# Closed-form evaluation of potential integrals in the Boundary Element Method

## TL;DR

This paper introduces an analytical method for evaluating singular and near-singular integrals in the Boundary Element Method for the Helmholtz equation, improving accuracy and guiding quadrature rule selection.

## Contribution

It develops a closed-form analytical approach for potential integrals, enhancing precision and efficiency in boundary element computations for wave problems.

## Key findings

- Achieves accuracy comparable to machine precision.
- Provides a criterion for quadrature rule selection.
- Demonstrates effectiveness in Helmholtz equation solutions.

## Abstract

A method is presented for the analytical evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green's function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision.