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

**Authors:** Justus Benad  
**arXiv:** [1809.00845](https://arxiv.org/abs/1809.00845)  
**Date:** 2018-09-04  
**Summary page:** https://www.summarizepaper.com/en/paper/1809.00845v1/

## Key points

- The Boundary Element Method (BEM) is a numerical technique used in engineering and physics.
- Justus Benad explores the possibility of using Fast Fourier Transformation (FFT) to obtain integrals of BEM on arbitrary shapes.
- The integral equations of BEM are convolutions over space that has one dimension higher than that of the boundary, which means FFT can still be used to calculate BEM integral equations on arbitrary shapes.
- However, this comes at the cost of increasing the dimension of FFT.
- Benad provides an example where he fully solves a BEM problem with a given closed boundary using only FFT approach without any hybrid techniques or other approximations.
- This work demonstrates how FFT can accelerate BEM computations for arbitrary shapes and presents a new way to tackle complex geometries in numerical simulations.

## Summary

The Boundary Element Method (BEM) is a powerful numerical technique used to solve problems in engineering and physics. In this work by Justus Benad, the possibility of using Fast Fourier Transformation (FFT) to obtain integrals of BEM on arbitrary shapes is explored. The idea behind this approach is inspired by the half-space approximation technique used in contact mechanics. For arbitrary shapes, this is not possible due to their complex geometries. However, Benad shows that although the integral equations of BEM are not convolutions over boundaries of arbitrary shapes, they are indeed convolutions over space that has one dimension higher than that of the boundary. This means that FFT can still be used to calculate BEM integral equations on arbitrary shapes but at the cost of increasing the dimension of FFT. To illustrate this concept further, Benad provides a small example where he fully solves a BEM problem with a given closed boundary using only FFT approach to obtain integral equations without any hybrid techniques or other approximations. Overall, this work demonstrates how FFT can accelerate BEM computations for arbitrary shapes and presents a new way to tackle complex geometries in numerical simulations.