The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements

This paper presents a new transformation termed as the distance sinh transformation for the numerical evaluation of nearly singular integrals arising in 3D BEM. The new transformation is an improvement of the previous sinh transformation. The original sinh transformation is only limited to planar elements. Moreover, when the nearly singular point is located outside the element, results obtained by the original sinh transformation combined with conventional subdivision method are not quite accurate. In the presented work, the sinh transformation combined with the distance function is proposed to overcome the drawbacks of the original sinh transformation. With the improved transformation, nearly singular integrals on the curved surface elements can be accurately calculated. Furthermore, an alternative subdivision method is proposed using an approximate nearly singular point, which is quite simple for programming and accurate results can be obtained. Numerical examples for both curved triangular and quadrangular elements are given to verify the accuracy and efficiency of the presented method.     

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.     

A general and effective way for evaluating the integrals with various orders of singularity in the direct boundary element method

A general and efficient technique is developed for the evaluation of the integrals with various orders of singularity, such as occur in the three-dimensional boundary element method (BEM). Generalized (extended) triangle, tetrahedron polar co-ordinate mappings together with two conditions are used to remove the singularity of the integrals, and to evaluate the corresponding non-singular ones in a new numerical space. Triangle and tetrahedron polar co-ordinates in Reference 1 are proved to be a special case of the generalized ones in this paper. With the developed idea, boundary element results converge rapidly towards the analytical solutions for the strongly singular integrals evaluated directly, and the analytical solutions can be gained in principle, even when employing higher order, triangular boundary elements and tetrahedral cells. The generality and practicability of the method are demonstrated in the case of higher order elements, discontinuous elements and large engineering problems.     

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.     

Unsymmetric extensions of Wilson's incompatible four-node quadrilateral and eight-node hexahedral elements

The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with Wilson's incompatible functions are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.     

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.     

A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method

The purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto-plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co-ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three-dimensional internal cells as occur in applications of the BEM to three-dimensional elasto-plasticity. For this purpose so-called tetrahedron polar co-ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the developed concept.     

New variable transformations for evaluating nearly singular integrals in 3D boundary element method

This work presents new variable transformations for accurate evaluation of the nearly singular integrals arising in the 3D boundary element method (BEM). The proposed method is an extension of the variable transformation method in Ref. [4] for 2D BEM to 3D BEM. In this paper, first a new system denoted as $(\alpha ,\beta )$ is introduced compared with the polar coordinate system. So the original transformations in Ref. [4] can be developed to 3D in $(\alpha ,\beta )$ or the polar coordinate system. Then, the new transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, a new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. To perform integration on irregular elements, an adaptive integration scheme combined with the transformations is applied. Numerical examples compared with other methods are presented. The results demonstrate that our method is accurate and effective.     

Analytical integrations in 2D BEM elasticity

In the context of two‐dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, finite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Integral and geometrical means in the analytical evaluation of the BEM integrals for a 3D Laplace equation

By using integral theorems and geometrical interpretations, the analytical formulas for the coefficients occurring in boundary element method (BEM) equations for a 3D Laplace equation were found for arbitrary planar polygonal boundary elements with constant approximation. Closed forms for the gradients of the coefficients were also obtained. In addition, an analytical formula for planar triangular boundary elements with a linear approximation of potential was given. The formulas obtained are appropriate especially in singular and nearly singular cases.     

广义协调平板型三角形壳元

本文构造了一种具有三个角点十八个自由度的平板三角形壳元GST18。其拉伸与弯曲部分分别由含旋转自由度的三角形膜元和薄板弯曲三角形元组成。广义协调方法的采用,使得该单元的收敛性得到保证。在结点上引入了平面内旋转自由度,从根本上克服了单元共面刚度矩阵出现奇异这一困难。对平面膜元采用了缩减积分方案,使该单元不会产生薄膜闭锁现象。数值算例表明,本文提出的GST18薄壳元是计算精度优于同类单元的可靠、实用的单元。     

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.     

The iterated sinh transformation

A sinh transformation has recently been proposed to improve the numerical accuracy of evaluating nearly singular integrals using Gauss–Legendre quadrature. It was shown that the transformation could improve the accuracy of evaluating such integrals, which arise in the boundary element method, by several orders of magnitude. Here, this transformation is extended in an iterative fashion to allow the accurate evaluation of similar types of integrals that have more spiked integrands. Results show that one iteration of this sinh transformation is preferred for nearly weakly singular integrals, whereas two iterations lead to several orders of magnitude improvement in the evaluation of nearly strongly singular integrals. The same observation applies when considering integrals of derivatives of the two‐dimensional boundary element kernel. However, for these integrals, more iterations are required as the distance from the source point to the boundary element decreases. Copyright © 2007 John Wiley & Sons, Ltd.     

A polar coordinate integration scheme with a hierachical correction procedure to improve numerical accuracy on the boundary element method

In order to calculate accurate physical values of interior region by the boundary element method, a new approach is proposed to singular kernel integration which is applicable to general isoparametric elements and never fails, no matter where the internal point under consideration may be located. The integration scheme consists of two parts.First, the singular kernel functions are assumed to be reciprocal of the distance types. Then the present scheme describes on the quadrate boundary element. The element is subdivided into four triangular regions for which Gauss-Legendre numerical quadrature is applied.Secondly, a method is proposed to reduce residual errors in the application of the above mentioned numerical scheme. The boundary integrals to calculate interior physical values are expressed formally with exact and numerical error terms, and boundary values in error terms are expanded by a Taylor series around the interior point. To evaluate the coefficient of each derivative in the series, a boundary integral form of an identity with respect to a vector from the interior point to the boundary surface is derived. Error resulting from numerical integration of the identity is found to coincide with the coefficient of the derivative in the Taylor series. Thus, the correction factor for numerical errors is obtained.The present scheme was verified to be quite effective such that both the numerical error and CPU time became 1/100 less than those by the double exponential quadrature. Moreover, the present numerical scheme is applicable to general curved elements.     

A SINGULAR INTEGRAL EQUATION SOLUTION FOR THE LINEAR ELASTIC CRACK OPENING DISPLACEMENT OF AN ARBITRARILY SHAPED PLANE CRACK: PART I FINITE-PART INTEGRAL SOLUTIONS

Abstract— The aim of the paper is to compute the local crack face displacements of a linear elastic body containing an arbitrarily shaped plane crack. From the crack face displacements the local stress intensity factors can be derived.
The boundary value problem for a plane crack of arbitrary shape, embedded in a linear elastic medium, has been treated by several authors by the singular integral equation (SIE) approach. Their computations lead to a set of hyper-singular integral equations for the Cartesian components of the unknown crack face displacements. To solve these equations the authors present a discretization procedure based on six-node triangular finite elements. A total set of 24 finite-part integrals defined over a triangular area can be developed. These 2D-finite-part integrals can be split into both a 1D-regular and a 1D-finite-part-integral by means of the polar coordinates so that they can be solved in closed form. Finally, the investigation of the SIEs is reduced to a discrete set of linear algebraic equations for the unknown nodal point values. The necessary steps will be demonstrated in detail. The derived closed-form solutions will be offered in the text and in the appendices.     

Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

For a long time, most of the current numerical methods, including the finite element method, have not been efficient to analyze stress fields of very thin structures, such as the problems of thin coatings and their interfacial/internal mechanics. In this paper, the boundary element method for 2-D elastostatic problems is studied for the analysis of multi-coating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt with efficiently by using a semi-analytical algorithm. The proposed semi-analytical integral formulas, compared with current analytical methods in the BEM literature, are suitable for high-order geometry elements when nearly singular integrals need to be calculated. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 10⁻⁶–10⁻⁹, which is sufficient for modeling most coated systems in the micro- or nano-scales.     

弹性力学的样条边界元法

本文对弹性力学问题提出一个样条边界元法。文中用三次B样条函数来逼近边界未知量,并且利用域外奇点法建立了样条边界元法的计算格式。这种计算格式十分简单,容易在微机或小机上实现。利用域外奇点法建立起来的计算格式,完全可以避免奇异积分。计算结果表明,样条边界元法是一个经济有效的计算方法。     

A non-linear transformation applied to boundary layer effect and thin-body effect in BEM for 2D potential problems

The accurate numerical evaluation of nearly singular integrals plays an important role in many engineering applications. In general, these include evaluating the solution near the boundary or treating problems with thin domains, which are respectively named the boundary layer effect and the thin-body effect in the boundary element method. Although many methods of evaluating nearly singular integrals have been developed in recent years with varying degrees of success, questions still remain. In this article, a general non-linear transformation for evaluating nearly singular integrals over curved two-dimensional (2D) boundary elements is employed and applied to treat boundary layer effect and thin-body effect occurring in 2D potential problems. The introduced transformation can remove or damp out the rapid variations of nearly singular kernels and extremely high accuracy of numerical results can be achieved without increasing other computational efforts. Extensive numerical experiments indicate that the proposed transformation will be more efficient, in terms of the necessary integration points and central processing unit-time, compared to previous transformation methods, especially for dealing with thin-body problems.     

The semianalytical analysis of nearly singular integrals in 2D potential problem by isogeometric boundary element method

Benefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density.