A Directional Multilevel Complex-Space Fast Multipole Algorithm for Electrically Large Problems of Various Dimensions

A directional multilevel complex-space fast multipole algorithm (DMLCSFMA) is proposed for solving electrically large problems of various dimensions. This algorithm implements a high-frequency generalization of the well-known mid-frequency multilevel fast multipole algorithm (MLFMA). It is established by exploring the fundamental connection between the conventional MLFMA and the recently developed directional fast multipole algorithms (D-FMAs), as well as the plane wave expansion induced from the complex source beam [Gaussian beam (GB)]. Different from the conventional MLFMA which exhibits the complexity of
$O({N^{2}})$
for certain situations such as the quasi-1-D elongated object, the proposed high-frequency generalized version is capable of achieving a stable complexity of
$O({N\log N})$
, irrespective of the dimensional features of the objects. Besides, the proposed algorithm also manifests itself as a spectral counterpart of the traditional D-FMAs. However, unlike the traditional D-FMAs which leverage the equivalent source-based sampling expansions, the proposed algorithm is established using the plane wave-based exponential expansions. Thus, the feasibility of building a D-FMA with analytically diagonalized translators is also demonstrated in this work. Several numerical examples are provided to illustrate the complexity and accuracy of the proposed algorithm.