Acceleration of the Boundary Element Method for arbitrary shapes with the Fast Fourier Transformation

Abstract: This work illustrates the possibility to apply the Fast Fourier Transformation to obtain the integrals of the Boundary Element Method (BEM) on arbitrary shapes. The procedure is inspired by the technique used with great success within the framework of the half-space approximation in contact mechanics. There, the boundary integral equations are given by simple convolutions over the boundary surface. For arbitrary shapes this is not the case. Thus, the FFT and the great reduction in computational complexity that comes with it cannot be utilized as easily. In this work, it is illustrated that although the integral equations of the BEM are not convolutions over boundary of arbitrary shapes, they are indeed convolutions over the space which is one dimension higher than that of the boundary. Therefore, the FFT can indeed be used to calculate the BEM integral equations on arbitrary shapes, only this comes at the cost of increasing the dimension of the FFT. A small example is given which illustrates how the concept can be used to fully solve a BEM problem with a given closed boundary using only the FFT approach to obtain the integral equations and no hybrid techniques or other approximations.