376632-2-T E. Topsakal M. Carr J. L. Volakis Simulation of 3D Non-Metallic Scatterers with Circuit Analog Surfaces and Matrix Compression Based on the Adaptive Integral Method Sikorsky Aircraft Corp. 1201 South Ave Mail Stop B101A Bridgeport, CT. 06604 June 1999 376632-2-T = RL-2521

PROJECT INFORMATION PROJECT TITLE: Fast integral equation algorithms for penetrable blade and hub scattering in the VHF and UHF bands REPORT TITLE: Simulation of 3D Non-Metallic Scatterers with Circuit Analog Surfaces and Matrix Compression Based on the Adaptive Integral Method U-M REPORT No.: 376632-2-T CONTRACT START DATE: END DATE: DATE: SPONSOR: May May 1998 1999 June 1999 Final Annual Report Daniel C. Ross Sikorsky Aircraft Corp. Mail Stop B101A 1201 South Ave. Bridgeport, CT. 06604 Phone: (203) 384-7010 Fax: (203) 384-6701 Email: dross@sikorsky.com SPONSOR CONTRACT No.: U-M PRINCIPAL INVESTIGATOR: John L. Volakis EECS Dept. University of Michigan 1301 Beal Ave Ann Arbor, MI 48109-2122 Phone: (313) 764-0500 FAX: (313) 647-2106 volakis @umich.edu http://www-personal.engin.umich.edu/-volakis/ PROJECT PEOPLE: Erdem Topsakal, Michael Carr and John L. Volakis

TABLE OF CONTENTS INTRODUCTION...............................4 CHAPTER I GENERAL THEORY.....................5 1.1 Surface Integral Equations...................5 1.2 Formulation for Different Boundaries.............8 1.2.1 PEC Boundary.........................8 1.2.1.1 EFIE Formulation...................8 1.2.1.2 MFIE Formulation.................. 9 1.2.1.3 CFIE Formulation...................9 1.2.2 Resistive Boundary................... 10 1.2.3 Dielectric Boundary.....................11 1.2.3.1 EFIE Formulation..................11 1.2.3.2 MFIE Formulation.................. 11 1.2.3.3 CFIE Formulation..................12 1.2.3.4 PMCHW Formulation.................12 1.2.4 Impedance Boundary...................12 1.2.5 Circuit-Analog Boundary.................13 1.2.5.1 Reduction to simple sheet condition....... 14 1.2.5.2 Surface Integral Equation..............15 II COMPOSITE STRUCTURES... 2.1 Composite Problem 1.......... 2.2 Composite Problem 2.......... 2.3 Composite Problem 3.......... 2.4 Composite Problem 4.......... 2.5 Composite Problem 5.......... 2.6 Composite Problem 6.......... 2.7 Composite Problem 7.......... 2.8 Composite Problem 8.......... 2.9 Composite Problem 9.......... 2.10 Composite Problem 10........ 2.11 Composite Problem 11......... 2.12 Composite Problem 12......... 2.13 Composite Problem 13......... 2.14 Composite Problem 14.......... 16........... 16........... 17........... 18........... 20........... 21........... 23............25............ 25............26............27............ 28.............29.............30............ 31 III MoM FORMULATION.................. 33 3.1 Basis Functions......................... 33 3.2 Testing Procedure....................... 33 2

IV ADAPTIVE INTEGRAL METHOD......... 35 V NUMERICAL RESULTS.................. 38 BIBLIOGRAPHY................................ 55 3

INTRODUCTION Integral equation methods such as method of moments (MoM) [1] have been extensively applied to electromagnetic simulations. It has been shown that, due to excessive memory requirements and high computational complexity, MoM is a hard to use method for the analysis of electrically large problems. When iterative solvers are used, fast fourier transform(FFT) can be successivelly utilize in the solution of the integral equations.[2] The common goal of using FFT is to construct matrix-vector product algorithms lower then the O(N2) complexity of the conventional method of moments. N is the number of unknown volume, surface, line elements. More recently several "fast" algorithms have been developed to reduce the memory and the complexity. Among these Adaptive Integral Method (AIM)[3] utilizes the Toeplitz property of the Green's function kernel to reduce the storage requirements of MoM and also accelerate calculations by the use of FFT. AIM is a very general method and can handle two and three dimensional geometries. Although there are several publications on AIM simulations of 2-D geometries, as far as we know there is a few puplished results on 3-D applications of AIM in the open literature and all of them are related to PEC structures. This report describes three dimensional MoM/AIM simulation of penetrable scatterers consisting of non-metalic and circuit analog sheets implemented in the code CADRISA. Another important aspect of this study is the inclusion of Circuit Analog sheets that is not available any other existing codes. This capability is combined with modeling capabilities for resistive sheets, metallic, dielectric and impedance surfaces, and combination of all. First chapter is devoted to the general theory on surface integral equations. In chapter II integral equations are derived in case of composite geometries. Chapter III describes the basis functions and the testing procedure that we used through out our calculations. Chapter IV gives a brief detail on AIM, and finally in Chapter V we give some numerical results regarding to some test geometries and some composite structures. 4

CHAPTER I 1 GENERAL THEORY 1.1 Surface Integral Equations Consider the general electromagnetic scattering problem depicted in Fig. 1. We are interested in a field point r, located in a closed volume V or on a Figure 1: Geometry showing the various boundaries. regular surface Si(i = 1,2,..., n). Beginning with the vector Green's theorem, J [Q.(V' x V' x T)- T.(V' x V' x Q)dv' + fn S[(T x V' x Q) - (Q x V' x T)]ds' = 0 / ~i=5 (1) where, Q(r), T(r) E C2; r E V, E=l Si. Here the primes refer to the primed (integration) coordinates. To derive the integral equations for the electric and magnetic currents on the surfaces Si, i = 1,2,..., n we set T = E(r) (electric field), and Q = iG(r, r) with e-jkir-r'I G(r, ' ) =4rr' k =WV (2) Note that when r -+ r' in V, G, VG and V2G have singularities. To overcome this singularity problem, when integrating G or its derivatives we exclude an infinitesimal sphere of volume V6 -+ 0 and centered at r = r'. We 5

deal with the spherical volume Va of radius 6 -+ 0 separately by invoking the divergence theorem. Next we introduce Maxwell equations V x E+ jwoH = -M (3a) V x H -jweE = J (3b) V.(/pH) =Pm (3c) V.(cE) = p. (3d) After some straightforward vector manipulations, (1) can be written as; [jwpJG + M x V'G - (p/e)V'G]dv' + I [j, ^(n' x H)G - (n' x EE x 'G(n )V'G]ds = 0 (4) where we have assumed that V in equation (1) is linear, isotropic and homogeneous. When r' is located on one of the Si(i = 1, 2,..) surfaces, we proceed to extract the integral singularity noted earlier. Refering to Fig. 2, we rewrite the surface integrals as, R V'=V-V8 n S'= ZS,+S., i=1 Figure 2: Geometry for singularity extraction., n' x E x V'G= lim [ + +1 ] (5) +Ss -0 o Si-Si,: Sis 2 6 limr + ] =-E(r)[1 - -] (6) Here, P is the absolute value of the solid angle subtended by Sis at r in the limit as 6 -4 0. P = elset (7) 1. 0, elsewhere. 6

Based on (6), (4) can now be written as E(r) = -0(r) [ v U[jwJG + M x V'G - (p/e)V'G]dv' + f w, jw(n' x H)G - (n' x E) x V'G - (n'.E)V'Gds']. (8) On invoking duality we also obtain the corresponding integral equation for H as; H(r) = 0(r) v,[-jweMG + J x V'G + (m/ls)V'G]dv' + f jwc(n' x E)G + (n' x H) x V'G + (n'.H)V'Gds'. (9) Here f denotes the Cauchy Principal Value and 0(r) can be given as follows 0(r)={ =EiSi (10) (0, elsewhere. () In (8) and (9) since all the sources are contained in the volume V', this volume integral can be refered to as the 'source term'. If there are no sources in V', this integral will be zero. We will assume that the sources are far from the scatterer and represent the source integral by (E',Hi) which later be set a plane wave incident in the scatterer. Equation (8) and (9) can then be written as E(r) = @(r) [Ei(r)+J [-jwp(n' xH)G+(n'xE)xV'G+(n'.E)V'G]ds'] (11) H(r) = 0(r) [H (r) + [awe(n'xE)G+(n'xH)xV'G+(n'.H)V'G]ds']. (12) Next we rewrite these in terms of surface current densities (J, M) where J = n x H (13a) M = E x n (13b) and it follows that n.E = - V.(n x H) =.-V.J (14a) 3we 3wc and n.H =. V.(n x E) = V.M. (14b) jw/A jw/ 7

Sustituting (13a, b) and (14a, b) into (11) and (12) gives the integral representation E(r) = O(r)[Ei(r) - AJ + QM] (15a) 1 [H) = (r)[H(r) - J - AM] (15b) where J and M are unknown surface current densities. Here, A and Q are the integro-differantial operators given bye, Ar(r) = swur(r') + V(V'.r(r'))]G(r - r')ds' (16a) and Qr(r) = r(r) x VG(r - r')ds', (16b) and 77 = V/i/e is the characteristic impedance of the medium. 1.2 Formulation for Different Type of Boundary Conditions 1.2.1 PEC Boundary(Metallic Surfaces) Different types of surface integral formulations have been developed for these kind of surfaces. We give here the very well-known EFIE(Electric Field Integral Equation), MFIE(Magnetic Field Integral Equation) and CFIE (Combined Field Integral Equation) formulations. 1.2.1.1 EFIE Formulation Consider the PEC surface depicted in Fig. 3. DIN EPH, Region AB(espj) / SH E HHl =O Region,/ Figure 3: Geometry for PEC boundary. Using (15a) and (15b) for the fields outside the PEC surface, we can write, El(r) = 0(r)[Ei(r) - AJ + QM], r E B1 U aB12 (17a) 8

Hi(r) = 0(r)[H'(r) - J -AM], r E BlUB2 (17b) E2(r) = H2(r) = 0, r E B2. (17c) To construct the integral equation, we note that on the metallic surface, n x [E1 - E2] = n x E1 = 0 (18a) n x [H - H2] = n x H1 = J. (18b) These imply, M = 0 and thus (17a) become, El(r) = E(r) - AJ(r), r E B1 (19a) and n x E'(r) = n x AJ(r), r 9B12 (19b) The above are the so called EFIE whose solution gives the unknown current on the surface. 1.2.1.2 MFIE Formulation From (17b) and (18b) with M = 0, the magnetic field on the boundary B12 is given by J( = n x Hi(r) - n x QJ(r), r aB12 (20) 2 (20) is the well-known MFIE for PEC structures. 1.2.1.3 CFIE Formulation The solution of EFIE and MFIE formulation can return non physical results at internal resonant frequencies. In this case we resort to the CFIE to overcome this problem[4]. CFIE combines the EFIE and MFIE equations in a linear fashion as aEi(r) + r7(1 - a)(n x Hi(r)) = aA - rq(1 - a)(n x Q + -J(r)). (21) 2 The coefficient a is arbitrary and possibly complex. Typical a is set to 1/2 but other choices can be made. Basically the CFIE shifts the resonances of the MFIE and EFIE outside the range of interest. 9

1.2.2 Resistive Boundary Consider the Resistive Boundary surface depicted in Fig. 4. For a resistive Region Bj(sF) aB, (2 R < Region B2(b^4) y2 M, Figure 4: Geometry for Resistive boundary. boundary, the boundary condition on dB12 are n x [E - E2] = (22a) n x [Ei + E2] = 2Re7rln x n x [H1 - H2] (22b) n x E =-M1; n x H = J1 (22c) n x E2 = M2; n x H2 =-J2. (22d) where Re refers to the normalized surface resistivity in ohms per square. Fields in the regions B1 and B2 can be given as E1 = 81(r)[Ei - AlJi(r) + QlMl(r)] (23a) H1 = 01(r)[Hi - QlJl(r) - 2AlMl(r)] (23b) E2 = 02(r)[-A2J2(r) + f2M2(r)] (23c) H2 = 02(r)[-Q2J2(r) - -A2M2(r)]. (23d) 12 In (23a - d), indices 1 and 2 are corresponding to the fields and currents in regions B1(e - q,, -+ 1,/ ), and B2(e -- e2,/L -+ /2), respectively. Using (22a - d) in (23a - d) we obtain Ai + Re - Re i J1 Ei lA + A2 + R -Q2 Mi = Hi. (24) Re Q2 A2 +Re J2J L 0 If (ei = E2, /1 = /2), (24) reduses to, 10

[A + 2Re]J = Ei, J=J1 +J2 (25) which refers to the case of a resistive sheet boundary in free space. 1.2.3 Dilectric Boundary A homogenous penetrable body is depicted in Fig. 5. Here we will give Region B1(el,^) aB12 Figure 5: Geometry for Dielectric boundary. four different integral equation formulations for dielectric boundaries. For all cases, we will be using the formulas (23a-d) for the fields E1,2 and H1,2. 1.2.3.1 EFIE Formulation The problem given in Figure 5 can be seperated into two sub-problems each gives the field in one of the regions. Boundary conditions for the E field for the external and the internal problems are; n x E1 =-M, and (26a) (26b) n x E2 = M2 respectively. From (23a - d) and (26a, b) with [J2 = -J1, M2 = -M], we obtain the EFIE equations, n x Ei = n x A1J1 -n x QlM1 -2 M1 0 = n x A2J1 - n x 2M1 + —. 2 (27a) (27b) 1.2.3.2 MFIE Formulation Boundary conditions for the Magnetic field for the external and internal problems are; 11

n x Hi = J1 (28a) and n x H2 = -J2 (28b) respectively, on the surface of the dielectric body. Using (28) in (23a - d), we arrive to MFIE. 1 J1 n x Hi = n x iJ1 + -2n x A1M1 +- (29a) 1 J2 = n x Q2J1 + -2n x A2M1 — (29b) 2 2 1.2.3.3 CFIE Formulation Combining the EFIE and MFIE formulations as outlined in (21), yields a[Eq.(27a)] + rt(1 - a)n x [Eq.(29a)] = CFIE1 (30a) a[Eq.(27b)] + 71(1 - a)n x [Eq.(29b)] = CFIE2 (30b) These are the most general integral equations to be solved for J1 and M1. They can in general be combined with similar integral equations from other dielectric boundaries for simulating rather complex geometries. 1.2.3.4 PMCHW Formulation The PMCHW formulation is another formulation which is also robust at interior resonant frequencies. The method relies on implying the continuity condition on the surface of the dielectric body. That is, n x [E1 - E2] = 0 (31a) and n x [Hi -H2] = 0. (31b) Employing these conditions to the fields given in (23a-d), we obtain the PMCHW equations A1+A2 -1 -Q2 1 fJ1 E (32)E fL1 + Q2 Ai1 + A2 iM1 l H (2) 1.2.4 Impedance Boundary The impedance boundary condition is of the form, n x E1 = Zn x (n x Hi). (33) 12

Substituting (33) into (23a, b), we get the surface integral equation. fAl+" - { - Q iiE] ' I[ Al ] [' H i [M]H' (34) In (34), r7 = J7 represents the characteristic impedance of the surrounding medium. 1.2.5 Circuit-Analog Boundary Consider a thin(penetrable or impenetrable) multilayered sheet(Fig. 6). tangential components on the two sides of the sheet(consistent with duality and reciproity) can be written as E+1 H+ E, HFigure 6: Geometry for CA Sheets. nx [E+ (r)+E-(r)] = Renxnx [H+(r)-H-(r)]-Rcnx nx [E+(r)-E- (r)] (35a) nx [H+(r)+H-(r)] = Rmnxnx [E+(r)-E-(r)]l+Rcnxnx [H+(r)-H-(r)] (35b) Here E+(r) and H~(r) represent the fields on the upper and the lower surfaces of the sheet, respectively; Re and Rm are the electric and the magnetic resistivities and Rc is a cross coupling term. Rewriting (35a) and (35b) in matrix form, we obtain [inxHJ] [X2u X22][nxH+ (36) where Xij(i,j = 1, 2) are given by Xii = [1 + (1/2 - Rc)2/ReRm]/[1 - {(1/4 - RC2)/ReRm}] (37a) X12 = -l1/[Rm[l - {(1/4- R2)/ReRm}] (37b) X21 =-l /[Re[l - {(1/4 - Rc2)/ReRm}] (37b) 13

X22 = [1 + (1/2 + Rc)2/ReRm]/[1 - {(1/4 - R2)/ReRm}] (37d) in which Rn(Re = 27iRe, Rm = -(2/7/)Rm, RE = 2Rc),(n = e,m,c) stand for the normalized resistivities. Assuming that the thin multilayered sheet can be characterized by its reflection and transmission properties, R,,(n = e, m, c) can be determined by relating them to the reflection and transmission coefficients of the sheet. For general layered structure we need two reflection (ri) and one transmission coefficient (T). The corresponding reflection and transmission coefficients from (35a, b) are r ) (2Recos8 - 2Rm/cosO + 4Rc) ) (4(ReRm + RC2) + 1 + 2 * ReCosO + 2Rm/os0) (38a) (4(R + ReRm) 1) _ 1 (4(ReRm + R2) + 1 + 2ReCosO + 2Rm/cos) ( ) where 0 stands for the incident angle. At normal incidence (8 = 0) the above can be inverted to yield Re = [T2(0)-(l +r+(o))(1+r- (o))]/[r+(o)r-(o)-(1- T2(o))2]/2 (39a) Rm = [T2(0)-(1-r+ ())(1-r-(o))]/[r+(o)r-(o)-(1 - T2(0))2]/2 (39b) R, = [r-(o) - r+(o)/[r+(o)r- () - (1 - T2())2]/2. (39c) Thus, upon having the reflection/transmission coefficients we can extract the corresponding RR(n = e, m, c) values. 1.2.5.1 Reduction to simple sheet condition The transmission line model can be used to relate the above Rn(n = e,m,c) parameters to the resistivity (Zp) and conductivity (Z8) values for simple sheets. Equating the reflection and transmission coefficients for the circuit with those from (38a,b) yields series: -+ Re + oo, Rm = -Z/Zo, Re = 0 (40a) parallel: - Rm -+ oo, Re =-Zp/Zo, Rc =0 (40b) and the Xij(i,j = 1, 2) matrices reduce to Xser 0 1 = [1 Zs/Zo] (41a) and Xpar -ZO/Zp 1] (41b) When (41a, b) is used in (36), we conclude that a single parallel impedance circuit represents a resistive boundary condition, whereas a single series imprdance circuit represent a magnetically conductive boundary. 14

1.2.5.2 Surface integral equations To construct a surface integral equation let us refer to Fig.5. In this case interior and the exterior fields can be expressed with (23a - d). The relation in between the tangential field components and the surface currents can also be given with (22c, d). Substituting (23a - d) into (36) with the identification that E1 = E+, E2 = E-, H1 = H+, H2 = H- we obtain the integral equation; Al-2 -21 -2X 0 - J1 - Ei' A1 + 2XlI 0 -2 | Ml Hi 2(X12- XlX2) 0 A2- 17 -Q2 J2 0 01 (X21- XiX22) 2 + X22.M2-..0 (42) 15

2 CHAPTER II 2.1 Composite Structures Consider the geometry depicted in (Fig. 7). In this chapter we will derive the integral equations for different composite structures that are implemented in CADRISA..' *.., i. Si S1 ~ ~ S2 Figure 7: Geometry for general composite structures implemented in CADRISA. 2.1 Composite problem 1 In our first problem S1 and S2 are dielectric and S3 and S4 are the perfectly conducting surfaces. Electric and the magnetic fields can be written as follows in three different regions; El(r) = 01(r)[E' - Al(Ji + J4) + QlM1] 1 Hi(r) = 01(r)[H' - Q(Ji + J4) - -AilM1] E2(r) = 02(r)[-A2(J2 - J1) + Q2(M2 - M1)] H2(r) = 02(r)[-22(J2 - J1) - 1A2(M2 - Mi)] i72 E3(r) = 03 (r)[-A3(J3 - J2) + Q3M2] H3(r) = 03(r)[- Q3(J3 - J2) - A3M2] 173 Boundary conditions on the surfaces S1 - S4 can be given as OnS: n x[E - E2] = 0; nx [H1 - H2] = OnS2: n x [E2-Es] =O; n x [H2-H] =O (43a) (43b) (43c) (43d) (43e) (43f) (44a) (44b) 16

OnS3: n x Es = O(EFIE); n x Hs = J (MFIE) (44c) OnS4: n x E1 = 0 (EFIE); n x H1 = J4 (MFIE). (44d) Using (44a - d) in (43a - f), and after some straight forward manipulations we find the surface integral equation as ZI = V (45a) A1 + A2 -Q1 - Q2 -A2 Q2 0 A1 Q1 + Q2 A1 + -A2 -Q2 -0 AQ ' 1?2 12 -A2 2 A2 + A3 - 23 - 2 -A3 0 Z= -Q2 -2 A2 Q2 + Q3 AA2 + A3 -Q3 0 '12 172 '13 0 0 -A3 23 A3 0. A1 -Qi 0 0 0 A1 (45b) -Ei(S)M1 Hi(S1) J2 0 I= V=. (45c) M2 0 J3 0 J4..E (S4) 2.2 Composite problem 2 In the second problem S1 is a dielectric, S2 is a resistive and S3 and S4 are the perfectly conducting surfaces. In this case fields can be expressed as El(r) = 0(r)[E' - A1(J1 + J4) + f21M1] (46a) H1(r) = O(r)[H' - Qi1(Ji + J4) - A1M1] (46b) E2(r) = 0(r)[-A2(J2 - J1) + 22(M2 - M1)] (46c) H2(r) = 0(r)[-Q2(J2 - J1) - -A2(M2 - M)] (46d) E3(r) = 0(r)[-A3(J3 + J2) - Q3M2]. (46e) 17

H3(r) = 0(r)[-Q3(J3 + J22) + ^A3M2] (46f) + 3 Boundary conditi ons on the surfaces S - S4 can be given as OnSl:n x [E-E2] = O; n x [H-H2] = (47a) OnS2: nx[E2 - E] = O; nx [E2 + Es] = 2r7Rnxnx[H2 - Hs] (47b) OnS3: n x E = O(EFIE); n x Hs = J3(MFIE) (47c) OnS4: n x E1 = O(EFIE); n x H1 = J4(MFIE). (47d) Using (47a - d) in (46a - f), and after some straight forward manipulations we find the surface integral equation as A1 + A2 -QI - Q2 -A2 Q2 0 0 Ai Q1 + 22 A1 + IA2 - -2 A2 0 0 -A2 Q2 A2 + -Q2 2 0 0 Z= — 11 2 - 2 2 A2+ 2A3+ R -Q3 -Q3 0 7-12 7A3 2' — 0 3 A3 + 7t A3 0 0 O 0 3 A3 A3 A1 -Q 0 0 0 0 A1 (48a) E1 (Ei(S1) M1 Hi(S1) J2 0 1= M2 =. (48b) J22 0 J3 0 J4. LEi(S4) 2.3 Composite problem 3 In the third problem S1 is a dielectric, S2 is a CA-boundary and S3 and S4 are the perfectly conducting surfaces. 18

In this case fields can be written as EI(r) = 0(r)[E' - Al(J1 + J4) + QIM1] (49a) 1 Hi(r) = 0(r)[Hi - i(J1 + J4) - — A1M1] (49b) E2(r) = 0(r)[-A2(J2 - J1) + Q2(M2 - Ml)] (49c) H2(r) = 0(r)[-f2(J2 - J1) - 1A2(M2 - Ml)] (49d) '2 0(r)E3(r) = -A3(J3 + J22) + Q3M22 (49e) 1 0(r)H3(r) = -il3(J3 + J22) - — AzM. (49f) Boundary conditions on the surfaces Si - S4 can be given as OnS i'nx [E1-E2]=O; nx[HI-H2]=0 (50a) OnS2: E 1 Xii Xi2 E2 (50b) x H8nxH L X21 X22 L r/n x H2 OnS3: n x E3= O(EFIE); n x H3 = J3(MFIE) (50c) OnS4 ~ n x E1 = O(EFIE); n x H1 = J4(MFIE). (50d) Using (50a - d) in (49a - f), and after some straight forward manipulations we find the surface integral equation as A1 + A2 -hi - f12 -A2 n2 0 0 0 Ai ill + f2 4Al +-4A2 1 2 -A2 0 0 0 Qh IT -72Y 1727 -A2 12 A2 - nX2-2 -2 ~ 0 0 0 2X21 2X21 2X2 0 -2 12 0 0 o o ~~~(x1~2-')qX1-2- - A 0f12 0 A2 02 -A22 ) 0 + 0 0 0 0 0 A3 -f3 A3 0 Al -fi 0 0 0 0 0 Ai (51a) 19

* J - *E(S1) M1 H'(S1) J2 0 M2 0 I = V = (51b) J22 0 M22 0 J3 0 J4 J Ei (S4) 2.4 Composite problem 4 In the fourth problem S1 and S2 are dielectric, 53 is a PEC and S4 is an impedance surface. In this case fields can be written as El(r) = 0(r)[E' - Al(Ji + J4) + Ql(Ml + M4)] (52a) 1 Hl(r) = O(r)[H' - Ql(Ji + J4) + — Al(M1 + M4)] (52b) E2(r) =)8(r)[-A2(J2 - J1) + Q2(M2 - M1)] (52c) H2(r) = 8(r)[-Q2(J2 - 1) - A2(M2 - M)] (52d) 72 E3(r) = 0(r)[-A3(J3 - J2) + 13M2] (52e) H3(r) = 0(r)[-Q3(J3 - J2) - 1A3M2] (52f) 7}3 and the boundary conditions on the surfaces S1 - S4 can be given as OnS: n x [E1-E] = 0; n x [H1-H2] = (53a) OnS2: n x [E2-Es] = 0; n x [H-Ha]=0 (53b) OnS3: n x E = 0(EFIE); n x H = J3(MFIE) (53c) OnS4:n x E1 = Zn x n x H1. (53d) Using (53a - d) in (52a - f), and after some straight forward manipulations we find the surface integral equation as 20

Al +A2 -Q1 - 22 -A2 Q2 0 Al -Q, Q1 + -' '1Al+-A2 -Q2 -- A2 0 Q Al 'i2 '12 '71 -A2 Q2 A2+ A3 -52 - Q3 -A3 0 0 z= -Q2 -1 A2 Q2 + Q3 -Al+-A3 -Q3 0 0 '12 172 '13 0 0 -A3 Q3 A3 0 0 Al -% 0 0 0 Al+4 -Q -17Al 0 0 0 Q1 oA,+ (54af '11 rE'(S1) ml H'(S1) J2 0 I M2 V= 0. (54b) J3 0 J4 Ei(S4) LM4 J LHi(S4) 2.5 Composite problem 5 In the fifth problem S1 is a dielectric, S2 is a resistive 53 is a PEG and S4 is an impedance surface. In this case fields can be given as EI(r) = 0(r)[Et - Al(J1 + J4) + Q1(M1 + M4)] (56a) 1 Hi(r) = 0(r)[H5 - Q1(J1 + J4) - lAl(M1 + M4)] (56b) 77~ E2(r) = 0(r)[-A2(J2 - J1) + Q2(M2 - M1)] (56c) 1 H2(r) = 0(r)[-Q2(J2 - J1) - -A2(M2 - M1)] (56d) E3(r) = 0(r)[-A3(J3 + J22) - Q3M21 (56e) 21

H3(r) = 0(r)[-Q73(J3 + J22) + I-A3M2]. 773 Boundary conditions on the surfaces S, -5S4 can be given as (56f!) OnSj: n x[El -E2] =O; nx[Hi-H2] =O (57a) OnS2 n x[E2 - E3] = 0; nx[E2 + E3]= 2771Rnx n x[H2 - H3S] ('57b) OnS3: n x E.3 = 0(EFIE); n x H3 = J3(MFIE) ('57c) OnS4:-n x E = Znx n xHi. (57d) Using (57a - d) in (56a - f ), and after some straight forward manipulations we find the surface integral equation as -Al + A2 fZl + f12 -A2 0n 0 Ai -111 - f2 0 I = -A2 A2 + 171 R 2 2 0 0 0 Ji 1 1,Z2 - A2 + -.A3 + -i 172 173 21 R 0 0 0 0 0 EI(S4) LHI (S4) 0 0 2 A3 + 17-R A3 0 0 0 0 0 A13 A3 03 0 Ai 0 0 0 z A1 + -z (58a) 0 0 0 07 (58b) J22 J3 J4 *M4 - 22

2.6 Composite problem 6 In the sixth problem Si is a dielectric, S2 is a CABC S3 is a PEC and S4 is an Impedance surface. In this case fields can be as Ei(r) = 0(r)[Ei - Ai(Ji + J4) + Ql(Ml + M4)] (59a) 1 Hi(r) = 0(r)[H' - Ql(Jl + J4) - -AI(Mi + M4)] (59b) E2(r) = 0(r)[-A2(J2 - J1) + Q2(M2 - M1)] (59c) H2(r) = 0(r)[-Q2(J2 - J) - A2(M2 - M)] (59d) E3(r) = 0(r)[-A3(J3 + J22) + Q3M22] (59e) H3(r) = 0(r)[-Q3(J3 + J22) - -A3M22]. (59f) U3 Boundary conditions on the surfaces Si - S4 can be given as OnSl: n x [E - E2] = O; n x [H -H2] = (60a) 2r 3 11 Xi 2][ 1(0 OnS2: x H3 21. nX2 X22H2 (60b) OnS3: n x E3 = O(EFIE); n x Hs = J3(MFIE) (60c) OnS4: n x E1 = Zn x n x H1. (60d) Using (60a - d) in (59a - f), and after some straight forward manipulations we find the surface integral equation as -A BZ = (61a) C D A1 + A2 -Q1 - 02 -A2 Q2 Q1 +Q2 1Ai + yA2 -Q2 -1A2 A= -A2 Q2 A2 - X22 -2 (61b) -Q2 - A( 2 1+ 2nx2 ~0 0 (X12- XX22) 0 2 X21x 23

0 0 0 A1 -Q, 0 0 0 %,1A T)1 2X21 0 0 0 0 (61d) 1 0 0 0 0 27lX12 A2 - TX1 -Q2 A3 0 0 0 0 0 1 X21 - X11X22) 0 0 0 0 c = (61e) Al -Ql 0 0 Q3 1 A3 +X22 Q3 0 0 A3 -Q3 A3 0 0 z (61f) 0 0 0 A, +2 -Q% 0 0 0 '~ ~ Al+~ F, 2Z ES(S1) ml Hs(S1) J2 0 M2 0 = J22 V= 0 (61g) M22 0 J3 0 J4 Es (S4) L M4 J LHI (S4)j 24

2.7 Composite problem 7 In the second problem S1 is a dielectric, S3 and S4 are PEC surfaces. In this case fields can be written as El(r) = 0(r)[Ei - Al(J1 + J4) + QiM1] (62a) Hi(r) = 0(r)[H' - Ql (Ji + J4) - AlMl (62b) r?) 1 E2(r) = 0(r)[-A2(J3 - J1) - Q2M1] (62c) H2(r) = O(r)[-Q2(J3 - J1) + A2M1]. (62d) Boundary conditions on the surfaces S1 - S4 can be given as OnS: n x [E-E2] = O; n x [H-H2] = (63a) OnS3: n x E2 = O(EFIE); n x H2 = J3(MFIE) (63b) OnS4: n x E1 = O(EFIE); n x H1 = J4(MFIE). (63c) Using (63a - c) in (62a - d), and after some straight forward manipulations we find the surface integral equation as A1 + A2 - -Q f2 -A2 A Q1 + Q2 A1 + -~A2 -Q2 Q1 Z= (64a) -A2 Q2 A2 0 A1 -l 0 A1. J1 -E'(S1) M1 H'(S1) I= V=. (64b) J3 0 J4. Ei (54) 2.8 Composite problem 8 In the eight problem S1 is a dielectric, S3 is a PEC and S4 is an impedance surface. In this case fields can be written as El(r) = O(r)[E - Al(J1 + J4) + %l(M1 + M4)] (65a) 25

Hl(r) = 0(r)[H' - Q1(J1 + J4) -- M1] (65b) 17i E2(r) = 0(r)[-A2(J3 - J1) - 2M1] (65c) H2(r) = 0(r)[-2(J3 - J1) + -A2M1]. (65d) r2 712 Boundary conditions on the surfaces S1 - S4 can be given as OnS: n x [E-E2] = O; n x [H-H2] = (66a) OnS3: n x E2 = O(EFIE); n x H2 = J3(MFIE) (66b) OnS4:n x E1 = Zn x n x H1. (66c) Using (66a - c) in (65a - d), and after some straight forward manipulations we find the surface integral equation as A1 + A2 -Q1 - Q2 -A2 A1 -Q Q1 +Q2 A1 + 1iA2 -Q2 Q1 eAi Z= -A2 Q2 A2 0 0 (67a) A1 -i1 0 A1 + Z -nQ Q. Ainl O Q1, + 2 'J - Ei(S1)M1 H*(Si) I J V= =. (67b) J4 E'(S4) M4 L Hi(S4) 2.9 Composite problem 9 In the eight problem S2 is a resistive, S3 and 54 are PEC surfaces. In this case fields can be expressed as El(r) = 8(r)[E' - Al(J2 + J4) + fiM2] (68a) Hi(r) = O(r)[Hi - Ql(J2 + J4) - r-AM2] (68b) E2(r) = 0(r)[-A2(J3 + J22) - Q2M2] (68c) 26

H2(r) = 0(r)[-02(J3 + J22) + — Q2M2] (68d) 1/2 Boundary conditions on the surfaces S1 - S4 can be given as OnS2: nx[E - E2] = 0; nx[Ei + E2] = 2r1RRnxnx[H2 - Hs] (69a) OnS3: n x E2 = O(EFIE); n x H2 = J3(MFIE) (69b) OnS4: n x E1 =0 (EFIE); n x H1 = J4 (MFIE). (69c) Using (69a - c) in (68a - d), and after some straight forward manipulations we find the surface integral equation as A1 + Ti2R R 0 Al 1 A + A A2 + -2 -Q2 Ql Z= R 2 A2 + 1R A2 0 (70a) 0 Q2 A2 A2 0 A1 -i 0 0 A1 J2 ' Ei(S2) M2 H'(S2) = J22 V= 0. (70b) J3 0 L J4. Ei(S4) 2.10 Composite problem 10 In the eight problem S2 is a resistive, S3 is a PEC and S4 is an impedance surface. In this case fields can be written as El(r) = 0(r)[E - A1(J2 + J4) + Ql(M2 + M4)] (71a) Hi(r) = 9(r)[H' - Q1(J2 + J4) - A1(M2 + M4)] (71b) E2(r) = 0(r)[-A2(J3 + J22) - Q2M2] (71c) 1 H2(r) = 0(r)[-fZ2(J3 + J22) + — Q2M2]. (72d) (2 27

Boundary conditions on the surfaces S1 - S4 can be given as OnS2: nx[E- E2] = 0; nx[E1 +E2] = 2r77Rnxnx[H1 - H2] (73a) OnS3: n x E2 = O(EFIE); n x H2 = J3(MFIE) (73b) OnS4: n x E1 = Zn x n x H1. (73c) Using (73a - c) in (72a - d), and after some straight forward manipulations we find the surface integral equation as -A + 3iR _-_ a ~ A1 - 0 2 2 Qn 'Al+ A2+ -Q2 -Q2 Q1 + A m/ Q2 A2 + MR A2 0 0 Z= 0 Q2 A2 A2 0 0 Al -Qi 0 0 A1 + z -Q1 Q eA1 0 0 Q1 rA +2Z (74a) *J2 Ei(S2)M2 H(S2) J22 0 1= V= (74b) J3 0 J4 Ei(S4) M4. Hi(S4) 2.11 Composite problem 11 In the eight problem S2 is a CA boundary, S3 and S4 are PEC surfaces. In this case fields can be given as El(r) = 0(r)[E' - Al(J2 + J4) + QiM2] (75a) Hi(r) = 0(r)[Hi - Q1(J2 + J4) - -A1M2] (75b) E2(r) = 0(r)[-A2(J3 + J22) + f2M2] (75c) H2(r) = 0(r)[-Q2(J3 + J22) - -Q2M221] (75d) 28

Boundary conditions on the surfaces S1 - S4 can be given as [E2 1 [X11 Xir 1X7\ OnS2 vn(7ba) 2 LnxH2J lX21 X22 [7n x H(76a) OnS3: n x E2 = O(EFIE); n x H2 = J3(MFIE) (76b) OnS4: n x E1 = O(EFIE); n x H1 = J4(MFIE). (76c) Using (76a - c) in (75a - d), and after some straight forward manipulations we find the surface integral equation as A1 X22 -Q1 -l 0 0 A1 - 2X21 2X21 '1 1 +2 0 2 0 Q ~ 2nXla2 2- x — (X12 - X11X22) 0 A2-X1 - 2 A2 0 Z= o0 (X21- lX22) Q2 1A2 + 2 2 0 0 0 A2 -Q2 A2 0 A1 -Qi 0 0 0 A1 (77a) J2 'S (E(2) M2 Hi(S2) J22 0 1= V= (77b) M22 0 J3 0 J4..Ei(S4) 2.12 Composite problem 12 In the eight problem S2 is a CA boundary, 53 is a PEC and 54 is an Impedance surface. In this case fields can be written as El(r) = 0(r)[E' - A1(J2 + J4) + Ql(M2 + M4)] (78a) 1 Hl(r) = 0(r)[H' - flQ(J2 + J4) - 7Al(M2 + M41 (78b) E2(r) = 0(r)[-A2(J3 + J22) + S12M22] (78c) 29

1 H2(r) = 0(r)[-f2(J3 + J22) - 22M22]. (78d) Boundary conditions on the surfaces S1 - 54 can be given as OnS2: 1 2 I 11 1 (7a) 7'n x H2 X21 X22 InxH] (79a) OnS3: n x Ea = O(EFIE); n x H2 = J3(MFIE) (79b) OnS4: n x E1 = Zn x n x H1. (79c) Using (79a - c) in (78a - d), and after some straight forward manipulations we find the surface integral equation as A1- 1X22 2-X2 1 -(Xl2 -, - XX2) 2 \~ X21 -Qi 1 0Al+ X 0 - 17 2X21 0 0 1 -2r7Xl2 — 2 0 0 A2 A2 - 2X A1 Qi 0 0 0 z = ~2(X21 - X,2 ) Q2 1A2 + 22 Q2 Z= 0X12) 2 A2+ 2q12 0 0 A2 -Q2 A2 A1 -Qi 0 0 0 A Qi1 4A 0 0 0 (80a) J2 Ei(52) M2 Hi(S2) J22 0 M = 2. (80b) J3 0 J4 E'(S4) M4..Hi(S4) 2.13 Composite problem 13 In the thirteenth problem 53 is a PEC and 54 is an Impedance surface. In this case fields can be written as -Q1 -;F 0 0 0 -QI 7^1+ 2Z 30

0(r)Ei(r) = E' - A1(J3 + J4) + Q1M4 (81a) 0(r)Hl(r) = Hi - Ql(J3 + J4) - 2AIM4. (81b) Boundary conditions on the surfaces S1 - S4 can be given as OnS3: n x E1 = O(EFIE); n x H1 = J3(MFIE) (82a) OnS4: n x Ei = Zn x n x H1. (82b) Using (82a, b) in (81a, b), and after some straight forward manipulations we find the surface integral equation as A1 A1 -Q1 Z= A1 A1+Z -Qi (83a) Ql QQ ~A1 + 2Z 'J3 -Ei(S3) I= J4 V= Ei(S4). (83b) M4. LHi(S4) 2.14 Composite problem 14 In the thirteenth problem S3 is an impedance and S4 is a PEC surface. In this case fields can be written as 0(r)Ei(r) = E' - A(J3 + J4) + fi1M3 (84a) 0(r)Hl(r) = H' - Q1(J3 + J4) - AIM3. (84b) Boundary conditions on the surfaces S1 - S4 can be given as OnS3: n x E1 = Zn x n x H1 (85a) OnS4: n x E1 = O(EFIE); n x H1 = J4(MFIE). (85b) Using (85a, b) in (84a, b), and after some straight forward manipulations we find the surface integral equation as A1 + z -Qi A1 Z = Q1 -A1 + Q1 (86a) A1 -Q1 A1 31

J3 - Ei(S3) - I= M3 V= Hi(S3). (86b) J4 E (S4) 32

CHAPTER III 3 MoM FORMULATION 3.1 Basis Functions Assume that for each of the surfaces mentioned in the previous chapter have a triangular facet model. For each surface electric and magnetic currents are expanded in terms of the RWG(Rao-Wilton-Glisson)[ 5] N J(r) = E Infn(r) (87a) n=1 N M(r) =-ro E Knfn(r) (87b) n=l where & (r- r+ r E T+ fn(r)= (r- ), r (88) 2A- n 0, elsewhere As it can clearly be seen from the (Fig. 8). Two triangles forming the edge 'n' denoted by T+ and Tn, with vertex points, r+ and r-. The length of the 'nth' edge is In and the area of the triangles T+ and T, are A: and A;, respectively. The minus sign and the factor o70 in the magnetic current expansion are included so that the resulting matrix equations are symmetric. The factor of rj0 is required since the H-field equation is normalized by )70. 3.1 Testing Procedure The MoM solution for a general 3-D body based on the CFIE formulation is obtained in terms of five generalized Galerkin(matrix) operators. These operators correspond to five integral operators A, 0, n x A n x f and fo. f0 is the operator which results from testing the current directly. Once these operators are defined it is easy to go directly from the integral equation to the MoM matrix equation. Assume that a complex inner product between vector functions A and B on a surface S as follows: (A, B)s = A*.Bds (89) Js 33

n+ Figure 8: Triangles making up a basis function. where the * indicates the complex conjugate. In the Galerkin form of MoM solution, testing functions are choosen to be equal to the basis functions. The electric field integral equations are tested with fn and the magnetic field integral equations are tested with t0fn. 34

CHAPTER IV 4 ADAPTIVE INTEGRAL METHOD 4.1 General Theory The Adaptive Integral Method,one of that have been recently developed, is one of the most powerful integral-equation solvers. The main idea of the fast solutions methods is to construct matrix-vector multiplication algorithms characterized by a complexity that is lower than the O(N2)(N is the number of unknowns). In the AIM methodology this reduction is achieved by generating a regular grid surrounding the structure. Evaluation of the Green's function on this regular grid produces a matrix with the Toeplitz property allowing the FFT to be used for the matrix-vector product which has a complexity of NlogN. Consider an arbitrary, perfectly conducting surface(B) illuminated by an incident field Ei. The scattered field Es is given by Es = -jwA - V0 (90a) where A is the magnetic potential defined as AIr e-jkR A(r)= -- J.(r') R dS', (90b) and the scalar potential 0 is given by 1 / e-jkR (r) = 4xrel ps(r') R dS'. (90c) In these equations R = Jr - r', J, and ps denote the surface current and charge densities, respectively. The continuity equation V,.J. = -jwup (91) provides the relation between the two densities. For a numerical solution of the integral equation one should discretize the surface into small triangular patches and expand the unknown current J, using a suitable set of basis functions Pn,(r). We let N J(r) E Inn (r) (92) n=l 35

where In are unknown coefficients In(r) defined by M3 n(r) = ( - Xnq)6(Y - Ynq)(z - Znq)[Anq* + Anq + Aq] (93) q=l M is equal to the expansion order, and rnq are the points on the grid surrounding the nth edge. For the divergence of the basis function following definition can be made M3 d(r) = E 6(x - Xnq)6(Y - ynq)6(Z - znq)Ad. (94) q=l Enforcing the boundary condition to the integral equation, we get / EiPmdS = f jwAPmdS + l V mdS m = 1,... N (95) by invoking the vector identity / Ei'PmdS = / jwA'mdS - j VmdS m =1,..., N. (96) By using (90 - 94) in equation (96) we get N M3 M3 Vm = jwlm In ( '( + Ay pq + nm + pAmq)G(rnp, rmq) n=l p=l q=l N M3 M3 +- In E E A>pXdG(rnp, rmq) (97) EW-n=1 p=lq=l where rp and rmq are the locations of the grid nodes and G(rnp rmq) exp{-jkrnp -r (98) 47rlrnp- rmql If we rewrite the (97) in a matrix form, we get [V] = [Z][I] (99) with 4 [z] = [L()[G][L() (100) 1=1 and [L()] = vjuCi[Ax] (lOla) [L(2)] = v/- [XA] (101b) [L(3)] = J/jA[XZ] (lOlc) 36

[L(4)] = V jW 'd (101d) The two sets of basis functions iPn(r) and fn(r)(RWGbasis) can become equivalent, by imposing their moments up to order M -+ oo with respect to the midpoint of the nth edge, r00 r00 J f Mqlqas = / J (r) (x - Xa)q (y - ya)(z - za)q3dxdydz -00 J -00 J -00 M3 = (xnq - xa)91 (Ynq - ya)2 - ( Za)3 [AXq + AX-Y + Anq z] q=1 (102) I00 o 00 O0 Mqq2 = 1 f fn(r)(x - xa)(y - ya)2(z - za)q3dxdydz. (103) J -00 J -00 J -00 Similar moments can be described for the divergences as follows. 00= r00 00 q2q d - d(r) (x ) (y - )q2 (z - za)q3dxdydz M3 = (Xn - Xa,) (yn - y()1(zn, - 0)"A (104) q=l r00 r00 r00 Dqqq3= Vfn(r)(x- Z)q (y - a)q2- za)q3dxdydz (105) -o J-00 -00 Mn = Mn qM q2q3 Mq1q2q3 Dn D)n Dq q2q3 Dqlq2q3 Solution of the last two equations gives the A coefficients. (106) (107) 37

CHAPTER V NUMERICAL RESULTS As benchmark tests, we ran the following geometries with three different methods: * Method of Moments (MoM) using direct factorization (LU) * Method of Moments using an iterative solver * Adaptive Integral Method (AIM) using an iterative solver Each chart characterizes the results from each of these methods, and various information about the run (such as number of unknowns, iterations, etc) is supplied. The tests were run on a 400 MHz Pentium II computer with 512MB of RAM. Intel supplied the numerical library used for LU factorization and Fast Fourier Transforms. In each test, the excitation was a planar wave. In the first graph, we analyzed monostatic backscatter of the NASA 14" PEC Almond at 1 GHz. The view angles for this simulation are along the phi-direction. Note that the AIM method did not converge with this geometry-this result is not surprising, because it is well known that the EFIE method is not optimum for closed bodies. On the second graph, we ran the same almond geometry using the MFIE method. In this case, both iterative MoM and AIM converged to the answer given by the LU method. Closer results could have been achieved with a smaller stopping residual. The third graph represents a combination of the EFIE and MFIE techniques. The CFIE technique gives convergence to a more accurate iterative answer than straight MFIE. On the fourth, fifth, and sixth graphs we simulated the monostatic RCS of a PEC box to examine the effects of very sharp corners on the AIM algorithm. The view angles were the same as those used previously in the case of the almond. We had convergence and very good agreement for each of the EFIE, MFIE and CFIE methods. The seventh graph shows the bistatic RCS of a dielectric sphere using the PMCHW formulation. The incoming wave travels in the -z direction while the scattering was taken along the 0-axis with (=0. The three methods showed good agreement between each other as well as published results.

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BIBLIOGRAPHY [1] Harrington R.F., Field Computation by Moment Methods, Macmillan, New York, 1968. [2] Volakis J.L. et al., Finite Element Method for Electromagnetics [3] Bleszynski E., M. Bleszynski and T. Jaroszewics, "AIM: Adaptive Integral method compression algorithm for solving large-scale electromagnetic scattering and radiation problems", Radio Sci., 31, 1225- 1251, 1996 [4] Mautz J.R. and R. F. Harrington, "H-field, E-field and combined filed solutions for conducting bodies of revolution," Arch. Elek. Utertragung., vol. 32, pp. 157-164, 1978. [5] Rao S. M., D. R. Glisson and A.W. Wilton, " Electromagnetic scattering by surfaces of arbitrary shape", IEEE Trans. Antennas Propag., 30, 409-418, 1982. 38