030601-12-T AWE TECHNIQUES IN FREQUENCY DOMAIN ELECTROMAGNETICS Y. E. Erdemli C.J. Reddy J.L. Volakis National Aeronautics and Space Administration Langley Research Center Hampton, VA 23681-0001 January 1997 30601-12-T = RL-2431

PROJECT INFORMATION PROJECT TITLE: REPORT TITLE: U-M REPORT No.: CONTRACT START DATE: END DATE: DATE: SPONSOR: Grant No. U-M PRINCIPAL INVESTIGATOR: Simulation of Conformal Spiral Antennas on Composite Platforms AWE Techniques in Frequency Domain Electromagnetics 030601-12-T January 1993 December 1997 January 1998 Fred Beck NASA Langley Research Center M/S 490 Hampton, VA 23681 Phone: (804)864-1829 NAG 1-1478 John L. Volakis EECS Dept. University of Michigan 1301 Beal Ave Ann Arbor, MI 48109-2122 Phone: (734) 764-0500 FAX: (734) 747-2106 volakis@umich.edu http://www-personal.engin.umich.edu/~volakis/ Y. E. Erdemli (UM), C.J. Reddy (Hampton Univ.) J. Volakis(UM). CONTRIBUTORS TO THIS REPORT:

AWE Technique in Frequency Domain Electromagnetics Yunus E. Erdemli*, C.J.Reddy# and John L. Volakis* Abstract: This paper presents fast radar cross section (RCS) computations using the Asymptotic Waveform Evaluation (AWE) technique in conjunction with Method of Moments (MoM) and hybrid Finite Element (FEM/MoM) implementations. In its traditional form, AWE constructs a reduced-order model of a given linear system by Taylor series expansion with respect to specific values of the system parameters (frequency, angle, etc.). Thus, AWE permits the prediction of the frequency response from a few frequency calculations. In this paper we modify AWE to instead employ a rational function (Pade approximation) representation for the system parameters. Using a Pade rational function instead of a Taylor series, the accuracy of the analysis is increased to a wider frequency range. AWE is also extended to allow monostatic RCS pattern prediction using again a few pattern values, thus eliminating a need to resolve the system when an iterative solver is employed. To demonstrate these extensions of AWE, numerical examples of three-dimensional metallic bodies and cavity-backed apertures are considered. Y. E. Erdemli and J.L. Volakis are with Radiation Laboratory, University of Michigan, Ann Arbor MI, USA. C. J. Reddy is with Department of Electrical Engineering, Hampton University, Hampton VA 23668, USA. 1

1. Introduction Numerical methods such as the Method of Moments (MoM), Finite Element Method (FEM) and hybrid FEM/MoM techniques have gained wide acceptance due to their flexibility to model arbitrarily shaped objects involving complex materials [1,2]. In all these methods, a matrix system is formed and solved to obtain the desired system parameters using a direct or an iterative solver. For electrically large problems, the solution of the matrix system is computationally intensive and must be repeated for each frequency. Also, certain analyses and designs may require both temporal and frequency responses placing additional computational burden in generating these responses. To speed-up computations for large-scale simulations, iterative techniques which incorporate fast algorithms such as the Fast Multipole Method (FMM) and Adaptive Integral Method (AIM) [3,4] have been introduced. When incorporated into MoM or hybrid FEM/MoM codes, these algorithms have allowed the solution of practical problems [5-7]. However, the solution must be independently carried out for each excitation (incidence angle). Monostatic RCS calculations are therefore computationally intensive and therefore for iterative methods, asymptotic waveform evaluation (AWE) or the extrapolation methods become attractive for CPU reduction. In this paper, we investigate the application of AWE in conjunction with MoM and hybrid FEM/MoM techniques for rapid frequency response calculations. We also present a new implementation of AWE for rapid monostatic pattern fill calculations when an iterative solver is employed. AWE provides a reduced-order model of a linear system that has already been successfully used in VLSI and circuit analyses to approximate the transfer function 2

associated with circuit networks [8,9]. We note that a similar method was used by Burke et al [ 10,11 ] under the name of Model Based Parameter Estimation (MBPE). Both AWE and MBPE are identical in nature. The AWE technique has also been applied for FEM based problems by Gong et al [12] and Polstyanko et al [13] used AWE for the efficient analysis of dielectric waveguides. In this paper, a detailed analysis of AWE applied to hybrid frequency domain EM techniques is presented for the first time. We also extend AWE to include a rapid monostatic pattern fill in the case of large problems which invariably depend on iterative solvers. In the AWE technique, the unknown variable (electric field/current) is expanded in a Taylor series about the frequency or angle. These moments of the Taylor series are then mapped to a Pade approximation. The latter has a much larger range of convergence. Thus, it provides a larger frequency or angular region of extrapolation. This paper is organized as follows. In section 2, the AWE implementation for MoM and the combined FEM/MoM technique is presented for obtaining a frequency response. The AWE procedure for monostatic RCS fill using iterative techniques is also described in section 2. Numerical results for various examples are presented in section 3. 2. Formulation AWE is an extrapolation tapproach which provides a reduced-order model of a linear system. On the basis of AWE, a Taylor series expansion of the matrix system (MoM or FEM/MoM) is generated about a specific value of the system parameter (frequency, angle etc.). The Taylor coefficients or moments are then used to extract poles and residues of the system yielding a rational function (Pade approximation) of the system 3

parameter. Pade representations have a larger circle of convergence and can therefore provide a broader extrapolation since it includes poles as well as zeros of the response. This representation provides an extension of the region of convergence (RoC) of the power series, thus increasing the accuracy of the analysis to a wider range. 2.1 Asymptotic Waveform Evaluation (AWE) To illustrate the AWE method, let us consider a one-variable complex function f(z). The Taylor series off(z) at zo is 0o f(z) = c,n(z-z,), c( = /(z ), (la) n=O f (.n.(Z d. nf zo cn fnI (0), f(")(zO) - ) (lb) _! dnfZ) This expansion is the basic starting point for the Pade approximation given by L al(z-zo)' (z) P(L/M) = = (2) i, (2) I+Ebr ( lo )m + (z) m=l in which L and AM are the orders of the zero and pole expansions. The coefficients al and bm of the numerator (QL) and denominator (RM) polynomials, respectively, are found by enforcing equality of (1) and (2), viz., o a,(z - zo)' n (Z-Z)(zL+ il) (3) n=-O l+ b,( - Zo) m=l 4

Upon cross-multiplying and equating equal powers of z in equation (3), we get the linear system [14] CLA+ CL~ CL~... CL bm ~ cLM c c c Vb (c 2L-M+l L-M+2 L-M+3 *** LL M L+1 CL-M+2 CL-M+3 CL-M+4 *** CL+ 1 M -\ CL+2 CL-M+3 CL-M+4 CL-M+5 *** CL+2 Al - 2 = CL+3 (4) CL CL+L CL+2.. CL+M-1 bI ) L+M which is of order M. This system refers only to higher powers of z, in particular z with iL+ 1,L+2,...,L+M As a result, all equations involving a, are eliminated. The a, coefficients are found after solution of (4) for bm and by equating powers of z less than L+1, giving [14] min(L,M) a = c, a = C + o, a1 = ~ + bac, + b2ca,..., aL = C + ~ bjC. (5) 1=1 Typically, there exist optimum values for L and M so that a Pade representation PPt(LM) best approximatef(z) around zo. Below we first describe the implementation of AWE into MoM for frequency extrapolation and monostatic pattern fill. We then proceed with a similar implementation for hybrid FEM/MoM systems. 2.2 Method of Moments (MoM) MoM is a popular tool for accurate prediction of radar cross section (RCS) calculations. Its implementation in connection with the Electric Field Integral Equation (EFIE) involves a solution of the electric current surface density using a direct or some 5

iterative solver. With frequency (f) as the parameter of interest for extrapolation, the discretized EFIE [15] results in the linear system [Zmn ()]NxN { n (k; (, )}N 1 {Vm (k 0, O)}NL, (6a) Zmn (k) jk70 jTm. JTTnGk (R) dds - i~ JJ(V. TM) JJ(V'. Tn)Gk(R) dsds, (6b) F,(k;q,?O) = JT,,. Ei(k;q,0) ds, En(k;,O 0) pi e, (6c) r xx + yy + zz, R= - r' = /(x - )2 + (y - y)2 + (z - ), (6d) p. = x(cos0cosqbcosa - sin ( sin a) + Y(cos0sin o cosa- cosossin a)- zsin 0cosa, k = (-xsin cosq + sin Osin + cosO)k, k - 2 f t00, Gk(R) -4R (6f) where we have expanded the unknown current density as N J(r)= I (k 0, 0) Tn (r) (7) n=\ in which Tn(r) denotes the subsectional basis function. Among the other parameters, V(k; AC) is the excitation vector and Zmn(k) represents the weighted integral used for generating the matrix. In(k; 0, 0) is the unknown current and N is the number of unknowns. Also, Eic denotes the incident electric field; pi is the unit vector representing the direction of the incident electric field; (0, 0) is the incidence angle; a is the polarization angle; k represents the propagation direction of the incident wave; rf0 is the intrinsic impedance of the medium; r and r are the vectors defining the observation and source points, respectively, and R is the distance between these two locations. 6

Equation (6a) is typically solved at a specific frequency Jo (with wavenumber ko) either by a direct or an iterative method. The advantage of a direct method is that [Zmn(ko)] needs to be decomposed or inverted only once and subsequently the much less CPU intensive forward/backward substitutions (LU decomposition) are performed to obtain the solution for J(ko;, 0) for each excitation. However, if J or the RCS response is needed over a frequency band, the evaluation of [Zm(k)] and its decomposition must be carried out repeatedly at each frequency. Since this may be an O(N3) operation for each frequency point, use of frequency extrapolation techniques, such as AWE, are very attractive for CPU reduction. 2.2.a Frequency Extrapolation In this section we introduce AWE to evaluate In(k; ~, 0) at multiple frequency points. We begin by expanding {I,(k)} as oJK i o L S(- I (S)), k (k dI(. (8) {In (k)}NI Z{I n |l(k-ko0. - I=)(k~) In(k0) i(k) (8):0s! dks k(k s=0 sk=ko The "moments" {AM, } can be evaluated in closed form using the relation {n = [Z (k0)r (9a) Si dm) _ Vm(k) Z() dZ(k) mk1 q 0(9b dk( k-k q-(k (9b) kk=ko k=ko Here Z(q)mn(k) is the qth derivative of Zmn(k) with respect to k. This evaluation is unfortunately a lengthy process but requires much less time than a direct decomposition of [Zmn(k)]. Actually, the derivatives can be obtained by successive differentiation of the 7

previous derivatives of Zmn(k). It can be shown that an explicit and compact representation of Z()mnm(k) is Z(q) (k) = jk0 JT, JTn (-j/)l - Gk (R) dsds (10) -10 JT(V TM ) (V' Tn)(R)Y L(q - P(jkR)p GR)dsds Similarly, the expression for the st derivative of Vn(k) with respect to k is readily given by Vms) (k) - JjlT pi (jk. r)s ekrds.11) Substituting equations (10) and (11) in (9a), we generate the AWE moments. These moments are subsequently introduced into (8) to yield {In(k)} at frequency points around ko. Note that this process requires the decomposition of [Zmn(k)] only at the reference frequency ko. However, the accuracy of (8) quickly deteriorates as k moves away from ko. Thus, we instead consider an alternative expansion based on Pade approximants. The Pade approximation can be obtained by using the procedure described in section 2.1. Each entry of the vector {In(k) }Nx in (8) can be thought of as a complex function of k and can be expanded as I (k)- Ms (k-k)s. (12) s=( Using this notation, the Pade representation for each In(k) (n= 1, 2,..., N) is then given by L Ea'(k - ko) PI(L /M)= - =o?, (13a) l + E- bm (k - ko )m m=l oo yM (k ko)s = P, (L /M) + 0(kL+M+). (13b) s=O0 8

The unknown coefficients, an and bmn appearing in the Pade approximation satisfy the linear system yMn'-jbj =-M L + I < I < L +M I > M: 'b = O (14a) n nnn J'=1 a =M, a', yM'-jb,I<<L,<n<N (14b) j=1 and these are similar relations to those found in (4) and (5) but stated in a different manner. The Pade coefficients are determined from a solution of (14). We can write {In(k)}NXI =- Pn(L /M)}NI. (15) 2.2.b Pattern Fill For monostatic pattern evaluations it is necessary to carry out the matrix vector product [Zmn(k)]-l{Vm(l,06)} for each excitation {Vm(Q~, )}. AWE can again be employed for calculating the angular variations of a monostatic pattern using only a few pattern points, thus eliminating a need to carry out the backsubstitution repetitively. For iterative solvers, the proposed pattern fill procedure eliminates repetition of the iterative solutions altogether except for a few pattern points. To show how AWE can be applied, we begin by restating the system {In (,O)}NL [Zmn(kO)] N {V' m(SO)}NI (16) where we observe that [Zmn(ko)]' is not a function of the incidence or observation angles. Assume now that {In({(0)} has already been computed at a given direction (0, 6o). Thus, we can use a Taylor series to express In((,0) as 9

.(0, o) -.(oo O + (I - io)+ W ^ A ------ (^ -^ ^- (^~ —~ ---dI- o de o V 2! ao =eo 6^00 69= 60=69 1 a2I,(,0) 1 ()200 (a-aoX o-o)~_. (17)) 0 ( Oo:aG o)+. (17) 2! d02 o ( 2! 0) a0 = o 0=-o =00o Next upon making use of (16) we can eliminate the derivatives of In(Q, 0) to obtain the more explicit expression { n(0, 0)}- [Zmn (k0)Jj {V. (0) 0)} q! }n (0+.r() 0o) 8a) q=l c} (18a) p=1 s=l (P ~ s) m( (q) (~0 oor Vm ' (, 0) r e, V m(t,)() (0 v- orO), ((P0+,s)(0) 0 o) = (18b) O=00 0=00 Note that the differentiations have now been transformed to the excitation column since [Zmn(ko)] is by its nature independent of (0,0). For one-dimensional calculations, the dependence in 0 can be further eliminated giving {I,(0)}= [Znl(k) {Vm(0)}+*L{ () - ), v( ) d)).d (19)q= Iq - z e=9o As an example, let us consider the case of an incident plane wave with Ein() = y F(0), F(O) = eJkoxsin, (20) obtained from (6c) by setting a= 90~ and = 0~. Inserting this into the expression for Vm(0) in (6c) yields the excitation vector whose qth derivative is needed in (19). We have Vm(q) (O= F-(q) dsq ( 0) ds () - (21 a)( doq(0 Tn Fd (21a) 18=80 10

q-I F(q) () jkoX S(q-p) (0 )( (0 )Cq, (2 1b) p=o P (q -1)! q- (q1), j -, (21c) S(Oo)=sin00, S(q-p)(() Re{ q-p sin 00+Im{} q- cosO,. (21d) Substituting these derivatives into (19), {In{(G} can be expressed in terms of the AWE moments {M"n} as {n (0)}= [Zrnn (k0) ] Vn (0 )} 0 - ), } [Zrnn (k4 V) (' r. (22) q=1 Note that the moments {Mn } can now be trivially calculated and therefore one could increase the order of the expansion as needed to extend the validity of the approximation to a greater angular sector. Moreover, the Pade representation of {In(0)} can instead be used to ensure better convergence. In this case equations (12)-(15) are applicable provided k is replaced by ko and 0 by 9. 2.3 Hybrid Method: FEM/MoM Being a frequency domain analysis, the hybrid FEM/MoM technique may not be appealing for broadband frequency computations. To obtain, for example, the frequencydependent antenna parameters, it is necessary to execute the analysis at very fine frequency increments. Needless to mention, this is a computationally intensive procedure which may be avoided using AWE. In this section, we describe how AWE can be applied to calculate the parameters of a cavity-backed aperture antenna over a band of frequencies using the combined 11

FEM/MoM technique. Although the hybrid analysis presented in this paper is not restricted to any specific input feed structure, we only present the formulation for the coaxial line feed structure as shown in Figure 10. In accordance with the FEMI/MoM procedure [16] the cavity-backed antenna in Figure 10 is formulated using the finite element method for the fields within the cavity and the boundary integral for mesh truncation across the aperture. For the fields within the cavity, we enforce the "weak form" of the equations, giving JJ(VxT). Vx E d-k TEdv-(T Hp ds K Pr sap (23) ico JJT (nix Hnp) ds Sinp Here, T is the vector testing function, V is the cavity volume, Hap and Hip are the magnetic fields at the aperture surface Sap and at the input surface Sinp, respectively, and n represents the outward unit normal for each surface; &r and /Pr denote, respectively, the relative permittivity and permeability of the cavity filling. For a solution of E using (23), it is necessary to eliminate Hap and Hinp. For Hap, this is done by introducing the integral equation J(Tx n) Hap d=- 2k2 JTs. JJM G (R) ds' d Sap Sap ap ( (24a) - 2 J(V TJ) JJ(V' M) Gk(R)d ds, Sap Sap - jkR T= T xn, G (R) k = 22T f ror, M = E x (24b) proving a relation between E and H on the aperture. 12

To eliminate Hinp we assume that only the dominant TEM mode and itlls reflection exist at the coaxial cable aperture Stnp. That is, the electric field across Sp can be expressed as Ep - ee + e ref e jk z (25a) inp Mc ref ein = i 2 rT p=xcoso+jysin, p= x2+y, (25b) eref. F, e, Fin, rc e ' J sds-e -2 'k r (25c) 27 n r, r,- P ) jp where r2 and r1 are the outer and inner radius of the the coaxial line, respectively; erc is the relative permittivity of the coaxial line and Fo represents the reflection coefficient of the TEM mode at z-z1 (i.e. at S=:Snp). On the basis of waveguide theory, Hinp = (j/co)V x Einp and therefore the right hand side integral in (23) can be written as JjJ n dsJT.(iixH E)ds —jk r $ T( U Er 2 i ( 2z1c, n(/r/)j p) S L p p T (27) 2jk ~e- jk 2, (^ 6rc 2z j PJds Discretization of (23) using tetrahedrals in the volume and triangles on the aperture [17] in conjunction with Galerkin's method yields the system A(k) e(k) g(k), (28a) g(k) -c JT fi ds^. (28b) ArC 2zrln r I,, S p ) 13

where A(k) is a partly sparse, partly dense complex symmetric matrix, g(k) is the excitation vector, and e(k) is the unknown electric field coefficient vector. We can express A(k) as the sum of four matrices: A(k) = A, (k) + A (k) + A3 (k) + A (k), (29a) Al (k) = (V xT).(V xE)dv-Kk2 JT.Edv (29b) V /r V A (k) = -2k2 IJ Ts I M G(R)ds' d, (29c) Sap SSap A3 (k) = 2J (V. Ts) J(V'. M) Gk (R) ds' ds (29d) Sap Sap A4(k lnI (k) - T rc P I ds }jE. ds (29e) 4 2/,Tp,1ln(r/r,)[ p Si P{ j Clearly, the solution of (28) must be repeated for each frequency / Using the solved fields on Si,, (z=0O) we can compute the reflection Fo = i- E. Pds-1 (30) /2rIn (r2 ip) { C which can then be used for the input admittance evaluation. We have 1 —r Y-n =yO (31) where Y,, is the input impedance and Yo=l/Zo is the free space admittance. Typically, of interest is the evaluation of Y,, over a frequency band. AWE can therefore be employed to achieve this using only a few frequency points. We describe AWE for this application in the next section. 14

2.3.a Frequency Extrapolation To apply AWE to the system (28a), we proceed in a similar manner as done in section 2.2.a. The unknown field vector e(k) is again expanded as "O e(k) = Mn(k-ko)n (32a) n=0 where the moments Mn given by Mn = A-(k) (,) j (32b) where A-(ko) is the inverse of A(ko) and for q>O, A(q)(ko) is the qh derivative of the matrix with respect to k evaluated at ko. Similarly, g9n)(ko) is the nh derivative of g(k) with respect to k evaluated at ko. As before, 8qo is the Kronecker delta function defined in (9b). Explicit expressions for the derivatives of A(k) and g(k) are readily obtained and are given by A-(k) = dkq = A, q 0, A m)(k) = Am(k); (33a) A()(k) = -2keCffT *E dv, A2)(k) = A)(k)k, A(q(k) = 0, q 3; (33b) v V A') (k) =-2JJT rJM [2k- jRk 2 Gk(R) ds' ds, (3 3 c) Sap Sap AWq) (k) -2 T5. M ( (R) G (R) ds' ds, q >1 (33d) Sap Sp Qk(R) (q 2)! (-R)2 + 2qk(-R)- + k2(-R); (33e) 15

Ag) (k) = 2 (V Ts) JJ(V'.M)(-jR)q Gk (R) ds' ds q; (33f) Sap ap A( )(k) = A4(k)/k, q) (k)= O, q > 2; (33g) g (k)- (-j z 1-k - g(k), n>0. (33h) As done in 2.2.a, once the AWE moments are obtained, the electric field coefficients at frequencies around the expansion frequency can be calculated by using (32a). This expansion or its Pade equivalent can in turn be used to compute the frequency response of antenna parameters. 3. Numerical Results In this section we present some numerical results to demonstrate the accuracy and efficiency of the AWE implementation in connection with MoM and hybrid FEM/MoM techniques. First, as frequency extrapolation applications, scattering by three-dimensional metallic bodies (a PEC plate, a three-PEC plate, a PEC ring) and radiation by a circular patch are examined. We then consider scattering by a metallic plate as an example of pattern fill implementation. Note that all numerical computations for the results presented here were done on an SGI-Indigo2 machine (150 MHz, IP22 processor). 16

3.1 Frequency Extrapolation We present below four frequency extrapolation examples' one radiation and three scattering problems. For the latter, the incident plane wave is edge-on incidence and only the monostatic (backscattering) case is considered, that is, a=900, "cC= 0fsca=90~, and lnc i=scat r-o. A. PEC Plate Figure 1 illustrates a metallic plate with dimensions 1 cm x 1 cm. This square PEC plate was discretized using triangular patches, thus yielding 603 unknowns. As a result, the dense full matrix of the MoM system was 603x603 in size. The RCS frequency response was calculated at 30 GHz and this was used as the expansion frequency to generate Pade approximations with L=M=1,2,3,4 in the frequency range 25-35 GHz. As shown in Figure 2, for L=M > 1, the convergence of this expansion is easily achieved. In Figure 3, the RCS frequency response is plotted for the MoM solution, Taylor series expansion with a order of 6, and Pade approximation with L=M=3. As seen, the AWE solution is in good agreement with the exact solution. It is also observed that away from the expansion point, the deviation from the exact response becomes apparent for the Taylor series solution. The Pade approximation with L=M=4 calculated at 0.1 GHz frequency increments resulting in 100 frequency calculations. In fact, AWE can virtually generate the response at any fine increment with almost no cost. The AWE computations were carried out in only 28 minutes while the exact MoM solution for 11 frequencies took 59 minutes (Table 2). This comparison proves that AWE can result in saving at least one half of the CPU time required for the exact solution. 17

B. Three-PEC Plate In Figure 4, a configuration of three PEC plates is depicted with dimensions specified in terms of the wavelengths (Ai, i=1,2,3) at the corresponding frequencies (fi=3 GHz, 2=-5 GHz, f3=7 GHz). For this geometry, the number of unknowns was 448 and the system matrix was therefore 448x448 in size. Pade approximations with L==M1,2,.,7 were obtained in the frequency range 1-9 GHz by expanding the exact solution about 5 GHz. Figures 5 and 6 show the convergence of Pade expansion. As seen, after L=M=5, the Pade solution converges and it agrees with the exact solution very well. However, Taylor series solution diverges as seen in Figure 7. In contrast to the Pade solution, as the order of expansion increases, the Taylor solution severely deteriorates as going away from the expansion point. Although both Pade and Taylor expansions use the same set of moments, the Taylor series fails to extract the dominant poles and residues of the system, unlike the Pade approximation, and therefore any additional term in the series representation does not improve the convergence of the solution. Table 3 showvs the CPU comparison of AWE and exact solutions. As seen, the exact solution took 88 minutes for 17 frequency points. However, the Pade approximation with L=5 and Al=4 at 80 frequencies was calculated in 19 minutes, which is about one fifth of the MoM solution time. This result demonstrates noticeable performance of AWE in terms of CPU time savings. 18

C. PEC Ring A metallic ring with inner radius and thickness of 3 cm and 0.3 cm, respectively, is shown in Figure 8. The number of unknowns for this case was 425, yielding a dense full system matrix of 425x425 in size. Pade approximations with L=M=4 were generated using multiple expansion points in order to recover the RCS response in the frequency band of 1-9 GHz. Figure 9 shows the Pade solution obtained using four expansion frequencies: f=2 GHz,f2=4 GHz. f3=6 GHz, andf4=8 GHz. The exact solution at each expansion point was used to extrapolate the solution in the corresponding frequency band designated by vertical lines as shown in Figure 9. These expansions frequencies and frequency bands in which the Pade approximations bmatch with the exact solution as optimum as possible were determined using the basic idea of the complex frequency hopping (CFH) algorithm [18]. As seen, the AWE solution again agrees very well with the exact MoM solution. For the CPU timing comparison as shown in Table 4, the frequency band (2.3-5.3 GHz) with the expansion frequency 4 GHz was chosen. The AWE solution for 30 frequency points was obtained in 18 minutes. On the other hand, in order to recover RCS response properly in that band, we needed to run the MoM code for 10 frequencies in that band, resulting in the CPU time of 44 minutes. This result again shows computational efficiency of the AWE implementation. 19

D. Circular Microstrip Patch Antenna A cavity-backed circular microstrip antenna radiating into an infinite ground plane is shown in Figure 10. The input plane Si,, is placed at z=0 plane and the radiating aperture at z-0.16 cm. The discretization of the cavity volume resulted in 6,325 unknowns of which 469 were on the aperture. Thus, the dense submatrix was 469x469 in size. The frequency response of the input impedance is calculated at 6 GHz and this was used as the expansion frequency to construct a Pade approximation with L=3 and M=2. The frequency response from 5 GHz to 7 GHz is plotted in Figure 11. As seen, a very good agreement is obtained between the Pade approximation and the exact solution over the frequency range. The Pade expansion was calculated at 0.01 GHz frequency increments resulting in 200 AWE frequency calculations. These were carried out in only 44 minutes whereas direct calculation of the input impedance at 13 frequencies required 5 hours as shown in Table 4. This comparison clearly demonstrates the distinct advantage of AWE for generating broadband frequency responses. 3.2 Pattern Fill In this section we present an example of pattern fill application described in section 2.2.b. Figure 12 shows a square PEC plate (o0 x Ao) upon which a plane wave with edgeon incidence is impinged. In this case, the order of the system was 408. The monostatic scattering pattern at a fix 0=0 cut was calculated by Pade approximations with L=M=2 and L=M=1 using three expansion angles of incidence, C"-=150,450,700. As seen in 20

Figure 13, each expansion best approximates the exact RCS pattern in the corresponding angular sector. 4. Conclusions In this paper, we presented the implementation of the AWE technique in connection with the frequency domain EM methods, namely MoM and hybrid FEMAMoM. AWE was employed as a frequency or an angular extrapolator in this implementation for the purpose of generating broadband frequency or angular r~eonses, respectively, using only a few of exact solutions. The formulation of this analysis was first time introduced. We presented some representative numerical results for scattering and radiation by threedimensional configurations to demonstrate the computational efficiency as wvell as the accuracy of the implementation. As observed from these results, AWE can generate the broadband responses quite accurately with a considerable CPU time saving. As expected, the Pade approximation brings about much reliable representation of these characteristics due to its larger circle of RoC when compared to the Taylor series expansion. In particular, using AWE for pattern fill applications will have the advantage of CPU time saving provided an iterative solver is used. Acknowledgement The authors would like to thank Dr. Jian Gong for his invaluable discussions at the beginning steps of the formulation of the analysis. 21

1 cm jkx -ye, A L. 0 k^ x Figure 1: Metallic square plate; edge-on incidence. m "o C3 cn u P4> 35 Frequency, GHz Figure 2: PEC plate; convergence of Pade solution (L=M= 1,2,3,4); expansion frequency, fo=30 GHz. 22

-10, -15 vct P -20 -25 -30 -35 25 26 27 28 29 30 31 32 33 34 35 Frequency, GHz Figure 3: PEC plate; MoM vs. AWE: Pade(L=M=3), Taylor(6). Problem Method Matrix Fill LU Factor Total Time (secs;;(seecs (secs)_; PEC Plate MoM (11 freq. points) 3,333 187 3,520 Frequency band Pade Approximation 25-35 GHz (L=M=4) 1,672 17 1,689 (/o=30 GHz) (100 freq. points) Table 1: CPU timings of MoM and AWE solutions for the square PEC plate. 23

fl=3 GHz f2=5 GHz f3-7 GHz inc = y e 3\/2 i, "~I-R"'"E'~'~ ~~~ ~ i~-'~~'''*~~;"~~! 2/2.4 - o3............; 3 3/4.................... x2/4.,/2 Figure 4: Three-PEC plate; edge on incidence. -20 UC Vd -100 2 3 4 5 6 7 8 9 Frequency, GHz Figure 5: 3-PEC plate; convergence of Pade solution; expansion frequency, fo=-5 GHz. MoM vs. Pade (L M= 1,2,3,4). 24

v -6o./ = 0 5 GHz I — 60 J-J \ -70 - Pade(7/7) ' Pade(6/6)! -80 - --- Pade(5/5),..... Pade(4/4) -90 0 mom -so - o MoM |' -100, ' ' I f I I 1 2 3 4 5 6 7 8 9 Frequency, GHz Figure 6: 3-PEC plate; convergence of Pade solution; expansion frequency, fo-=5 GHz. MoM vs. Pade (L=M-4,5,6,7). Problem Method Matrix Fill LU Factor Total Time (secs) (secs) (sees) 3-PEC plate MoM (17 freq. points) 5,168 119 5,287 Frequency band Pade Approximation 1-9 GHz (L=5, M-4) 1,110 7 1,117 (o=5 GHz) (80 freq. points) Table 2: CPU timings of MoM and AWE solutions for the 3-PEC plate problem. 25

-100 1 2 3 4 5 6 7 8 9 Frequency, GHz Figure 7: 3-PEC plate; divergence of Taylor solution; expansion frequency, fo-5; MoM vs. Taylor (N=1,2,...,10). Table 3: CPU timings of MoM and AWE solutions for the PEC ring problem. 26

t= ( Figure 8; edge-on incidence. Figure 8: PEC ring; edge-on incidence. 10 5 0 -5 q -10 -15 -20 -25 -30 -35 -40 1 2 3 4 5 6 7 8 9 Frequency, GHz Figure 9: PEC ring; MoM vs. Pade (L=M=4 ) approximation using multiple expansion points, fi=2 GHz, f2=4 GHz, 3=6 GHz, f4=8 GHz 27

Problem Method Matrix Fill LU Factor Total Time (secs) (secs) (secs) Circular Hybrid FEM/MoM Microstrip (13 freq. points) 16,900 1,144 18,044 Patch Antenna Frequency band Pade Approximation 5-7 GHz (L-3, M=2) 2,535 88 2,623 (f=6 GHz) (200 freq. points) Table 4: CPU timings ofFE/MoM and antenna problem. AWE solutions for the circular microstrip patch 0.54cm ->. Substrate ( r = 2.4) (2cmX2cm) Circular Patch at z=O016cm (radiLu=0.84cm)... w.. I& —...........: e:.77.M.. I e. - -. - -. - -. - -..:...:= — m.... r- 1 0.16cm 50 oaal fee 50QL coaxial feed Figure 10: Cavity-backed circular microstrip patch antenna. 28

0 C) -C N nO3 2 1.5 1 0.5 0 -0.5 _1.. * * FE/BI Solution.. __ Pade Approximation........................................... ~_. ' Resistance. J- i -:................................................................................Rectance * *- * fo=6GHz' - I 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 Frequency(GHz) Figure 11: Normalized input impedance of a cavity-backed circular microstrip patch antenna. 29

m L E c(O)- j=,5:oXsinO III PEG P ae iiiPate^^^^^^^^^^^^^^^ Figure 12 ~ Metallic square plate. 15 --- —------------------— ^ ---- ------- 1 Pade(2/2) Pade(2/2) 5 0.. -5 -- - Pade I -20 - MoM I:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::...........................:::::::::::::... m...I.................................... - 5............................................... 1........................................................................................................~~~ ~ ~ ~ ~..................................................................... -.................................................................... -25 0 10 20 30 40 50 60 70 80 90 Incidence Angle, 0 (deg.) Figure 13: PEC plate; pattern fill using multiple expansion points, 31=15~, 02=45~, 03=70~. MoM vs.Pade (L=M=2 and L=M= 1). 30

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