A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals

Abstract

SUMMARYA new transformation technique is introduced for evaluating the two-dimensional nearly singular integrals, which arise in the solution of Laplace's equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one-dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two-dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of b values considered (0<b 10), and for flux integrals where b>0.01. For smaller values of b, the use of the L −1/5 1 method is recommended for flux integrals.

On the computation of nearly singular integrals in 3D BEM collocation

“…Precisely, in the section of the numerical tests we chooseq = 2 when |c| = 0.1, 0.01 andq = 4 when |c| = 0.001. When |c|>0.01 the numerical results obtained by using (14) and those by (16) and (18) are quite similar. However, when |c| 0.01 the transformations (16) and (18) are more effective than (14).…”

“…When |c|>0.01 the numerical results obtained by using (14) and those by (16) and (18) are quite similar. However, when |c| 0.01 the transformations (16) and (18) are more effective than (14). Indeed, they allow one to obtain good accuracy in the computation of (6) by means of the n-point Gauss-Legendre rule already for small values of n.…”

“…which, similar to (14), allows one to eliminate the nearly singular term 1/ 2 +c 2 through the Jacobian. Indeed, we have…”

“…In the literature (see, for instance, [7][8][9][10][11]14]), the numerical strategies for nearly singular integrals generally have been defined and tested on triangles or quadrangles of unit size (h ≈ 1) and for small values of dist(x, ), such as 0.1, 0.05, 0.03, 0.01, 0.001 (in some case even 0.0001). Indeed, in these cases (see [14]), the straight Gaussian quadrature performs very poorly. Also here, to define our strategy and to compare our numerical results with those of the above-mentioned papers, we consider the triangle of unit size and dist(x, ) 0.1.…”

On the use of non‐linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues

“…Specifically, it plays a key role in the boundary element method (BEM) in which the appearance of singular kernels (of weak or Cauchy-type) constitutes an important problem. Owing to its noticeable influence, various contributions within the BEM community place emphasis on minimizing the impact of the inaccurate integration [16,[19][20][21][22][23]. The use of non-linear transformations for the modification of the integration quadratures has been studied in depth, and several rules have been proposed that might be roughly classified into polynomial and sigmoidal [24].…”