031307-9-T FAST FINITE ELEMENT-BOUNDARY INTEGRAL IMPLEMENTATION FOR SLOT ANTENNA ANALYSIS USING THE AIM ALGORITHM Sunil Bindiganavale John L. Volakis July 1997 31307-9-T = RL-2445

Theory and code manual for AIM-Plate and AIM-Prism Sunil S. Bindiganavale and John L. Volakis Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 This report describes the theory and execution procedure for AIM-Plate and AIM-Prism. AIM-Plate performs the same functions as a standard moment method code for analysis of planar conducting scatterers but with drastically reduced memory requirement and solution time. This reduction is accomplished by incorporating the Adaptive Integral Method (AIM) in an iterative solution of the Electric Field Integral Equation (EFIE). AIM-Prism performs the same functions as a Finite Element-Boundary Integral (FE-BI) program for radiation and scattering analysis from planar cavity-backed antennas. However, incorporation of AIM in boundary integral computations alleviates memory and execution time requirements considerably thus enabling the analysis of antenna configurations which result in large numerical systems. 1 Working principle of AIM-Plate A metallic scatterer can be considered as a special case of a resistive body with the resistivity Rr = 0. A resistive body is modeled using the resistive boundary condition [1] n x (Ei + ES) = roRJ (1) Consider a resistive body illuminated by an incident plane wave of unit amplitude given by E = (0cos a + $)sin ) eijk(xsin icosi+ysin ini) (2) where ko is the free space wavenumber, a is the polarization angle and (0i, qi) indicate the direction of incidence. The scattered field Es can be determined from the surface current J according to E5 =-jwA - V (3) where the magnetc vector potential A is given by [r eor S -JkoR A(r) = J(r) eR dS' (4) 1

with S being the surface of the body. The scalar potential 4 is given by 1 f -jkR q(r) = 47rC s(') dS' where R is the distance between observation and source points, viz. R= Jr - r' = /(x- x')2 + (y - y)2+ (z - The continuity equation is used to relate the surface charge density and the current Vs J = -jw Enforcing (1) on S yields the electric field integral equation for J Etan = (jA + Vq>)tan + rToRr J r E S (5) (6) (7) (8) To model the current, the scatterer is discretized into triangular patches. The current is then expanded in terms of vector basis functions [2] which are especially suited for triangular domains. Each basis function is associated with an interior (nonboundary) edge, and is nonzero only on the two triangles sharing that edge. Figure 1 shows the nth interior edge shared by triangles T+ and T,- of area A+ and A- respectively. A point in the triangle pair th n edge In Figure 1: Local coordinates for the nth edge can be designated by either the global position vector r, or local position vectors ~p- = r-rn. The basis function fn(r) for the nth edge is defined as In __ n fn(r) = 2A- Pn, 0, r in T+ r in Totherwise (9) The current J on S is approximated by N J - Infn(r) n=l (10) 2

where N is the number of interior edges and the unknown coefficients In represents the current density flowing across the nth edge of the mesh shared by the T+ and T- triangles. To solve for the basis coefficients, Galerkin's technique is applied to (8) giving sEi fm dS = j J A fm dS - s vs J fm dS + o J RrJ -fm dS m = 1,..., N. (1]1) Using (9) in (11) yields the N x N system of linear equations, V = ZI where In is the Nth basis coefficient, Zmn is the impedance matrix whose elements are computed from ljf+r e-jkoR Z JJI J AA Pm(r)Pn (r ) dS'dS - Jfff EmA eR dS'dS + II A7mA~ pm(r) ~ p(r)dS} (12) JJT4 AAmA +1 r in T+ 7 -inT (13) m n and C +1 r' in T+ -1 r in T;Efl={1-1 ([4) -I r' in TnThe elements of the interaction matrix can be computed directly from (12). However, a more convenient way of evaluating these elements is to consider a pair of faces and compute all nine interactions between edges contained by this pair. This enables the loops for assembly of the matrix elements to be over faces, instead of edges, thus speeding up the assembly process. For an observation face p paired with a source face q, the quantity Zq is computed for all mn edge pairs as 4APAq [ IT T Pm(r) P (r) dS'dS 2'2 JTP IIT R dS'dS 27 2 P RP R + R p(r) n(r)dS} (15) The positive current reference signs, Cm and E,n, are now assigned according to P +1, if TP is T- (16) -1, if TP is T and - +1, if Tq is T+ n - -1, (17) -1, if Tq is T 3

The integrals in (15) are evaluated for near and self cells by the techniques detailed in [3]. It should be noted that in (15) Tp = T+ + Tp and Tq = T + + T-, thus computation of ZPn involves summation over four triangles. The elements of the excitation vector are given by V m fpm (r) ( vm ' A. (0 cos a + ( sin a) Vmm =Ay ejko(xsinO cos i+ysinOi sin i) dS (18) The N x N linear system can be solved either by direct methods such as matrix factorization (which would mean an execution time of O(N3)) or iterative methods involving an operation count of O(N2)/iteration. The Adaptive Integral Method is an algorithm which reduces the computationally complexity of moment method solutions. In the case of AIM, the CPU reduction is achieved by mapping the original 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 and as can be understood, this calculation is very time consuming. For planar scatterers the dimensionality of the FFT is reduced by one, thereby significantly accelerating the solution. In this report, we examine the benefits of AIM when the body is not electrically large, but is highly tessellated owing to its intricate construction, thus leading to a large unknown count. We show that significant savings in CPU and memory can be achieved by AIM and examine its accuracy for near field and far field computations. 1.1 AIM for Planar Scatterers In this report, we describe the application of AIM to planar scatterers. Following the standard moment method discretization procedure, we begin with the linear system [Z]{I} = {V (19) with [Z] being the elements interaction matrix, whereas {I} is the vector of the unknown coefficients and {V} is the excitation vector. The matrix [Z] is fully populated, demanding O(N2) storage, and each [Z]{I} matrix-vector product requires O(N2) multiplications. Fast algorithms such as FMM and AIM are used to reduce the operation count from N2 down to NV, where a < 1.5. Both algorithms work on approximating the far zone interactions. In the case of AIM, the CPU reduction is achieved by first splitting the matrix as [z] = [Znear] + [Zfar] (20) based on a threshold distance referred to as the near-zone radius. The matrix [Znear] contains the interactions between elements separated less than the threshold distance, whereas [Zfar] contains the remaining interactions. The elements of [Znear] are evaluated with the exact MM while those of [Zfar] and the product [Zfar]{I} are evaluated in an approximate manner as prescribed by the AIM procedure [4]. Application of AIM requires that the whole geometry be enclosed in a regular rectangular grid. Basically, the fields of each interior edge is re-expressed using a new expansion based 4

on delta sources located at the nodes of the uniform AIM grid as depicted in Figure 2. For the mth edge, this new expansion has the form M2 Om = 6(X - Xmq)6(Y - Ymq)[hAxqr + AqY] (21) q=1 where rmq are the position vectors of M2 points on the square surrounding the center of the edge and 6(x) is the usual Dirac delta function. The coefficients AXmy are suitably chosen so that the new expansion is equivalent to the original representation using triangular elements. A similar expansion is used for the divergence of the basis functions M2 m = 6(X - Xmq)6(Y - ym)Amq (2 2) q=1 To find a relation between the Axy and In coefficients, we equate moments of the two expansions up to order M. Specifically, we set Mm = Fm (23) qMq2 q,q2 ( where Mq = J m(- Xa)q, (y - ya)q2dxdy for 0 < ql, q2 < M M2 = (xmq -a)q(Ymq - ya)2[Axmx + Amqy] with q = q + q2 (24) q=1 FO= f00 Fm = fm(x- Xa)q(y-ya 2 dxdy (25) -00 -00 Similarly, by equating moments of Vs Js with the new expansion (22), we establish a relation between Am and In. That is, we set Dqlq2 = Hql,q2 (26) where 0 0o M2 Dmq2 = d( - x) (y- ya)q2dxdy = ](xmq - Xa)q(ymq - ya)2Amq (27) -o -0c q=1 Hqm - J V v fm(X - Xa )ql(y - ya)q2dxdy (28) (23) and (26) give three M2 x M2 systems yielding the equivalence coefficients as the solution. This process is depicted pictorially in Figure 2. Were we to use the equivalent expansions to represent the currents everywhere, the resulting impedance matrix will be of the form 3 [Z] = EA[G[A] (29) i=l In this, [A]i are the sparse matrices containing the coefficients of the expansion (21 ) and (22) whereas [G] is the Toeplitz matrix whose elements are the free space Green's function 5

Original MM discretization AI grid (section) m2* m* Y Y L *tY m1$'m 7 z X *Am 9 m7 * iAIM Representation for edge * in." (Note: Each of the * * * * * original basis functions is */ /* * * * * * * * * * *.. defined on triangle pairs) Original MM discretization 2 M =3 r> AIM coefficients = M =9 Figure 2: The process of transformation from the original MM grid onto the AIM grid evaluated at the grid points. It has been shown in [4] that [Z~t1o] is not of sufficient accuracy for modeling the interactions between the nearby current elements. To take advantage of element interactions. However, we will retain the exact interaction matrix elements for the near element interactions. That is, we rewrite [Zfto] as vitotal - [zinear ar ( J0) [z]o = [TZ] + [Z]fA (30) AIM - [ZAIM + LZ]AIM Comparing this to (20) and setting [Z]fa' [Z]aM we can rewrite the original [Z] matrix as r7l- [ r7~\near p[ near \ I + rpltotal (31) [Z] - ([Z]= - [Z]\M) + [Z]f~'l (31) VI — Q 1 AI IM or 3 [Z] - [S] + [A][G][A]]T (32) i=1 where [S] = [Z]n' - [Z]near is a sparse matrix corresponding to the difference between the near field interactions computed by MM and AIM. The Toeplitz property of the Green's function, defined on the regular grid, enables use of the FFT to accelerate the computation of the matrix-vector product. The sequence of operations involved in the construction of the coefficient and Green's function matrices are indicated in Figure 3(a); those for the matrix-vector product execution are outlined in Figure 3(b). In the computation of the matrix-vector product, the initial step of transforming the currents from the original MM grid onto the uniform AIM grid is comparable to the grouping operation of the FMM. While the FMM relies on grouping to reduce the number of scattering centers, the sequence of operations in AIM can be interpreted as a realignment of scattering centers onto a regular grid. Although, this process does not reduce the number of scattering centers, the regularity of their location enables use of the FFT for fast computation of matrix-vector products. 1.2 Results When examining the merits of a fast integral algorithm such as AIM, of importance is the memory and CPU requirements, both contrasted to the delivered accuracy. Although 6

(onstruct cartesian grid which extends over the object Calculate moments of the i original basis functions | (RWGs)................................................................................................... Solve for equivalence coefficients Calculate Toeplitz Green's function over the cartesian grid................................................................................................... J Compute the near-field matrix (interactions between elements closer than threshold) (a................................................................................................... (a) Compute = A I ( Performs transformation of current from MM grid to AIM grid) Compute I=:' I (Compute the Fourier transform of the discrete current distribution).................................................................................................. Compute V = G I (Far-field Fourier Transform - G is the Green's function) Compute V = I V (Inverse transform to AIM grid) i far ] Compute V = AV (Far field back to MM representation) I 'r near - I Compute nV = In (Near field with exact MM) i -- Add to form total field far near V = V + V (b) Figure 3: (a) Matrix build operations and (b) Matrix vector product computation in AIM approximate analytical expressions have been derived in [4] for some of these parameters, these refer to implementations involving cubical grids and the three-dimensional FFT. Our goal in this chapter is to assess the accuracy of AIM in treating small details within an aperture/surface and to provide the reader with quantitative measures on the performance of AIM when implemented with the two dimensional FFT. The near-zone radius or threshold distance has a dramatic impact on the CPU requirements since it controls the non-zero element population of the system matrix. In the case of AIM, because of the inherent mapping to a uniform grid, we are highly interested in examining its suitability to model small and fine details embedded in much larger scale structures. The calculations for the plate configurations given next are intended to address this issue by examining the method's 7

performance for a number of representative and practical situations. All of the included results were generated using single precision arithmetic on an HP9000/C-110 workstation with a rated peak speed of 47 Mflops (the level 4 optimization option was also used). In all cases, a third order (M=3) multipole expansion was used with a grid spacing of 0.05A. Figures 4-8 depict the 00 and 0q5q polarization radar cross section patterns (q = 0~ cut) as calculated by AIM for the different threshold distances indicated on the figures. The first circular plate has no holes and was used to validate the method. From the pattern comparisons, it is clear that AIM recovers the exact result very well. As given in Table 1 and 2, AIM achieves this with at least a factor of five less memory thanthe traditional MM, even though the geometries are still rather small to demonstrate the full impact of AIM. Also, Table 2 shows that a near zone radius of 0.3A is sufficient to maintain good accuracy (below one dB in RMS error [5]). The advantage of AIM is more pronounced when gaps are inserted into the plate's surfaces and this is the primary reason that one may prefer AIM over other fast integral methods for planar structures. As depicted in Figures 5 and 6, AIM maintains its accuracy for the same threshold criterion even though the gaps/slots have a dominant effect on the RCS pattern as shown in Figure 5. In the case of narrow slots (or thin ridges in the plates)case of narrow slots (or the 0.03A, the memory requirements of the traditional MM increase quickly due to the higher element density. For the geometry in Figure 7, AIM yields memory saving of 79% and the CPU time is reduced by a factor of 12 while retaining the monostatic pattern accuracy to within a tenth of a dB. This is achieved by using a uniform AIM grid density of 20 points per linear wavelength even though the cell density of the original plate mesh is much greater due to the narrow slot. One may assume that this change in grid density will affect the near zone field. However, our observations indicate that the surface current is equally accurate. For the configuration in Figure 7 the average current density error is 7.3% for a threshold distance of 0.2A and 6% for a threshold distance of 0.4A. The currents for the geometry in Figure 7 along the center narrow strip are plotted and compared in Figure 9. These results demonstrate the important feature that the near zone threshold criterion is not affected by the specific geometrical details, leading to tremendous memory savings. Moreover, the accuracy of the results provide a convincing argument that AIM can efficiently handle highly irregular and resonant (i.e. antenna) geometries as well as smooth scatterers. At the same time, the convergence rate of the AIM system is unaffected indicating that the system condition is unchanged. This is of critical importance for fast iterative solutions, since an increase in the iteration count would annul the faster computation of the matrix-vector product. Figure 8 shows the monostatic RCS pattern for a grating structure which acts as a "polarization filter". The thin ridges in the grating cause a strong specular return for the (fq polarization (almost 10 dB above the return in the absence of the gratings) as is evident from the results in Figure 8(d). Of importance is that the MM triangular mesh in Figure 8 required a cell size of 0.02A per linear dimension because of the narrow grating. However, the overlaid rectangular AIM grid could be selected to have a much coarser discretization. More specifically, we chose grid spacings of 0.05A and 0.1A for the AIM grid and, thus, computational requirements of AIM were much lower. For the 0.1A grid spacing the solution time was reduced from 2.75 minutes down to only 12 sees at the expense of some accuracy (fraction of a dB). To further increase in accuracy, we employed a 0.05A grid spacing and as shown in Figure 8(b) the AIM curve is now indistinguishable from the reference MM result (within 0.1 dB). From Tables 1 and 2, the AIM computational and memory requirements are 8 times and 9 times less, respectively, without loss of accuracy. This is a significant 8

observation and we have found that both the convergence rate and condition of the AIM system remains essentially unchanged from the original moment method system. The original discretization for the geometry in Figure 8 and the equivalent AIM grids are pictorially depicted in Figure 10. It should be noted that even though the size of the discretization is very small, retaining the self-cell term alone in the moment method system introduces huge error (Figure 11), thus emphasizing the importance of non-self terms. 1.3 Summary The performance of AIM is much improved when applied to scattering from flat complex scatterers and scatterers with high discretization rates. Thus, the reduction of solution time is considerably more for the geometries depicted in Figure 12(a) and 13(a) than for the geometries in Figure 12(b) and 13(b). A memory reduction of 5 to 10 times over traditional MM was observed without compromise in accuracy when using a threshold radius of 0.2A. This CPU reduction is achieved without resorting to parallelization or optimization techniques (as is known AIM is particularly amenable to such improvements). More importantly, the AIM algorithm is capable of modeling very small details in large bodies with a high degree of accuracy, while simultaneously saving considerable memory. This is of importance when modeling broadband antennas (spirals or log-periodics) and gratings which are both large in overall size but can contain features as small as A/100 in size. Application of AIM for analysis of cavity-backed antennas is described in the next section. Discretization Geometry Facets Edges Unknowns MM memory (MB) MM solution time 00 pol (0 = 0~ inc.) Figure 4 586 908 850 5.51 32 secs Figure 5 554 890 772 4.54 29 sees Figure 6 1130 1806 1584 19.14 4 mins 50 secs Figure 7 1036 1667 1441 15.84 4 mins Figure 8 1038 1957 1157 10.21 2 mins 45 secs Table 1: Solution CPU time and memory requirement of the moment method 1.4 AIM-Plate execution The execution of AIM-Plate is done in a three step process 1. Convert the meshed geometry file from IDEAS Master Series 2.1 into a format required by the code. This is done with the help of two mesh-processing programs - ms21-u2c.f and c2p-fast.f and the transcript of a session with these programs with reference to the geometry of Figure 8 is indicated below [ 271 ] PlateFreqAIM.dir -: ms21_u2c Name of universal file? P1118slotss025.unv 9

AIM Data Geometry Threshold Non-Zeros Memory Solution time RMS Error(dB) (A) in Near Z (MB) 00 pol (0 = 0~ inc.) 00 pol | pol 0.3 59928 0.68 23 secs 0.1718 0.0755 Figure 4 0.4 100182 1.14 25 secs 0.1490 0.0693 0.7 257390 2.94 28 sees 0.0728 0.0490 Figure 5 0.4 79030 0.9 21 sees 0.0728 0.0583 0.6 157994 1.8 27 secs 0.0721 0.0520 Figure 6 0.7 283774 3.24 3 mins 32 sees 0.8017 0.5185 Figure 7 0.2 296250 3.39 20 sees 0.1063 0.0949 0.4 649556 7.43 31 sees 0.0548 0.0632 Figure 8 0.2 120220 1.37 18 sees 0.0469 0.0469 Table 2: Solution CPU time, memory requirement and RMS error of AIM (all entries in this table were computed with an AIM grid spacing of 0.05A) Name of converter file? cnv Encountered header There are 912 nodes. There are 1038 elements. [ 273 1 PlateFreqAIM.dir -: c2pfast Name of data file? P1118slotssO25Dat Finished reading in data There are 912 nodes There are 1038 elements Be patient #*!?/#@*!!! 100 elements done 200 elements done 300 elements done 400 elements done 500 elements done 600 elements done 700 elements done 800 elements done 900 elements done 1000 elements done Max. no. of edges emanating from a node = 6 1957 1957 Edge count = 1957 Finding free edges There are 800 free edges 2. Run the AIM preprocessor program to determine the appropriate dimensions to be set in the file dim. inc. The transcript of a session with PreProc.f is indicated below 10

again with reference to the geometry of Figure 8 [ 262 ] PlateFreqAIM.dir -: PreProc Input name of mesh file (Note: Dimensions are assumed to be in CENTIMETER)! P1118slotssO25Dat maxx=.5 minx= -.5 maxy=.5 miny= -.5 Input frequency at which the structure will be analyzed (GHz) 30 Enter AIM grid step in WAVELENGTHS (0.05 suggested) Note: The main AIM code has been hard-wired for 0.05 lambda but can be easily changed to 0.1 lambda by changing the variable step.05 nx in main progam to be= 25 ny in main progam to be= 25 FFT order along x (iFFTx in main pgm) 64 FFT order along y (iFFTy in main pgm) 64 Total number of elements= 1957 maxnod= 2700 maxedg= 8100 maxtri= 5400 nmax= 4 nintedg= 1157 ntris= 1038 Order of system = 1157 No. of triangular elements = 1038 Enter near field threshold in CENTIMETER Note: This is related to the Maximum number of non-zeros in the near-field matrix Recommend this to be 0.2-0.7 times the wavelength in centimeter. This is a empirical quantity and needs to be determined by trial & error 0.2 Set number of nonzeros in Near Z = 60110 3. Now the AIM program is executed and a sample session is indicated below 11

[ 259 ] PlateFreqAIM.dir -: FltrcAIM Enter mesh file name: P1118slotssO25Dat Enter surface type (1-pec): 1 Enter output file name: Opfile Enter pattern (1-bistatic, 2-backscatter): 2 Enter E-field polarization angle alpha (in degrees): 0 Number of cuts 1 Enter cut specifications... Fix (1-phi, 2-theta): 1 Enter fixed observation angle phi (in degrees): 0 Enter start observation angle theta (in degrees): 0 Enter stop observation angle theta (in degrees): 90 Enter number of observation points: 91 Enter Frequency (GHz) 30 --------------- Files --------------- Mesh: P1118slotssO25Dat Output: Opfile ------------ Surface Type ----------- PEC ------------ Pattern Type ----------- Backscatter -------- Observation Angles --------- Number of cuts: 1 Cut # 1 ----------- Phi:.00 deg. Start Theta:.00 deg. Stop Theta: 90.00 deg. -- Number of Observation Points --- 12

91 -— Frequency of analysis: 30.0 GHz — Above data O.K. (1-Yes, 2-No)? 1 Finished reading nodes Finished reading elements Interior edges = 1157 No. of triangular facets = 1038 50 triangles done 100 triangles done 150 triangles done 200 triangles done 250 triangles done 300 triangles done 350 triangles done 400 triangles done 450 triangles done 500 triangles done 550 triangles done 600 triangles done 650 triangles done 700 triangles done 750 triangles done 800 triangles done 850 triangles done 900 triangles done 950 triangles done 1000 triangles done Equivalence coefficients computed now 50 elements done 100 elements done 150 elements done 200 elements done 250 elements done 300 elements done 350 elements done 400 elements done 450 elements done 500 elements done 550 elements done 600 elements done 650 elements done 700 elements done 750 elements done 800 elements done 13

850 elements done 900 elements done 950 elements done 1000 elements done 1050 elements done 1100 elements done 1150 elements done Toeplitz G calculation Enter near field threshold in CENTIMETER Note: This is related to the Maximum number of non-zeros in the matrix Recommend this to be 0.2-0.7 times the wavelength in centimeter. This is a empirical quantity and needs to be determined by trial & error 0.2 The output will then appear in the following format indicating the angles of observation (Phi & Theta), Backscatter RCS and number of BCG iterations for convergence. Phi Theta Alpha=0 BCG Iter --- ----- ------- -------- 0.00 0.00 0.074222 484 0.00 1.00 0.068417 148 0.00 2.00 0.047878 168 0.00 3.00 0.013304 161 0.00 4.00 -0.036448 161 0.00 5.00 -0.097169 161 0.00 6.00 -0.171789 171 0.00 7.00 -0.258932 185 0.00 8.00 -0.356277 178 0.00 9.00 -0.463876 172 0.00 10.00 -0.581200 191 0.00 11.00 -0.704883 194 0.00 12.00 -0.832557 179 0.00 13.00 -0.972245 203 0.00 14.00 -1.111443 198 0.00 15.00 -1.254094 184 0.00 16.00 -1.397070 189 0.00 17.00 -1.541648 177 0.00 18.00 -1.684258 141 0.00 19.00 -1.823769 140 0.00 20.00 -1.961649 142 0.00 21.00 -2.096322 146 14

0.00 22.00 -2.226939 147 0.00 23.00 -2.354322 149 0.00 24.00 -2.481832 177 0.00 25.00 -2.606813 156 0.00 26.00 -2.730812 148 0.00 27.00 -2.855004 147 0.00 28.00 -2.980468 146 0.00 29.00 -3.109120 148 0.00 30.00 -3.241405 148 0.00 31.00 -3.379283 147 0.00 32.00 -3.521605 171 0.00 33.00 -3.672821 172 0.00 34.00 -3.834239 189 0.00 35.00 -4.003928 167 0.00 36.00 -4.180166 132 0.00 37.00 -4.379130 180 0.00 38.00 -4.581637 150 0.00 39.00 -4.801592 150 0.00 40.00 -5.035196 148 0.00 41.00 -5.283309 150 0.00 42.00 -5.542431 150 0.00 43.00 -5.821033 155 0.00 44.00 -6.115244 194 0.00 45.00 -6.425240 162 0.00 46.00 -6.751153 150 0.00 47.00 -7.092030 185 0.00 48.00 -7.454947 172 0.00 49.00 -7.830948 173 0.00 50.00 -8.226508 179 0.00 51.00 -8.638626 172 0.00 52.00 -9.069971 178 0.00 53.00 -9.519878 191 0.00 54.00 -9.988374 185 0.00 55.00 -10.476139 168 0.00 56.00 -10.982099 138 0.00 57.00 -11.510056 170 0.00 58.00 -12.058821 171 0.00 59.00 -12.628474 178 0.00 60.00 -13.220427 178 0.00 61.00 -13.832880 173 0.00 62.00 -14.473590 177 0.00 63.00 -15.137205 178 0.00 64.00 -15.827974 178 0.00 65.00 -16.545870 188 0.00 66.00 -17.293951 186 0.00 67.00 -18.073139 195 0.00 68.00 -18.887245 184 15

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Phi 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 69.00 70.00 71.00 72.00 73.00 74.00 75.00 76.00 77.00 78.00 79.00 80.00 81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00 Theta 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 -19.737566 -20.627750 -21.561594 -22.543802 -23.579075 -24.673714 -25.835037 -27.072275 -28.397873 -29.824045 -31.369724 -33.056828 -34.916584 -36.988480 -39.333023 -42.032391 -45.216888 -49.106750 -54.116230 -61.165161 -73.211105 -121.335602 Alpha=90 9.583014 9.568883 9.518059 9.430880 9.306488 9.144302 8.945677 8.706966 8.431199 8.115008 7.755467 7.357507 6.914128 6.427933 5.893502 5.314169 4.680663 3.998905 3.262678 2.468545 1.620582 0.712998 -0.246950 186 185 191 190 179 215 211 304 258 297 231 297 227 304 258 298 251 297 273 340 810 810 BCG Iter 654 313 361 267 379 185 342 320 346 309 715 233 357 227 227 616 244 259 237 645 240 414 355 16

0.00 23.00 -1.256379 585 0.00 24.00 -2.294207 291 0.00 25.00 -3.348589 303 0.00 26.00 -4.366242 563 0.00 27.00 -5.294078 298 0.00 28.00 -6.052021 300 0.00 29.00 -6.583304 333 0.00 30.00 -6.852431 471 0.00 31.00 -6.862307 336 0.00 32.00 -6.692267 299 0.00 33.00 -6.396056 437 0.00 34.00 -6.071772 367 0.00 35.00 -5.737149 333 0.00 36.00 -5.425569 461 0.00 37.00 -5.151746 302 0.00 38.00 -4.928895 436 0.00 39.00 -4.758909 281 0.00 40.00 -4.633798 261 0.00 41.00 -4.560894 461 0.00 42.00 -4.530453 273 0.00 43.00 -4.537940 469 0.00 44.00 -4.588966 444 0.00 45.00 -4.671724 373 0.00 46.00 -4.792987 370 0.00 47.00 -4.940162 474 0.00 48.00 -5.117064 261 0.00 49.00 -5.299952 485 0.00 50.00 -5.510026 381 0.00 51.00 -5.739635 455 0.00 52.00 -5.962952 257 0.00 53.00 -6.203727 481 0.00 54.00 -6.439026 549 0.00 55.00 -6.685131 361 0.00 56.00 -6.921620 539 0.00 57.00 -7.147860 495 0.00 58.00 -7.370139 256 0.00 59.00 -7.572776 505 0.00 60.00 -7.765730 473 0.00 61.00 -7.939365 448 0.00 62.00 -8.096338 511 0.00 63.00 -8.241356 553 0.00 64.00 -8.370316 965 0.00 65.00 -8.479603 455 0.00 66.00 -8.574388 394 0.00 67.00 -8.656987 531 0.00 68.00 -8.726567 477 0.00 69.00 -8.788289 403 17

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Cut # idone 70.00 71.00 72.00 73.00 74.00 75.00 76.00 77.00 78.00 79.00 80.00 81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00 -8.845181 -8.884315 -8.905354 -8.954330 -8.979956 -9.005004 -9.016927 -9.042066 -9.054319 -9.071721 -9.083326 -9.100692 -9.104582 -9.114861 -9.124923 -9.128168 -9.142673 -9.138368 -9.140660 -9.148211 -9.147070 486 549 387 564 502 541 242 540 244 459 242 461 650 595 241 388 238 151 151 331 140 2 Working principle of AIM-Prism In this section, we review a finite element - boundary integral formulation for analyzing three dimensional cavity-backed antennas. The finite element discretization is in the form of triangular prisms. Such prisms are the element of choice for modeling planar antennas with fine detail (as small as 50th or 100th of a wavelength) as they require only surface discretization information. In contrast to tetrahedral elements [6], this eliminates the need to generate volume meshes which could be tedious and also removes the possibility of ill-conditioned systems due to degraded mesh quality. In general, for modeling planar configurations the prism element also requires lesser number of unknowns than tetrahedral elements. However, very small details and consequently dense meshes can still lead to boundary integrals with extremely large computational requirement. In the previous section it was shown that AIM reduces the computational requirement considerably. In this section, we present the key elements of a three dimensional finite element - boundary integral formulation with emphasis on the boundary integral computation. For details on the prism element the reader is referred to [7]. Consider a cavity-backed antenna recessed in a ground plane as depicted in Figure 14. To solve for the E-field inside and on the aperture of the cavity, it is necessary to extremize a functional, which for radiation and scattering problems may be generalized as F(E)= - J {(V x E).r-1- (V x E)- OkE r~ E}dV 2j J Ij r 0 E 18

+ J E. (jkoZoJ' + V x l-. Mi) dV + jkoZo II E (H x n)dS (33) where 46 and /r denote the relative tensor permittivity and permeability of the cavity filling, So represents the non-metallic portions of the aperture and Sf denotes the junction opening to the feeding structures. The volume V, refers to the volume occupied by the impressed sources J; and M;. Also, H denotes the magnetic field on So or Sf and ni is the outer normal to these surfaces. For a unique solution of E we require knowledge of H over So and Sf. In the case of Sf, H is determined by the feed excitation while that over the non-metallic portions of the aperture is determined by the boundary integral equation H = Hs~ + 2jkoYo Jj G(r, r') (z x E(r')) dS' (34) JJ(4 where G is the electric dyadic Green's function of the first kind such that n x G = 0 is satisfied on the metallic platform. For the cavity recessed in a ground plane, G becomes the half space dyadic Green's function ) e1koR G (I+ 1EV) -V] {E} (35) with R = Ir - r'j and I is the unit dyad. For this problem, Hgs is equal to the sum of the incident and reflected fields for scattering computations and zero for antenna analysis. To discretize (33) the volume region is subdivided using prismatic elements. The field in each prism is approximated using a linear edge-based expansion as 9 Ee = E;vj = [V]{Ee*} (36) j=1 where [V]e = [{V}, {V}], {V,}] and {Ee} = {El, E,..., E}T. On the aperture, since the top and bottom faces of the prism are triangles, we have a corresponding representation for the aperture fields as 3 Es(r) = ZE'S'(r) = [S]Tj{E} (37) i=l where [S], = [Sr, Sy]. To generate a linear system for the solution of En, (36) and (37) are substituted into (33). Subsequent minimization of the functional yields f AFv Nv Ns Nv Ns OEe- = E[A ]{Ee} + E[BS]{Es} + E{K} + {L} = 0 (38) e=l s=1 e=l s=l1 where Nm and Ns indicate the number of volume and surface elements, respectively. The matrix elements are given by Ae = J///{(V x vi ).- (v xVi - Vj)-k i.. Vj}dV (39) J -j Pr:9 19

Ke = ffVf Vi' [jkoZoi + V x r-1 V x Mi] dV (40) B = -ff/ f 2k0S(r) S(r')Go(r,r')dSdS' Bj S- S 0 ~ + 2f [V x S^(r)]z[V' x S(r')]zGo(r,r')dSdS' (41) L = 2jkoZo ff S (Hi x z)dS (42 The boundary integral equation in (41) is discretized using basis functions defined on the top face of the prism as Si= 2Ae x (r- ri) (43) similar to the function defined in (9). Substitution of (43) into (41) gives the discretized boundary integral which is treated using the procedure outlined in Section 1. Several cavity-backed antennas contain small features and details which may necessitate high discretization. This could take the form of very narrow slots which may be a fiftieth or hundredth of a wavelength in width. Discretization of such geometries could lead to very large numerical systems even if the size of the antenna is not electrically very large. To efficiently treat such systems, the properties of an algorithm based on an iterative solver, should include the following * It is of paramount importance that the "threshold" distance (distance beyond which interactions are treated as of the far zone variety) is as small as possible. * It should be capable of characterizing small perturbations in an otherwise smooth surface. * It should be capable of modeling near fields accurately. * If the algorithm incorporates a process by which very small discretization details can be "mapped" onto a different domain which is less dense than the original, computation of the matrix vector product in this domain would simulate the effect of a reduced number of unknowns. Figures 8 and 9 depict two planar configurations analyed by the AIM from which it can be gleaned that all the above criteria are met. Unlike FMM, which carries out the matrix vector product on the original moment method discretization, the ability of AIM to map the small details onto a sparse grid and still retain accuracy makes it the method of choice to analyse such antennas. For efficient modeling of the cavity we employ FEM with its low O(N) storage and execution time. Triangular prisms are used for discretization of the cavity volume for the reasons described in [7]. 2.1 Implementation The FE-BI formulation for three dimensional cavity-backed antennas using prismatic elements is described in the previous section. Substitution of (43) in (41) gives a discretized boundary integral of the form in (15). The near and far zone terms are treated as outlined in 1.1. The FEM matrix and the near zone interactions of AIM are stored in a sparse storage format, thus affecting significant savings in memory. 20

2.2 Results Figure 15 shows the radiation pattern for an annular slot computed in the elevation plane, X = 50~. The reference FE-BI solution [7] is contrasted with computations of BI using AIM (indicated as FE-AIM). It is seen that for this example, the threshold distance in AIM can be reduced to 0.25A without significant loss of accuracy. This enables the reduction of matrix entries stored in the near field portion by a factor of three resulting in a corresponding savings in memory as indicated in the tabulation of the near-zone non-zero entries. Figure 16 shows the radiation pattern for the same antenna in the 5b = 900 elevation plane. The normal direction in this plane, reveals the characteristic separation between co-polarization and cross-polarization levels for the annular slot at observation angles close to normal in the elevation plane. From this figure, it is gleaned that the threshold distance in AIM can be reduced down to even 0.15A if an average error of a dB could be tolerated. From the computation of near-zone matrix entries, such a threshold would result in a factor of five saving in memory. Figure 17 shows a scattering cross-section for the same slot but at a frequency of 0.73 GHz (at which the antenna is electrically even smaller) instead of the previous 1 GHz. It should be noted that for a threshold of 0.4A (larger than the diameter of the BI contour) the near-zone and far-zone entries for AIM cancel each other in accordance with (31), thus yielding a very small error (0.00086 dB) in comparison to the FE-BI solution. A quantity of vital importance in antenna computations is input impedance. Figure 18 depicts the input impedance of a very narrow probe-fed annular slot, computed using FE-BI and FE-AIM. The probe is placed at y = 0. It is seen that evaluation of the boundary integral with AIM enables the reduction of the near-zone non-zeros by more than half. Computation of input impedance demands very high accuracy and the threshold distance was held constant at 10.5 cm (corresponding to 0.35A at 1 GHz and 0.49A at 1.4 GHz - the corresponding diameter of the entire BI contour varying from 0.513A to 0.718A). While, Figures 15-18 demonstrate the ability of AIM to translate very fine details such as a narrow slot onto a coarser equivalent grid, Figure 19 and 20 indicate the importance of a low threshold distance in modeling cavity-backed antenna arrays. Figure 19 and 20 indicate that for an average error of less than a dB in scattering and radiation patterns it is possible to reduce the number of non-zeros in the near-zone part of the impedance matrix by a factor of six, resulting in substantial saving in memory. This is a consequence of employing a threshold distance of 10 cm, which is about a fifth of the cavity diameter. It is necessary to note that employing such a threshold distance results in a majority of the interactions between different slots being treated with the AIM procedure. This is of paramount importance in modeling antenna arrays and spiral antennas. While Figures 15-19 compare spatial domain FE-BI and FE-AIM solutions, Figure 21 compares the spatial domain FE-AIM solution with a spectral-domain FE-BI solution presented in [8] for the scattering by a cavity-backed patch antenna. 2.3 Summary AIM, with its low threshold distance, and ability to translate to an equivalent grid is capable of saving a significant amount of memory and solution time for bodies which are finely discretized even though they may not be electrically large. Its accuracy is preserved even while performing radiation computations thus making it the method of choice for analyzing antennas with intricate details. 21

2.4 AIM-Prism execution Computation of radiation with AIM-Prism is a three step process, however for the geometry in Figure 18 step one and two have already been executed and are listed merely to aid future development 1. Mesh the antenna geometry of choice. AIM-Prism requires just a surface mesh since it employs prismatic elements. The surface mesh needs to have the following details * Triangles in the slots need to be grouped. * Nodes in the slots nees to be grouped. * Boundary nodes belong to both metal and aperture groups. * Corner nodes need to be grouped. * Nodes between which probes are connected need to be grouped A universal file (level 6 IDEAS) meeting these specifications is ring-slot.unv which contains the geometry depicted in Figure 18. It is processed with the pre-processor shellilevel6. f to extract the above information. A dimension file DIM. INC along with other subsidiary files is written as a result of the pre-processing operation. DIM. INC needs to be augmented with information from the AIM pre-processor executed in step 2 before it is complete. A transcript of the session with the IDEAS level 6 pre-processor is indicated below. [ 412 ] temp -: shelllevel6 NAME OF UNIVERSAL FILE? ring.slot.unv ENCOUNTERED HEADER THERE ARE 270 NODES. THERE ARE 512 ELEMENTS. THERE ARE 96 NODES ON THE SLOTS THERE ARE 96 ELEMENTS ON THE SLOTS THERE ARE 26 NODES ON THE EDGE OF TOP THERE ARE 2 PROBES IN THE SYSTEM BE PATIENT!!! COUNTING EDGES... EDGE COUNT = 781 PROCESSING SLOT FOR ON-SURFACE EDGES... 96 SLOT SURFACE-TRIANGLES. THERE ARE 192 SLOT EDGES AND 96 NONPEC'S 685 EDGES ON THE PEC SURFACE 2. The AIM pre-processor determines dimension parameters related to the boundary integral. This program PreProcAnt.f produces the following output [ 343 ] AIMPrism.dir -:!! PreProcAnt maxx= 8.075 22

minx= -8.075 maxy= 8.06903 miny= -8.06903 Input frequency at which the structure will be analyzed (GHz) 1.35 Enter AIM grid step in WAVELENGTHS (0.05 suggested) Note: The main AIM code has been hard-wired for 0.05 lambda but can be easily changed to 0.1 lambda by changing the variable step.05 nx in main progam to be= 19 ny in main progam to be= 19 FFT order along x (iFFTx in main pgm) 64 FFT order along y (iFFTy in main pgm) 64 maxnod= 96 maxedg= 192 maxtri= 96 nmax= 50 nintedg= 96 ntris= 96 Order of system = 96 No. of triangular elements = 96 Enter near field threshold in CENTIMETER Note: This is related to the Maximum number of non-zeros in the near-field matrix Recommend this to be 0.2-0.7 times the wavelength in centimeter. This is a empirical quantity and needs to be determined by trial & error 10.5 Set number of nonzeros in Near Z = 2245 As a result of this the dimension file DIM. INC is augmented by the following few lines Parameter Parameter Parameter Parameter Parameter Parameter (nonzero=2245) (nx=19) (ny=19) (iFFTx=64) (iFFTy=64) (ordermax=10) 3. AIM-Prism is then executed and an example which produces the input impedance at 23

1.35 GHz for the annular slot ring is depicted below [ 347 ] AIMPrism.dir -: NewPrism ****** USER-ORIENTED DATA INTERFACE ****** ******************************************* INPUT CAVITY HEIGHT AND NUM. OF SGMTS. ALONG Z 3 2 INPUT SEGMENT SIZE ALONG Z AXIS FROM TOP TO BTM. ENTER THE HEIGHT FOR SEGMENT 1(1 REAL) 1.5 ENTER THE HEIGHT FOR SEGMENT 2(1 REAL) 1.5 INPUT NUMBER OF DIELECTRIC LAYERS (1 INTGR) 1 ASSUME THE LAYERS ARE COUNTED FROM THE BOTTOM, THUS EP,EU (2 CMPLX) & NUM. OF SEMS. (1 INTGR) FOR LAYER 1 (1.35,0.) (1.,0.) 2 ENTER PATTERN (1-BISTATIC,2-BACKSCATTER,3-RADIATION): 3 ENTER FEED STYLE: (1-VERTICAL,2-HORIZONTAL) 2 ENTER FREQ. (IN GHz): 1.35 ENTER: 1-RADIATION PATTERN,2-INPUT IMPEDANCE,3-GAIN 2 ENTER TOLERANCE (eg. 0.0001) FOR BICG ITERATIONS:.001 ABOVE DATA O.K. (1-YES, 2-NO)? 1 TOTAL PEC EDGES: 1544 TOTAL NUMBER OF NON-PEC EDGES: 1339< 2400 TOTAL NUMBER OF EDGES: 2883< 4000 DONE WITH FEM MATRIX FILLING! Equivalence coefficients computed now 50 elements done Toeplitz G calculation Enter near field threshold in CENTIMETER Note: This is related to the Maximum number of non-zeros in the matrix Recommend this to be 0.2-0.7 times the 24

wavelength in centimeter. This is a empirical quantity and needs to be determined by trial & error 10.5 50 elements done DONE WITH BI MATRIX FILLING! NS 2 3 FINISH COMBINING! Iteration number Iteration number Iteration number Iteration number START BICG 1 Residual 2 Residual 3 Residual 4 Residual ITERATION....9794996 1.17442 1.21015.8353357 Iteration Iteration Iteration Iteration Iteration Iteration Iteration number number number number number number number 1146 1147 1148 1149 1150 1151 1152 Residual Residual Residual Residual Residual Residual Residual 4.96248E-03 7.29750E-03 2.34599E-03 2.00533E-03 1.48691E-03 2.24077E-03 1.99295E-03 1152 TIMES ITERATIONS! 1.35 59.84542 -67.166 2.4.1 Cavity-backed slot array analysis For the slot array of Figure 20, the procedure for the single slot discussed above differs in a few respects. The universal file is an IDEAS Master Series 2.1 which is converted with a new pre-processor shellMSC.f. Also, for the radiation pattern each of the four slots is fed and the file containing the probe feeds ESOURCE is correspondingly augmented. The transcript of the slot array run is indicated below ****** USER-ORIENTED DATA INTERFACE ****** ******************************************* INPUT CAVITY HEIGHT AND NUM. OF SGMTS. ALONG Z 1.5 1 INPUT SEGMENT SIZE ALONG Z AXIS FROM TOP TO BTM. ENTER THE HEIGHT FOR SEGMENT 1(1 REAL) 1.5 INPUT NUMBER OF DIELECTRIC LAYERS (1 INTGR) 1 ASSUME THE LAYERS ARE COUNTED FROM THE BOTTOM, THUS EP,EU (2 CMPLX) & NUM. (2,0) (1,0) 1 25

OF SEMS. (1 INTGR) FOR LAYER 1 ENTER PATTERN (1-BISTATIC,2-BACKSCATTER,3-RADIATION): 1 ENTER FEED STYLE: (1-VERTICAL,2-HORIZONTAL) 2 ENTER FREQ. (IN GHz): 1 ENTER E-FIELD POLARIZATION ANGLE ALPHA (IN DEGREES): 0 ENTER ANGLES OF INCIDENCE... PHI (IN DEGREES): 0 THETA (IN DEGREES): 0 ENTER CUT SPECIFICATIONS... FIX (1-PHI, 2-THETA): 1 ENTER FIXED OBSERVATION ANGLE PHI (IN DEGREES): 90 ENTER START OBSERVATION ANGLE THETA (IN DEGREES): 0 ENTER STOP OBSERVATION ANGLE THETA (IN DEGREES): 90 ENTER NUMBER OF OBSERVATION POINTS: 90 ENTER TOLERANCE (eg. 0.0001) FOR BICG ITERATIONS: 0.001 ABOVE DATA O.K. (1-YES, 2-NO)? 1 TOTAL PEC EDGES: 4546 TOTAL NUMBER OF NON-PEC EDGES: 983< 2500 TOTAL NUMBER OF EDGES: 5529< 6000 Equivalence coefficients computed now 50 elements done 100 elements done 150 elements done 200 elements done Toeplitz G calculation Enter near field threshold in CENTIMETER Note: This is related to the Maximum number of non-zeros in the matrix Recommend this to be 0.2-0.7 times the wavelength in centimeter. This is a empirical quantity and needs to be 26

determined by trial & error 10 50 elements done 100 elements done 150 elements done 200 elements done FINISH COMBINING! Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number Iteration number START BICG 1 Residual 2 Residual 3 Residual 4 Residual 5 Residual ITERATION... 1.22629 2.06872 2.49931 4.67545 10.21917 210 211 212 213 214 215 216 Residual Residual Residual Residual Residual Residual Residual 1.20682E-03 1.37842E-03 1.62599E-03 1.42027E-03 1.17562E-03 1.14276E-03 1.03158E-03 216 TIMES ITERATIONS!.0 1.0 3.33673 3.33345 -27.88 1.01124 2.02247 3.03371 4.04494 5.05618 6.06742 7.07865 8.08989 9.10112 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 3.32997 3.30988 3.27648 3.22976 3.16974 3.09644 3.00988 2.91008 2.79706 3.32668 -27.8791 3.30659 -27.9024 3.27321 -27.95 3.2265 -28.0221 3.16652 -28.1189 3.09325 -28.2408 3.00673 -28.3883 2.90698 -28.5618 2.79403 -28.7618 10.11236 1.0 2.67086 2.6679 -28.9893 11.12359 1.0 2.53152 2.52864 -29.2448 12.13483 1.0 2.37908 2.37628 -29.5295 13.14606 1.0 2.21356 2.21086 -29.8444 14.1573 1.0 2.03503 2.03243 -30.1909 15.16853 1.0 16.17977 1.0 17.191 1.0 1 18.20224 1.0 19.21347 1.0 20.22471 1.0 21.23595 1.0 22.24718 1.0 23.25842 1.0 24.26965 1.0 1.84355 1.84106 -30.5706 1.63916 1.63679 -30.9851 42192 1.41967 -31.4368 1.19191 1.1898 -31.9281.9492078.9472271 -32.4619.6938675.6920296 -33.042.4259951.4243047 -33.6726.1456651.1441254 -34.359 -.147019 -.148405 -35.1076 -.451955 -.453187 -35.9264 27

25.28089 1.0 -.769034 -.770111 -36.8255 26.29212 1.0 -1.09814 -1.09906 -37.8173 27.30336 1.0 -1.43913 -1.43991 -38.9179 28.31459 1.0 -1.79191 -1.79254 -40.1475 29.32583 1.0 -2.15631 -2.15681 -41.5314 30.33706 1.0 -2.53218 -2.53256 -43.0985 31.3483 1.0 -2.91937 -2.91965 -44.8732 32.35954 1.0 -3.31771 -3.3179 -46.8398 33.37077 1.0 -3.72701 -3.72714 -48.8261 34.38201 1.0 -4.14708 -4.14719 -50.2623 35.39325 1.0 -4.57773 -4.57784 -50.333 36.40449 1.0 -5.01873 -5.0189 -49.1148 37.41572 1.0 -5.46987 -5.47015 -47.442E 38.42696 1.0 -5.9309 -5.93134 -45.8266 39.4382 1.0 -6.40157 -6.40226 -44.4074 40.44944 1.0 -6.88162 -6.88264 -43.191E 41.46067 1.0 -7.37077 -7.37221 -42.152E 42.47191 1.0 -7.86873 -7.87072 -41.261; 43.48315 1.0 -8.37518 -8.37785 -40.4927 44.49438 1.0 -8.88983 -8.89334 -39.826E 45.50561 1.0 -9.41234 -9.41686 -39.246E 46.51685 1.0 -9.94237 -9.9481 -38.7407 47.52809 1.0 -10.4796 -10.4867 -38.2981 48.53932 1.0 -11.0236 -11.0325 -37.910E 49.55056 1.0 -11.574 -11.5849 -37.5716 50.5618 1.0 -12.1305 -12.1438 -37.2752 51.57304 1.0 -12.6927 -12.7088 -37.016E 52.58427 1.0 -13.2602 -13.2795 -36.7921 53.59551 1.0 -13.8327 -13.8557 -36.598e 54.60675 1.0 -14.4097 -14.437 -36.4314 55.61799 1.0 -14.9909 -15.0233 -36.289: 56.62923 1.0 -15.5761 -15.6141 -36.169c 57.64046 1.0 -16.1648 -16.2094 -36.07 58.65169 1.0 -16.7569 -16.809 -35.9887 59.66293 1.0 -17.352 -17.4127 -35.9241 60.67417 1.0 -17.9499 -18.0206 -35.874L 61.6854 1.0 -18.5507 -18.6325 -35.8384 62.69664 1.0 -19.1541 -19.2488 -35.814% 63.70788 1.0 -19.7601 -19.8696 -35.8017 64.71912 1.0 -20.3689 -20.4952 -35.798( 65.73035 1.0 -20.9807 -21.1262 -35.804 66.74159 1.0 -21.5957 -21.7632 -35.817L 67.75283 1.0 -22.2143 -22.4071 -35.837L 68.76406 1.0 -22.8371 -23.059 -35.8631 69.77529 1.0 -23.4647 -23.7203 -35.8937 70.78653 1.0 -24.0979 -24.3926 -35.928; 71.79777 1.0 -24.7377 -25.078 -35.9661 I L 3 c ) 5 3 5 5 3 3 2 3 5 5 5 T 3 28

72.80901 73.82025 74.83148 75.84272 76.85396 77.86519 78.87643 79.88767 80.89891 81.91014 82.92137 83.93262 84.94385 85.95509 86.96633 87.97756 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 -25.3852 -25.779 -36.0064 -26.0418 -26.4986 -36.0485 -26.7087 -27.2406 -36.0916 -27.3875 -28.0095 -36.1351 -28.0799 -28.811 -36.1784 -28.7873 -29.6521 -36.2209 -29.5111 -30.5419 -36.2621 -30.252 -31.4919 -36.3015 -31.0102 -32.5172 -36.3385 -31.7837 -33.6386 -36.3728 -32.5682 -34.8851 -36.404 -33.3545 -36.2989 -36.4318 -34.1269 -37.9457 -36.4558 -34.8596 -39.9356 -36.4758 -35.5149 -42.4744 -36.4915 -36.0433 -46.0249 -36.5029 88.9888 1.0 -36.3905 -52.0628 -36.5098 90.00004 1.0 -36.5121 -80.0 -36.5121 141.03u 0.22s 2:33.22 92.1% References [1] T.B.A. Senior and J.L. Volakis, Approximate boundary conditions in electromagnetics. IEE press: London, 1995. [2] S. Rao, D. Wilton, and A. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Transactions on Antennas and Propagation, vol. 30, no. 3, pp. 409-418, 1982. [3] D.R. Wilton, S.M. Rao, A.W. Glisson, D.H. Schaubert, O.M. Al-Bundak, and C.M. Butler, "Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Transactions on Antennas and Propagation, vol. 32, pp. 276 -281, March 1984. [4] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, "AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems," Radio Science, vol. 31, no. 5, pp. 1225-1251, 1996. [5] S. S. Bindiganavale and J. L. Volakis, "Guidelines for using the fast multipole method to calculate the RCS of large objects," Microwave and Optical Technology Letters, vol. 11, no. 4, pp. 190-194, 1996. [6] J. Gong, J.L. Volakis, and A.C. Woo, "A hybrid finite element-boundary integral method for the analysis of cavity-backed antennas of arbitrary shape," IEEE Transactions on Antennas and Propagation, vol. 42, no. 9, pp. 1233-1242, 1994. [7] J. Gong, J.L. Volakis, and H.T.G. Wang, "Efficient finite element simulation of slot antennas using prismatic elements," Radio Science, vol. 31, no. 6, pp. 1837-1844, 1996. 29

[8] A.C. Polycarpou, M.R. Lyons, J. Aberle, and C.A. Balanis, "Analysis of arbitrary shaped cavity-backed patch antennas using a hybritization of the finite element and spectral domain methods," in 1996 IEEE Int. Symp. on Antennas and Propagation Digest, July 1996. 30

Plate orientation 1 reference YA z x 00 Polarization Polarization referencey 00 Polarization X 25 1-15 '0 1-i3 wv -5 C.0 Cu =-15 -25 - ADIM - Threshold = 03.R 'A AIM - Threshold = 0O4x - * AIM - Thrveshold = 0.7x. Cu 25 20 15 10 S 0 -5 ID AIMN- Threshold =0.3?X A AIM - Threshold = 0.4~ * * AIM~ -'Threshold = 0.7k 0 15 30 45 60 Observation Angle 0 (deg.) 75 - lI - - - - - - - - - - - - - - 90 0 15 30 45 60 Observation Angle e (deg.) 75 90 Fig ure 4: Monostatic RCS for a circular plate of diameter 2A; Comparison of the standard MM & AIM 31

Plate orientation reference Z Z x 0.685 0.05 25 m 15 1-1 - 5 u -15 -25 1.6 A (a) 00 Polarization 0 15 30 45 60 75 90 Observation Angle 0 (deg.) (b) )) Polarization 25 as 15 p - 5 -15 -25 1....., I. I M - AIMr - Threshold = 0.4i, A AIM - Threshold = 0.6k, j-.aa I..... I..... I..... I..... 0 15 30 45 60 Observation Angle 9 (deg.) (c) 75 90 0 15 30 45 60 r Observation Angle e (deg.) e_ II(d) 75 90 Polarization reference / x Figure 5: Monostatic RCS for a circular plate of diameter 2A with three slots computed with standard MM & AIM (a) Geometry (b) Effect of the slots on the RCS (c) 00 polarization backscatter RCS for the plate with slots (d) bq polarization backscatter RCS for the plate with slots 32

Plate orientation reference Y Z X 30,. 25, 20 - -10 -15 - -1L ***n Va~ Z --- A/ arization reference y /. Px 0) Polarization X 00 Polarization 0 15 30 45 60 75 90 0 15 30 45 60 Observation Angle 0 (deg.) Observation Angle 0 (deg.) 75 90 Figure 6: Monostatic RCS for a square standard MM & AIM plate of side 4A with three holes computed with 33

00.6 Plate orientation reference YA Z X 0.03X ' 0 ( Smaller than the AIM grid size) Polarization reference. y 00 Polarization ~ Polarization X 0 10.......... MM 105 E ALM - ThMeshold = 0.29 -a -10 A AIM - Threshold = 0.44 0 C -20 -30:AIM - Threshold = 0.2 -10 A AIM - Thteshold = 0.4", -40. i. -15.... i 0 15 30 45 60 75 90 0 15 30 45 60 75 90 Observation Angle 0 (deg.) Observation Angle e (deg.) Figure 7: Monostatic RCS for a circular plate of diameter 1A sampled at 0.03 A (smaller than the AIM grid spacing) due to the narrow center ridge 34

te"""Yw"""""""""""""" 0.O5 'I I ) Plate orientation reference Z X 0. - 0.04A ice X, 1. 0.02 ) (b) 0.04X (a) Polarization referen 00 Polarization tx Polarization 'Ni..... I.... I - -................... 0 -O -5 0-10 CA. -15 32 I. I" I ^ ^I - - ' I. - - ALM - I- Grid spacing = 0.05x (For Figi a - A- - - ALM - Grid spacing = 0.1 (For Fig (a) \ - M (For Fig (b)) 13 10 32 1-11 <<; t) rA u L. CP u m 1m 5 0 MM (For Fig (a)) - - - AIM - Grid spacing = 0.O05~ (For Fig (a) AII - Grid spacing = 0.1. (For Fig (a)' MM (For Fig (b)) -N\ i/ \.-. \ \ /.. \ —/ \ / \/..... I... I..... I. I... I..... I..... -5 1 -25 1 -10 -15 -30 0 15 30 45 60 Observation Angle 0 (deg.) (c) 75 90 0 15 30 45 60 Observation Angle 0 (deg.) (d) 75 90 Figure 8: (a) Geometry and mesh of the grated plate (b) Geometry and mesh of the "Groove" plate without gratings (c) 00 polarization backscatter RCS computed by AIM and MM (d) Oqf polarization backscatter RCS computed by AIM and MM 35

Plate orientation reference YA z x 0.6 /A Currents at the centroid of the shaded triangular patches on the narrow ridge It -! - 0.03 a ( Smaller than the AIM grid size) - MM -- AIM - Threshold = 0.4 X 1n-4 A L', 4. 1- _ —_ I, - 0 -6. I.n 0 rA *& ^3 Ca3 4.3 4.1 3.9 3.7 3.5 3.3 3.1 2.9 2.7 2.5 I I I I I I I I I I I I I I --— r- — ---- ----v --- —-------- I I I - I I I I I - I I I I I I I I I - I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I - I I I I! -0.2 -0.1 0 0.1 y in wavelengths 0.2 0.3 Figure 9: Electric currents (Solution coefficients) on the narrow ridge for the geometry of Figure 7 36

0.1)~ -- o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 4 O O O O O O O O O O O O O 0 0 0 C, 0 0 C, 0 0 0 0 0a 0 0 0 00 0 0 0 0 0 a 0.05S 0 0 0 0 0 00000000000000000000 0 o a 0 0 0 000000000000000000000 000000000000000000000 0 0 0 0 0 00000000000000000000 o ea A I 00g0000000000000000000 o o o o 0000000000000000 0 0 000 o o o o o 0 0 000000000000000000 O o o o o o 00 00 0000000000000000 o o o o o 000000000000000000000 o o o o o 000000000000000000000 o o o o o 000000000000000000000 Overlaid.AIM grids Figure 10: Original discretization and equivalent AIM grids for the geometry of Figure 8 Standard moment method (0 pol) -------- Standard moment method (0 pol) A Self-cell only (0 pol) [Avg. Error = 36.81 dB] v Self-cell only (4 pol) [Avg. Error = 31.96 dB] 10 Cl U U U cn ~r 04 1 - 0 -10 -20 -30 -40 -50 0 15 30 45 60 75 90 Observation angle 0 (deg.) Figure 11: Error introduced by retaining only the self-cell interactions of the moment method 37

B ~ 0 -0.5 -1 -".1.5 '-2.5 4 -4 -2 2 -2 2 -4,2 -4 4 y 4 4 X X (a) (b) Figure 12: (a) Flat and (b) Curved plate with equal side lengths and discretization rates, resulting in equal number of unknowns. While the moment method yields equal solution time for both geometries, AIM would accelerate the solution for the geometry in (a) considerably more than that for the geometry in (b) Both geometries contain 1681 nodes, 3200 elements, 4800 edges Gridded at tJ40 Gridded at 2J10 -.. 445- - -. -2.-.5C-%-0.5 0.5. ~ 1.5,. X2 X ftgo r ini>t'.7/~ (a)7 r/7779~$ itt4T 0 / 0./ 4 4 -0A ttzt4 A -I27 7' 0...?-.y. ^*^.:4:"':-". '/ —'....^. '","-...., '....Y ^' -(.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2 -1.5 1 -0.5 0 0.5 1 1.5 2 (a) (b) Figure 13: (a) Geometry of a IX square plate gridded at A/40 and (b) 4A square plate gridded at A/10. While the moment method results in equal solution times for both geometries since they have equal number of unknowns, AIM would accelerate the solution for the geometry in (a) considerably more than that for the geometry in (b) owing to the smaller FFT pad for the geometry in (a) 38

Annular Slot I Ground plane I ~~~~e.. --- —-------- Cavity f Feed Sf Figure 14: Geometry of a cavity-backed annular slot antenna in a ground plane Probe y a = 12.35 cm b = 0.75 cm \P =p 7.7 cm x Frequency = 1 GHz 0 FE-BI FE-AIM (0.53.) FE-AIM (0.35.) FE-AIM (0.25 X) Non-zeros in near Z matrix 4656 4656 2245 1537 -5 4 -10 - I -15 v -20 0 c -25.-" & -30 N -35 ~ -40 z -45 Elevation plane pattern at 4=50 _[ x I I I I I I I I I I I -- II -50... -90 -75 -60 -45 -30 -15 - FE-BI (Co-pol) * l *. l...il l l ll. l. i l Average Error (dB) Co-pol X-pol 0.4402 0.2998 0.6343 0.5334 0 15 0 (deg.) 30 45 60 75 90 I FE-AIM (0.35 2) FE-AIM (0.25 X) -- FE-BI (X-pol) FE-AIM (Threshold = 0.53X) (Co-pol) FE-AIM (Threshold = 0.53X) (X-pol) -- FE-AIM (Threshold = 0.35X) (Co-pol) - - - FE-AIM (Threshold = 0.35X) (X-pol) FE-AIM (Threshold = 0.25X) (Co-pol) FE-AIM (Threshold = 0.25X) (X-pol) Figure 15: Radiation pattern from an annular slot in the q = 00 elevation plane 39

a = 12.35 cm b = 0.75 cm p = 7.7 cm Frequency = 1 GHz Frequency = 1 GHz Non-zeros in near Z matrix FE-BI 4656 FE-AIM 915 (0.15 X) FE-AIM 1537 (0.25 X) FE-AIM 2245 (0.35 X) Average Error (dB) Co-pol X-pol FE-AIM 1.0524 1.0373 (0.15 ) FE-AIM 0.6142 0.9105 (0.25 X) FE-AIM 0.876 0.2506 (0.35 k) (a) 0 "0 -S3 0 -"0 0 z — -10 -20 -30 -40 -90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 Elevation angle 0 (deg.) FE-BI (0 pol) (Co-pol) ---- FE-BI (0 pol) (X-pol) --- FE-AIM (0 pol) (Co-pol) (Threshold = 0.15 X) - - - FE-AIM (0 pol) (X-pol) (Threshold = 0.15 X) * FE-AIM (H pol) (Co-pol) (Threshold = 0.25 X) * FE-AIM () pol) (X-pol) (Threshold = 0.25 X) FE-AIM (0 pol) (Co-pol) (Threshold = 0.35 X) FE-AIM (0 pol) (X-pol) (Threshold = 0.35 k) (b) Figure 16: Radiation pattern from an annular slot in the o = 90~ elevation plane 40

Non-zeros in near Z matrix FE-BI 4656 x I - a = 12.35 cm b = 0.75 cm p = 7.7 cm Frequency = 0.73 GHz FE-AIM (0.25 X) FE-AIM (0.33 X) FE-AIM (0.4 X) 2184 3272 4656 FE-AIM (0.25 X) FE-AIM (0.33 X) FE-AIM (0.4 X) Average Error (dB) Co-pol 1.3124 1.0666 0.00086 (a) 7:3 E 4-. ct CT P 0 -10 -20 -30 -40 -50 -60 -70 -80 FE-AIM ( pol) (Threshold = 0.53 A) 0 15 30 45 0 15 30 45 60 75 90 Elevation angle 0 (deg.) (b) Figure 17: Bistatic scattering pattern from an annular slot; Normal incidence in the q = 0~ plane and observation is in the s = 900 elevation plane 41

Prohp y a = 12.35 cm b = 0.75 cm p = 7.7 cm x0 / Near Z non-zeros Near-zone threshold / 45 held constant at 10.5 cm AIM: 2245 80(,,' Lines- FE-BI 6 ------ --- -- ----- - I)ots- FE-AIM 40(0- - - 4 --- --------- I '- - I — - - +4 ---I - -- _ --- I I '' I \I I, I I I I I\I I I I 42( ----- - ------- --- T — 1 --- T,.(,., \ I i \ l 0-4 0 oo-r -. -r- --— 7- - -__ _ _. -\ _ _ — ______.. __^ I I I, I I I I I - ' I I I I I ~; I I I I I -20 ------------------------- ----— T --- — II I I I I I 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Frequency (GHz) Frequency (GHz) FE-BI FE-AIM 1.0667 71.74 + j259.04 71.46 + j268.51 1.1333 248.51+j422.88 258.03 + j453.73 1.2 708.59-j170.31 710.39- j269.71 1.25 238.47-j259.27 206.96 - j266.05 1.3 109.77-jl46.96 95.48- jl142.82 1.35 67.63 - j71.61 59.84 - j 67.16 1.4 49.7- jl16.48 44.42-j 12.95 Figure 18: Input impedance of a very narrow annular slot computed with FE-BI and FE-AIM 42

! ~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 = cm Slotwi248(Non-Pec)th = 1.5 cm 496 (Slot) 248 (Non-Pec) Frequency = 1 GHz (wavelength = 30 cm) e1 n..,..,..,..,.. 1U 0 -10 - -20 t -30 L) X -40 a -50.CD -60 -70 -80 F I I I II -- AIM (Non-zeros in near Z = 5008) / - BI (Non-zeros in near Z = 30876) Average error = 0.8 dB I) I F l,........ 0 15 30 45 60 Observation angle 0 (deg.) 75 90 FE-BI ----- FE-AIM (10 cm threshold) Figure 19: Bistatic RCS at normal incidence (O = 900 plane) from a cavity-backed slot array computed with FE-BI and FE-AIM 43

Total Edges: 5529 Non-PEC: 983 Surface Edges: 2110 (PEC) 496 (Slot) 248 (Non-Pec) 0 -10 -20 ' -30 a.t 4 Inner radius of each slot = 7.325 cm Cavity diameter = 49.4 cm Cavity depth = 1.5 cm Cavity filling r= 2 Slot width = 1.5 cm Frequency = I GHz (Lambda = 30cm) Co-pol (FE-BI) -75 -60 -45 -30 -15 0 15 30 45 60 75 Observation angle (deg.) Figure 20: Radiation from a cavity-backed slot array computed with FE-BI and FE-AIM in the $ = 900 plane 44

z'A y Patch - ~; fI.................................. 0.5 cm I Y L x (a) "0 m 0-1 -/ ED (X 0 -10 -20 -30 -40 -50 -60 1 2 3 4 5 Frequency (GHz) (b) Figure 21: (a) Geometry and surface discretization of a cavity-backed patch antenna (b) Monostatic RCS at normal incidence versus frequency - cavity filling has a Cr = 2.2 - jO.002 and ur = 1 45