Analytical Galerkin boundary integrals of Laplace kernel layer potentials in $\mathbb{R}^3$

TL;DR

This paper introduces an analytical method for computing boundary integrals of Laplace layer potentials in 3D boundary element methods, using recursive dimensionality reduction and symmetry to handle singularities.

Contribution

It presents a novel recursive dimensionality reduction technique for analytical computation of layer potential integrals in 3D BEM, including singular cases, with validation against benchmarks.

Findings

Accurate analytical expressions for all source-receiver configurations.

Effective handling of singular integrals through symmetry and reduction.

Validated results with benchmark comparisons and convergence studies.

Abstract

A method for analytical computation of the double surface integrals for all layer potential kernels associated with the Laplace Green's function, in the Galerkin boundary element method (BEM) in $\mathbb{R}^3$ using piecewise constant flat elements is presented. The method uses recursive dimensionality reduction from 4D ($\mathbb{R}^2\times\mathbb{R}^2$) based on Gauss' divergence theorem. Computable analytical expressions for all cases of relative location of the source and receiver triangles are covered for the single and double layer potentials and their gradients with analytical treatment of the singular cases are presented. A trick that enables reduction of the case of gradient of the single layer to the same integrals as for the single layer is introduced using symmetry properties. The method was confirmed using analytical benchmark cases, comparisons with error-controlled computations of regular multidimensional integrals, and a convergence study for singular cases.