Multiscale analysis of large complex arrays

Array modeling issues are challenging, since they involve large structures (in terms of the wavelength), but also fine details that require much-smaller than wavelength discretizations, and that dominate the frequency response of input parameters. The Integral Equation (IE) approach is largely used to attack these problems, through the Method of Moment (IE-MoM) discretization scheme, and/or the Generalized Admittance Matrix method (GAM). It is well known, however, that standard techniques are severely limited by the matrix size and condition number involved in the problems of interest. In these problems, the structure of the solution exhibits very different scales of variation; for examples, local interactions in a geometry, like sub-wavelength details, edges and discontinuities, generate small-scale details of high spatial frequency, while distant interactions as well as resonant lengths are responsible for the low-frequency, slow spatial variations. One is typically forced to choose mesh cells of size comparable to the smallest foreseen scale of the solution, i.e. with the highest possible spatial resolution, or likewise if waveguide modes are used as expansion functions. Unfortunately, this leads to a large number of unknowns, densely populated MoM matrices with a poor condition number, and renders the direct approach of large problems numerically intractable. A number of techniques have been presented in the past years to overcome the above difficulties, whose review is outside the scopes of this work; references to relevant work on the broad and specific topics dealt with here can be found in the cited literature, and omitted here for the sake of conciseness. We will focus here on techniques that attempt to keep explicit information about the multi-scale nature of the solution directly into the representation of the unknown fields/currents.