying this principle, a suitable dyadic Green’s function which effectively transforms the tangential surface electric field to an exterior radiated magnetic field in the geometric optics region must be derived. The surface topology in the immediate vicinity of an aperture situated on an electrically large prolate spheroid may be regarded as locally planar. Hence, a planar approximation may be used to determine the exterior magnetic field radiated by a magnetic current distribution over the aperture in the geometrical optics region of an electrically large prolate spheroid. The geometrical optics region is of primary interest since the antennas under investigation in this dissertation radiate primarily in the geometrical optics region. From image theory, the transformation of a magnetic surface current source on a PEC plane to an exterior magnetic field is given by twice the free- space dyadic Green’s function =far = = VV e-ij Ge rr' =2G = 1+— 3.127 2( i ) o i k: JZIZ'R ( ) where r and r' are position vectors to the observation and source points, respectively, I is the dyadic unit vector (or idem factor) given by Law/9422', (3.128) 95 and the distance between the source and observation points is given by R = Ir — r '|. In the far zone, (3.119) may be expressed as M M e-Jko' . . . (xx'+ w'+ 22')——e"‘°"' (3.129) 27rr Since the far field is evaluated in spherical coordinates, a near field to far field transformation for a magnetic surface current distribution over a quasiplanar patch may be found be expressing the source vector in (3.121) in prolate spheroidal coordinates, while expressing the observation vector in spherical coordinates. Hence, the dyadic Green’s function which effectively transforms a surface magnetic current to an exterior geometrical optics far-zone magnetic field may be written as = far 6.. (19¢ 6'.n'.¢')=éé'Gf§' +6663" +6603 +6665? +60'G£"' (3.130) +7393 '02”? where each of the components are given by G 9,. = (a cos6sin 6'cos((o - 6) ') —b sin 6 'cos 6 ') e"’°' e jk0[bsin6'sin6cos(¢—¢')+acos6cos6'] ’2 x/asin’6'+bcos’6' 27” (3'131) 652,7. = _ (b COS 0 C05 6 ' COS(§0 - (0') + a Sin 6'sin 9 ') e-jkor ejk0[bsin6'sin6cos((v—¢')+acos6cos6'] (3.132) Jasin’6'+bcos’6' 27" 9,,1 . , e-jkor jk0[bsin6'sin 6cos(¢—¢')+acos6cos6'] GeZ =COS€Sln(¢—¢ )Ee (3.133) _ a Sin 6 'Sin(¢ '— ¢) e-jkor ejk0[bsin6'sin 6cos(¢-¢’)+acos6cos6‘] G”"— 62 ' 2 r 2 r x/asrn 6+bcos 6 27" (3.134) c — k G8". _ bCOSg'Sln((0 —(0') e I or ejk,,[bsin6'sin6cos(¢-qr')+acos6cos6'] e2 _ (3.135) Jasin’ 6'+bcos’ 6' 27" 96 W e-jkor G... =cos< E E ii" :2: E 5.10102: . - m 1‘4 ‘2:>:4D" 'A' X .3 £014 ' ‘ E7 5:03 54:50". 4 i 74.:- v - < - ‘ - l>:<>.'I [Deg.] Figure 4.28 Comparison of the azimuthal plane radiated field of a 2.5 x 2.5 cm patch antenna in the geometrical optics region of a 200 x 100 cm prolate spheroid with the radiated field of a patch residing on a planar surface. 138 Gain [dB] 50 25 I N 01 I U'I o I 1 40-090 -60 —3o 0 30 so 90 e [Deg.] Figure 4.29 Comparison of the azimuthal radiated field of a 2.5 x 2.5 cm patch antenna mounted on a 200 x 8 cm prolate spheroid at specified elevational angles with the azimuthal field of an identical patch antenna mounted on a circular cylinder with an 8 cm radius. 139 Gain [dB] 50 r I I . . cyL _ PS: e=eo° 25~ Ps:e=80° . - ..... PSI 9:700 0 -25 _5o_ _ -75- J _10 J 1 l 1 1 150 -60 -30 0 30 60 90 ¢ [Deg-l Figure 4.30 Comparison of the azimuthal radiated field of a 2.5 x 2.5 cm patch antenna mounted on a 20 x 8 cm prolate spheroid at specified elevational angles with the azimuthal field of an identical patch antenna mounted on a circular cylinder with an 8 cm radius. 140 Gain [dB] 50 . I I . . cyl. _ PSI 0:900 25- PS:e=80° - - ----- PSI 6:700 0 -25 ' -50 -75~ - _100 1 r l l L -90 —60 -30 0 30 60 90 ¢ [099-] Figure 4.31 Comparison of the azimuthal radiated field of a 2.5 x 2.5 cm patch antenna mounted on a 10 x 8 cm prolate spheroid at specified elevational angles with the azimuthal field of an identical patch antenna mounted on a circular cylinder with an 8 cm radius. 141 CHAPTER 5 EXPERIMENTAL RESULTS 5.1 Introduction In order to verify the FE-BI simulation results presented in the preceding chapter, the measured input impedance of a patch antenna mounted on a ground plane and on a prolate spheroid are presented in this chapter. For the case of a patch antenna radiating on a ground plane, the purpose of the experiment is to assess the accuracy of the resonant frequency and magnitude of the input impedance as predicted by the prolate spheroidal FE-BI routine in the planar limit (e.g. large axial and azimuthal radii of curvature). For the case of the patch antenna radiating on a prolate spheroid, the purpose of the experiment is to assess the effect of surface curvature variation on the input impedance of the patch at various elevational positions on the spheroid surface. The lack of published experimental data on the input impedance of patch antennas conformal to prolate spheroidal surfaces may be due to the considerable difficulty involved in constructing this type of configuration. The presentation of the experimental results is preceded by a discussion of the antenna fabrication and experimental setup. 5.2 Antenna Fabrication The fabrication of the patch antennas to be used in these experiments is discussed in this section. The dimensions of the first antenna to be considered are as follows: 3.0 x 3.0 cm patch within a 6.0 x 6.0 cm aperture. The face of the antenna is milled from GML 1100 copper clad laminated board with a thickness of 0.0236 cm. The laminated board consists of a layer of dielectric material with a permittivity 8:3.29— j0.0132 at 2.5 GHz sandwiched between two copper layers. The feed configuration consists of a female SMA 142 connector soldered to 3.2 cm of semi-rigid coaxial cable. The coax center conductor provides the probe feed for the antenna. As shown in Figure 5.1b, a 0.914 mm (0.0360 in) hole through which the center conductor of the coaxial cable connects to the patch is drilled 0.98 cm from the bottom edge and 1.48 cm from the lefi edge of the patch. To prevent a short circuit between the patch and back antenna surfaces, the center conductor is encased by a teflon tube to insulate it from the walls of the top and bottom copper layers of the laminated board before it is fed through the probe feed hole. The center conductor is soldered to the patch while the outer conductor is soldered to the metallic back surface of the antenna. The 4.0 x 3.0 cm patch is fabricated from the same material using the same procedure, except that the feed-through hole for the center conductor is located 0.52 cm from the bottom edge and 1.99 cm from the left edge of the patch as shown in Figure 5.1. The fabrication of the patch antenna to be mounted on a metal foil covered bowl, which simulates a PEC prolate spheroid, is described next. The maximum radius of the bowl is 14.74 cm and its height is 17.0 cm. The dimensions of the conformal patch antenna are as follows: 3.0 x 3.0 cm patch within a 6.0 x 6.0 cm aperture. Since the antenna must conform to the doubly curved surface of the bowl, it is fabricated from GML 1100 copper clad laminated board with a thickness of 0.014 cm in order to minimize buckling along the surface of the bowl. The thinness of the board necessitates a different fabrication technique than was used for the thicker patch antenna. In view of the thinness of the metallic layer, the face of the antenna is chemically etched from the GML 1100 board using a full-strength ferric chloride solution. In Figure 5.2 cross-sectional and top views detailing the construction of the patch antenna are provided. The probe feed 143 consists of an SMA connector soldered to 4.55 cm of semi-rigid coaxial cable. A 0.914 mm hole through which the center conductor is fed is drilled 1.01 cm from the bottom and 1.54 cm from the left edge of the patch. A caveat of constructing a patch antenna out of such thin board is that it is quite difficult to ensure electrical isolation across the dielectric when solder is applied near the probe feed hole to electrically bond the center and outer conductors to the patch and back surfaces, respectively, of the antenna. Hence, in order to ensure electrical isolation between the back and patch surfaces, the following technique is used. First, nonconductive epoxy (Stycast 2850FT) is used to structurally bond the coaxial outer and center conductors to the bottom and patch surfaces, respectively, and also to prevent metallic debris from entering the hole and shorting across the top and bottom surfaces. Next, to prevent a short circuit between the patch and back antenna surfaces, the center conductor is encased by a teflon tube to insulate it from the walls of the top and bottom copper layers of the laminated board before it is fed through the probe feed hole. Finally, instead of solder, a silver coating obtained from the evaporation of a colloidal silver solution is applied to furnish a low impedance electrical connection between the coax outer conductor and the bottom surface and between the center conductor and patch surface as shown in Figure 5.2. 144 Center conductor Dielectric 5 Solder Teflon tube substrate V Copper bottom surface ‘5— Outer conductor of semi-rigid coax .-.-.--m r.- .JLJ’" ) in terms of the Cartesian unit vectors (x ,y,z )is given by [5] H'/— ”7 ——nz-§cosgoi+/§L;%— ——4fsin(py+ ’52:] 772 (13.5) Substituting (B. 1) and (B2) into (8.5) 170 cosh Q . .. cosh Q srn 6 cos q) x + \lcosh2 Q — cos2 6 \lcosh2 Q — cos2 6 J cosh2 Q —1 . + 2 2 cos 6 z cosh Q - cos 6 é= sin6singoy Expressing coshQ in terms of a and c via (3.3) and substituting into (B6) 8 = a/c sin6cosgoi+ a/c \l(a/c)2—cosz6 \/(a/c)2 —cos2 6 2 +\/ (a/c) —l cos6i (a/c)2 — cos2 6 sin6sin¢y _ asin6cosq) i+ asin6sin¢2 .+J a2-—c2 c0362 ‘ 2 a 2 2 Jaz— —c2 cos 26 J02 —c2 cos 26 -6 cos 6 c0362 asin6cosq) i+ asin6sinq) .+\/ 02—02 2 a = y \/a2— —02 cos 26 \[a2 —02 cos 26 -czcosz6 1 z. a: \la2 sin26+b2 c0826 (asin6cosrpi+ asin6singoy +bcos6i) Following the same procedure for i] , beginning with [6] 2 I 2 —1 o A 1'.” A rysrngoy+ éz 2_”2 62‘772 substitute (B. 1) and (8.2) into (8.8) and simplifying 2 2 fi=\/ (a/c) -1 cos6cos¢i—J( (ale) _1 cos6sin¢y (a/ 6)2 -cos2 6 a/ c)2 —cos2 6 2rycosgofr— a/c . . + srn6z \[(a/c)2 -cos2 6 - J 02—62 cos6cos¢i\/ 02—02 cos6singoy a2 -c2 cos6 a2 —c2 cos2 6 171 (8.6) (13.7) (B.8) asin6 . 2 2 2 2 Ja —c cos 6 1 :> f] = \/ (12 sin2 6 +b2 cos2 6 (—bcos6cosgoi—bcos6sin(py +asin6z) (B.9) Finally, the azimuthally directed unit vector ([1 is the same in prolate spheroidal and spherical coordinates and is given by ¢=—sinq)i+cos¢y (B.10) Conversely, the Cartesian unit vectors may be expressed in terms of prolate spheroidal unit vectors. Beginning with [5] 1-772 . 52—1 . . . ézwzécowé- Wncown-smw (13.11) g: Following the same procedure as before, substitute (B.1) and (3.2) into (B.11) and simplify which results in . l—cos2 6 .. cosh2 Q—l . , .. X = cosh 0 cos — cos 6 cos — srn . 12 \lcosh2 Q — cos2 6 (05 cosh2 Q - cos2 0 $11 (P (P (B ) Expressing cosh!) in terms of a and cas before leads to asin6cos¢ \[a2 —c2 cos2 6 i: A 02—62 A A §+ 2 2 cos6cosgon—singoq) (3.13) a —c cos6 After further simplification by means of the Pythagorean relationship between a , b , and c the final expression for i is obtained i= 1 (asin6cosgoé—bcos6cosg0fi—singo(f)) (B.14) \/a2 sin2 6 +b2 cos2 6 In the same manner, 9 initially is given by [6] 172 _ 2 A 2— 1 :7” .fsin¢§— fiqsinwfi-t-coswé (B.15) which, upon the substitution of (B.1) and (B2), and simplification becomes 1 (asin6singoé—bcos6sin(pfi+cosgo(j)) (B.16) Jazsin26+b2cosz6 $1: The unit vector i is written initially as [6] . 2-1 . — 2 . za/finén/fién (3.17) which, by the same procedure, becomes 2: l (bcos6&+asin6fi) (B.18) J02 sin2 6 + b2 cos2 6 Summarizing, the prolate spheroidal unit vectors parameterized in terms of the spherical coordinates 6 and (p and expressed in terms of the Cartesion unit vectors are given by a: 1 (asin6cos¢i+asin6singpy+bcos62) (B.19) x/a2 sin2 6 +b2 cos2 6 f] = l (-—b cos6cos¢i—bcos6sin¢y + asin6i) (B20) J02 sin2 6 + b2 cos2 6 ¢=—sin¢i+cos¢y (B21) The Cartesian unit vectors, parameterized in terms of the spherical coordinates 6 and g0 and expressed in terms of the prolate spheroidal unit vectors, are given by i= 1 (asin6cosgoé—bcos6cos¢fi—sin(o(j)) (B22) J02 sin2 6 + b2 cos2 6 1 $1: \/a2 sin2 6 + b2 cos2 6 (asin6singoE—bcos6singofi+cosgo) (B23) 173 . 1 ~ . . z = \la2 sin2 0 +b2 cosz 6 (bcos6§ + asrn6 1]) (B24) For the case of a sphere where a = b l 12,: (asin6cosgoi+asin6singoy+bcos6i) \/a2 sin2 6 + b2 cos2 6 =sin6cosgoi+sin6sinq2y+cos6i=R (B25) {1 = 1 (-bcos6cos¢i—bcos6sin¢y+asin6i) x/a2 sin2 6 + b2 cos2 6 = —cos6cosrpi—cos6sin¢y+sin6i=—0 (3.26) «i = «i» (B27) The Cartesian unit vectors expressed in terms of the prolate spheroidal unit vectors now become 1% = 1 (asin6cosgoé—bcos6cosgpr’]—sin¢(j)) \laz sin2 6 +b2 cos2 6 =sin6cos¢R+cos6cos¢0—sin¢rjr (B28) 1 Ja2 sin2 6 + b2 cos2 6 =sin6sin¢R+cos6sinrp0+cosgoqi (B29) 9: (asin6singoa—bcos6sin¢fi+cos¢(jr) i: 1 (bcos6&+asin6f]) J02 sin2 6+b2 cos2 6 = cos6R—sin60 (B30) which express the Cartesian unit vectors in terms of the spherical coordinate unit vectors. 174 APPENDIX C DERIVATION OF THE EXACT EIGENFUNCTION SERIES FOR THE CIRCULAR CYLINDER DYADIC GREEN’S FUNCTION The second-kind electric dyadic Green’s function for the infinite perfectly conducting circular cylinder is derived most expediently from the free-space magnetic dyadic Green’s function (=lmo. Beginning with the dyadic form of the vector wave equation, expressed in terms of the free-space electric dyadic Green’s function, a relationship between the free-space electric dyadic Eeo and the free-space magnetic dyadic Green’s function Emo may be derived [10]. Beginning with VxVero(R|R')—kZEeo(R|R')=I5(R—R') ((3.1) where R and R' are three-dimensional position vectors to the field and source points, respectively. Employing the relationship Vero(R|R')=Emo(R|R') (02) results in VxEmo(R|R')=I6(R—R')+k23eo(RIR') (03) Since (=} m0 is piecewise continuous with a discontinuity at p = O , it may be decomposed into two components Ema =E‘LOU(p- p')+E=;.oU(p'- p) (C.4) where the unit step functions are defined by 1, p > p' U — ' = (p p ) {0’ p < p. 175 U(p'—p)={1’ ’0”, (CS) Taking the curl of (C4) and invoking an appropriate dyadic identity from [10] yields VXEmo = (ano)U(p—p')+VU(p—p')xao +(VXE;0)U(p'-p)+VU(p'—p)xE;o (C.6) From the theory of distributions, the following relationships can be derived [10,49] VU(p-p')=136(p-p') VU (p'-p)=-135(p-p') (07) Substituting (C.7) into (C.6) gives VxEmo =(VxE;ojU(p—p')+(VxE;o)U(p'—p) +f)5(p-p')x(E;o-E;zo) (C.8) The boundary condition on tangential magnetic fields across an interface may be expressed in dyadic form as 1343,20 -E—;o) = i.6(r-r') (09) where I. = I—fifi is the two-dimensional idem factor, I is the three-dimensional idem factor, r , and r' are position vectors from the origin to the field point and source point on the surface, respectively, and 11 is the outward unit normal vector to the interface. Evaluating (C9) in cylindrical coordinates yields fix(=(—;=;.o—E,—no)=(I—fifi)5(¢—¢')5(z—z') (C10) and rewriting (C8) in terms of (010) yields 176 Vx(=}mo =(VxE;o)U(p—p')+(VxE;o)U(p'—p) +(I—pfi)§(go—(o')6(z-z')6(p-p') (C.11) Rewriting (C3) in terms of (Cl 1) and solving for Geo (VXE;0)U(,0—p')+(VxE;o)U(p'—p)+(I—fifi)6((p—¢')5(z—z')§(p—p') = I6(R-R')+k2(=}eo :>[vxET...)u(p—p')+[vxE‘;o)z/(pv—p)+(i—aa)a(¢—¢')a(z—z')5(p—p') =I6(¢2-(o')6(z—z')6(p-p')+k28eo :> Eco(RIR')=%[(VxGLoJU(p—p')+(VxE;0)U(p'—p) _ mummy] (012) Thus, (C.12) expresses the free-space electric dyadic Green’s function Ego in terms of the free-space magnetic dyadic Green’s function (=}mo which satisfies the dyadic form of the wave equation vxvxao (RlR')—k25mo (R|R') = Vx[I§(R—R')] (C.13) At this point, the method of Ohm-Rayleigh is employed whereby the source term in (C.13) is expanded in the orthogonal basis of solenoidal vector wave functions M and N and manipulated according to the procedure in [10] resulting in 1Nm(k )M'—k( 2,)+M(2)(k )N' (—k), p> p' GM RR) jk °° °° c.14 ()l -;_M_£ 211;]ch {N(kz )M(2)r(_ kzz)+M(k )N(2).(_ wk) p

(x) 5.21;“ (x)(jr}kpe’""’e"‘=’ (C.15) x 177 M'(—kz) = {.2111 (x')fr'—J '(x')¢'}kpe-M'e-M (C.16) k N‘2’(k‘)= {ijm (x)1)—::—'—H(2)(x)q)+H(2)(x)kpz}—f—e"“’e"‘zz (C.17) N'(kz)= {—jka 'x( )p-——J (x) +J,,(xv)k,2'}%e-We-M (C.18) where x = k p p , x' = k p p’, and H f,” (x) is the second-kind Hankel function representing outgoing cylindrical waves. =+ Taking the curl of each component (i. e. Gmo or Gmo) and exploiting the symmetrical property of the vector eigenfunctions V x M = kN V x N = kM (C. 19) the following expressions are obtained map—13.11. Z k—, i,)[M<2>(k M ( —k ,2)+N‘2)(k )N ( —k )], p>p (C20) w ngr-w: k—2[M(kz—)M”"( km)+N‘2’(k)N‘2"( k)], pp' k’ M"’(kz)M“>'(—k.)+N'(-k.), p(k )M<2>(— k,)+b,,N<2>(k,)N(2"(—k,)] (C24) fl—w "=2” kp After enforcing the Neumann boundary condition on the cylindrical surface fix vx[M +a,,M(2) +N +bnN] : 0 (C25) p=a and defining y = x' = k pa the expansion coefficients are given by ’J" andb —'J" (7) = ___—___ _. —— C26 0 Substituting (C26) into (C24) and evaluating (C23), the electric dyadic Green’s function of the second kind for the perfectly conducting infinite circular cylinder is obtained = n — '21: z -jnHr(r ”(1') j" H512) '(X) c *1 G92: 2 2 e] 3 :[dk e 1 —("T—+— (2). mp (2”)n nyn (7) 72 k—O Hn (7) kzka§2"(x) H;2>'(x) nk, 2 H§2>(x) . ., nkzkai’Kx) —j 2 (2). 92+ (2) _ — T ("9+ 2 (2). (p2 kaHn (7) 7Hn (Y) k0}, an (7) yxkOHn (7) kk Hf,”(x) .. 1 k 2 H”) x .. + 2 (3.1—‘1" — —p (bf) 21' (C27) 70" H. (7) 7 k0 H" (7) where a=(p-(o', E=z-z',and x=kpp. 179 APPENDIX D FOCK FUNCTIONS In his investigations into the phenomenon of diffraction by convex bodies, F ock encountered certain recurring canonical integrals. These canonical integrals take the form of a contour integral whose integration path encloses the complex poles of Airy functions or their derivatives and are known as Fock functions [23]. Two varieties of Fock functions are encountered in this work: the on-surface and far—zone. Furthermore, these types of F ock functions occur in two forms: hard and soft. The hard Fock ftmctions arise from canonical problems where the Neumann boundary condition has been enforced, while the soft Fock functions arise in cases where the Dirichlet boundary condition has been enforced. The on-surface Fock fiinctions are given by — I'M/4 5.3/_2 00 W2 '(7) ~14r u (5) _ 8 7;? 1m ——ew2 (T) dr (D.1) _ l jfl/4 g °° W2 (T) —j§r V(§) _ 2 e x —[2x/3 W dz. (D.2) where u(§) is the soft type and v(§) is the hard type. The Fock-type Airy function of the second kind, denoted by, W2 (1) , and its derivative, W2 '(2') , are defined as w2(z') = 71; 1 emf/3612 (0.3) F: 1ze’z'z3/3dz (13.4) r2 . _L W2 (7)—J; 180 where the integration contour F2 in the complex 1 plane is depicted in Figure DJ. The relationship between the Fock-type Airy function and Miller-type Airy function, Ai(-) , is expressed as w2 (2') = 25e'1”/6Ai(—re’”/3) (D5) The asymptotic expansions of (DI) and (D2) for small arguments (if < 0.6) are given by [2] J— - ,, .5 5 -7: 11(5) ~ 1.0—351: I “53” +1553 JET/E‘s 1 “59/2 +... (D.6) Jr? , 7 7 . ~1.0—— /2+ '— 3+—— "’7‘ 9/2+... D.7 V03) 4 6 1606 512‘Ee 6 ( ) The asymptotic expansions of (DI) and (D2) for large arguments (6 >06) take the form of rapidly converging pole residue series and are given by [2] 11(5) ~ 2e’”/’J7;§3/2§;(rn)-le""" (D.8) v(§) ~ e-er/4 J7EE(2,=) : 2.061473 _1.3 g g s 0,5 : g(°)(5) =1.39937+Z6:E£mT)(K§)m m=l m° 10 e[m'(m)§l 0.5 < 5 s 4.0 : gm) (9”) = 2:; a'(m)Ai(m) —(0.8823—10.5094)5-)53/3] 5 > 4.0:g(°) (5) =1.8325e[ for g(”(5): g < -2.8: g‘” (5) = - 120(52 + j%_%§)e-M% —2.8 s 5 s 0.5: g(1)(5)= Z6:— m=l m! 0.5 < 5 s 4.0: g“) (g) : x: Aim) —(0.8823— 10.5094)¢- 1.53 /3 )1 .5 > 4.0: g“) (.5) : —1.8325(O.8823 — j0.5094 + 15214 for f‘o) (5): 182 (D.12) (D.13) (D.l4) (D.15) (D.16) (D17) (D.l8) (D19) 5 < —1 .1 : f‘°> (5) : 125(1— 0135 + g)?” (D20) -113 5 s 0.5: f<°> (5) : 0.77582 + e’11/3;3%)-(x5)”’ (D21) (0) 1 / 10 e[m(m)€] 22 < . = -M 3 0.5 <5 _ 4.0 .f (5) e :4“: Ai'(m) (D. ) 5 > 4.0: f”) (5) : 0.0 (D23) In (D.12)-(D23), K = 2’1””6 , the coefficients used in (D.12)-(D.19) are listed in Table D2, and the coefficients used in (D20)-(D23) are listed in Table D.3. 1m(1’) Figure D.1 Integration contour for w2(r). Im‘(r) Figure D2 Integration contour for the far-zone Fock functions. 183 Table D.1 Zeros of the Fock-type Airy function of the second kind 192(2) and of its derivative W2 (7). Note that 7,, = 1,, e””/3 and 2": I1 1 -j7r/3 T n e Table D2 Constants for gm) (5) and g(')(5). Aim 184 Table D.3 Constants for f (0) (5). 185 APPENDIX E BiCONJU GATE GRADIENT PSEUDOCODE An iterative solver approach is employed to solve the FE-BI system of equations. Iterative solvers are more efficient than direct solvers at solving the large sparse matrices that arise from PDE based techniques in that direct solvers employ matrix fill-in, whereas iterative solvers do not. Consequently, iterative solvers preserve the sparseness of the system. The Biconjugate Gradient (BiCG) iterative solver employing Jacob’s algorithm has proven to be readily applicable to the solution of sparse linear systems [46]. Initialize r0=b—Axo P0 = 1'0 Do until (res. S tol.) (r1341) ak = (P; 2 Apr) 1‘1+1 = x1: +akpk r1+1 = r1 —akApk . _ (rk+19rk+1> :61 — (17:33,) Pk+1 = r1+1 + flkpk End Do where x is the unknown solution vector for which an accurate estimate is to be determined, p is the search vector which points in the direction in the n-dimensional space that the algorithm must move in order to improve upon the solution estimate, and r is the residual vector. The subscript 0 denotes an initial guess which for x0 can be set 186 equal to {0} , subscripts k and k +1 denote the previous and current estimates, respectively. 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