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

Abstract

This paper builds upon the results of a recent study which illustrates how the Fast Fourier Transformation (FFT) can be used to accelerate the Boundary Element Method (BEM) for arbitrary shapes. In the present work, we further deepen this understanding and focus especially on implementation details in order to calculate the boundary integrals with the FFT. Different numerical techniques are compared for an exemplary shape. Also, additions to the concept are mentioned such as the introduction of a high-resolution grid close to the boundary and a low-resolution grid farther away.

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

“…Again, the results are compared on a rough qualitative level to finite element results. In the last main section of this paper, we build on two recent studies [11] and [12], where the FFT-based BEM is performed for completely arbitrary shapes. We present exemplary results of this technique for the two-dimensional NAVIER equation.…”

“…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. This makes it possible to align a regular two-dimensional grid on which the FFT is performed with the domain (see Fig.…”

“…11b). It is shown in [11] and [12] how one has to appropriately set zeros at grid points which are not in close proximity to the actual surface so as not to distort the results one obtains with this technique. Using the FFT in such a fashion as to obtain the boundary integrals on arbitrary shapes lowers the computational complexity from O(n 4 ) to O(n 3 log n 1.5 ).…”

“…Using the FFT in such a fashion as to obtain the boundary integrals on arbitrary shapes lowers the computational complexity from O(n 4 ) to O(n 3 log n 1.5 ). While this is still for one dimension higher than the O(n 2 log n) complexity of the half space, the technique opens up opportunities for further reduction in computational complexity through a variable grid (see [12]), and offers the advantage of utilizing highly efficient implementation of the FFT, which can be accelerated even further on parallel systems such as the Graphics Processing Unit (GPU). We now build upon the results found in [11] and [12] where the FFT is used to accelerate the calculation of the boundary integrals on arbitrary shapes for the LAPLACE equation.…”

“…(7), the two-dimensional fundamental solution of Eq. (5) is for this case given with [33] 2 00 * (3 4 ) (12) providing through Eq. (9) the stress vector t j * of the fundamental solution on the boundary for a certain 0…”