Physics‐based iterative scheme for computing antenna array embedded element patterns

Rigorously accounting for mutual coupling effects in antenna array synthesis, generally requires that embedded element patterns (EEPs) be determined. The method of moments (MoM) is typically used. Multiple linear system solutions are required, corresponding to every excitation vector associated with an EEP. This paper presents the direct coupling technique (DCT) to obtain the MoM solution for arrays with disjoint elements, by iteratively solving a series of localized subproblems, one associated with each array element. It utilizes a physics‐based, direct‐coupling approximation to incorporate global coupling into the local problems. Local matrices do not change between iterations or with excitation vectors, hence they can be inverted as a preprocessing step, allowing for efficient solution of multiple excitation configurations. Numerical results demonstrate rapid convergence. The convergence rate can be controlled via local problem domain sizing. Since the DCT is ideally suited to parallelization, it is applicable to very large arrays.


| INTRODUCTION
Neglecting the effects of mutual coupling in analytical antenna array synthesis methods, generally leads to suboptimal designs. 1 However, accounting for these effects may lead to expensive synthesis processes relying either on global optimization methods, 2 or analytical methods where the embedded element patterns (EEPs) for all array elements must be determined for multiple configurations. 3 The method of moments (MoM) formulated using the electric field integral equation (EFIE) and Rao-Wilton-Glisson (RWG) basis functions and Galerkin testing, is the typical approach for the electromagnetic analysis of arrays consisting of perfect electrical conductor (PEC) antennas in free space, 4 yielding the following system of linear equations: In (1), Z J , , and V denote the impedance matrix, the column vector of unknown basis function coefficients and the known excitation vector, respectively. To obtain the EEPs for all array elements, (1) must be solved for each antenna excited in turn, with all others being passive. 5 In direct solution methods the system matrix is factorized once and then used to obtain any EEP through back-substitution. Since direct solution runtime scales unfavorably, iterative solvers not requiring factorization, are of interest. Each iteration requires a matrix-vector product (MVP) involving the global impedance matrix. Matrix compression schemes such as the multilevel fast multipole method (MLFMM) 4 or adaptive cross approximation (ACA) 6 can achieve reduced MVP cost complexity. However, iterative solvers require a repeated iteration sequence for each new excitation vector, as well as expensive preconditioners with generally unpredictable numbers of iterations. Other methods, such as those using generalized scattering matrices (GSMs) 7 or macro basis functions (MBFs), 8 attempt to reduce the number of degrees of freedom (DoFs) in the global problem to make direct solution more feasible. For GSM methods, accuracy depends on the number of modes included and there is a restriction on the minimum inter-element distance. With MBFs, accuracy depends on the validity of the reduced basis set, for representing the true solution vector. There is also a recently proposed iterative method relying only on elemental far-field data, 9 which is extremely fast, but yields approximate results of which the error relative to the full-wave solution cannot be controlled.
This work introduces a formulation to obtain the full array solution to user-specified accuracy (measured with the normalized residual as is standard for iterative linear system solvers), by iteratively solving a series of localized subproblems, one associated with each array element. It utilizes a physics-based, directcoupling technique (DCT) to incorporate global coupling into the local problems. Local problem matrices do not change between iterations or with excitation vectors, allowing for efficient solution of multiple excitation configurations. These properties make the formulation ideally suited to parallelization. Note that the domain Green's function method (DGFM) for array analysis has similar broad characteristics, however, the DGFM in its standard form, 10,11 is not suitable for excitation configurations where many elements are passive. Also, DGFM local solution matrices are excitation dependent. Section 2 presents the new DCT formulation, followed by numerical results in Section 3 to assess its performance. Section 4 concludes the paper.

| DCT FORMULATION
For an array consisting of N identical, disjoint elements, each with M degrees of freedom (DoFs), (1) can be blockpartitioned as (2) , denotes the M M × coupling submatrix between source element q and observer element p. J p and V p are the M-dimensional current and excitation subvectors, respectively, for element p.
To obviate the need to directly solve (2), an iterative method is formulated wherein each iteration's solution consists of a series of localized solutions.

| Physics-based, direct-coupling approximation of nonlocal currents
Define the series of localized solutions as solving the current on each element individually. With the goal of formulating an approximation of nonlocal currents to incorporate global coupling in the local solution for element p, consider the linear expression In (3), the vector β and matrix Λ are known coefficients depending on physical coupling and the previous iteration's solutions on all elements except p. To achieve a linear relationship as in (3), consider the q-th block row of the partitioned MoM system (2): Suppose that there exists a previous k ( − 1)-th iteration solution and use it to approximate the effect of the third term in (4), due to all antennas apart from q and p. This allows (4) to be expressed in an approximate form as with the approximated term defined as -th global solution vector and p ext is the set of all array elements external to element p (thus all excluding element p). Rearranging (5) yields as the direct-coupling based approximation of a nonlocal elemental current. The coefficients in (7) are defined as and the current on element p is solved by substituting (7) into the p-th block row equation of (2), as Solving (10) for every ∈ p N {1, …, } yields J k ( ) , the current solution for the entire array after iteration k.

| Extended local domains
The local domains are now extended to consist of clusters of antennas rather than single elements. This serves two purposes: (i) it more accurately incorporates strong local coupling by formulating the MoM on the focus element and its closest neighbors; and (ii) it improves the accuracy of the far-current approximation in (7)  ; and let p ext now denote the set of all elements external to S p . The MoM is rigorously formulated on domain S p and the DCT formulation is obtained by generalizing the approximation (7) to model nonlocal currents through direct coupling to all elements in S p , as where the active impedance matrices and excitation vectors are defined as . For each local MoM problem, the current on element p is considered to be the most accurately solved and only its solution is retained. Hence a global solution J k ( ) is obtained by solving (12) for every element in the array.

| Summary of the DCT algorithm
The normalized residual error at the end of the k-th iteration is calculated in the standard way, as where    F denotes the Frobenius norm. The iteration scheme continues until the stopping criterion of  μ ϵ k ( ) is met (with μ = 10 −3 in this paper). The formulation consists of the following steps: 1. Choose the number of nearest neighbors, Q, to be included in each cluster.

| Application to EEP calculations and computational cost
To obtain multiple EEPs for a given array, (1) must be solved for multiple excitation vectors. As a preprocessing step, set up all local problem matrices (12)   , respectively. Clearly, the biggest strength of the method is MVP cost, which scales better with N than that of the MLFMM, which is MN . The comparatively expensive preprocessing, active excitation vector computations, and convergence testing costs may all be greatly reduced by employing matrix compression schemes and by parallelization. The DCT is clearly ideally suited to the latter. However, these aspects are beyond the present scope. Finally, this discussion relies on the assumption that few iterations are required, which is assessed next.

| Irregular LPDA arrays
This section presents convergence results for large, irregular log-periodic dipole antenna (LPDA) arrays. Figure 2 describes the element. The arrays have 225, 400, and 625 elements, respectively, with the 625-element layout shown in Figure 3. The smaller arrays are concentric subsets of the largest array. Three test excitation configurations are considered for each array. These are the uniform excitation case and two singleexcitation cases (as required for EEP computations), where a central and an edge element are excited in turn. The EEP excitation elements for the 625-element array are marked in Figure 3  First, consider the 225-element array. Figure 4A shows the convergence results. Convergence is reached in all cases, with the convergence rate increasing with increased cluster size. For a given cluster size the convergence rate is seen to be very similar for all three excitations. To further investigate the excitation vector's effect upon convergence, Figure 4B shows convergence results for Q = 2 with uniform excitation and with EEP excitations at 12 uniformly distributed elements. The convergence trails form a narrow band, confirming that the DCT convergence rate is practically independent of the array excitation. Figures 4C,D show the convergence results for the 400-and 625-element arrays, respectively. It is seen that the DCT convergence rate does decrease as the array size is increased, but that this effect is less pronounced for the larger cluster size values of Q = 4, 8, and particularly, the required number of iterations for convergence grows slower than the number of array elements, for these Q values. This is an important beneficial characteristic for the method's application to very large arrays.

| Regular patch antenna arrays
To assess performance under different circumstances, consider a regular array of 11 × 11 circular pin-fed patch antennas with varied spacing, as shown in Figure 5. The operating frequency is 5 GHz. The patch diameter is 22.4 mm, with PEC-backed substrate height of 1.2 mm and ε = 2.3 r . The layered-media Green's function MoM is used, with system matrices generated by FEKO. 12 Two small inter-element spacings of ∕ λ 8 and ∕ λ 4 are considered (free space wavelength); and again with three excitations: uniform, single middle element, and single corner element.
The convergence results in Figure 5 confirm the observations made on algorithmic performance, for the LPDA array example. Additionally, they show that convergence deteriorates with decreased element spacing and that divergence can occur (as in the Q = 0 case at very tight spacing), but that convergence can be very effectively retrieved by increasing Q. Rapid convergence is observed for spacings of ∕ λ 4 and above, for all Q values.

| CONCLUSION
The DCT iterative scheme for the MoM solution of large antenna arrays consisting of identical disjoint elements is developed, whereby a global solution is obtained through a series of local solutions. For a given array, the local problem matrices remain unchanged between iterations and over different global excitation vectors, hence they can all be inverted once and reused for all desired excitation configurations, such as required for EEP calculations. Rapid iterative convergence is demonstrated for arrays of different sizes, elemental constructions (PEC structures and microstrip patches) and inter-element spacings, using modest local domain sizes. The DCT's convergence rate does decrease with increased array size and reduced inter-element spacing, but can be readily controlled by adjusting the local domain size.
Localization means that the DCT is amenable to very efficient parallelization, which together with its attractive convergence behavior, means that it has the potential to be an effective tool for solving very large arrays. However, to realize its full potential, accelerating the setup of local system matrices and the calculation of local problem excitations, for example, through ACA compression, 6 is necessary. Such implementations could be considered in future, to compare computational costs of the DCT and other array solver schemes.
The DCT does not require all elements to be identical. The implications of this could be considered in future.

ACKNOWLEDGMENTS
Funding by the South African Radio Astronomy Observatory (SARAO) toward this research is hereby acknowledged (www.ska.ac.za). This work was supported by the National Research Foundation of South Africa under grant 75322.

DATA AVAILABILITY STATEMENT
All results data is included in the form of graphs in the paper.