boundary elements for cracks and notches in three dimensions plane and curved cracks are modelled by boundary elements of geometry defined by conforming quadratic and hybrid quadratic hermitian cubic shape functions displacement and traction are interpolated by the quadratic functions supplemented by singular functions by which are multiplied stress intensity factors corresponding to each of the three modes of crack opening displacement for the first three eigenvalues of the williams eigenfunction expansion and its equivalent for antiplane strain singular and hypersingular boundary integral equations are taken at nodes of elements and auxiliary collocation points singular and hypersingular components of integrals are evaluated by consideration of trial displacement fields simple solutions for subdomains lying to either side of the crack examples are shown of buried and surface cracks and computed results compared with those obtained by other methods for surface cracks the computed results reveal the cause of significant discrepancies between values given by well established empirical and other formulae the modelling of notches is demonstrated by an analysis of stress in rock near a tunnel intersection computational efficiency is discussed and improvements and extensions of the analysis are proposed copyright 2005 john wiley sons ltd

an implementation of the boundary element method for zoned media with stress discontinuities the boundary element method as it applies to two dimensional elastostatic problems is implemented with quadratic variation in boundary variables and geometry additional parameters are associated with nodes at segment intersections to allow for discontinuous tractions supplementary equations consistent with linear elasticity are used to augment the regular boundary integral equations for certain mixed and displacement boundary conditions zoning or subregionalization capabilities are included to provide solutions to piecewise nonhomogeneous problems or problems on regions of irregular shape example problems are given to demonstrate the accuracy of the technique

the dual boundary element formulation for elastoplastic fracture mechanics in this paper the extension of the dual boundary element method dbem to the analysis of elastoplastic fracture mechanics epfm problems is presented the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation in order to avoid collocation at crack tips crack kinks and crack edge corners both crack surfaces are discretized with discontinuous quadratic boundary elements the elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield this region is discretized with internal quadratic quadrilateral and or triangular cells this formulation was implemented for two dimensional domains only although there is no theoretical or numerical limitation to its application to three dimensional ones a centre cracked plate and a slant edge cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature j type integrals are calculated

singular boundary elements for the analysis of cracks in plane strain straight and curved cracks are modelled by direct formulation boundary elements of geometry defined by hermitian cubic shape functions displacement and traction are interpolated by the hermitian functions supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip and subdominant modes in which it is proportional to r1 5 the singular functions extend over many boundary elements on each crack face a nodal collocation scheme is used in which additional boundary integral equations are obtained by differentiation of the equation obtained from betti s theorem the hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack examples are shown of the analysis of buried and edge cracks to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements

on the non singular traction bie in elasticity the work reported herein develops a generalized traction bie formulation which involves only weakly singular integrals in the three dimensional problem or totally regular integrals in the two dimensional problem the first step deals with the terms in the somigliana displacement identity and then the derivatives of these terms the only conditions required for the existence of the traction bie and the related somigliana stress identity are weak continuity of the in plane derivatives of the surface displacements and of the surface tractions it is shown that the cauchy principal value cpv interpretations so commonly used in bie developments are unnecessary the formulation is established not only at a smooth boundary point but also at a corner point the extension of the non singular formulation to discontinuous boundary tractions and tangential derivatives of the boundary displacements applicable to a generalized problem statement as well as the usual bem implementations is also shown in the demonstrated formulation the source points are located directly at the boundary nodes and non conformal elements are not needed

on the treatment of corners in the boundary element method abstract a new technique for treating surface discontinuities within boundary element calculations is proposed multiple nodes are used to represent the geometry and the necessary additional equations are obtained by differentiating the usual boundary element integral equation in deriving expressions for the resulting singular integrals it is found that constraints must be placed on the functional approximation at the discontinuity an algorithm for incorporating these constraints is developed and numerical tests for an exactly solvable three dimensional laplace equation problem are presented

evaluation of nearly singular integrals in boundary element contour and node methods for three dimensional linear elasticity this article addresses the issue of nearly singular integrals that arise in boundary element methods bems when one tries to evaluate displacements and stresses at points inside a body that lie close to its bounding surface an efficient new approach is proposed for the evaluation of these integrals the same issue is then discussed in the context of the boundary contour method bcm and the boundary node method bnm finally numerical results from both the bcm and the bnm are presented for an illustrative problem

direct computation of cauchy principal value integrals in advanced boundary elements this paper presents a new method for the direct computation of strongly singular integrals existing in the cauchy principal value sense it can be usefully applied in the solution by the direct boundary element method of many different problems initially some considerations are provided in order to summarize the state of the art on this issue then the features of the proposed method are reported the procedure allows the direct calculation of cauchy principal value integrals with first order singularity and it is applicable even in advanced boundary element methods employing high order elements it requires only the use of standard gaussian quadrature formulae plus the computation of a logarithmic term some examples show the effectiveness and efficiency of the procedure

evaluations of hypersingular integrals using gaussian quadrature a gaussian quadrature formula for hypersingular integrals with second order singularities is developed based on previous gaussian quadrature formulae for cauchy principal value integrals the formula uses classical orthonormal polynomials and the formula is then specialized to the case of legendre and chebyshev polynomials numerical experiments are carried out using the current formula and a previous formula developed by kutt it is found that the two methods generally give similar results and in some cases the current method works better it has also been shown that the current method allows the choice of an appropriate weight which can increase the convergence rate and the accuracy of the results copyright 1999 john wiley sons ltd

error analysis and adaptivity in three dimensional linear elasticity by the usual and hypersingular boundary contour method abstract two related topics are addressed in this article the first part of the article proves that for a certain admissible class of problems in linear elasticity the hypersingular boundary contour method hbcm can be collocated at all boundary points on the surface of a three dimensional 3 d body including those on boundary contours edges and corners because the hbcm shape functions satisfy a priori all the smoothness requirements for collocation at these points in contrast the hypersingular boundary element method needs in general relaxation of some of these smoothness requirements for its shape functions even for collocation at regular points that lie on the boundaries of boundary elements a hypersingular residual obtained from the standard and hypersingular boundary integral equations hbies has been recently proposed as a local error estimator for a boundary element for the boundary integral equation the second part in the present article is concerned with a definition of an analogous local error estimator for the boundary contour method for 3 d linear elasticity this error estimator is then used to drive an h adaptive meshing procedure numerical results are presented to demonstrate adaptive meshing for selected example problems

cpv and hfp integrals and their applications in the boundary element method the subject of this paper is hadamard finite part hfp and cauchy principal value cpv representations of certain singular integrals of primary concern here are hfp interpretations of strongly singular integrals for which cpv interpretations also exist a relationship between the two is stated and proved this matter is then illustrated by two examples and also discussed in the context of singular boundary integral equations bies for linear elasticity

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

kurtosis of garch and stochastic volatility models with non normal innovations both volatility clustering and conditional non normality can induce the leptokurtosis typically observed in financial data in this paper the exact representation of kurtosis is derived for both garch and stochastic volatility models when innovations may be conditionally non normal we find that for both models the volatility clustering and non normality contribute interactively and symmetrically to the overall kurtosis of the series

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

a hybrid micro macro bem with micro scale inclusion crack interactions local analysis schemes capable of detailed representations of micro features of a problem are integrated with a macro scale bem technique capable of handling finite geometries and realistic boundary conditions this paper focuses on micro scale interactions among cracks and inclusions as well as their ramifications on macro scale damage evaluations the micro scale effects are introduced into the macro scale bem computations through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows complete decomposition of the real problem into two sub problems one residing entirely at the micro level while the other resides at the macro level this allows for investigations of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the structural component under consideration as a first attempt dilute inclusion densities with strong inclusion crack and crack crack interactions are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can be easily extended to investigate the effects of other micro features e g interfaces short or continuous fibers in the context of linear elasticity on macro scale failure modes observed in structural components

evaluation of hypersingular integrals in the boundary element method by complex variable techniques strongly singular and hypersingular integrals that occur in many two dimensional applications of the boundary element method e g potential theory and liner elasticity can be treated in a unified manner by complex analysis this paper presents an elegant approach for the evaluation of such hypersingular integrals the limit to the boundary method where a source point is moved from the inside or outside of a body to its boundary is carried out using complex variables the connection of this limit with the hadamard finite part of hypersingular integrals and principal value integrals is established many examples are worked out throughout the paper to illustrate the theory

a weakly singular sgbem for analysis of cracks in 3d anisotropic media abstract a weakly singular symmetric galerkin boundary element method sgbem is developed for analysis of fractures in three dimensional anisotropic linearly elastic media the method constitutes a generalization of that by li et al s li m e mear l xiao symmetric weak form integral equation method for three dimensional fracture analysis comput methods appl mech engrg 151 1998 435 459 for isotropic media and is based upon a pair of weak form displacement and traction integral equations recently established by rungamornrat and mear j rungamornrat m e mear weakly singular weak form integral equations for cracks in three dimensional anisotropic media int j solids struct 45 2008 1283 1301 the formulation involves only weakly singular kernels and as a consequence standard c 0 elements can be employed in the numerical discretization the kernels possess a relatively simple form for general anisotropy which involves an equatorial line integral like that associated with the displacement fundamental solution despite their simple form it is still necessary to evaluate these kernels efficiently in the numerical implementation to do so certain symmetry properties associated with the specific material type under consideration are exploited and a simple yet accurate interpolation strategy is introduced another important aspect of the numerical treatment is the use of a special crack tip element which allows highly accurate mixed mode stress intensity factor data to be extracted as a function of position along the crack front this crack tip element was originally introduced by li et al s li m e mear l xiao symmetric weak form integral equation method for three dimensional fracture analysis comput methods appl mech engrg 151 1998 435 459 in their treatment of isotropic media and here it is extended to allow evaluation of the stress intensity factors for generally anisotropic media the element involves extra degrees of freedom associated with the nodes along the crack front with these quantities being directly related to the stress intensity factors and it is capable of representing the asymptotic behavior in the vicinity of the crack front to a sufficiently high order that relatively large crack tip elements can be utilized various examples are treated for cracks in an unbounded domain and for both embedded and surface breaking cracks in a finite domain and it is demonstrated that very accurate results can be obtained even with relatively coarse meshes

degenerate scale problem when solving laplace s equation by bem and its treatment in this paper laplace problems are solved by using the dual boundary element method bem it is found that a degenerate scale problem occurs if the conventional bem is used in this case the influence matrix is rank deficient and numerical results become unstable both the circular and elliptical bars are studied analytically in the continuous system in the discrete system the fredholm alternative theorem in conjunction with the svd singular value decomposition updating documents is employed to sort out the spurious mode which causes the numerical instability three regularization techniques method of adding a rigid body mode hypersingular formulation and cheef combined helmholtz exterior integral equation formulation concept are employed to deal with the rank deficiency problem the addition of a rigid body term c in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e c the torsion rigidities are obtained and compared with analytical solutions numerical examples including elliptical square and triangular bars were demonstrated to show the failure of conventional bem in case of the degenerate scale after employing the three regularization techniques the accuracy of the proposed approaches is achieved copyright 2004 john wiley sons ltd

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

the meshless standard and hypersingular boundary node methods applications to error estimation and adaptivity in three dimensional problems the standard singular boundary node method bnm and the novel hypersingular boundary node method hbnm are employed for the usual and adaptive solutions of three dimensional potential and elasticity problems these methods couple boundary integral equations with moving least squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter the hypersingular residuals developed for error estimation in the mesh based collocation boundary element method bem and symmetric galerkin bem by paulino et al are extended to the meshless bnm setting a simple a posteriori error estimation and an effective adaptive refinement procedure are presented the implementation of all the techniques involved in this work are discussed which includes aspects regarding parallel implementation of the bnm and hbnm codes several numerical examples are given and discussed in detail conclusions are inferred and relevant extensions of the methodology introduced in this work are provided copyright 2001 john wiley sons ltd

a broadband fast multipole accelerated boundary element method for the three dimensional helmholtz equation the development of a fast multipole method fmm accelerated iterative solution of the boundary element method bem for the helmholtz equations in three dimensions is described the fmm for the helmholtz equation is significantly different for problems with low and high kd where k is the wavenumber and d the domain size and for large problems the method must be switched between levels of the hierarchy the bem requires several approximate computations numerical quadrature approximations of the boundary shapes using elements and these errors must be balanced against approximations introduced by the fmm and the convergence criterion for iterative solution these different errors must all be chosen in a way that on the one hand excess work is not done and on the other that the error achieved by the overall computation is acceptable details of translation operators for low and high kd choice of representations and bem quadrature schemes all consistent with these approximations are described

a hybrid micro macro bem formulation for micro crack clusters in elastic components local analysis schemes capable of detailed representations of the micro features of a problem are integrated with a macro scale bem technique capable of handling complex finite geometries and realistic boundary conditions the micro scale effects are introduced into the macro scale bem analysis through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows decomposition of the complete problem into two sub problems one residing entirely at the micro level and the other at the macro level this allows for investigations of the effects of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the component or structure under consideration as a first attempt elastic fracture mechanics problems with interacting cracks at close spacings are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can easily be extended to investigate the effects of other micro features e g interfaces short or continuous fibre reinforcements voids and inclusions in the context of linear elasticity on macroscopic failure modes observed in structural components

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

hypersingular and finite part integrals in the boundary element method abstract a new definition of the hadamard finite part hfp of hypersingular integrals is proposed in this paper this definition does not involve a limiting process it is completely general and is valid for one as well as higher dimensional integrals on closed as well as on open surfaces it reduces respectively to the cauchy principal value cpv and riemann integral respectively for the special cases of strongly singular and weakly singular integrands of course suitable symmetric exclusion zones must be chosen to realize cpv integrals starting with this new definition of the hfp of certain hypersingular boundary integral equations hbie that arise in potential theory and in wave scattering a regularization method is carried out in order to express the hypersingular integrals in terms of ones that are at most weakly singular the regularized versions are completely consistent with those available in the recent literature where a different definition of the hfp was employed

global exponential convergence of cohen grossberg neural networks with time delays in this paper we derive a general sufficient condition ensuring global exponential convergence of cohen grossberg neural networks with time delays by constructing a novel lyapunov functional and smartly estimating its derivative the proposed condition is related to the convex combinations of the column sum and the row sum of the connection matrices and also relaxes the constraints on the network coefficients therefore the proposed condition generalizes some previous results in the literature