Investigation of the Spectral Properties of Generalized Equivalence Integral Equations

Method of moments integral equation solvers for electrically large problems require the use of fast methods to avoid an O(N²) computational complexity (N being the number of problem unknowns, which is assumed here to scale with the frequency, for a fixed mesh density). To this end, it is convenient and common to use algebraic compression that exploits the ε-rank deficiency (ε being the best low-rank approximation error bound) of the off-diagonal moment matrix block, such as the adaptive cross approximation (ACA). If the blocks, which are organized to represent interactions between well-separated source and testing sub-domains of the scatterer Ω, exhibit ranks that scale slow with the increase in number of domain basis functions, the complexity can reach O(N log N). However, for electrically large problems, this is guaranteed only for reduced dimensionality (i.e., elongated, quasi-planar) geometries, whereas for broadside interactions the ε-rank scales, asymptotically, linearly with O(N). Fortunately, for an essentially-convex Ω, this limitation can be overcome by formulating the problem as a generalized integral equation, with a modified kernel that includes an auxiliary component that significantly reduces the broadside interactions and highlight reduced-dimensionality interactions, thus enabling the design of fast solvers (A. Sharshevsky, Y. Brick, and A. Boag, “Direct solution of scattering problems using generalized source integral equations,” IEEE Transactions on Antennas and Propagation, vol. 68, no. 7, pp. 5512-5523, Jul. 2020. DOI: 10.1109/TAP.2020. 2975549). The auxiliary components are produced by sources placed inside Ω, which can be designed in various manners. Their design may have influence on the spectral properties of the generalized integral equation and the conditioning of the corresponding moment matrix that, to date, have not been studied.