NSPWMLFMA: A low frequency stable formulation of the MLFMA in three dimensions

The iterative solution of integral equations containing the Green function of the Helmholtz equation as the integration kernel requires repeated matrix-vector products. These products can be accelerated by means of a so-called fast multipole method (FMM). Of the many fast multipole methods in use today, the multilevel fast multipole algorithm (MLFMA) is arguably among the most successful ones. It allows the simulation of electrically large structures that are intractable with direct or unaccelerated iterative solvers. Testimony to the MLFMAs myriad uses is its implementation in various commercial EM software packages such as FEKO and CST Microwave studio. However, the MLFMA has one big drawback: a numerical instability prevents the method from being used on low frequency (LF) interactions, i.e. interactions between sources and observers that are less than approximately one wavelength apart. As a consequence configurations containing significant sub-wavelength geometrical detail cannot be efficiently treated using the MLFMA alone and a hybrid method is necessary. However, the LF methods in use today are generally less efficient due to non-diagonal translation operators (multipole methods) or the need for six radiation patterns (spectral methods). In this contribution a novel algorithm, called the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA) [1], will be presented that is stable at LF, exhibits diagonal translation operators and requires only one radiation pattern. The method is based on an analytical expression for a translation operator in the z-direction. This translation operator is made numerically stable using a shift of the integration path into the complex plane. It even has a DC-limit. A QR-based method is then used to extend the applicability to all the other translation directions. The algorithm has also been parallelized using open FMM [2]. Finally some numerical results will be shown.