# A robust and non-singular formulation of the boundary integral method for the potential problem

## TL;DR

This paper introduces a non-singular boundary integral method for the Laplace equation that eliminates singularities analytically, improving accuracy and efficiency in multi-scale potential problems.

## Contribution

It presents a new formulation of the boundary integral method that removes singularities, enabling standard quadrature and robust near-boundary potential calculations.

## Key findings

- 60% reduction in coding effort
- Enhanced accuracy near boundaries
- Applicable to 2D, 3D, and axisymmetric problems

## Abstract

A non-singular formulation of the boundary integral method (BIM) is presented for the Laplace equation whereby the well-known singularities that arise from the fundamental solution are eliminated analytically. A key advantage of this approach is that numerical errors that arise due to the proximity of nodes located on osculating boundaries are suppressed. This is particularly relevant in multi-scale problems where high accuracy is required without undue increase in computational cost when the spacing between boundaries become much smaller than their characteristic dimensions. The elimination of the singularities means that standard quadrature can be used to evaluate the surface integrals and this results in about 60% savings in coding effort. The new formulation also affords a numerically robust way to calculate the potential close to the boundaries. Detailed implementations of this approach are illustrated with problems involving osculating boundaries, 2D domains with corners and a wave drag problem in a 3D semi-infinite domain. The explicit formulation of problems with axial symmetry is also given.