Fast Multipole Method For 3-D Helmholtz Equation In Layered Media

In this paper, a fast multipole method (FMM) is proposed to compute long-range interactions of wave sources embedded in 3-D layered media. The layered media Green's function for the Helmholtz equation, which satisfies the transmission conditions at material interfaces, is decomposed into a free space component and four types of reaction field components arising from wave reflections and transmissions through the layered media. The proposed algorithm is a combination of the classic FMM for the free space component and FMMs specifically designed for the four types reaction components, made possible by new multipole expansions (MEs) and local expansions (LEs) as well as the multipole-to-local translation (M2L) operators for the reaction field components. { Moreover, equivalent polarization source can be defined for each reaction component based on the convergence analysis of its ME. The FMMs for the reaction components, implemented with the target particles and equivalent polarization sources, are found to be much more efficient than the classic FMM for the free space component due to the fact that the equivalent polarization sources and the target particles are always separated by a material interface.} As a result, the FMM algorithm developed for layered media has a similar computational cost as that for the free space. Numerical results validate the fast convergence of the MEs and the $O(N)$ complexity of the FMM for interactions of low-frequency wave sources in 3-D layered media

Exponential convergence for multipole and local expansions and their translations for sources in layered media: 2-D acoustic wave

In this paper, we will first give a derivation of the multipole expansion (ME) and local expansion (LE) for the far field from sources in general 2-D layered media and the multipole-to-local translation (M2L) operator by using the generating function for Bessel functions. Then, we present a rigorous proof of the exponential convergence of the ME, LE, and M2L for 2-D Helmholtz equations in layered media. It is shown that the convergence of ME, LE, and M2L for the reaction field component of the Green's function depends on a polarized distance between the target and a polarized image of the source

Fast multipole method for 3-D Laplace equation in layered media

In this paper, a fast multipole method (FMM) is proposed for 3-D Laplace equation in layered media. The potential due to charges embedded in layered media is decomposed into a free space component and four types of reaction field components, and the latter can be associated with the potential of a polarization source defined for each type. New multipole expansions (MEs) and local expansions (LEs), as well as the multipole to local (M2L) translation operators are derived for the reaction components, based on which the FMMs for reaction components are then proposed. The resulting FMM for charge interactions in layered media is a combination of using the classic FMM for the free space components and the new FMMs for the reaction field components. With the help of a recurrence formula for the run-time computation of the Sommerfeld-type integrals used in M2L translation operators, pre-computations of a large number of tables are avoided. The new FMMs for the reaction components are found to be much faster than the classic FMM for the free space components due to the separation of equivalent polarization charges and the associated target charges by a material interface. As a result, the FMM for potential in layered media costs almost the same as the classic FMM in the free space case. Numerical results validate the fast convergence of the MEs for the reaction components, and the O(N) complexity of the FMM with a given truncation number p for charge interactions in 3-D layered media.Comment: arXiv admin note: text overlap with arXiv:1902.0513

Exponential convergence for multipole and local expansions and their translations for sources in layered media: three-dimensional Laplace equation

In this paper, we prove the exponential convergence of the multipole and local expansions, shifting and translation operators used in fast multipole methods (FMMs) for 3-dimensional Laplace equations in layered media. These theoretical results ensure the exponential convergence of the FMM which has been shown by the numerical results recently reported in [9]. As the free space components are calculated by the classic FMM, this paper will focus on the analysis for the reaction components of the Green's function for the Laplace equation in layered media. We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex plane. Then, using the Cagniard-de Hoop transform and contour deformations, estimate for the remainder terms of the truncated expansions is given, and, as a result, the exponential convergence for the expansions and translation operators is proven