Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half space

Multilevel Fast Multipole Algorithm for Radiation Characteristics of Shipborne Antennas Above Seawater

“…The half-space dyadic Green's function can be split into a termĪg i representing the "direct" radiation between source and observation point (as in free space, but using in general a complex wave number k i ) and a remaining dyadic ∆Ḡ Aii accounting for interactions with the interface (i.e., here ∆ is not an operator) [9][10][11].…”

“…For a half-space FMM/MLFMA, it is essential to include the effects of the far interface interactions. However, that the number of terms L required for convergence can be prohibitively large for general complex source points, undermining the efficiency of using (4) for far interface interactions in the context of the discrete compleximage technique [9][10][11].…”

“…While the half-space dyadic Green's function is rigorously accounted for in the near interaction matrix, far interactions are often less sensitive to approximations in the Green's function [9][10][11]. The FMM has been successfully extended to the scattering from a PEC object above or buried in a half space by employing the asymptotic form of the Green's function [11] for far interactions.…”

“…In [9][10][11], Geng and colleagues developed an approximate means of handling the dyadic half-space Green's function, with application to the fast multipole method (FMM) and the multilevel fast multipole algorithm (MLFMA), which were originally developed for targets in free space [12][13][14][15][16][17][18][19][20][21][22]. In their works, the near MLFMA terms are evaluated via the use of the exact dyadic Green's function, the latter evaluated efficiently via the complex-image technique [23].…”

“…The far terms, evaluated efficiently via the clustering algorithm, employ the asymptotic form of the dyadic Green's function. As elucidated further in the following, each component of the approximate Green's function is expressed in terms of the direct-radiation term plus radiation from an image source in real space [9][10][11]. The former accounts for the radiation of currents into the medium in which it resides, while the latter accounts for interactions with the half-space interface.…”