Three dimensional multilevel fast multipole algorithm at very low frequencies

In the simulation of electromagnetic phenomena in microwave integrated circuits where the simulated object can be a tiny fraction of the wavelength but the number of unknowns may be very large, neither the static MLFMA nor the dynamic MLFMA can be used. Therefore, a fast 3-D algorithm which can work at very low frequencies must be developed to bridge the gap from static to electrodynamic. In this paper, based on the original undiagonalized-form 3-D MLFMA, a normalized-form 3-D LF-MLFMA is developed for very low frequency problems by using the normalization idea from the 2-D LF-MLFMA and 3-D problems at low frequencies.