A document discusses a method for simple and efficient numerical integration (quadrature) of integrals characterized by a near-hypersingularity. Near-hypersingular integrals arise in gradients of Newton-type potentials when the fields at a desired observation point are close to the source. The method designed here provides for accurate and efficient integration for both the strongly and weakly singular terms. Suitable cancellation techniques for numerical computation of weakly-singular integrals have been identified, and this method is a cancellation type and handles multiple singularity types.
Rather than splitting a triangular source element into three sub-triangles, as is described in a much referenced paper by M.G. Duffy, this method proposes splitting into a disk centered at projection of the observation point onto the surface containing the source. The three sub-triangles become truncated at the disk. This new splitting, in conjunction with a highly efficient Riemann quadrature for azimuthal integration over the disk and a specialized, weighted quadrature over the radial direction on the disk, enables extremely efficient integration of the nearly-hypersingular terms. A table of specialized quadrature coordinates and weights is tailored to specific disk radii. These coordinates exactly integrate exponentials that arise from a radial-directed transformation. The disk radius is selected to match one of the pre-computed quadrature rules contained in the table.
This method assumes a mathematical transformation of variables that the authors have published previously.
This work was done by Patrick W. Fink and Michael A. Khayat of Johnson Space Center. MSC-24332-1