## 1. Introduction

[2] The numerical simulation of the electromagnetic behavior of large finite antenna arrays remains a challenge [*Holter et al.*, 2000; *Lee et al.*, 2005], especially when the elements of the array have complex shapes and involve finite dielectric parts. This is also true for the analysis of periodic bandgap materials [*Crocco et al.*, 2007] or metamaterials [*Shelby et al.*, 2001]. For this type of structures, methods based on Finite Elements, or on a combination of Finite Elements and Integral Equations [see, e.g., *Vouvakis et al.*, 2006] are well suited. For large structures, as long as the materials involved are piecewise homogeneous, the method of moments (MOM), together with its fast extensions, remains an excellent candidate, mainly because unknowns are limited to the interfaces between homogeneous regions and because of its ability to accurately handle dielectric materials with high contrast.

[3] Over the last few years, noniterative solution techniques have been increasingly investigated. Such methods are known under different names, like the Macro Basis Functions approach (MBF) [*Suter and Mosig*, 2000], or the Characteristic Basis Functions approach (CBF) [*Prakash and Mittra*, 2003], Aggregate Basis Functions [*Matekovits et al.*, 2001], or Standard distributions [*Craeye et al.*, 2004]. In each case, slight variants appear in the way to determine such functions. In the following, we will use the name “Macro Basis Functions” (MBFs), while the functions used to test the fields will be named “Macro Testing Functions” (MTFs). MBFs are generally obtained through the solution of smaller problems; for instance very small arrays, or a single element under a large number of excitations. In a first class of methods [*Crocco et al.*, 2007; *Matekovits et al.*, 2007], MBFs are obtained from excitation by a number of auxiliary sources distributed over an equivalence surface and then selected based on the level of their linear independence, with the help of a SVD procedure. In another class of methods, a “primary” MBF is obtained by direct excitation of a given antenna, while “secondary” MBFs can be obtained by solving for the field on neighboring elements, excited by the fields radiated by the primary solution [*Yeo et al.*, 2003]. Infinite-array solutions can also be exploited as MBFs, as explained by *Craeye and Sarkis* [2008], where the Array Scanning Method [*Munk and Burrell*, 1979] is exploited to select the (real and imaginary) scan angles to be considered for the infinite-array solutions.

[4] The two main contributions of this paper are the efficient computation of interactions between MBFs and MTFs and the fast evaluation of embedded element patterns. The interactions will be obtained with a complexity proportional to *N*, or even less for elements with complex shapes, where *N* is the number of elementary basis functions on a single antenna within the array. This method can be regarded as an extension of the approach presented by *Craeye* [2006] to unit cells containing dielectric material. Further time saving is also obtained by eliminating redundant terms in the formulation proposed by *Craeye* [2006] for metallic antennas. Time savings obtained by combinations of MBFs and multipoles can also be found in the work of *Lu et al.* [2007], where scattering by metallic periodic structures is considered in an iterative scheme. Using the MBF approach, the number of unknowns can be reduced by about one to two orders of magnitude. This also enables the fast computation of the array impedance matrix, which is obtained, as will be shown below, by a further reduction of the reduced system of equations.

[5] Other important characteristics of finite arrays are the “embedded element patterns”, which correspond to the radiation pattern obtained when one element is excited, while the other elements of the array are passively terminated. As a result of array truncation, these patterns differ between all elements of the array. Even if current distributions can be rapidly obtained for all possible excitations at individual elements, the computation of embedded element patterns in all directions can be very time consuming and it can become dominant in the overall computation time. We will show that, thanks to the MBF approach, the embedded element patterns, as well as the patterns for any excitation law, can be obtained as a finite series of pattern multiplications, which allows the exploitation of the fast Fourier transform in the pattern computation. Some of the results presented in this paper have already been shown in recent conferences [*Craeye*, 2007; *Craeye and Dardenne*, 2007; *Craeye et al.*, 2007], generally with focus on some subtopics and without mathematical proof of formulas, because of limited available space. The present paper provides full details of mathematical proofs, along with additional examples for larger arrays. Comparisons with brute-force solutions for a 4 × 4 array will also show the convergence of solutions when the number of MBFs is increased. The MBFs are constructed with the help of primary, secondary and higher-order scattering products; the relationship between these MBF and Krylov subspaces will be briefly highlighted in Appendix A.

[6] This paper is organized as follows. In section 2, the MBF approach is recalled and comments are made on the approach adopted for the determination of the set of MBFs. The method is then particularized to arrays of complex elements involving finite dielectric material. A formulation for the subsequent computation of the array impedance matrix is provided in section 3. Section 4 explains how the computation of the MoM impedance matrix can be accelerated with the help of the Multipole approach in the case of dielectric volumes containing metallic surfaces. Fast pattern evaluation is detailed in section 5, and numerical examples for the case of complex elements are provided in section 6. Finally, comments, concluding remarks and further prospects are provided in section 7.