Bi-directional sinh transformations based on the generalized Duffy space for nearly singular integrals

Abstract

This paper focuses on the sinh transformation in the generalized Duffy space for accurate computation of the nearly singular integral. The integral patch is normalized through the Duffy transformation. Near-singularity has been found in both directions of the transformed parametric space. Based on the complex function theory, these near-singularities of both variables in the integral are canceled by the bi-directional sinh transformation method. Comparing with the sinh transformation in a single direction, the bi-directional transformation achieves a much more accurate result. To further improve the computational accuracy, an iterative sinh transformation is applied. In the application of the bi-directional iterative sinh transformation, it has been found that the results are much less sensitive to the location of the nearest point. Thus, high accuracy and stability can be kept in most cases.