Abstract
In this paper, we present a purely numerical procedure to evaluate strongly near-singular integrals involving the gradient of Helmholtz-type potentials for observation points at finite, arbitrarily small distances from the source domain. In the proposed approach the source domain is subdivided into a disc plus truncated subtriangles, and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity. A novel feature of the proposed angular transform is that required discrete values of the inverse transform, which is transcendental, are determined via a root-finding procedure; the same idea can also be applied to other transforms that arise in singularity cancellation methods. The resulting integral may then evaluated via a low order Gauss-Legendre quadrature scheme.
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