Potential integrals on triangles

TL;DR

This paper presents a general, exact, and easily implementable method for evaluating potential integrals on planar triangular elements using polar coordinate decomposition, applicable to potential and gradient calculations with constant or linear source terms.

Contribution

It introduces a novel polar coordinate decomposition approach for potential integrals on triangles, handling principal value and finite part integrals with high accuracy.

Findings

Method is accurate and convergent for potential and gradient evaluations.

Applicable to both constant and linearly varying source terms.

Tested on a single triangular element outside a unit cube, demonstrating effectiveness.

Abstract

The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a field point lying in the plane of the element. Results are presented for the evaluation of the potential and its gradients, where the integrals must be treated as principal values or finite parts, for elements with constant and linearly varying source terms. These results are tested by application to a single triangular element to the evaluation of the potential gradient outside the unit cube. In both cases, the method is shown to be accurate and convergent.