Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method

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

“…The semi-analytical method is accurate but it is not available for a general distributed function and curved boundary surface because closed-form integrations of the boundary function are not possible [6]. The most popular techniques used currently to calculate the nearly singular integrals are various nonlinear transformations, for example, the rational transformation [21], the optimal transformation [24], the distance transformation [28], the sinh transformation [22,[29][30][31], and the exponential transformation [32]. The common feature of these transformations is to remove or smooth out the near singularities of the integrand using a nonlinear function before the conventional Gaussian quadrature is applied.…”

“…Inspired by the pioneering work mentioned above, we present an extension of the previously published sinh transformation [30,33] for the evaluation of nearly singular integrals, which arise in general 3D anisotropic boundary element analysis. It will be demonstrated that the nearly singular integrals with anisotropic properties can be always transformed to regular ones through simple, although rigorous, manipulations.…”

“…In addition, almost all the methods developed so far share the common feature of evaluating nearly singular integrals that only exist in isotropic BEM formulations. To date, very few studies on the calculation of nearly singular integrals arising in general anisotropic media have been reported in the BEM community [30].…”

A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM

“…Recently, Mehdi and Hossein [21,22] presented a semianalytical algorithm to calculate the nearly strongly singular boundary integrals in BEM of 2-D Poisson's equation based on the complex space. Lv [23] and Gu [24], respectively, gave the function transformations for r to successfully compute the nearly strong and hypersingular integrals on curved elements in 2-D potential BEM.…”

Calculation of three-dimensional nearly singular boundary element integrals for steady-state heat conduction

“…Effective computation of nearly singular integrals has received intensive attention in recent years. Various numerical techniques have been developed to remove the near singularities, such as nonlinear transformations [19], Taylor series expansion algorithm [22], global regularization [6,23], optimization transformation [24], semi-analytical or analytical integral formulas [25][26][27][28][29], the sinh transformation [30][31][32][33][34], polynomial transformation [35], adaptive subdivision method [7,36], distance transformation technique [21,[38][39][40], the PART method [41][42][43], and the exponential transformation [44][45][46]. Most of them benefit from the strategies for computing singular integrals [6,22,23].…”