A fast approach is developed here to analyze large finite planar periodic arrays of elements in a grounded multilayered medium with relatively arbitrary element truncation boundaries. An array shape matrix is introduced which contains information on the array element truncation boundary, and it can be easily incorporated into preconditioned iterative method of moments matrix solvers without significantly increasing computational resources. The performance of the present approach is illustrated through numerical results.
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 Large, finite, periodic, phased antenna arrays are of considerable interest, particularly, in the context of arrays of printed elements, or slots, in grounded multilayered media, which are attractive for aerospace, satellite and shipboard applications, since such applications require relatively high gain, low profile, and electronic beam steering capability. It is therefore useful to develop an efficient tool which can analyze the performance of large arrays and help the design process. A variety of numerical methods have been developed in the past for this purpose. Among all these methods, the method of moments (MoM) is considered the most suitable tool for this class of problems, since by using a special grounded multilayered dyadic Green's function as the kernel of the governing integral equation, one is able to restrict the unknowns to be solved to only the equivalent array element currents thereby resulting in the fewest number of unknowns. However, when the array size becomes very large, the conventional MoM can become highly inefficient or even intractable, despite the use of the special Green's function, due to the exorbitant computational cost and memory storage requirements for solving the corresponding MoM matrix equation.
 A number of techniques have been developed to improve the efficiency of the MoM-based full-wave solvers in order to facilitate the handling of very large array problems. These techniques can be divided into two major categories, namely the first category for rapidly evaluating the multilayered media Green's functions to drastically reduce the MoM operator matrix filling time, and the second category which is concerned with the implementation of efficient iterative solvers to accelerate the solving process. The first category includes two main approaches, one of which is incorporated in the present work, and it develops an asymptotic closed form approximation to the special Green's function for source-observation point separations which are larger than about one free space wavelength [see Jackson and Alexopoulos, 1986; Marin et al., 1989; Barkeshli et al., 1990], while the singularity extraction technique of Jackson and Alexopoulos  that is extended in this work to allow for the presence of multilayered (of more than one or two layered) grounded media is employed for smaller separations. The second approach in this first category employs a closed form spatial domain Green's function technique based on the discrete complex image method (DCIM) as, for example, in the work of Fang et al. , Chow et al. , and Wang et al. , respectively. The second category which deals with fast iterative solvers involves, for example, an application of the fast matrix-vector multiplication in MoM using FFT to accelerate the iteration process [see Sarkar et al., 1986; Peters and Volakis, 1988] together with the use of an effective preconditioner [see Canning and Scholl, 1996; Lee et al., 2003; Zhao and Lee, 2004; Janpugdee et al., 2006].
 Although the methods mentioned above can significantly improve the efficiency of the MoM solver, most of them typically assume finite planar periodic arrays with rectangular element truncation boundaries, i.e., an M × N array consists of N rows with M elements in every row. However, arrays in practical applications can have nonrectangular element truncation boundaries, such as circular or hexagonal arrays, etc.; thus it is desirable to develop a full-wave solver which can handle these kinds of arrays as well. Fasenfest et al.  developed an algorithm using a method similar to the adaptive integral method (AIM), which can be applied to efficiently compute the impedance matrix for the case of arrays with nonrectangular boundaries. This method is shown to be effective, but it involves a certain level of complexity due to the Lagrange-Green's function interpolation process. In this work, a simpler yet efficient approach is developed to extend a conventional MoM solver such that it can analyze finite planar periodic arrays with relatively arbitrary element truncation boundaries, which for convenience will later be referred to as nonrectangular arrays, as depicted in Figure 1. Of course, the present approach remains valid also for a rectangular element truncation boundary which is a special case. Although this work deals primarily with finite planar periodic arrays of printed elements (or even slot elements) in grounded multilayered media, the approach developed in this work can be easily incorporated into any existing MoM solver for finite planar periodic arrays with only minor modifications. It is noted, as indicated above, that an additional significant increase in the computational speed of the MoM approach has been achieved in this work due to a drastic drop in the fill time of the MoM operator matrix resulting from the use of an asymptotic closed form evaluation of the special, grounded, multilayered dyadic Green's function that has been developed recently and which is being reported separately by Mahachoklertwattana . The latter is new in that the previous related work of Marin et al.  and Barkeshli et al.  was restricted to only a single or double layer medium, however, the extension to the multilayered case is not trivial, but fortunately, the final result is not correspondingly complicated because of an interesting mathematical simplification which occurs in the end.
 The present paper is organized as follows. Section 2.1 precedes the array shape matrix and the corresponding shaping operator which will be used to obtain the MoM operator matrix for a nonrectangular array in terms of the one for a corresponding circumscribing rectangular array. The application of the array shape matrix in the preconditioned iterative solvers is discussed in section 2.2 and is followed by some numerical results in section 3. A boldface capital letter is used to denote a matrix, and an underlined boldface lower case letter is used to denote a column vector throughout the paper. Also, an ejωt time dependence is assumed and suppressed for the fields and sources in the following development.
2.1. Array Shape Matrix
 Consider first the case when only one expansion mode is used to represent the equivalent current distribution on each element of an M × N rectangular array. Testing each expansion mode and enforcing the boundary condition results in the system of equations Zi = v, where i is a column vector with the MN unknown expansion coefficients and v is a column vector resulting from the testing of the impressed field. Z is the MN × MN method of moments (MoM) operator matrix. By exploiting the periodicity of the array, Z can be arranged such that it is a NM × NM “block Toeplitz with Toeplitz blocks” (BTTB) matrix, i.e., a N × N block Toeplitz matrix with M × M Toeplitz blocks [see Janpugdee et al., 2006]. However, if the number of elements in each row are not the same, the Toeplitz blocks will not be uniform, and thus the BTTB structure will be destroyed. This is highly undesirable since the MoM operator matrix cannot be represented by only one row as is typically the case for finite planar periodic arrays, and furthermore the fast matrix-vector multiplication method using FFT also cannot be applied directly in this situation. In contrast, one can follow the procedure described below to overcome the above-mentioned limitations.
 Now, let ZA be the MoM operator matrix for the original finite array with a nonrectangular element truncation boundary, which is circumscribed by a somewhat larger M × N rectangular boundary. Assume that the total number of elements in the original nonrectangular array is K, then obviously K < NM. It is evident that ZA is a K × K non-BTTB matrix and it must contain information regarding the element truncation boundary. However, one can observe that all elements in ZA must be included in Z, which is the MoM operator matrix for the circumscribing M × N rectangular array. In other words, ZA can be constructed from Z.
 The observation made above leads to the introduction of an array shape matrix A which is an NM × K matrix which contains the element truncation boundary information. Its elements are defined as
where the function index(k) maps the kth index in the nonrectangular array into the corresponding double index (n, m) for the equivalent circumscribing rectangular array or in other words, it relates the kth element in the nonrectangular array to the (n, m)th element in the rectangular array. It is also worthwhile noting that (n − 1)M + m corresponds to the single index obtained by simply reordering the double indices of the two-dimensional array by row. Figure 2b shows an example of an octagonal array and its element indices. Also shown in Figure 2a are the double indices used for the conventional M × N rectangular array which encloses (or circumscribes) the octagonal array. For the example shown in that figure, K = 36, M = 8, N = 6, and index(1) = (1, 3), index(15) = (3, 5), index (36) = (6, 6), etc. Thus, a3,1 = 1, ak,1 = 0 for ∀k ≠ 3. Likewise, a21,15 = 1, ak,15 = 0 for ∀k ≠ 21 and a46,36 = 1, ak,36 = 0 for ∀k ≠ 46. Using the array shape matrix, one can obtain ZA from Z via the relation
The operation ATZA simply represents the selection of appropriate rows and columns in Z to construct a new matrix according to the information specified in the array shape matrix, and thus AT(•)A can be considered a ‘shaping’ operator. It is noted that ATA = K, where K is the K × K identity matrix. Figure 2c shows the form of the A-matrix for the above example.
 The approach mentioned above can be easily extended to the case of multiple expansion modes by first noticing that the MoM matrix operator matrix in this case becomes a block matrix with each block being a BTTB matrix. Thus, by applying the shaping operator AT(•)A given in (2) to each BTTB submatrix, the MoM operator matrix for a nonrectangular array can be obtained. Now, introducing A such that
where Nmode denotes the number of expansion modes per array element, and the operator denotes the Kronecker product, which can be given by
where A is a K × L matrix, and ai,j denotes the (i, j)th element of A. Using (3), the MoM operator matrix for the nonrectangular array using multiple expansion modes can be given by
Clearly, the K × K submatrix ZAi,j represents the coupling between the ith and the jth expansion modes.
 It is evident from (2) and (5) that by introducing the array shape matrix, the block-Toeplitz property of the MoM operator matrix can still be exploited and therefore the storage requirement can be kept minimal. Also, it is worthwhile noting that the shaping operator can be applied without explicitly constructing the matrix A; it only needs to store the information regarding the element truncation boundary, thus a large amount of memory is not required. Furthermore, the product Ai can be performed by just adding a certain number of zero elements to i without actually doing the multiplication, thus the computational cost for this would be extremely small. Likewise, the product ATi can be obtained by just removing a certain number of elements in i, thus it would require a minimal computational cost as well.
2.2. Application of the Array Shape Matrix in Iterative Solvers
 It is well-known that the computational cost of iterative solvers is proportional to the cost of matrix-vector multiplication. For an NM × NM BTTB matrix, the fast matrix-vector multiplication method using FFT can be applied to compute a matrix-vector product, which requires only ��(NM log NM) operations instead of ��((NM)2) operations. Since the MoM operator matrix for a M × N finite planar periodic array becomes a BTTB matrix for one expansion mode case and a block matrix with BTTB blocks for the case of multiple expansion modes, this method can be applied to improve the efficiency of iterative solvers. By using the array shape matrix defined in the previous section, it is evident that this fast multiplication method can be applied to nonrectangular array problems as follows. For the one expansion mode case, using (2) yields
where y = Ai. Since Z is a BTTB matrix, the matrix-vector product in the parentheses can be computed using FFT. It is noted that the multiplications involving the matrix A can be performed without actually computing the product, thus the additional cost due to the multiplications of A and AT will be extremely minimal. This approach can be easily extended to the case of multiple expansion modes by noticing that the matrix-vector of each submatrix ZAi,j can be computed in the same way as
where ij is the coefficient vector for the jth mode and yj = Aij. Therefore, the fast matrix-vector multiplication using FFT is also applicable for nonrectangular array problems with additional shaping operations that are not expensive.
 In general, the efficiency of iterative solvers depends greatly on the convergence rate of their solutions, which in turn is a function of the condition number of the pertaining matrix. In this work, a DFT-based preconditioner developed by Janpugdee et al.  is implemented to accelerate the convergence of iterative solvers. This preconditioner is considered effective since it is a good approximation to a block matrix with BTTB blocks and can be computed efficiently by FFT without explicitly constructing large matrices. It can be obtained by first noticing that the impedance matrix in the transform domain, denoted here by and given by
is highly sparse and near diagonal. Here, FNM denotes the two-dimensional NM × NM Fourier matrix given by
where FN, FM denote N × N, M × M Fourier matrices, respectively. Therefore, one can approximate the impedance matrix in the transform domain by retaining only diagonal elements, i.e.,
and d is the vector that contains all diagonal elements of . Hence, the inverse transform of d should well approximate Z, i.e.,
The DFT-based preconditioner can then be obtained by performing the inverse transform of (FNMHdFNM)−1, which is given by
 As discussed earlier, the MoM operator matrix for the nonrectangular arrays can be obtained by simply applying the appropriate shaping operator to the matrix for the enclosing rectangular array. This observation leads to the choice of a preconditioner matrix, MA−1, for nonrectangular arrays, where
and M−1 is the preconditioner for the rectangular array. It can be observed that if the number of elements in the nonrectangular arrays is comparable to the number of elements in the enclosing rectangular array, i.e., K ≈ NM, the array shape matrix will approach the identity matrix, thus MA−1 ≈ M−1. The preconditioned system for the one expansion mode case can be given as follows:
It is noted that AATZ on the left hand side of (16) will only ‘remove’ certain rows and columns which correspond to elements not included in the nonrectangular arrays from Z, and thus M−1AATZ should have the spectral properties comparable to those of M−1Z, which in turn makes the spectral properties of MA−1ZA comparable to those of M−1Z. Therefore, this modified preconditioned system should work reasonably well compared to the original one. Finally, the modified preconditioned system for the case of multiple expansion modes can be simply obtained by using A instead of A in (16), and it can be given by
and M−1 is the preconditioner for the rectangular array as before.
3. Numerical Results
 In this section, several numerical results are presented to illustrate the effectiveness of the approach described above. All results have been computed on a desktop PC (3 GHz Pentium 4 with 512 MB memory). The iterative solvers used in this work are preconditioned bi-conjugate gradient stabilized (PBiCGSTAB) and conjugate gradient normal residual (CGNR), and the convergence condition is when the relative residual falls below 10−4. Also, the basis functions used in this work are based on the overlapping piecewise sinusoidal basis function.
 The first example deals with radiation from printed dipole arrays in a grounded multilayered medium with hexagonal and rectangular array boundaries. The elements and substrate as well as the array element spacings used in this calculations are shown in Figure 3. Since the printed dipole is assumed to be thin, only the -directed current is excited; thus, only -directed expansion modes are required. Three expansion modes per one element are used in this calculation. The hexagonal truncation boundary is shown in Figure 4, which is circumscribed or enclosed by a 100 × 87 boundary, and the total number of elements in the hexagonal array is 6,500, while that of the circumscribing array is 8,700. Figure 5 shows the calculation of radiation from the array of printed dipoles with the hexagonal truncation boundary and also for reference, the radiation from the rectangular array which encloses the hexagonal array. As can be seen from the figure, the radiation pattern of the hexagonal array has lower sidelobes in the principal plane (ϕ = 0° plane) while it has higher sidelobes in the ϕ = 45° plane. It is noticed that in the ϕ = 0° plane, the hexagonal array has no edge perpendicular to the pattern cut while the rectangular array has two such edges, which leads to more significant edge diffraction effects resulting in higher sidelobes. It is also noted that the opposite occurs in the ϕ = 45° plane. Table 1 shows the number of iterations and computational solution times required for solving this printed dipole array problem when using different iterative solvers. As can be seen from the table, the preconditioner implemented in this work can help accelerate the convergence process for both rectangular and hexagonal arrays. It is noted also that the number of iterations for the hexagonal array is smaller; this is because the number of unknowns is smaller than that of the circumscribing rectangular array.
Table 1. Number of Iterations and Solve Times for CGNR and PBiCGSTAB When Solving the Array of Printed Dipoles
Number of Iterations
Solve Time (s)
100 × 87 rectangular array
(Program stopped before end)
 The second example deals with the plane wave scattering from patch antenna arrays on a grounded single-layered medium with elliptic and rectangular boundaries. The dimensions of the patch antenna used here are (L, W) = (3.66, 2.6) cm and the spacing in both directions is 5.5517 cm. The substrate is 0.158 cm thick and its dielectric constant is 2.17. The incident plane wave comes from (θi, ϕi) = (30°, 0°) at 3.7 GHz. Since both -directed and -directed currents are excited by the incident field, both -directed and -directed expansion modes are needed and thus the number of expansion modes per element used here is 6. The elliptic truncation boundary is shown in Figure 6, which is enclosed by a 200 × 100 rectangular boundary, and the total number of elements in the elliptic array is 15,708, while that in the enclosing array is 20,000. Figure 7 shows TM bistatic RCS pattern in the E-plane of these patch antenna arrays with elliptic and rectangular element truncation boundaries. As can be seen from the figure, the RCS patterns of both elliptic and rectangular arrays have almost the same peak levels, but the elliptic array has significantly lower sidelobes, which is same as the trends seen in the radiations from printed dipole arrays with hexagonal and rectangular boundaries. Table 2 summarizes the number of iterations and computational solution times required for solving this patch antenna array problem when using different iterative solvers. The same trend as that observed in the printed dipole array case can be seen here as well.
Table 2. Number of Iterations and Solve Times for CGNR and PBiCGSTAB When Solving the Array of Patch Antennas
Number of Iterations
Solve Time (s)
200 × 100 rectangular array
 The last example is the radiation from probe-fed patch antenna arrays on a single-layered substrate with circular and square boundaries. The dimensions of the patch antenna are (L, W) = (2.0, 3.0) cm. and the spacing in both directions is 4.0 cm. The probe feeds are located 0.65 cm from the long edge of each patch and its inner, outer radii are 0.043 and 0.14 cm, respectively. The substrate is 0.127 cm thick and its dielectric constant is 10.2 − j0.051. The circular boundary is shown in Figure 8, which is circumscribed or enclosed by a 128 × 128 square element truncation boundary, and the total number of elements in the circular array is 12,892, while that in the circumscribing array is 16,384. The unknown current for each element is expanded by -directed patch modes, -directed patch modes, wire modes and an attachment mode to enforce the continuity of the current at the probe-patch junction. The total number of expansion modes per element is 27. The square array is excited by an exponential excitation taper, while the circular array is excited by the uniform taper. Figure 9 shows the E-plane radiation patterns of both arrays when scanned at (θs, ϕs) = (30°, 0°) and operating at 2.3025 GHz. As can be seen, the circular array has lower sidelobes and thus in certain cases shaping of array element truncation boundaries can lower the sidelobe levels in the principal plane. Table 3 summarizes the number of iterations and solve times required for solving this patch antenna array problem when using PBiCGSTAB. It is noted that the solution cannot be obtained without a preconditioner in this case. As can be seen from the table, the overall trend observed here is similar to those seen in other cases.
Table 3. Number of Iterations and Solve Times PBiCGSTAB When Solving the Array of Probe-Fed Patch Antennas
Number of Iterations
Solve Time (s)
128 × 128 square array
 An approach is presented which extends existing array MoM-based full-wave solvers for finite planar periodic arrays to also treat finite planar periodic arrays with relatively arbitrary element truncation boundaries. It can be implemented without many modifications while maintaining the efficiency of the solvers. In fact, it is found that it takes shorter time to solve nonrectangular arrays than corresponding circumscribing rectangular ones due to smaller numbers of unknowns. Although only results from arrays of printed elements in grounded multilayered media are shown in this paper, the present approach is expected to be easily incorporated in any MoM-based full wave solver code for finite planar periodic arrays.