| The integral equation method at low frequency is further studied and the fast-multipole algorithm is improved over the previous version. First, a new type of loop basis is reported when the object is not simply connected. The frequency scaling of the integral equation will change. The analytical expression of the interaction is provided and tested.; Second, a new loop-loop interaction is proposed to remove the inaccurate static substraction. Also a new multiregion low-frequency fast-multipole frame is proposed to remove the unnecessary construction of FMA trees and translators. To further improve the speed, the rotational translator technique together with rank-reduction technique is implemented and tested. A new setup of radiation/receiving pattern is also implemented to cut down the matrix-vector-multiplication by another factor of two. The saving of CPU time and memory cost is also provided.; Then we study two methods to improve the convergence of integral equation method at low frequency. The three-step iteration scheme is physics-based and shows great improvement when the loop and tree currents are decoupled. The flexible GMRES is a new type of iterative method that ideally fits the fast-multipole scheme. It has two loops of iteration: the inner iteration, which works like a right preconditioner, can be updated with FMA with less accuracy; and for the outer iteration loop, FMA with acceptable accuracy is used.; Finally, the new fast-multipole scheme is used to model the seabed logging problem. The popular method to model the problem is the finite-difference time-domain (FDTD) method, which suffers from very large number of marching steps because the time step is small and the operating frequency is low. Our numerical results show several orders magnitude of saving, both in CPU time and memory. |
