The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer.

In this paper, we present an accurate method of moments (MoM) solution of the combined field integral equation (CFIE) using the multilevel fast multipole algorithm (MLFMA) for scattering by large, three-dimensional (3-D), arbitrarily shaped, homogeneous objects. We first investigate several different MoM formulations of CFIE and propose a new formulation, which is both accurate and free of interior resonances. We then employ MLFMA to significantly reduce the memory requirement and computational complexity of the MoM solution. Numerical results are presented to demonstrate the accuracy and capability of the proposed method. The method can be extended in a straightforward manner to scatterers composed of different homogeneous dielectric and conducting objects.

In the solution of an integral equation using the conjugate gradient (CG) method, the most expensive part is the matrix‐vector multiplication, requiring O(N2) floating‐point operations. The fast multipole method (FMM) reduced the operation to O(N15). In this article we apply a multilevel algorithm to this problem and show that the complexity of a matrix‐vector multiplication is proportional to N (log(N))2. © 1994 John Wiley & Sons, Inc.

ABSTRACT|-FISC (Fast Illinois Solver Code) is designed to compute RCS of a target described by a triangular facet le. The problem is formulated by the method of moments (MoM), where the RWG (Rao, Wilton, and Glisson) basis functions are used. The resultant matrix equation is solved iteratively by the conjugate gradient (CG) method. The multilevel fast multipole algorithm (MLFMA) is used to speed up the matrix-vector multiply in CG. Both complexities for the CPU time per iteration and memory requirements are of O(N logN), where N is the number of unknowns.

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Abstract-In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.Index Terms-Electromagnetic scattering, Krylov methods, multilevel fast multipole algorithm (MLFMA), sparse approximate inverse (SAI) preconditioning.

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A fast far‐field approximation (FAFFA) is developed to estimate the RCS of conducting scatterers. This method accounts for the interaction between subscatterers in two ways, depending on the electrical distance between the subscatterers. The interactions of subscatterers separated by a large electrical distance are computed in three stages: (1) aggregation, which computes the total field at a group center due to the subscatterers of the group; (2) translation, which translates the field from one group center to another; and (3) disaggregation, which distributes the field in a group center to each subscatterer in the group. Two strategies are employed to accelerate the computation in the above three stages. One is the use of far‐field approximation to simplify the computation in the translation stage; the other is the use of interpolation and smoothing techniques, which reduces the complexity of aggregation and disaggregation. The overall computational complexity for a matrix‐vector multiplication is of the order of N1.33, and the memory requirement is of order N. Numerical results show that this method can predict a RCS that is very close to exact solution, and that the method can be applied to objects with very large electrical sizes. © 1995 John Wiley & Sons. Inc.