Evaluation of Biot-Savart integrals on tetrahedral meshes

TL;DR

This paper introduces a simple, efficient method for evaluating Biot-Savart integrals on tetrahedral meshes using Gaussian quadrature and ray tracing, improving computational speed and accuracy.

Contribution

The paper presents a novel approach that replaces complex analytical formulas with Gaussian quadrature and ray tracing for Biot-Savart integrals on tetrahedral meshes.

Findings

The method achieves second order convergence.

It demonstrates near-linear speedup on parallel systems.

It eliminates the need for complex mathematical operations.

Abstract

An arithmetically simple method has been developed for the evaluation of Biot--Savart integrals on tetrahedralized distributions of vorticity. In place of the usual approach of analytical formulae for the velocity induced by a linear distribution of vorticity on a tetrahedron, the integration is performed using Gaussian quadrature and a ray tracing technique from computer graphics. This eliminates completely the need for the evaluation of square roots, logarithms and arc tangents, and almost completely eliminates the requirement for trigonometric functions, with no operation more costly than a division required during the main calculation loop. An assessment of the algorithm's performance is presented, demonstrating its accuracy, second order convergence and near-linear speedup on parallel systems.