Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half space

Abstract

Abstract-An extension of the multilevel fast multipole algorithm (MLFMA), originally developed for targets in free space, is presented for the electromagnetic scattering from arbitrarily shaped three-dimensional (3-D), electrically large, perfectly conducting targets above or embedded within a lossy half space. We have developed and implemented electric-field, magnetic-field, and combined-field integral equations for this purpose. The nearby terms in the MLFMA framework are evaluated by using the rigorous half-space dyadic Green's function, computed via the method of complex images. Non-nearby (far) MLFMA interactions, handled efficiently within the multilevel clustering construct, employ an approximate dyadic Green's function. This is expressed in terms of a direct-radiation term plus a single real image

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“…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].…”

“…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.…”

“…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. The computational complexity of O(N log N ), both in RAM and CPU (per iteration), remains unchanged compared to the free-space version [11]. However, Geng only applied the half-space FMM/MLFMA to scattering problems.…”