A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity

Abstract

A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the near singularity of integrands resulting from the very small distance between the source and the integration points. By taking advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae, for both the 2D and 3D elastic problems. IntroductionThe accurate numerical evaluation of nearly singular boundary integrals has challenged researchers since the advent of the boundary element method. The problem generally refers to the boundary layer effect, a phenomenon that the boundary integrals cannot be computed accurately with the ordinary Gaussian quadrature if the computing point is located closely adjacent to the boundary [1-3]. Similar to the case of singular boundary integrals, the boundary layer effect comes from the properties of fundamental solutions and their derivatives, having the denominator of the Euclidean distance, r, between the source and the field points of various orders [4]. In consequence, the values of the integrals over a boundary element have a reverse correlation with this distance relative to the size of the element, so the system matrices are well behaved so enabling the boundary integrals to be evaluated accurately.The accurate numerical evaluation of nearly singular boundary integrals is crucial for many engineering problems, including, for example, the analysis of the thin or shell-like bodies as occur in many structural applications [5][6][7], the crack problems when the crack tip is deformed with an opening displacement [8], the contact problem [9] when the distance between the two contacting bodies is very small, as well as the sensitivity problem [10]. The importance of this subject area is considered second only to the singular case of the boundary element method and this explains the great attention and effort that has been focused on it in recent years [11][12][13][14][15][16][17][18][19][20][21][22][23].The difficulty encountered in the numerical evaluation results mainly from the fact that the integrands of nearly singular boundary integrals vary drastical...

Boundary integral equation supported differential quadrature method to solve problems over general irregular geometries

“…This is a limitation in the application of the BIE-DQM. However, these boundary integrals can be computed indirectly with the techniques of nearly singular boundary integrals [22][23] without any difficulty.…”

“…For the evaluation of hypersingular boundary integrals with other kernels in (14), the singular integrals are approximated by the mean values of the two corresponding nearly singular boundary integrals with distance transformation techniques [22][23]. In the numerical examples, four-point Gauss quadrature is used in general for evaluation of ordinary integrals and eight-point Gauss quadrature is used for singular integrals, but at most 16-point Gauss quadrature is used for nearly singular integrals according to the distances between x and y.…”

Evaluation of Singular and Nearly Singular Integrals in the BEM with Exact Geometrical Representation

“…The direct algorithms are calculating the nearly singular integrals directly. They usually include, but are not limited to, interval subdivision method [12][13], special Gaussian quadrature method [14][15], the exact integration method [16][17][18], and nonlinear transformation method [19][20][21][22][23]. Although great progresses have been achieved for each of the above methods, it should be pointed out that the geometry of the boundary element is often depicted by using linear shape functions when nearly singular integrals need to be calculated [22].…”

A quadrature formula for integrals with nearby singularities

“…The other way to proceed with these integrals is based on quadrature rules for each of these integrals. We can distinguish procedures based on transformation, which remedy the singularities [12][13][14], and on procedures based on general quadrature rules [15][16][17][18][19][20][21][22] (special quadratures rules). The latter are able to give the exact result with only few integration points.…”