## 1. Introduction

[2] 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.

[3] 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* [1986] 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.* [1988], *Chow et al.* [1991], and *Wang et al.* [1998], 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].

[4] 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.* [2004] 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* [2007]. The latter is new in that the previous related work of *Marin et al.* [1989] and *Barkeshli et al.* [1990] 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.

[5] 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 *e*^{jωt} time dependence is assumed and suppressed for the fields and sources in the following development.