Taylor expansion based fast multipole method for 3-D Helmholtz equations in layered media

Abstract

In this paper, we develop fast multipole methods for 3D Helmholtz kernel in layered media. Two algorithms based on different forms of Taylor expansion of layered media Green's function are developed. A key component of the first algorithm is an efficient algorithm based on discrete complex image approximation and recurrence formula for the calculation of the layered media Green's function and its derivatives, which are given in terms of Sommerfeld integrals. The second algorithm uses symmetric derivatives in the Taylor expansion to reduce the size of precomputed tables for the derivatives of layered media Green's function. Numerical tests in layered media have validated the accuracy and O(N ) complexity of the proposed algorithms.

A Matrix Basis Formulation For The Green's Functions Of Maxwell's Equations And The Elastic Wave Equations In Layered Media

“…Remark 3.1. In the previous work of the Helmholtz equations [13,14,15], the separation of variable z ′ was also executed, so that each reaction component was further divided by the "propagating direction" of z ′ . Then, we re-format the restricting equations of the tensor potential G A , including the interface conditions and the radiation conditions, using the matrix basis…”

“…These linear equations are solvable, from the knowledge of the acoustic wave equation in layered media:• −iωb 1 is exactly the reflection/transmission coefficient in the frequency domain of the Green's function of the Helmholtz equation in layered media, with piecewise constant material parameters 1/ε. Thus we can solve b 1 in the frequency domain like solving the known scalar layered Helmholtz problem[14]. • Similarly, −iωµ j b 2 is exactly the one with piecewise constant parameters 1/µ.…”

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

“…Let us first review the integral representation of the layered Green's function derived in [32]. Consider a layered medium consisting of L interfaces located at z = d , = 0, 1, • • • , L − 1 as shown in Fig.…”

“…In a different approach, an inhomogeneous plane wave fast method was developed [19] by approximating the Green's function with plane waves sampled along a steepest descent path in the complex wave number space. So far, to handle the wave interaction of sources embedded in general layered media using layered Green's functions, some other fast methods have been proposed such as FMM using Taylor expansion-based low-rank representation of Green's function [32] [31], and cylindrical wave decomposition of the Green's function together with 2-D FMMs for cylindrical waves [8]. On the other hand, kernel independent compression techniques [37] [33] could also be considered for the layered Green's functions.…”

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

“…Sheng and Sun (2014) discussed the application of an exponential splitting method, and the method can solve the n-dimensional paraxial Helmholtz equation with highly oscillatory condition, which introduced an eikonal transformation for the decomposition of oscillation-free platforms and matrix operators. A Taylor expansion based on a fast multipole method by Wang et al (2020) was developed for implementing the low-frequency three-dimensional Helmholtz Green's functions in layered media. Li (2019) presented a fast singular boundary method for solving 3D Helmholtz equation, and this approach is a meshless method with boundary type, which combines the advantages of boundary element method and fundamental solutions method.…”