A Family of Augmented Duffy Transformations for Near-Singularity Cancellation Quadrature

A new family of systematically constructed near-singularity cancellation transformations is presented, yielding quadrature rules for integrating near-singular kernels over triangular surfaces. This family results from a structured augmentation of the well-known Duffy transformation. The benefits of near-singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher-order basis function and curvilinear settings. Six specific transformations are constructed for near-singularities of orders one, two and three. Two of these transformations are found to be equivalent to existing ones. The performance of the new schemes is thoroughly assessed and compared with that of existing schemes. Results for the gradient of the scalar Green function are also presented. For simplicity, static kernel results are shown. The new schemes are competitive with and in some cases superior to the existing schemes considered.