Optimal transformations of the integration variables in computation of singular integrals in BEM

Abstract

SUMMARYThe paper deals with numerical integrations of singular integrals in BEMs. It is shown that from the point of view of numerical integrations, the only serious problem which can arise is due to weakly singular and nearly singular integrals. We pay attention to the study of numerical integrations of nearly singular integrals by using transformations of the integration variables. Theoretical considerations and numerical experiments are performed for the integrals occurring in 2-D BEM formulations. The use of optimal transformation is confronted with the optimization of a polynomial transformation.

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

“…Burton and Miller showed that the two integral equations share only one common solution and thus their combination will overcome the mathematical uniqueness difficulty [22]. However, the normal derivative integral equation contains a hyper-singular integral kernel which is numerically formidable (the integral does not exist because of divergent terms in the primitive function of the integrand [23]). A large body of literature exists on the regularisation and computation of these integrals, which remains a somewhat problematic task [24][25][26].…”

“…This may be used in conjunction with Telles adaptive transform to adjust the optimisation parameter automatically. However, previous investigations have illustrated the strong sensitivity of the integration accuracy to the optimisation parameter [23], and recently other transformations suitable for computing nearly weakly singular integrals have also been proposed [54,55]. In particular, the sinh transformation by Johnston and Elliot provides a useful alternative [50,55,56].…”

A general algorithm for evaluating nearly singular integrals in anisotropic three-dimensional boundary element analysis

“…Apart from pure analytical integration, which has obvious limitations (low order elements), many other methods have been devised. The methods developed so far include, but are not limited to, element subdivision methods [15][16][17], semi-analytical methods [6,18,19] and various nonlinear transformations [20][21][22][23][24][25][26][27]. The element subdivision method is appealing, stable, and accurate but is costly because the number of sub-elements and their sizes are strongly dependent on the order of the near singularity and the dimension of the element.…”

Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

“…However, from the point of view of numerical integrations, these integrals can not be calculated accurately by using the conventional numerical quadrature since the integrand oscillates seriously within the integration interval. In the past decades, tremendous effort was devoted to derive convenient integral forms or sophisticated computational techniques for calculating the nearly singular integrals, such as the interval subdivision method [27,28], special Gaussian quadrature method [29] and various nonlinear transformation methods [30][31][32][33][34][35][36][37][38]. Among all the available methods, analytical integration as an alternative way to improve the calculation accuracy of the nearly singular integrals has received a great amount of attention.…”