A Nondirective Plane Wave MLFMA Stable at Low Frequencies

A novel method, called the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA), is presented to evaluate the low-frequency (LF) interactions that cannot be handled by the multilevel fast multipole algorithm (MLFMA). It is well known that the MLFMA cannot be used for LF interactions, since it suffers from numerical instability. Contrary to current techniques, the proposed technique is not based on the spectral representation of the Green function. Instead the addition theorem of the MLFMA is manipulated into a form that allows numerically stable translations along the
z
axis. The translation operator for these translations is derived in closed form. A QR-based method is devised to allow stable translations in all the other directions. Interpolations and anterpolations are also provided, allowing a full multilevel algorithm. Since the NSPWMLFMA is based on the same mathematical foundations as the MLFMA, it requires limited adaptations to existing MLFMA codes. The fact that a QR is needed limits this algorithm to LF interactions. However, a coupling with the MLFMA is straightforward, allowing the easy construction of a broadband algorithm. The DC limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems. Finally, it is shown that the error introduced in the different steps of the algorithm is controllable, and a single-level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity.