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.     

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

This paper presents a study of the performance of the non‐linear co‐ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non‐linear polynomial transformations is presented for two‐dimensional problems. Effectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more efficiently handled with transformations valid for end‐point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non‐linear transformation has been proposed. This composite approach is found to be more accurate, efficient and robust than the singularity subtraction method and the non‐linear transformation methods. Copyright © 2001 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 sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals

A new transformation technique is introduced for evaluating the two‐dimensional nearly singular integrals, which arise in the solution of Laplace's equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one‐dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two‐dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of b values considered (0<b?10), and for flux integrals where b>0.01. For smaller values of b, the use of the Lmethod is recommended for flux integrals. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

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.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

A fast boundary cloud method for 3D exterior electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least‐squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non‐singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Generic domain decomposition and iterative solvers for 3D BEM problems

In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary‐element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi‐singular boundary‐element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non‐Hermitian matrices. The former fact has implied in CPU‐time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle‐polar‐co‐ordinate transformation and the Telles co‐ordinate transformation, applied in previous works independently for evaluating singular and quasi‐singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrix‐assembly routines. In addition, the Jacobi‐preconditioned biconjugate gradient (J‐BiCG) solver is used to develop a generic substructuring boundary‐element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted. Copyright © 2006 John Wiley & Sons, Ltd.     

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd.     

Three-dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies

A frequency domain boundary element methodology of solving three dimensional electromagnetic wave scattering problems by dielectric particles is reported. The method utilizes a computationally attractive surface integral equation containing only weakly and strongly singular integrals in the contrast to most formulations involving not only strongly singular but hypersingular integrals as well. The main advantage of this integral equation is the fact that its strongly singular part is similar to the one appearing in the corresponding integral equation of dynamic elasticity. Thus, well known advanced integration techniques used successfully in elastic scattering problems can be directly applied to the present analysis. Both continuous and discontinuous quadratic elements are employed in order to accurately treat dielectric scatterers with smooth and piecewise smooth boundaries. Numerical examples dealing with three dimensional electromagnetic wave scattering problems demonstrate the accuracy and efficiency of the proposed boundary element formulation.     

Boundary element formulation for 3D transversely isotropic cracked bodies

The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three‐dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter‐point crack front elements. No change of co‐ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three‐dimensional quadratic and quarter‐point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

A direct traction BIE approach for three-dimensional crack problems

A boundary element (BE) approach based on the traction boundary integral equation for the general solution of three-dimensional (3D) crack problems is presented. 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 of any change of coordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. In order to show the generality, simplicity and robustness of the proposed approach, different flat and curved crack problems in infinite and finite domains are analyzed. A simple BE discretization strategy is adopted. The results obtained using rather course meshes are very accurate. The emphasis of this paper is on the effective application of the proposed BE approach and it is pretended to contribute to the transformation of hypersingular boundary element formulation in something as clear, general and easy to handle as the classical formulation but much better suited for fracture mechanics problems.     

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.     

Complete semi‐analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method

A complete semi‐analytical treatment of the four‐dimensional (4‐D) weakly singular integrals over coincident, edge adjacent and vertex adjacent triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, is presented. The overall analysis is based on the direct evaluation method, utilizing a series of coordinate transformations, together with a re‐ordering of the integrations, in order to reduce the dimensionality of the original 4‐D weakly singular integrals into, respectively, 1‐D, 2‐D and 3‐D numerical integrations of smooth functions. The analytically obtained final formulas can be computed by using typical library routines for Gauss quadrature readily available in the literature. A comparison of the proposed method with singularity subtraction, singularity cancellation and fully numerical methods, often used to tackle the multi‐dimensional singular integrals evaluation problem, is provided through several numerical examples, which clearly highlights the superior accuracy and efficiency of the direct evaluation scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Galerkin boundary integral analysis for the axisymmetric Laplace equation

The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.