The fast multipole method (FMM) for electromagnetic scattering problems

Abstract

The fast multipole method (FMM) was developed by Rokhlin to solve acoustic scattering problems very efficiently. We have modified and adapted it to the second-kind-integral-equation formulation of electromagnetic scattering problems in two dimensions. The present implementation treats the exterior Dirichlet (TM) problem for two-dimensional closed conducting objects of arbitrary geometry. The FMM reduces the operation count for solving the second-kind integral equation (SKIE) from O(n 3) for Gaussian elimination to O(n 4/3) per conjugated-gradient iteration, where n is the number of sample points on the boundary of the scatterer. We also present a simple technique for accelerating convergence of the iterative method: "complexifying" k, the wavenumber. This has the effect of bounding the condition number of the discrete system; consequently, the operation count of the entire FMM (all iterations) becomes O(n 4/3). We present computational results for moderate values of ka, where a is the characteristic size of the scatterer. Comments

The Fast Multipole Method: Numerical Implementation

“…Since the early papers by Rokhlin [21,26,50,51], the fast multipole method (FMM) has proven to be a very powerful and efficient scheme for solving the Maxwell equation with an integral formulation (see, for example, [56][57][58]). However, implementation of the FMM proved to be a rather difficult task in part because of its complexity (many lines in a complex and long code) and because of the need to optimize all the steps of the FMM.…”

A portable parallel particle program

“…In addition, we note applications in molecular dynamics [29], computational fluid dynamics [30,31] and partial differential equations relevant to biology [32]. Fast multipole methods have been used to address twodimensional problems in potential flows [33], and electromagnetic scattering [34]. Much chemistry is done with small N , and these may not benefit much from treecodes.…”

A Tdie/Tdpo Hybrid Method for the Analysis of Tm Transient Scattering From Two-Dimensional Combinative Conducting Cylinders

“…Thus, it is necessary to develop fast algorithm for TDIE. So far, there are two kinds of fast algorithm based on TDIE: one is the plane wave time-domain algorithm (PWTD) [20][21][22] proposed by Shanker et al, which is actually the time-domain version of fast multi-pole method (FMM) [8][9][10]. This method could decrease computational complexity significantly, but the method itself is very complex and not easy to program.…”

“…Integral equation (IE) is widely used for the numerical analysis of electromagnetic radiation and scattering [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Especially the time domain integral equation (TDIE) has been receiving much interest in studying many practical transient electromagnetic problems [15][16][17][18][19].…”