The Spherical Multipole Expansion of a Triangle

TL;DR

This paper presents an analytical method for computing multipole moments of charge distributions on triangles, enhancing the efficiency and accuracy of boundary element methods for solving Laplace's equation.

Contribution

The authors introduce a recursive integration technique for multipole moments on triangles, applicable to non-uniform densities and integrated with FMBEM.

Findings

Accurate for low aspect ratio triangles up to degree 32

More efficient than 2D Gauss-Legendre quadrature

Applicable to non-uniform charge densities

Abstract

We describe a technique to analytically compute the multipole moments of a charge distribution confined to a planar triangle, which may be useful in solving the Laplace equation using the fast multipole boundary element method (FMBEM) and for charged particle tracking. This algorithm proceeds by performing the necessary integration recursively within a specific coordinate system, and then transforming the moments into the global coordinate system through the application of rotation and translation operators. This method has been implemented and found use in conjunction with a simple piecewise constant collocation scheme, but is generalizable to non-uniform charge densities. When applied to low aspect ratio ($\leq 100$) triangles and expansions with degree up to 32, it is accurate and efficient compared to simple two-dimensional Gauss-Legendre quadrature.