The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements

This work presents an improved approach for the numerical evaluation of nearly singular integrals that appear in the solution of two-dimensional (2D) boundary element method (BEM) using parabolic geometry elements. The proposed method is an extension of the sinh transformation, which is used to evaluate the nearly singular integrals on linear and/or circular geometry elements. The new feature of the present method is that the distance from the source point to parabolic elements is expressed as r2=(ξ−η)2g(ξ)+b2 where g(ξ) is a well-behaved function, η and b stand for the position of the projection of the nearly singular point and the shortest distance from the calculation point to the element, respectively. The sinh transformation therefore can be employed in a straight-forward fashion. The proposed method is shown to have the same advantages as the previous sinh transformation, in that it is straight-forward to implement, very accurate and can be applied to a wide class of nearly singular integrals.