Scattering of SH wave by a crack terminating at the interface of a bimaterial

A general method for solving the scattering of plane SH wave by a crack terminating at the interface of a bimaterial is presented. The crack can terminate at the interface in an arbitrary angle. In order to solve the proposed problem, the Greens function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Greens function and the Betti-Rayleigh reciprocal theorem, the total scattered field of the crack is constructed. The total scattered field of the crack is divided into a regular part and a singular part. The hypersingular integral equation of the crack is obtained in terms of the regular and singular scattered field as well as the free wave field. The stress singularity order and singular stress at the terminating point are analyzed by the hypersingular integral equation and the singular scattered field of the crack. The dynamic stress intensity factor (DSIF) at the terminating point is defined in terms of the singular stresses at the terminating point. Numerical solution of the hypersingular integral equation gives the DSIFs at the crack tips. Comparison of our results with known results confirms the proposed method. Some numerical results and corresponding analysis are given in the paper.Constructive advice from the anonymous reviewers is acknowledged.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.     

Differentiability of strongly singular and hypersingular boundary integral formulations with respect to boundary perturbations

In this paper, we establish that the Lagrangian-type material differentiation formulas, that allow to express the first-order derivative of a (regular) surface integral with respect to a geometrical domain perturbation, still hold true for the strongly singular and hypersingular surface integrals usually encountered in boundary integral formulations. As a consequence, this work supports previous investigations where shape sensitivities are computed using the so-called direct differentiation approach in connection with singular boundary integral equation formulations. Communicated by T. Cruse, 6 September 1996     

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

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

Self-regular stress integral equations for axisymmetric elasticity

The stress hypersingular integral equations of axisymmetric elasticity are considered. The singular and hypersingular integrals are regularized using the imposition of auxiliary polynomial solution, and self-regular integral equations are obtained for bounded and unbounded domains. The presented numerical examples show high efficiency of the proposed approach. The boundary layer effect is completely eliminated, and stresses and deformations can be calculated in the whole domain continuously up to the boundary.     

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

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

Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants

For a potential problem, the boundary integral equation approach has been shown to yield a nonunique solution when the geometry is equal to a degenerate scale. In this paper, the degenerate scale problem in boundary element method (BEM) is analytically studied using the degenerate kernels and circulants. For the circular domain problem, the singular problem of the degenerate scale with radius one can be overcome by using the hypersingular formulation instead of the singular formulation. A simple example is shown to demonstrate the failure using the singular integral equations. To deal with the problem with a degenerate scale, a constant term is added to the fundamental solution to obtain the unique solution and another numerical example with an annular region is also considered.     

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

Stress intensity factors for semi-elliptical surface cracks in plates and cylindrical pressure vessels using the hybrid boundary element method

At first, a hybrid boundary element method used for three-dimensional linear elastic fracture analysis is established on the basis of the first and the second kind of boundary integral equations. Then the concerned basic theories and numerical approaches including the discretization of boundary integral equations, the divisions of different boundary elements, and the procedures for the calculations of singular and hypersingular integrals are presented in detail. Finally, the stress intensity factors of surface cracks in finite thickness plates and cylindrical pressure vessels are computed by the proposed method. The numerical results show that the hybrid boundary element method has very high accuracy for the analysis of surface crack.     

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

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

A fast multipole boundary integral equation method for crack problems in 3D

This paper discusses a three-dimensional fast multipole boundary integral equation method for crack problems for Laplace's equation. The proposed implementation uses collocation and piecewise constant shape functions to discretise the hypersingular boundary integral equation for crack problems. The resulting numerical equation is solved with GMRES (generalised minimum residual method) in connection with FMM (fast multipole method). It is found that the obtained code is faster than a conventional one when the number of unknowns is greater than about 1300.     

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.     

Determination of the natural frequencies and natural modes of a rod using the dual BEM in conjunction with the domain partition technique

In this paper, the dual BEM in conjunction with the domain partition technique is employed to solve both natural frequencies and natural modes of a rod. In this new approach, there exists no spurious eigenvalue using the complex-valued singular or hypersingular equation alone. In the derivation of the singular and hypersingular integral equations, if only the real parts of the kernel functions are chosen, the resulting eigenequations have spurious eigenvalues. Such spurious eigenvalues stem from adding the dummy links into the interior structures considered. Although the spurious eigenvalues exist in this approach which uses the real-valued kernel functions, the possible indeterminacy of eigenmodes using the conventional real-valued singular or real-valued hypersingular equations disappears when the domain partition technique is adopted. The conventional real-valued multiple reciprocity BEM results in spurious eigenvalues for the mixed boundary conditions and indeterminacy of eigenmodes owing to insufficient rank of the leading coefficient matrix for the Dirichlet and Neumann boundary conditions. Such problems can be solved by combining the singular and hypersingular equations together; however, they also can be treated by using the real-valued singular or hypersingular equation alone if the domain partition technique is adopted. Three examples including the Dirichlet, Neumann and mixed type boundary conditions are investigated to show the validity of current approach.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

On the evaluation of hyper-singular integrals arising in the boundary element method for linear elasticity

The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.     

Boundary integral equations for thin bodies

A boundary integral equation formulation for thin bodies which uses CBIE (conventional BIE) only is well known to be degenerate. A mixed formulation for a thin rigid scatterer which combines CBIE and HBIE (hypersingular BIE) is motivated by examining the discretized form of the integral equations, and this formulation is shown to be non-degenerate for thin non-rigid inclusion problems. A near-singular integration procedure, useful for singular integrals as well, is presented. Finally, numerical examples for acoustic wave scattering from rigid and soft scatterers are presented.     

Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity

Summary A principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. As is the case with the corresponding definitions of Cauchy-type one-dimensional and two-dimensional principal value singular integrals, as well as Mangler-type one-dimensional principal value hypersingular integrals, the present definition is based on the special consideration of an appropriate region around the singular point. The cases of circular, square and equilateral triangular regions are considered in some detail.     

HYPERSINGULAR BEM FOR TRANSIENT ELASTODYNAMICS

The topic of hypersingular boundary integral equations is a rapidly developing one due to the advantages which this kind of formulation offers compared to the standard boundary integral one. In this paper the hypersingular formulation is developed for time-domain antiplane elastodynamic problems. Firstly, the gradient representation is found from the displacement one, removing the strong singularities (Dirac's delta functions) which arise due to the differentiation process. The gradient representation is carried to the boundary through a limiting process and the resulting equation is shown to be consistent with the static formulation. Next, the numerical treatment of the traction boundary integral equation and its application to crack problems are presented. For the boundary discretization, conforming quadratic elements are tested, which are introduced in this paper for the first time, and it is shown that the results are very good in spite of the lesser number of unknowns of this approach in comparison to the non-conforming element alternative. A procedure is devised to numerically perform the hypersingular integrals that is both accurate and versatile. Several crack problems are solved to show the possibilities of the method. To this end both straight and curved elements are employed as well as regular and distorted quarter point elements.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

The boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known. To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels. Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.