Fast Fourier transform on multipoles (FFTM) algorithm for Laplace equation with direct and indirect boundary element method

Abstract

In this paper, the fast Fourier transform on multipole (FFTM) algorithm is used to accelerate the matrixvector product in the boundary element method (BEM) for solving Laplace equation. This is implemented in both the direct and indirect formulations of the BEM. A new formulation for handling the double layer kernel using the direct formulation is presented, and this is shown to be related to the method given by Yoshida (Application of fast multipole method to boundary integral equation method, Kyoto University, Japan, 2001). The FFTM algorithm shows different computational performances in direct and indirect formulations. The direct formulation tends to take more computational time due to the evaluation of an extra integral. The error of FFTM in the direct formulation is smaller than that in the indirect formulation because the direct formulation has the advantage of avoiding the calculations of the free term and the strongly singular integral explicitly. The multipole and local translations introduce approximation errors, but these are not significant compared with the discretization error in the direct or indirect BEM formulation. Several numerical examples are presented to compare the computational efficiency of the FFTM algorithm used with the direct and indirect BEM formulations.

Efficient Calculation of the Bem Integrals on Arbitrary Shapes With the FFT

“…In order to test the method, we use a simple analytical solution for the LAPLACE equation. Here, we use the exemplary solution a 1 / 2 ux  (13) of Eq. (1).…”

“…Many valuable methods have been proposed in the past to reduce the computational complexity of both the FEM and the classical BEM for arbitrary shapes [10][11][12][13][14]. A recent study [15] illustrates how the classical BEM for arbitrary shapes can be accelerated with the FFT in a similar way as it is done with great success in contact mechanics within the framework of the half-space approximation.…”

Numerical Methods for the Simulation of Deformations and Stresses in Turbine Blade Fir-Tree Connections

“…In 1991, GAO introduced a first-order perturbation method for the half-space approximation to take into account stress concentration effects of slightly undulating surfaces [24]. In later years, various methods were developed to accelerate the classical BEM for completely arbitrary shapes which, in part, make use of the low computational complexity of the FFT (see for example [25], [26] and [27]), or utilize other techniques such as hierarchical matrices (see for example [28]) to accelerate the calculation. Recently, it was illustrated in [11] and [12] that the integral equations of the BEM for completely arbitrary shapes (no half-space) can be obtained in a manner very similar to the FFT-based half-space approach: For the case of the half-space, the boundary integral (1) is evaluated in the plane of the two coordinates x and y which perfectly aligns with the even half-space surface.…”

Kassiopeia: a modern, extensible C++ particle tracking package

“…However, when combined with fast multipole techniques [29,30] they are a powerful tool which can greatly reduce the time to solution and reduce memory cost. While a full description of the specific algorithm used by the KEMField library is beyond the scope of this paper, KEMField provides a modified multipole method which is a hybrid of the traditional fast multipole method (FMM) [31] and a Fourier transform based approach known as the fast Fourier transform on multipoles (FFTM) [32,33], which is described in detail in [34]. Krylov subspace methods benefit greatly from preconditioning when dealing with the three-dimensional Laplace BEM and KEMField provides several simple choices such as Jacobi and Block-Jacobi, as well as an implicit preconditioner which acts by solving the same problem at reduced accuracy at each iteration in order to very effectively reduce the number of full accuracy iterations needed.…”