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

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

“…The derived fundamental solutions u * k (x, y), q * (x, y) and q * k (x, y) appearing in the integral representations (6) and (7) are given by [4] …”

“…One of the typical and most significant issues of almost all BEM analyses is the accurate evaluation of nearly singular boundary element integrals [2][3][4][5][6]. These integrals are nearly singular in the sense that the calculation point is approaching towards, but not on, the boundary element.…”

A multi-patch nonsingular isogeometric boundary element method using trimmed elements

“…One approach uses special techniques to accurately evaluate these integrals, e.g., analytical methods [27,28], degenerate mapping methods [41], radial integration methods [22,23] and non-linear transformation methods [71,81] for singular integrals, and domain division methods [9,24], semi-analytical and analytical methods [54,82] and a series of transformation methods [35,49,50,79,80] for the nearly singular integrals. These methods only work for simple shapes such as triangular and quadrilateral elements, and therefore they cannot be used in irregular trimmed elements.…”

Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

“…To date, very few studies on the calculation of nearly singular integrals over a curved surface have been reported in the BEM community [23]. Johnston and Elliott [8][9][10] have proposed an efficient strategy to numerically compute the nearly singular integrals over the 9-point Lagrangian curved surface element in 3D BEM using the sinh transformation, and Miao and his co-workers [12] also have done further researches on this topic. However, it is worth noting that, in their studies, only simple test integrals have been examined.…”

“…transformation [4], sigmoidal transformation [5,6], sinh transformation [6][7][8][9][10][11][12], rational transformation [13], distance transformation [14][15][16][17], exponential transformation [18][19][20][21][22][23] and combinations of the polar coordinates approach and variable transformations [6]. In this work, we focus on the sinh transformation, based on the sinh function, proposed by Johnston and his collaborators [7].…”