A transformation for near strongly singular integrals
A simple transformation using a singularity cancellation technique is presented that enables accurate and efficient computation of near strongly singular integrals involving the gradient of Helmholtz-type potentials for projected observation points that lie external to the source element. The coordinate reference is offset from the projection of the observation point onto the plane containing the source element. In many cases, this permits highly accurate results without the splitting of the source element.
A Simple Transformation for the Numerical Evaluation of Near Strongly Singular Integrals
A simple transformation using a singularity cancellation technique is presented that enables accurate and efficient computation of near strongly singular integrals involving the gradient of Helmholtz-type potentials for projected observation points that lie external to the source element. Unlike previous Duffy-like transformations, the coordinate reference is offset from the projection of the observation point onto the plane containing the source element. In many cases, this permits highly accurate results without the splitting of the source element. When splitting is required, high-aspect-ratio subtriangles are easily avoided.
Direct numerical evaluation of near strongly-singular integrals
In this work we present a simple and efficient numerical procedure to evaluate near strongly-singular integrals involving potential gradients for observation points at finite arbitrary small distance from the source domain. The proposed approach is based on a singularity cancellation scheme, subdividing the source domain into a disc plus truncated sub-triangles. After proper variable transformations, the kernel singularity is exactly cancelled, and the resulting integral is evaluated through a low order Gauss-Legendre quadrature scheme.