Comparison of Integral-Equation Formulations for the Fast and Accurate Solution of Scattering Problems Involving Dielectric Objects with the Multilevel Fast Multipole Algorithm

Abstract

We consider fast and accurate solutions of scattering problems involving increasingly large dielectric objects formulated by surface integral equations. We compare various formulations when the objects are discretized with Rao-Wilton-Glisson functions, and the resulting matrix equations are solved iteratively by employing the multilevel fast multipole algorithm (MLFMA). For large problems, we show that a combined-field formulation, namely, the electric and magnetic current combined-field integral equation (JMCFIE), requires fewer iterations than other formulations within the context of MLFMA. In addition to its efficiency, JMCFIE is also more accurate than the normal formulations and becomes preferable, especially when the problems cannot be solved easily with the tangential formulations.Index Terms-Dielectrics, iterative solutions, multilevel fast multipole algorithm (MLFMA), surface integral equations.

Analysis of photonic-crystal problems with MLFMA and approximate Schur preconditioners

“…Among many choices, CTF and JMCFIE are usually most suitable formulations in terms of accuracy and efficiency [7]. CTF is a modified and more stable version of the well-known Poggio-MillerChang-Harrington-Wu-Tsai (PMCHWT) formulation [3].…”

“…This method is based on the calculation of interactions between basis and testing functions in a group-by-group manner in a multilevel scheme. In the case of dielectric problems, MLFMA must be applied for both inner and outer media [7].…”

“…APPROXIMATE SCHUR PRECONDITIONERS Both CTF and JMCFIE lead to non-hermitian and indefinite systems, whose iterative solutions may not converge easily without preconditioning. Being a second-kind integral equation, JMCFIE leads to diagonally-dominant matrices so that simple block-diagonal preconditioners can be effective [7]. However, CTF does not provide diagonally-dominant matrices, and its efficient solutions may require strong preconditioners constructed from all available interactions in MLFMA, i.e., near-field matrices.…”

“…However, a good approximation to y is required in (7). Consequently, we also need to develop a good approximation to the Schur complement matrix.…”

“…2 depicts the solution of periodic-slab problems at 300 MHz. We compare iteration counts when solutions are accelerated with a four-partition block-diagonal preconditioner (4PBDP) [7], ILU-type preconditioners (ILU(0) for JMCFIE and ILUT for CTF) [16]), and IASP, in addition to the no-preconditioner (NP) case. Solutions employing 4PBDP are omitted for CTF since they do not converge.…”