A sinh transformation for evaluating nearly singular boundary element integrals

Abstract

SUMMARYAn implementation of the boundary element method requires the accurate evaluation of many integrals. When the source point is far from the boundary element under consideration, a straightforward application of Gaussian quadrature suffices to evaluate such integrals. When the source point is on the element, the integrand becomes singular and accurate evaluation can be obtained using the same Gaussian points transformed under a polynomial transformation which has zero Jacobian at the singular point.A class of integrals which lies between these two extremes is that of 'nearly singular' integrals. Here, the source point is close to, but not on, the element and the integrand remains finite at all points. However, instead of remaining flat, the integrand develops a sharp peak as the source point moves closer to the element, thus rendering accurate evaluation of the integral difficult. This paper presents a transformation, based on the sinh function, which automatically takes into account the position of the projection of the source point onto the element, which we call the 'nearly singular point', and the distance from the source point to the element. The transformation again clusters the points towards the nearly singular point, but does not have a zero Jacobian. Implementation of the transformation is straightforward and could easily be included in existing boundary element method software.It is shown that, for the two-dimensional boundary element method, several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of Gaussian quadrature. Asymptotic estimates for the truncation errors are also quoted.

Fast Ewald summation for Stokesian particle suspensions

“…We will here detail how the method can be applied to the stresslet sum (25), essentially by following their derivation step by step and introducing the higher-rank tensors of the stresslet. We begin by introducing an arbitrary parameter η to split the Gaussian in the k-space sum (24),…”

“…The sums converge rapidly in terms of r c and K, but not in terms of computational time; the stresslet sum (25) has O N 2 complexity and the stokeslet sum (26) has O (N p N ) complexity, both with constants that grow rapidly as we increase the cutoffs. Evidently, a fast method of computing these sums is required.…”

“…When we compute the Ewald sums for the stresslet (25) and stokeslet (26), we need to truncate them at some point to avoid adding an infinite number of terms. To make our computations efficient, we would furthermore like to truncate the sums where the truncation error falls below a certain tolerance.…”

“…We then create a rotated polar m p × n p grid with the north pole at x p on the spheroid, using the same quadrature rule as on the original grid, but with the θ direction quadrature points clustered closer to the pole by a sinh transformation [25]. This is to resolve the rapidly varying integrand in the nearly singular region 4 .…”

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

“…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].…”