Physical preconditioning for modeling 2D large periodic arrays

We investigate diffraction effects on truncated periodic structures at the field operator level. A physically based approximate operator inverse of the electric field integral equation (EFIE) is used either to obtain an accurate first estimate of element currents, or as a preconditioner for a more rigorous, iterative solution. To illustrate, we analyze linear arrays of cylindrical elements constituting antennas or periodic structures with arbitrary element geometry and excitation, though the method can be extended to 3D problems. We show the reduction in the matrix solution time that is achieved, and also discuss the physical insight into array truncation effects that the method provides. The MoM matrix and preconditioner, once constructed for a given array, are excitation independent and may be used repeatedly to obtain or analyze element current distributions, element tapering and scanning effects, element input impedances, active input impedances, element-failure impact on the radiating system, or coupling between element pairs in a finite array.


Introduction
The number of applications for large array antennas and periodic struchires grows increasingly, thus accentuating the need for efficient and accurate means to analyze these structures. A method of moments (MOM) approach provides the required accuracy, although direct MOM analyses require long matrix fill and solve times as well as large amounts of memory to store the MOM matrices. The array's intrinsic translational symmetry along with asymptotic estimates of mutual coupling between array element pairs can be used to reduce the fill time. To reduce the solution time, various algorithms have been developed that use periodicity-induced physical properties. For example, [l], [2] use an a priori estimate of the fields scattered by truncated arrays, which behave as Floquet-modulated-diffracted fields [3], to construct global basis functions. Here, instead, we investigate diffraction effects on truncated periodic structures at the field operator level. A physically-based approximate operator inverse of the electric field integral equation (EFIE) is used either to obtain an accurate first estimate of element currents, or as a preconditioner for a more rigorous, iterative solution. To illustrate, we analyze linear arrays of cylindrical elements constituting antennas or periodic structures with arbitraty element geometry and excitation, though the method can be extended to 3D problems. We show the reduction in the matrix solution time that is achieved, and also discuss the physical insight into array truncation effects that the method provides. The MOM matrix and preconditioner, once constructed for a given array, are excitation independent and may be used repeatedly to obtain or analyze element current distributions, element tapering and scanning effects, element input impedances, active input impedances, clement-failure impact on the radiating system, or coupling between element pairs in a finite array.

Formulation
Consider a finite periodic array with ID periodicity (see figure) whose elements arc invariant in z with an arbitrary cross scction in the x-y plane. Consider also the virtual array formed by cxtending the actual array to form an infinite array. Superscripts a and v denote the actual and virtual array, respectively (see figure below). The standard EFIE for the actual finite array is written as Z""I" = V" ( 1) which provides the current I" on the actual array when solved. The moment ma.trix Z"", whose elements represent interactions between basis-and testing function pairs, is assumed to be partitioned into blocks representing interactions between pairs of array elements. Consider now the actual and virtual portions comprising the infinite array. For this new problem, after proper partitioning, thc EFIE and its inverse can be symbolically written as  (2) by requiring that the current r on the virtual array vanish. Combining the impedance and admittance matrices for the infinite array we derive the exact identities to which we give two interpretations. First, when P" is used as a preconditioner for z"" in (l), it regularizes the hypersingular part of 2"" since YZ"" is equal to the identity operator U plus a diffracted operator r"%" that contains no self interactions. Indeed, as shown in the figure below, when Y"z" operates on a current Z of the actual array, it produces voltage excitations V' = Z " I on the virtual array which in tum produce changes in current Y""V" back on the actual array. Furthermore, the current Y"'2""I decays away from the array truncation as a diffracted current (ignoring, for simplicity, the possible presence of modes supported by the structure). Second, (3b) relates the admittance (matrix inverse) operator [Z""]-' for the finite array to the sub-block Y'" of the admittance matrix for an infinite array, with truncation effects represcnted by physical interactions between the actual and virtual array.
Note that P" IS the restriction of the infinite array admittance matrix to the finite array. Away from the edges of the array, the term Y " P I is small since it represents a diffractive term that decays away from the cdgcs. To avoid computing a matrix inverse, we therefore approximate [U-r""Z'"]-' =: U +P%", which, again, approaches U for currents far from the array edges.
The invcrsc of the MOM matrix thus may be approximated as Further approximations may be made to any of the matrices on the RHS to simplify the preconditioncr. Practically, for example, it is likely that the domains and/or ranges of these matrix operators must be windowed. In the following we suppose the array consists of Narray all the interactions since the significant influencc of an array element usually extends to just a fcw nearby elements. Thus only a few interactions need be taken into account in constructing the terms P" and Y%". In the next section we provide a possible algorithm to construct these terms, though other ways may still be advantageous.

Construction of the Preconditioner from the Infinite Array Solution
If thc array is formcd by N array elements, the matrix P" is formed of N sub-blocks, though in practicc we use a smaller number 2E+1 of sub-blocks, accounting only for the interactions between the nearest E array dements to the left and right of an element. The matrix r"' has infinite dimension, howcvcr, we approximate the virtual array using only F elements nearest the array edges. In practice, we may choose F=E so that all the sub-blocks of r"' replicate those of P"; E and F can be as small as two or three. Analogous arguments hold for the evaluation of Z"", which reuses previously evaluated blocks of 2". A key feature of the approach is the fact that matrices P" and P" can be evaluated from the solution of the infinite array problem as follows. We denote by [z"(a) i.e. the element mutual admittance matrices for infinite and finite arrays are not the same. We

Numerical Examples
In the following examples, the current on each array element is evaluated by first solving the standard EFIE P I " = P without preconditioning, then using the two simple preconditioners YZ"" I" = P" P and f P I" = f P, in which the preconditioning matrices P" and y a r e defined in Sec. 2 and Eq. (4), respectively, and computed as outlined above. Example 1 -Array of 31 Cylinders. The test array is an array of 31 conducting cylinders illuminated by a normally incident TE, plane wave as shown in the figure. The array period is d=l.3m; the radius of the cylinders is OSm, and the frequency is 100MHz. In the Table, the spectral condition number (CN) and the number of BiCGstab iterations needed to solve the lincar system with a relative error less than are reported for the EFIE P I " =V, and for various forms of its preconditioned counterpart, Y " P P = Y", in which P" is constructed using different numbers E of sub-blocks. Note that the number of iterations decreases with increasing E, but no significant improvement occurs for E >4.

I 7 6 Y S l 64Y
Further improvement is obtained using the preconditioned equation (U-Y"'Z'")P= YV' (see identity (3a)) since the replacement of Yz"" by U-Y'Z" eliminates the computation of the self interactions; effectively, we analytically extract thc identity matrix U from YT'. In a second test for this array of cavitics, plane wave excitation was replaced by single elenlent excitation using a voltage generator placed on top of the 41h element. Again, the system was solved in only two iterations using the previously computed matrices Tu and P". Example 3 -Inverse, Approximated by Preconditioner. The two prcconditioners P" and are used here to approximate the inverse matrix [Z'"]-'. Currents dircctly evaluated as I" =P"V', or as I" = P V', without solving a linear systcm, arc compared with thc "exact" solution provided by the EFIE z""P =v" solved numerically. The two previous gcomctries arc investigated for plane wave illumination (with unity magnetic field), and comparison is made with a current sample on each array elemcnt (one very sensitivc to edge effects). In the first example (array of 3 1 cylinders), wc sample the current on the right-most part of the cylinders.
Both approximations I" =P"V' and I" = YP V' providc good results. Note that the current stabilizes away from the edge where cdge effects arc negligible (in this example no guided modes are excited by the truncation). In the second example (array of I O cavities) we plot current samples on the top-most point of each clement. Again both approximations P =P"V and I" = P P provide good results. In both geometries, use of the simple preconditioner Y"' , which contains low-order diffraction effects, provides good results, while thc more elaborate preconditioner P slightly improves the current estimate. It is expected that arrays that support guided modes (surface waves) would require the more elaborate P to accurately estimate mode reflection at the edges.