Tangential derivative of singular boundary integrals with respect to the position of collocation points

This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper definition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the definition of the integral itself. This is done here by esorting to a shape sensitivity approach, considering a particular class of infinitesimal domain perturbations that ‘move’ individual points, and especially the singular point, but leave the initial domain globally unchanged. This somewhat indirect strategy provides a proper mathematical setting for the analysis. Moreover, the resulting sensitivity expressions apply to arbitrary potential-type integrals with densities only subjected to some regularity requirements at the singular point, and thus are applicable to approximate as well as exact BEM solutions. Quite remarkable is the fact that the analysis is applicable when the singular point is located on an edge and simply continuous elements are used. The hypersingular BIE residual function is found to be equal to the derivative of the strongly singular BIE residual when the same values of the boundary variables are substituted in both SBIE and HBIE formulations, with interesting consequences for some error indicator computation strategies. © 1998 John Wiley & Sons, Ltd.     

A Galerkin symmetric and direct BIE method for Kirchhoff elastic plates: formulation and implementation

A variational Boundary Element formulation is proposed for the solution of the elastic Kirchhoff plate bending problem. The stationarity conditions of an augmented potential energy functional are first discussed. After addressing the topic of the choice of the test functions, a regularization process based on integrations by parts is developed, which allows to express the formulation in terms of double integrals, the inner being at most weakly singular and the outer regular. Standard integration procedures may then be applied for their numerical evaluation in the presence of both straight and curved boundaries. The normal slope and the vertical displacement must be C⁰ and C¹ continuous, respectively. Numerical examples show, through comparisons with analytical solutions, that a high accuracy is achieved. © 1998 John Wiley & Sons, Ltd.     

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.     

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.     

Green's functions for Kirchhoff plates of irregular shape

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B, the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.     

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

A new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case. Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.     

A boundary spectral method for solving exterior acoustical problems with hypersingular integrals

A boundary spectral method is developed to solve acoustical problems with arbitrary boundary conditions. A formulation, originally derived by Burton and Miller, is used to overcome the non‐uniqueness problem in the high wave number range. This formulation is further modified into a globally non‐singular form to simplify the procedure of numerical quadrature when spectral methods are applied. In the present approach, generalized Fourier coefficients are determined instead of local variables at nodes as in conventional methods. The convergence of solutions is estimated through the decay of magnitude of the generalized Fourier coefficients. Several scattering and radiation problems from a sphere are demonstrated with high wave numbers in the present paper. Copyright © 1999 John Wiey & Sons, Ltd.     

AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS

An indirect Boundary Element Method is employed for the static analysis of homogeneous isotropic and linear elastic Kirchhoff plates of an arbitrary geometry. The objectives of this paper consists of a construction and a study of the resulting boundary integral equations as well as a development of stable powerful algorithms for their numerical approximation. These equations involve integrals with high-order kernel singularities. The treatment of singular and hypersingular integrals and a construction of solutions in the neighborhood of the irregular points on the boundary are discussed. Numerical examples illustrate the procedure and demonstrate its advantages. © 1997 by John Wiley & Sons, Ltd.     

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated. This revised version was published online in August 2006 with corrections to the Cover Date.     

On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials

A symmetric Galerkin formulation and implementation for heat conduction in a three‐dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co‐ordinate, is used to develop a boundary‐only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct ‘limit to the boundary’ approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

Electrostatic BEM for MEMS with thin conducting plates and shells

Micro-electro-mechanical systems (MEMS) sometimes use beam or plate shaped conductors that can be very thin-with h/L≈O(10⁻²−10⁻³) (in terms of the thickness h and length L of the side of a square pate). Conventional Boundary Element Method (BEM) analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface tractions) separately on the top and bottom surfaces of such plates. A new boundary integral equation (BIE) is derived in this work that, when used together with the standard BIE with weakly singular kernels, results in a powerful technique for the BEM analysis of such problems. This new approach, in fact, works best for very thin plates. This thin plate BEM is derived and discussed in this work. Numerical results, from several BEM based methods, are presented and compared for the model problem of a parallel plate capacitor.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

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.     

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

The symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy. Copyright © 2009 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.     

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.     

Error estimation of quadrature rules for evaluating singular integrals in boundary element problems

The efficient numerical evaluation of integrals arising in the boundary element method is of considerable practical importance. The superiority of the use of sigmoidal and semi‐sigmoidal transformations together with Gauss–Legendre quadrature in this context has already been well‐demonstrated numerically by one of the authors. In this paper, the authors obtain asymptotic estimates of the truncation errors for these algorithms. These estimates allow an informed choice of both the transformation and the quadrature error in the evaluation of boundary element integrals with algebraic or algebraic/logarithmic singularities at an end‐point of the interval of integration. Copyright © 2000 John Wiley & Sons, Ltd.     

Dynamic and static analysis of cracks using the hypersingular formulation of the Boundary Element Method

A new methodology for computing dynamic stress intensity factors in the frequency domain based on the mixed boundary element method, a combination of the equations corresponding to the integral representations of displacements and tractions, is proposed and analysed. The expressions of hypersingular fundamental solution are presented and their singular parts extracted. Also, a discontinuous Singular-Quarter-Point element is constructed. Finally, various parametric computations and applications are described in order to illustrate the simplicity and accuracy of the proposed method as applied to both static and dynamic problems. © 1998 John Wiley & Sons, Ltd.     

A new fundamental solution for boundary element analysis of thin plates on Winkler foundation

This paper introduces an efficient boundary element approach for the analysis of thin plates, with arbitrary shapes and boundary conditions, resting on an elastic Winkler foundation. Boundary integral equations with three degrees-of-freedom per node are derived without unknown corner terms. A fundamental solution based upon newly defined modified Kelvin functions is formulated and it leads to a simple solution to the problem of divergent integrals. Reduction of domain loading terms for cases of distributed and concentrated loading is also provided. Case studies, including plates with free-edge conditions, are demonstrated, and the boundary element results are compared with corresponding analytical solutions. The presented formulations provide a very accurate boundary element solution for plates with different shapes and boundary conditions.     

HYPERSINGULAR RESIDUALS—A NEW APPROACH FOR ERROR ESTIMATION IN THE BOUNDARY ELEMENT METHOD

This paper presents a new approach for a posteriori ‘pointwise’ error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.