375458-2-T FINITE ELEMENT-FAST INTEGRAL METHODS FOR SOLVING SINGLE AND PERIODIC ANTENNA ARRAY PROBLEMS T. Eibert, K. Sertel, D. Filipovich and J. Volakis Hewlett-Packard Company 1400 Foutaingrove Pkwy Santa Rosa, CA 375458-2-T = RL-2540

PROJECT INFORMATION PROJECT TITLE: REPORT TITLE: Algorithms for Phased Array Simulations Finite element-fast integral methods for solving single and periodic antenna array problems U-M REPORT No.: CONTRACT START DATE: END DATE: DATE: 375458-2-T July 1997 May 1999 February 1998 SPONSOR: David Wilson Hewlett-Packard EEsof Division M/S 2US-H 1400 Fountaingrove Pkwy Santa Rosa, CA 95403-1799 SPONSOR GRANT/CONTRACT No.: Round Table Agreement Dated 6/01/97 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) 747-2106 volakis @umich.edu http://www-personal.engin.umich.edu/-volakis/ Thomas Eibert, K. Sertel, D. Filipovich and John Volakis CONTRIBUTORS TO THIS REPORT:

Finite Element - Fast Integral Methods for Solving Single and Periodic Antenna Array Problems Thomas Eibert, Kubilay Sertel, Dejan S. Filipovich, John L. Volakis Radiation Laboratory Department of Electrical Engineering and Computer Science, The University of Michigan Ann Arbor, MI 48109-2212 email volakis@eecs.umich.edu February 28, 1998

Contents 1 Introduction 4 2 Theory 6 2.1 FE-BI Formulation............................................... 6 2.2 Fast Integral Methods............................................ 10 3 Adaptive Integral Method 11 3.1 Theory and Implementation........................... 11 3.2 Results: Single Antenna Problem........................ 14 3.3 Results: Periodic Array Antennas........................ 16 4 Fast Multiple Method 18 4.1 Theory and Implementation........................... 18 4.2 Results: Single Antenna Problem........................ 21 4.3 Results: Finite Periodic Arrays......................... 24 5 Conclusion and Future Work 25 2

Abstract The introduction of fast integral techniques into the finite elementboundary integral method is investigated in the report. The resulting finite element- fast integral (FE-FI) method permits a reduction of the overall computational requirements of the boundary integral down to O(NJ) or even down to 0(Nlog(N)). Coupled with usual 0(N) requirements of the finite element technique, the FE-FI can permit an overall 0(Nlog(N)) CPU and memory requirement without degradation accuracy or convergence as is the case with truncations based on the absorbing boundary conditions or artificial absorbers. Here, we consider the application of FE-FI methods for the planar and doubly curved periodic arrays. Both, the fast multiple method (FMM) and adaptive integral method (AIM) are considered. 3

1 Introduction Fast integral equation methods were introduced in computational electromagnetics in the early 1990s and have been shown effective in accelerating the computation of matrix-vector products in iterative solvers. The Adaptive Integral Method (AIM) [1] and the Fast Multipole Method (FMM) [2] belong to this class of fast integral solution techniques. Both AIM and FMM reduce the solution time and memory requirement of the moment method solutions. The application of hybrid finite element (FE) - fast integral (FI) methods to single antennas and infinite periodic antennas (or frequency selective surfaces) is very attractive because it provides full 3D modeling flexibility and exactness in mesh termination. Basically, the finite element method is used to model the dielectric section of the unit cell (or antenna), whereas the boundary integral provides a rigorous mesh truncation. This successful combination is unfortunately limited by the size of the problem, i.e. by the number of the unknowns casted from its discretization (dominant influence comes from the boundary integral part). To alleviate the CPU and memory botllnecks, acceleration and compression techniques, like AIM and FMM can be used. FMM achieves its CPU reduction by grouping the far-zone unknowns and interacting their weighted contributions. In the case of AIM, the CPU reduction is achieved by mapping the original moment method (MM) discretization onto a rectangular grid and exploiting the Toeplitz property of the Green's function on this grid. That is, the Fast Fourier Transform (FFT) is invoked to compute the matrix-vector products in the iterative solver. For an arbitrary three dimensional body, a three dimensional FFT is required. For planar radiators, the dimensionality of the FFT is reduced by one, thereby significantly accelerating the computation of the FFT. 4

In this report, we consider the implementation of AIM and FMM in connection with a finite element-boundary integral solutions for single and periodic antennas and structures. Although our formulation and discussion is on an antenna radiation, it can be easily extended to scattering. Chosen examples and geometries are both, flat and curved, and and thus allow us to obtain some general conclusions. The report is organized as follows: In section 2, we review some basic theoretical concepts for array modeling. In chapter 3, we present results based on the AIM implementation. Annular slot cavity backed antennas and infinite periodic array of strip dipoles are used as examples. The FMM theoretical concepts and implementations are examined in chapter 4. Examples computations for cavity backed antennas and arrays mounted on curved surfaces are given showing speed-up using the FMM. 5

2 Theory 2.1 FE-BI Formulation The FE-BI formulation for cavity backed antennas recessed in a ground plane is well known and is given in [3, 4, 5]. Bellow we present the corresponding formulation for periodic arrays. The differences are in Green's function, and in the necessity to apply periodic boundary conditions at the volume and surface boundaries to simulate array. The considered periodic array structure is illustrated in Fig. 1. For time harmonic electromagnetic fields (ewt time convention) the pertinent finite element functional is F(E Eaad) ( V x Ead) E)- kEEad+jkZo Ea' Jint] dv +jkoZo JJEad * (H x n) ds, (1) s where Ead is the solution of the adjoint field problem, Jint denotes an excitation current within the FE domain, S represents the planar surfaces on the top (and/or bottom) of the FE domain, and nF is the unit surface normal directed out of the FE domain. Also, ko and Zo are the free space wave number and wave impedance, respectively. Invoking the surface,Z' ---Ii -,, ---,/- — '...-, —,, Dyp I,, --- r-e'- - - r ---,T I' ' ' " I i /- - - - -_" - - - ' - - - - - Dxp Figure 1: Infinite periodic structure 6

equivalence theorem, the magnetic field in the surface S of (1) can be expressed as H=-2j [|g(rr')(E x t) ds + V g(r,r')V' (E x )ds +Hin, (2) where g(r,r') is the appropriate periodic Green's function and HinC is the incident wave. Assuming an array periodic in the xy-plane, with Dxp and Dyp as the periods in the x- and y-directions, respectively, the periodicity condition for E in a rectangular lattice is given by E(x + m Dxp, y + n Dyp) -= E(x, y) e-j3m Dxp e-j&,n Dyp (3) where A: = ko sin 90o cos 990, Ay = ko sin t9o sin po0 (4) and (Xo, po) are the chosen scan directions of the array. Using the periodicity condition and the boundary integral representation (2), the computational domain can be reduced to a single unit cell of the array. In the implementation of the hybrid FE-BI method, this requires the imposition of appropriate phase boundary conditions in the FE and in the BI portions of the method. Moreover, in (2) g(r, r') must be replaced by the Green's function for an infinite array of 6-sources. In our implementation, the infinite series representation of periodic Green's function is calculated using Ewald's transform which subdivides series into spatial and spectral domain components in a manner that optimal convergence is achieved. To construct a linear set of equations from (1) and (2), we must first tessellate the volume and introduce expansions for each of the tessellation elements. For this application, the chosen tessellation elements can be of constant depth but must be more adaptable for surface modeling. The edge-based prismatic elements presented in [4, 5] allow for this type of flexibility. 7

z 21 Z=Zc +AZ 3:9 7 8 Z=Z c 6 1=1,2,3 J=4,5,6 k=7,8,9 Figure 2: Right angled prism Using the same type of tessellation elements (see Figure 2), the field is expanded within the cavity volume as 9 Ee ZE;W = [W]T{Ee} (5) j=1 where [W]e = [{We}, {Wy}, {Wz}] and {El} = {Ee, E,..., E)T are specified explicitly in [5]. On the aperture, since the top and bottom faces of the prism are triangles, these are reduced to 3 Es(r) EES (r) [S]TJ{Es} (6) i=1 where [S]s = [S., Sy] and Si = 2A x (r- ri) (7) To generate a linear system for Ej, (5) and (6) are substituted into (1) and (2) and 8

Galerkin's method (setting T = W) is employed to yield the assembled system [ {EV} 0 [0] [ {E1'} f {b1} (8) [A] + = (8) {ES} J [0 [13] {ES} {b} In this system, {EV} denotes the field unknowns within the volume enclosed by So, whereas {ES} represents the corresponding unknowns on the boundary So. The excitation column {bv} are due to internal antenna sources and {bS} is associated with the incident field excitations (for scattering). 9

2.2 Fast Integral Methods The FE-BI system (8) is partly sparse and partly dense. More specifically [A] is sparse, whereas [B] is dense. Thus, although [B] is much smaller in rank than [A], it is usually responsible for most of the CPU and memory when an iterative algorithm is used for the solution of (8). To alleviate the CPU and memory requirements, fast integral methods have recently been introduced [1, 2, 9, 10] to perform the matrix-vector product [B]{Es} much faster and with much less memory. Two of these approaches are the adaptive integral method (AIM) [1], and the fast multipole method (FMM) [2, 10]. Both FMM and AIM reduce the CPU time and memory requirement from O(N2) down to O(NV) where a < 1.5. The main feature of AIM and FMM is the decomposition of the matrix as [B]= [B]near + [B]far (9) based on some threshold distance referred to as the near-zone radius. The matrix [3]near contains the interactions between elements separated less than the threshold distance, whereas [Bl]ar contains the remaining interactions. The elements of [L]near' are evaluated without approximation. However, the product [B]farEs is evaluated in an approximate manner leading to a much faster execution. Next, we describe how the evaluation of [B]fa{ES} is performed for each of the fast algorithms. 10

3 Adaptive Integral Method 3.1 Theory and Implementation The basic idea of AIM is to split [Z] (equivalent to [B] in (9) as [Z] = [znear] + [Zfar], (10) where [Znear] contains the elements of [Z] that are near the self-cell and correspondingly [Zfar] contains the remaining "far-zone" elements of [Z]. AIM reduces the CPU time and memory of the iterative solver by exploiting convolutional properties of the Green's function for the evaluation of the matrix-vector products associated with the mostly full matrix [Zfar] [1]. That is, the far-zone matrices are not explicitly generated and the matrix-vector products are performed in the discrete Fourier domain (DFT) utilizing appropriate 2D FFT algorithms [18]. However, the convolutional properties can only be exploited on a regular grid and therefore, the basis functions on the original and possibly irregular triangular mesh must first be mapped onto a regular grid. To do so, we introduce auxiliary basis functions on a regular rectangular grid which is coincident with the original triangular grid [19]. The auxiliary expansion for the magnetic surface currents is given by M N M ux [M Zx + M y] (x - xo - mAx) S(y - yo - nAy), (11) m=1l n=1l where (xo, Yo) is the lower left corner of the regular grid and Ax, Ay are the sample distances in the x and y directions, respectively. Since we work with a mixed potential integral equation (MPIE) formulation for the BI implementation, we must introduce an additional expansion for the divergence of the magnetic surface currents, M N V. Mu= E E qmn (x - xo - mAx) 6(y - yo - nAy) (12) m=l n=1l 11

In mapping the original basis functions onto the regular grid we relate each basis function to the auxiliary J-basis functions on the nine closest grid points of the regular grid. This is done by equating a sufficiently large number of the corresponding moments of basis functions from the two coincident grids. All moments can be evaluated analytically or numerically, and for each basis function of the original mesh, three linear algebraic systems of order 9 must be evaluated. This leads to the construction of the sparse mapping matrices [A], [A]y, and [A]q with n, rows and nine columns. The Z matrices for the rectangular grid can be expressed as [Z]IM- = G[A] A]+ [A]y[G][A]T + [A]q[G][A]Tq. (13) t total _flnear ryl/ar Here, [C] is a Toeplitz matrix whose elements stem from the periodic Green's function evaluated on the grid points of the regular grid. For the implementation of AIM it is assumed that [Z]ta is sufficiently accurate for the far zone elements. For the near zone interactions we keep original matrix elements. Therefore, we decompose the AIM matrices as [ztt= [Z]ina + [Z]a (14) lAIM AIM + LZ AIM (14) and when this is combined with (10), we can rewrite the original Z matrices as [Z]approx _ ([]near [Z]nAear) + [z]total (15) -Zj - — J I~I^\AIM) -— AIM, (15 ) With this representation of [Z], the near zone interactions are evaluated without compromise in accuracy. However, since the majority of [Z]approx consists of the Toeplitz kernel [G], the associated matrix-vector products can be performed using only 0(ns) memory and (9(nslogns) CPU time. In the final numerical implementation, a near-zone threshold is defined so that [Z]approx is sufficiently accurate representation of [Z]. The threshold distance is controlled by the quasi-static singularities of the Green's function. In the case of the 12

infinite periodic Green's function, we must also account for singularities associated with those elements which neighbor the boundaries of the unit cell since the latter are adjacent to the next periodic cell. For calculation of matrix-vector products in an iterative solver, [Z]fM is not computed explicitly. After mapping the actual source distributions onto the regular grid through A matrices, the pertinent matrix-vector products are performed in DFT domain using FFT algorithm for the corresponding transformation. After transformation of the results back to the spatial domain, the fields on the original mesh are obtained by reverse mapping between the auxiliary unknowns and the original grid unknowns. However, for a computation of [Z]Aee it is advantageous to collect contributions of the regular grid in the spatial domain before the Toeplitz Green's function is transformed into the DFT domain. 13

3.2 Results: Single Antenna Problem We analiyzed the planar cavity backed annular slot antenna with one, two, four and twelve elements. To increase the BI unknowns, we considered bistatic scattering from the complementary structures (annular strips) obtained by inverting metalic and dielectric part of the antenna aperture. ax Inner radius of each slot = 7.325 cm Cavity diameter = 49.4 cm Cavity depth = 1.5 cm Total Edges: 5529 Non-PEC: 983 Cavity filling E= 2 Surface Edges: 2110 (PEC) Slot width = 1.cm 496 (Slot) 248 (Non-Pec) Frequency = 1 GHz (Lambda = 30cm) Co-pol (FE-BI) 0 -. X... I --- X-pol (FE-BI) A Co-pol (FE-AIM).10 — v X-pol (FE-AIM) -20 wro-30 -A\ -40 -.50 -90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 - Observation angle (deg.) Figure 3: Geometry, parameters and radiation pattern (X = 900) for cavity backed 4-elements annular ring slot antenna 14

These scatterers may not have practical importance, but are nevertheless quite appropriate for the case studies of interest. The aperture surface was discretized in a nonuniform triangle mesh using SDRC I-DEAS. The radiation pattern, aperture discretization and parameters for a 4 element annular slot array are given in Fig. 3, with corresponding memory requirements and CPU time (per iteration) given in Fig. 4. Clearly, large memory savings are attained by AIM. We should also point out that the problem 12 annular strips with 5695 surface unknowns, required 360MBytes and could not be therefore executed on a typical workstation. AIM required only 95MBytes in addition to the CPU reduction. x 4 MEMORY REQUIREMENTS FOR BI AND AIM SOLUTION TIME PER ITERATION FOR BI AND AIM - — I i —...........-.-...... ----------------- --- -—. --- —-.9. -------------- --- -------- ---- \4.... \..... ---- --------- _ --- — ---------- 42.5 ----- - ------- - _ - - -I-_- --- - 4 -----— _ --- —-- __ ---- — _ —_ ----- ---- 3 ------- I --------- 2-_ —_ --— __ ---_. 3 ------- ' --- -, —, ---,-' --- —-- ------------- ---.5 -------------- ------------, ----- " I 2.- ---- — 1 - -_11 --- - 3.5. --- —- ---- -. ---0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 NUMBER OF THE SURFACE UNKNOWNS NUMBER OF THE SURFACE UNKNOWNS (a) (b) Figure 4: (a) Comparison of the memory requirements, (b) Comparison of the CPU time consumed per one iteration., / 00,00.',0,00 I0,0 /::. '~030 00 00 60,UBE /,,H JUFC,NNON ',BE,F jH /, RAC J' 'T — t)...1 -,.../~....~... Figure~t —'...-.-4......~.....no temmr eqieet,() oprsnofteCUt consued pr on.......... 15

3.3 Results: Periodic Array Antennas To show the speed improvements and storage reduction due to AIM, we consider a relatively simple strip dipole frequency selective surface (FSS) as shown in Fig. 5 (see also [17]). We calculated the reflection coefficient for three different FE meshes as described in Table 1 using triangular surface meshes for discretizing the planar surface of the unit cell Irightangled prisms were used in the volume). The BI was applied on the top and bottom of the cell for mesh truncation, resulting in two fully populated BI submatrices. 1.0 O-O — FE/BI 20x20.o -0-. - FE/BI 32x32 U -6-A- FE/BI 44x44 ~ x- MOM 9x18 0.6 -— MOM 3x8 0.6 0.4 0.2 0.0 12 14 16 18 20 22 24 Frequency (GHz) Figure 5: Resonance curves for strip dipole FSS Since BI matrices were identical, only one matrix was computed. Table 1 shows model parameters and the size of the system matrices with and without AIM for BI matrix computations. The number of matrix elements, (columns 5 and 6 of this table), comprise both, the top and bottom BI matrices. As noted earlier, the AIM implementation require that only near-elements of the BI matrices be calculated explicitly. This results in memory savings of up to a factor of 15 for model 3 and these savings will increase further for surface meshes with a larger number of unknowns. The CPU times for different models are summarized 16

in Table 2. The FE matrix fill time is, of course, the same for AIM and the conventional implementation of the hybrid method. Model Surf. mesh Surf. unk. Vol. unk. Matrix elements AIM Conv. 1 20x20 2*1240 7435 588 079 3 000 000 2 32x32 2*3136 35432 2 183 120 20 000 000 3 44x44 2*5896 67000 4 672 610 71 000 000 Table 1: Unknowns and storage for different discretization models Mod. FE fill BI fill Solver Iterations Total AIM Conv. AIM Conv. ] AIM Conv. AIM jConv. 1 3 736 1 400 228 166 227 234 967 1 569 2 38 1 930 7 330 1 081 1 446 415 364 3 049 8 814 3 70 3821 5073 676 81964 Table 2: CPU times (in sec) for the different discretization models For the investigated problem the largest CPU savings were obtained due to the reduced BI matrix fill time. For small BI subsystems (as is the case with modell),AIM did not result in CPU speed-ups. However, for larger BI subsystems, the lower complexity of AIM resulted in considerable CPU reduction (see model 2). For model 3, we performed only calculations based on AIM since the storage requirements were 850 MByte using traditional implementation. AIM reduced storage down to 70 MBytes. 17

4 Fast Multiple Method 4.1 Theory and Implementation The FMM [2, 9, 10] is based on two elementary identities. One of them is the expansion [8] of the scalar Green's function appearing in moment matrix elements as eiklr+dl oo Ird = ik (-1)1(21 + 1)j1(kd)hl) (kr)Pj(d. r). (16) Ir +dl 1=0 Here ji is the spherical Bessel function, h(1) is the spherical Hankel function of the first kind, PI is the Legendre polynomial, and d < r is the condition for the valid of the expansion. In the FMM formulations, where the source point is denoted by x' and the observation point by x, r will be chosen to be close to x - x' so that d will be small as depicted in Fig. 6. The second identity is the expansion of the product jlP1 appearing in Eq. (16) as a sum of propagating plane waves (spectral integral): 47ril'j(kd)P (d~ r) = J d2keiki(k r). (17) Using this identity, the expansion (16) can be rewritten as eklr+d ik = d2 ked(21 + 1)h1) (kr)P(k, (18) Jr + d[ 47r 1=o where the orders of summation and integration are interchanged. The speed-up of FMM is derived from the observation that the sum L TL(kr, k. ) = (21 + l)hll)(kr)P (k r) (19) 1=0 is independent of kd and can thus be computed for various values of kr which can be reused in the computation of the Green's function. The number of terms, L+1, kept for approximating the sum depends on the maximum value of kd, ard the desired accuracy. 18

source field center rn lm' r center r center m Figure 6: The geometry constructions used in FMM formulations, illustrating the relation between source point, field point and the group centers. Noting that the direct path from the source to the field point can be decomposed into three parts (see Fig. 6) where rji = rm + rmm' - rim', (20) the Green's function can be rewritten as 47rG(rj, ri) [- -2VV'] rji d2k [ - 2 v'] eik'(rm-rm)TL(krmm,, k* rmm') = J d2k [I- kk] eik(rim -- )TL(krmm,, k rmm'). (21) The matrix entries can then be approximated as Bij = 2 dsSi(r). ds' [S,(r') + '. (r')] R ik 2 J d2kVfmj(k) TL(krmm, k rmm')V, i(k), (22) with Vsmi(k) = dseikr' [I-] Si(rim), Vfmj(kc) = dseikr" [I- kk Sj(rjm) (23) 19

in which S, and Sj denote the expansion and testing functions for the unknown magnetic surface current density. For out computations, we used the well known RWG [6] basis functions. Note also that * denotes complex conjugation. To realize the FMM speed-up the computational domain is divided into M groups. The total memory storage needed is O(N2/M) + O(KN) + O(KLM2) where K is the number of wave vector directions used in the numerical evaluation of the outermost integral in Eq. (22). Using the proportionalities K oc L2, D2 oc N/M, and L oc D, this expression can be simplified to C1 (N2 /M) + C2(NM N/M), where Ci and C2 are machine (and implementation) dependent constants. The coefficient C2 is actually quite small compared to C\ and thus the memory is dominated by the O(N2/M) term. The CPU requirement of this FMM implementation is O(NM) + O(N2/M)[9, 10]. This can be minimized by choosing M = W and the result is an O(N1'5) algorithm. The memory required for the FMM also becomes O(N1'5). In practice, both the operation cost and the memory requirement of FMM is less than those of standard MoM formulation for problem sizes larger than 1000, making the FMM more suitable for the solution of large problems. 20

4.2 Results: Single Antenna Problem One concern with the approximations introduced by FMM is accuracy for antenna simulations. Thus for validation we first apply FMM for the analysis of a 3.5cm.x3.5cm. cavitybacked patch mounted on a cylindrical surface of radius 14.95cm. The patch is placed on the aperture of a cavity 0.3175cm. deep filled with a dielectric layer having Er=2.32. Fig. 7 shows the layout of the curved antenna obtained by wrapping the flat antenna on the cylindrical surface and corresponds to the one considered in [7]. As explained in [7], two different radiation modes (axial and circumferential) exist depending on the feed location. The radiation patterns for the axial and circumferential excitation modes are shown in Fig. 8(a). As expected, the FMM calculations are in agreement with measurements in the broadside region. Toward the shadow directions (around 0=-90o and 0=900), the agreement deteriorates since we used the half-space Green's function where in [7] the exact cylindrical Green's function was employed. In the region 60 < 0 < -60 the measured and calculated patterns agree to within 1 dB. The input impedance dependence on antennas curvature is plotted in Fig. 8(b) for the circumferentially polarized patch. The results are again in good agreement with those presented in [7] and the resonant frequency can be accurately predicted since it is a more local phenomenon. Figures 9(a) and (b) show the comparison of the timings for the conventional FE-BI method with and without FMM. The speed up of FMM is observed for all problem sizes of practical interest. Fig. 9(a) depicts the trend for the total solution time and Fig. 9(b) shows the time consumed per iteration in the iterative solver. In essence, Fig. 9(a) includes the matrix fill time, where Fig. 9(b) does not. Clearly, for small number of BI unknowns (less then 2000), the CPU time speed-up is primarily due to the faster matrix fill. We can conclude that the FMM implementation is both robust and requires much 21

- \ curved e ' I < \ platform \ r-h / conformal coaxial feed patch antenna \ location Figure 7: Microstrip patch antenna on a curved platform. less resources for large scale implementations. We are currently investigating more accurate methods applicable to arbitrarily curved antenna geometries. 22

()Ir t li Ox c -5 m 2-10 co z -15.N o -20 z -25 0.'. 0:0 v: 0 / *1....;/...... -- Compu;I - - Compu -- Compu. 0 Kempe ~,-I * Kempe 0 Measur I.... Measul - >-A -......... ~ *0~, ted E-Plane ted H-Plane \ et.al.[7] I et.al.[7] red [Sohtell] red [Sohtell] \: * ' \ " o o \........ -. \: \.......\ -30 -80 -60 -40 -20 0 20 40 60 Angle (8) [degrees] (a) 80 (b) Figure 8: (a) The radiation pattern of the curved antenna, (b) The input impedance for the circumferentially polarized antenna. 0 D E 14 0 1500 2000 2500 3000 3500 Number of BI Unknowns 1500 2000 2500 3000 3500 Number of BI Unknowns (a) (b) Figure 9: (a) Comparison of the total CPU time consumed for the solution, (b) Comparison of the CPU time consumed per iteration. 23

4.3 Results: Finite Periodic Arrays The greatest advantage of the FMM implementation is its capability for an efficient computation of 3D problems. The necessity of using 3D-FFT's in AIM implementations may not render AIM as attractive without suitable optimizations on the implementation of the FFT. Thus, for doubly curved arrays we consider implementation of the FE-FI methods with FMM rather then AIM. Below, we show some results for the analysis of printed dipole array on a curved surface. The radiation pattern for such an array is shown in Fig. 10. These are the preliminary results. -40 0 -20 -20 20 -80 -60 -40 -20 0 20 40 60 80 (a) (b) Figure 10: (a) Curved array geometry, (b) Radiation patterns for flat and curved arrays. At this point, we simply present a comparison between the calculations for the flat and curved arrays. As expected, the two patterns agree near broadside and deviate only near endfire grazing. The curved array does not has a null at 0=90~ and -90~ since the array continues to radiate in the shadow region of the cylindrical surface. 24

5 Conclusion and Future Work Hybrid FE-BI method is indeed a powerful and flexible approach for EM analysis. Use of fast integral methods such as AIM and/or FMM can considerably decrease CPU time and reduce memory. Both, AIM and FMM accelerate and compress the solution requirements thus overcoming the usual bottlenecks in considering large scale problems. The most attractive feature of F1 methods is that speed-up and memory reduction are obtained irrespective of the machine or computational platform. Efficient programming and parallelization will only provide further speed-ups and more efficient use of resources. For planar geometries AIM is certainly preferred since it naturally provides an (.(Nlog(N)) CPU requirements as is the case with CG-FFT. Implementations of the FMM is however easier to implement for curved arrays. To better address the speed-up and efficiency of the FMM and AIM in the future we will: 1. Examine more optimal implementation of the AIM algorithm (for planar antenna geometries); 2. Complete FMM formulation and validation for arbitrary 3D geometries; 3. Investigate applicability of AIM (and FMM) in domain decomposition for more general applications. 25

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