A modified Gauss quadrature formula with special integration points for evaluation of Quasi-singular integrals

It is well known in the boundary element method that integration rules fail when the integrand presents a nearby singularity. This drawback arises when the field point is near the source point, i.e. in the case of a domain with very narrow boundaries or when the field point where we try to calculate stresses or any other field variables, is near the boundaries.In the present paper a quadrature formulas for ℓ isolated singularities near the integration interval, based on ordinary or special Langrange interpolatory polynomials, is obtained. This interpolatory formulas present similarities with known formulas for the numerical evaluation of singular integrals. Quadrature formulas for regular and singular integrals with conjugate poles are also derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy.     

The tanh transformation for the solution of singular integral equations

Using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained. The formula has the same step length h as the formula for regular integrals derived by F. Stenger. These quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity. Using these formulae the solution of singular integral equations does not depend on the order of the end point singularities. Furthermore the collocation points are given by a very simple equation and, in the case of constant coefficients, by a closed-form formula.     

A general algorithm for accurate computation of field variables and its derivatives near the boundary in BEM

A general algorithm was proposed in the paper for the accurate computation of the field variables and its derivatives at domain points near the boundary in attempt to solve the so-called boundary layer effect in the boundary element method. The algorithm is based on the parameter, including modified Gauss–Tschebyscheff quadrature formula with the aid of the approximate distance function introduced, where the parameter is defined as the ratio of the minimum distance of the domain point to the boundary and the length of the boundary element. The algorithm is not only numerically stable because the singular part of the integrand serves as the weight function in the modified Gauss–Tschebyscheff quadrature formula but also independent of the kind of boundary elements. The method can be extended to the three-dimensional case with little modifications.Numerical examples of the potential problem and the elastic problem of plane strain were given by using the cubic and the quadratic boundary elements, respectively, showing the feasibility and the effectiveness of the proposed algorithm.     

Direct Gauss quadrature formulae for logarithmic singularities on isoparametric elements

Weakly singular logarithmic integrals occur in two-dimensional (2D) BEM problems when the integration is over an element which includes the current source node. For boundary elements with curved geometry such as the quadratic and cubic element then numerical integration must be used. Recently developed direct Gauss quadrature schemes which implicitly consider the integral to include a sum of singular and non-singular terms appear to be superior to conventional schemes which represent these separate terms explicitly. This paper discusses these direct quadratures and introduces new Gauss schemes which integrate exactly the logarithmic singularities on any one of the 3 nodes of a quadratic element using a single formula. This new quadrature may also be used for linear and constant elements. Similar quadrature formulae for the 4 singularities on a cubic element are also discussed. This new approach is both accurate and simple, reducing the size of the computer program and allowing the use of the same quadrature for several isoparametric types.     

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

In 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/r² and 1/r³, 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. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical evaluation of arbitrary singular domain integrals based on radial integration method

In this paper, a new approach is presented for the numerical evaluation of arbitrary singular domain integrals based on the radial integration method. The transformation from domain integrals to boundary integrals and the analytical elimination of singularities can be accomplished by expressing the non-singular part of the integration kernels as polynomials of the distance r and using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some numerical examples are provided to verify the correctness and robustness of the presented method.     

Application of sigmoidal transformations to weakly singular and near‐singular boundary element integrals

Accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method. For a source point distant from a particular element, standard Gaussian quadrature is adequate, as well as being the technique of choice. However, when the integrals are weakly singular or nearly singular (source point near the element) this technique is no longer adequate. Here a co‐ordinate transformation technique, based on sigmoidal transformations, is introduced to evaluate weakly singular and near‐singular integrals. A sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration. The degree of clustering is governed by the order of the transformation. Comparison of this new method with existing co‐ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained. Based on observations of several integrals considered, guidelines are suggested for the order of the sigmoidal transformations. Copyright © 1999 John Wiley & Sons, Ltd.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity

Singular integrals occur commonly in applications of the boundary element method (BEM). A simple mapping method is presented here for the numerical evaluation of two-dimensional integrals in which the integrands, at worst, are O(1/r) (r being the distance from a source to a field point). This mapping transforms such integrals over general curved triangles into regular 2-D integrals. Over flat and curved quadratic triangles, regular line integrals are obtained, and these can be easily evaluated by standard Gaussian quadrature. Numerical tests on some typical singular integrals, encountered in BEM applications, demonstrate the accuracy and efficacy of the method.     

The direct boundary element method in plate bending

The direct boundary element method based on the Rayleigh-Green identity is employed for the static analysis of Kirchhoff plates. The starting point is a slightly modified version of Stern's equations. The focus is on the implementation of the method for linear elements and a Hermitian interpolation for the deflection w. The concept of element matrices is developed and the Cauchy principal values of the singular integrals are given in detail. The treatment of domain integrals, the handling of internal supports, the properties of the solution and the effect of singularities are discused. Numerical examples illustrate the various techniques. In the appendix the influence functions for the second and third derivatives of the deflection w are given.     

A direct approach for boundary integral equations with high‐order singularities

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several fields of applied mechanics, particularly when second‐ and third‐order derivatives of the primary variable are required. However, their definition and numerical treatment pose several problems. In this paper, it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to efficiently and reliably compute all the singular integrals. This is done by extending in full generality the so‐called direct approach. Only for definiteness, the method is presented for the analysis of the deflection of thin elastic plates. Numerical results concerning integrals with singularities up to order r⁻⁴ are presented to validate the proposed algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics

Summary A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.With 1 Figure     

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.     

A two-dimensional time-domain boundary element method for dynamic crack problems in anisotropic solids

A time-domain boundary element method (BEM) for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented in this paper. Strongly singular displacement boundary integral equations (DBIEs) are applied on the external boundary of the cracked body while hypersingular traction boundary integral equations (TBIEs) are used on the crack-faces. The present time-domain method uses the quadrature formula of Lubich for approximating the convolution integrals and a collocation method for the spatial discretization of the time-domain boundary integral equations. Strongly singular and hypersingular integrals are dealt with by a regularization technique based on a suitable variable change. Discontinuous quadratic quarter-point elements are implemented at the crack-tips to capture the local square-root-behavior of the crack-opening-displacements properly. Numerical examples for computing the dynamic stress intensity factors are presented and discussed to demonstrate the accuracy and the efficiency of the present method.     

Numerical quadrature over triangles with weak singularities

The present paper studies in a unified way the numerical computation of integrals over triangles with weak singularities arising from the ln r and 1/r kernels present in the boundary element method by mapping the triangle into a square and thus obtaining a regularizing effect. Three types of co‐ordinate transformations are evaluated and their performance with respect to numerical quadrature is assessed. Copyright © 1999 John Wiley & Sons, Ltd.     

Semi‐sigmoidal transformations for evaluating weakly singular boundary element integrals

Accurate numerical integration of line integrals is of fundamental importance to reliable implementation of the boundary element method. Usually, the regular integrals arising from a boundary element method implementation are evaluated using standard Gaussian quadrature. However, the singular integrals which arise are often evaluated in another way, sometimes using a different integration method with different nodes and weights. Here, a co‐ordinate transformation technique is introduced for evaluating weakly singular integrals which, after some initial manipulation of the integral, uses the same integration nodes and weights as those of the regular integrals. The transformation technique is based on newly defined semi‐sigmoidal transformations, which cluster integration nodes only near the singular point. The semi‐sigmoidal transformations are defined in terms of existing sigmoidal transformations and have the benefit of evaluating integrals more accurately than full sigmoidal transformations as the clustering is restricted to one end point of the interval. Comparison of this new method with existing coordinate transformation techniques shows that more accurate evaluation of weakly singular integrals can be obtained. Based on observation of several integrals considered, guidelines are suggested for the type of semi‐sigmoidal transformation to use and the degree to which nodes should be clustered at the singular points. Copyright © 2000 John Wiley & Sons, Ltd.     

A sinh transformation for evaluating nearly singular boundary element integrals

An 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. Copyright © 2004 John Wiley & Sons, Ltd.     

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

This work contains an analytical study of the asymptotic near‐crack contour behaviour of stresses obtained from the self‐regular traction‐boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The flat crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self‐regular traction‐BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near‐crack contour self‐regular traction‐BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log‐singular stresses are identified and extracted from the BIE. Log‐singular stress terms are revealed for the free integrals written as contour integrals and for the self‐regularized integral with the integration region divided into sub‐regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the final BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identified, and the singular information is all contained in the region near the crack contour. Copyright © 2003 John Wiley & Sons, Ltd.     

Limiting forms for surface singularity distributions when the field point is on the surface

Scalar and vector mathematical identities involving an integral of singularities distributed over a surface and sometimes over a field can be employed to define field values of a quantity of interest. As the volume excluding the singular point from the field tends to zero, the field value is derived. The expressions that result become singular as the point of interest in the field approaches the boundary. Derivation of limiting integral expressions as the field point tends to the surface having a distribution of first and second degree singularities is the main task reported. The limiting expressions for vector values require evaluation as generalized Cauchy Principal-Value Integrals for which some aspect of symmetry in a local region excluding the singularity is required. A contribution from the integral over the local region doubles the value of the identities at a point on the boundary. For a doublet distribution, a singular term arises from the local-region integration that cancels a similar singularity in the integral over the remaining surface. This local contribution for doublets depends explicitly upon the shape of the local region as well as non-orthogonality of the surface coordinate axes. The resulting expressions for surface integrals reproduce known relations for line integrals in two-dimensional fields.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.