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

Abstract

SUMMARYIn this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R 3 arising in boundary element method collocation. The strategy is based on a proper use of various non-linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r , 1/r 2 and 1/r 3 , r = x−y being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive.

A new smoothing strategy for computing nearly singular integrals in 3D Galerkin BEM

“…In this paper we use the approach proposed in [21] for the computation of the inner integral (1.3), and we propose a new smoothing strategy, based again on polynomial transformations, for the computation of the outer integral (1.2). Indeed, to improve the accuracy given by the canonical product Gauss-Legendre rule when computing integrals (1.2), where the distance between ∆ i and ∆ j is very small, we preliminarily introduce proper changes of variable of polynomial type.…”

“…The author herself has recently proposed in [21] an efficient and highly competitive strategy to compute the nearly singular integrals (1.3) over planar triangles. It is based on a proper use of various nonlinear transformations; among them, the transformations of polynomial type have played a fundamental role.…”

“…By means of some comparisons we show that, when the distance between the triangles is very small with respect to their sizes, the preliminary introduction of proper polynomial transformations in the double outer integral allow improving the accuracy of the standard product Gauss-Legendre quadrature rule. To approximate the inner integral in (1.1) with a high precision, the technique in [21] or that in [6] is recommended. They give quite comparable numerical approximations of (1.1), even if a direct comparison between them just for the double inner integral shows that the approach in [21] generally gives more accurate numerical results; moreover, it uses a smaller number of quadrature nodes and, consequently, a lower computation time.…”

Analysis and estimation of quadrature errors in weakly singular source integrals of the method of moments

“…Examples of more recent schemes are previous studies. 5,[10][11][12][13][14][15] The schemes from these references are each tailored to either weak or stronger near-singularities (relating to the three-dimensional, scalar Green function and its gradient, or components thereof).…”

A Novel Smoothing Scheme of Temporal Basis Function Independent Method in Mot Based Tdie

“…This proposed technique can eliminate the singularity (or smooth) of the integrated functions by introducing the idea of variable substitution [22,23]. It is applicable to more types of T (t) compared with the analytical convolution method [19].…”