The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements

This paper presents a new transformation termed as the distance sinh transformation for the numerical evaluation of nearly singular integrals arising in 3D BEM. The new transformation is an improvement of the previous sinh transformation. The original sinh transformation is only limited to planar elements. Moreover, when the nearly singular point is located outside the element, results obtained by the original sinh transformation combined with conventional subdivision method are not quite accurate. In the presented work, the sinh transformation combined with the distance function is proposed to overcome the drawbacks of the original sinh transformation. With the improved transformation, nearly singular integrals on the curved surface elements can be accurately calculated. Furthermore, an alternative subdivision method is proposed using an approximate nearly singular point, which is quite simple for programming and accurate results can be obtained. Numerical examples for both curved triangular and quadrangular elements are given to verify the accuracy and efficiency of the presented method.