Numerical evaluation of the Kirchhoff-Helmholtz integral outside a sphere

TL;DR

The paper introduces a fast, accurate method for evaluating transient acoustic fields outside a sphere using Lebedev quadratures and advanced time algorithms, significantly improving computational efficiency for large source sets.

Contribution

It presents a novel numerical approach combining Lebedev quadratures and interpolation techniques for efficient transient field evaluation outside a sphere.

Findings

Achieves near machine-precision accuracy.

Provides a computational speed-up over direct evaluation.

Effective for large numbers of source points.

Abstract

A method is presented for the fast evaluation of the transient acoustic field generated outside a spherical surface using surface data on the sphere. The method employs Lebedev quadratures, which are optimal integration on the sphere, and Lagrange interpolation and differentiation in an advanced time algorithm for the evaluation of the transient field. Numerical testing demonstrates that the approach gives near machine-precision accuracy and a speed-up in evaluation time which depends on the order of quadrature rule employed but breaks even with direct evaluation at a number of field points about 1.15 times the number of surface quadrature nodes, making the method an efficient means of evaluating the field generated by a large number of sources.