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

 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. Received: 20 November 2001 / Accepted: 4 June 2002 The work was supported by the Science Foundation of Shanghai Municipal Commission of Education.     

Hypersingular integrals at a corner

For a smooth boundary, hypersingular integrals can be defined as a limit from the interior, the approach direction being taken, for convenience, normal to the surface. At a boundary corner, the limit process, with a necessarily non-normal approach direction, provides a valid definition of the hypersingular equation, as long as the same direction is employed for all integrations. The terms which are potentially singular in the limit are shown to cancel, provided the function approximations at the corner are consistent. The analytical formulas for the singular integrals are more complicated than for a smooth surface, but are easily obtained using symbolic computation. These techniques have been employed to accurately solve the ‘L-shaped domain’ potential considered by Jaswon and Symm (Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, New York, USA, 1977).     

New variable transformations for evaluating nearly singular integrals in 2D boundary element method

This work presents a further development of the distance transformation technique for accurate evaluation of the nearly singular integrals arising in the 2D boundary element method (BEM). The traditional technique separates the nearly hypersingular integral into two parts: a near strong singular part and a nearly hypersingular part. The near strong singular part with the one-ordered distance transformation is evaluated by the standard Gaussian quadrature and the nearly hypersingular part still needs to be transformed into an analytical form. In this paper, the distance transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. With the new formulation, the nearly hypersingular integral can be dealt with directly and the near singularity separation and the cumbersome analytical deductions related to a specific fundamental solution are avoided. Numerical examples and comparisons with the existing methods on straight line elements and curved elements demonstrate that our method is accurate and effective.     

Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE,and piecewise-constant and piecewise-linear approximations

In this article the hypersingular integrals that arise when boundary integral equation (BIE) methods are used to solve fracture mechanics problems are considered. An approach for hypersingular integral regularization is based on the theory of distribution and Green's theorems. This approach is applied for regularization of the hypersingular integrals over triangular boundary elements (BEs) for the case of piecewise-constant and piecewise-linear approximations. The hypersingular integrals are transformed into regular contour integrals that can be easily calculated analytically.     

Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.     

Finite parts of singular and hypersingular integrals with irregular boundary source points

The subject of this paper is the evaluation of finite parts (FPs) of certain singular and hypersingular integrals, that appear in boundary integral equations (BIEs), when the source point is an irregular boundary point (situated at a corner on a one-dimensional plane curve or at a corner or edge on a two-dimensional surface). Two issues addressed in this paper are: an unified, consistent and practical definition of a FP with an irregular boundary source point, and numerical evaluation of such integrals together with solution strategies for hypersingular BIEs (HBIEs). The proposed formulation is compared with others that are available in the literature and interesting connections are made between this formulation and those of other researchers.     

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A 3-D hypersingular Boundary Integral Equation (BIE) of elastoplasticity is derived. Using this formulation the displacement rate gradients and the complete stress tensor on the boundary can be evaluated directly as opposed to the classical approach, where the shape functions derivatives are to be calculated. The regularization of strongly singular and hypersingular boundary integrals, as well as strongly singular domain integrals for a source point positioned on the boundary is carried out in a general manner. Arbitrary types of elements and arbitrary positions of the source point with respect to continuity requirements can be used. Numerical 3-D elastoplastic examples (notch and crack problems) illustrate the advantages of the proposed method.     

The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements

This work presents an improved approach for the numerical evaluation of nearly singular integrals that appear in the solution of two-dimensional (2D) boundary element method (BEM) using parabolic geometry elements. The proposed method is an extension of the sinh transformation, which is used to evaluate the nearly singular integrals on linear and/or circular geometry elements. The new feature of the present method is that the distance from the source point to parabolic elements is expressed as r²=(ξ?η)²g(ξ)+b² where g(ξ) is a well-behaved function, η and b stand for the position of the projection of the nearly singular point and the shortest distance from the calculation point to the element, respectively. The sinh transformation therefore can be employed in a straight-forward fashion. The proposed method is shown to have the same advantages as the previous sinh transformation, in that it is straight-forward to implement, very accurate and can be applied to a wide class of nearly singular integrals.     

Regularization of nearly hypersingular integrals in the boundary element method

A critical aspect in all implementations of the boundary element method is an accurate computation of the kernels' integration. These kernels are singular or hypersingular when the collocation point belongs to the integration element, and different techniques have been devised to tackle this problem. Another important issue is the integration of the kernels when the collocation point is close to but not in the integration element. The ensuing integrals although regular are termed quasi-singular or nearly singular, and quasi-hypersingular or nearly hypersingular since the integrand varies rapidly within the integration interval, and cannot be accurately computed by standard procedures. A kernels' complex regularization procedure is presented in this paper, which leads to a decomposition of the quasi-singular and quasi-hypersingular integrals in a series of simpler terms. The method is applied to the stress boundary integral equation for two-dimensional bodies, and it is tested in both curved and straight elements. For straight elements, the method leads to closed-form formulas, which are included in the paper.     

Using the iterated sinh transformation to evaluate two dimensional nearly singular boundary element integrals

Recently, sinh transformations have been proposed to evaluate nearly weakly singular integrals which arise in the boundary element method. These transformations have been applied to the evaluation of nearly weakly singular integrals arising in the solution of Laplace's equation in both two and three dimensions and have been shown to evaluate the integrals more accurately than existing techniques.More recently, the sinh transformation was extended in an iterative fashion and shown to evaluate one dimensional nearly strongly singular integrals with a high degree of accuracy. Here the iterated sinh technique is extended to evaluate the two dimensional nearly singular integrals which arise as derivatives of the three dimensional boundary element kernel. The test integrals are evaluated for various basis functions and over flat elements as well as over curved elements forming part of a sphere.It is found that two iterations of the sinh transformation can give relative errors which are one or two orders of magnitude smaller than existing methods when evaluating two dimensional nearly strongly singular integrals, especially with the source point very close to the element of integration. For two dimensional nearly weakly singular integrals it is found that one iteration of the sinh transformation is sufficient.     

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co‐ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical treatment of finite part integrals in 2-D boundary element analysis with application infracture mechanics

For the solution of problems in fracture mechanics by the boundary element method usually the subregion technique is employed to decouple the crack surfaces. In this paper a different procedure is presented. By using the displacement boundary integral equation on one side of the crack surface and the hypersingular traction boundary integral equation on the opposite side, one can renounce the subregion technique.An essential point when applying the traction boundary integral equation is the treatment of the thus arising hypersingular integrals. Two methods for their numerical computation are presented, both based on the finite part concept. One may either scale the integrals properly and use a specific quadrature rule, or one may apply the definition formula for finite part integrals and transform the resulting regular integrals into the usual element coordinate system afterwards. While the former method is restricted to linear or circular approximations of the boundary geometry, the latter one allows for arbitrary curved (e.g. isoparametric) elements. Two numerical examples are enclosed to demonstrate the accuracy of the two boundary integral equations technique compared with the subregion technique.     

The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements

This paper presents a further development of the Boundary Contour Method (BCM) for two-dimensional linear elasticity. The new developments are: (a) explicit use of the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor, (b) quadratic boundary elements compared to linear elements in previous work and (c) evaluation of stresses both inside and on the boundary of a body. This method allows boundary stress computations at regular points (i.e. at points where the boundary is locally smooth) inside boundary elements without the need of any special algorithms for the numerical evaluation of hypersingular integrals. Numerical solutions for illustrative examples are compared with analytical ones. The numerical results are uniformly accurate.     

Hypersingular quarter-point boundary elements for crack problems

The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.     

Numerical solution of convection–diffusion problems at high Péclet number using boundary elements

This paper describes a boundary element scheme for solving steady-state convection–diffusion problems at high Péclet numbers. A special treatment of the singular integrals is included which uses discontinuous elements and a regularization procedure. Transformations are performed to avoid directly evaluating Bessel functions for Cauchy principal value and hypersingular integrals. Test examples are solved with values of Péclet number up to 10⁷ to assess the numerical scheme. © 1998 John Wiley & Sons, Ltd.     

Distance transformation for the numerical evaluation of nearly singular integrals on triangular elements

The accurate numerical evaluation of nearly singular boundary integrals is a major concerned issue in the implementation of the boundary element method (BEM). In this paper, the previous distance transformation method is extended into triangular elements both in polar and Cartesian coordinate systems. A new simple and efficient method using an approximate nearly singular point is proposed to deal with the case when the nearly singular point is located outside the element. In general, the results obtained using the polar coordinate system are superior to that in the Cartesian coordinate system when the nearly singular point is located inside the element. Besides, the accuracy of the results is influenced by the locations of the nearly singular point due to the special topology of triangular elements. However, when the nearly singular point is located outside the element, both the polar and Cartesian coordinate systems can get acceptable results.     

Coupled 2D electric and mechanical fields in piezoelectric solids containing cracks and thin inhomogeneities

This paper develops integral equations and boundary element method for determination of 2D electro-elastic state of solids containing cracks, thin voids and inclusions. It proves that stress and electric displacement field near the tip of thin inhomogeneity possesses square root singularity. Thus, for determination of electromechanical fields near thin defects new special base functions are introduced. The interpolation quadratures along with the polynomial transformations are adopted for efficient numerical evaluation of singular and hypersingular integrals. Presented numerical examples show high efficiency and accuracy of the proposed approach.     

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.     

3D acoustic wave simulation using BEM formulations: Closed form integration of singular and hypersingular integrals

Among the obstacles to applying boundary element techniques to three-dimensional wave propagation problems is the difficulty of accurately representing the singular and hypersingular terms at the points of application of the virtual loads. This paper presents the analytical evaluation of the singular and hypersingular integrals for constant boundary elements. First, the singular integral results are compared with those evaluated by means of a Gaussian quadrature scheme, which uses an enormous amount of sampling points. In the case of hypersingular integrals the comparison makes use of the results provided by the method presented by Terai [T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods, J Sound Vib 69 (1980) 71–100.]. An additional verification is performed by comparing the boundary element method (BEM) results with known analytical solutions for cylindrical inclusions.     

Optimal transformations of the integration variables in computation of singular integrals in BEM

The paper deals with numerical integrations of singular integrals in BEMs. It is shown that from the point of view of numerical integrations, the only serious problem which can arise is due to weakly singular and nearly singular integrals. We pay attention to the study of numerical integrations of nearly singular integrals by using transformations of the integration variables. Theoretical considerations and numerical experiments are performed for the integrals occurring in 2‐D BEM formulations. The use of optimal transformation is confronted with the optimization of a polynomial transformation. Copyright © 2000 John Wiley & Sons, Ltd.