Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals

A new method of direct numerical evaluation of hypersingular boundary integrals has been applied to the differentiated form of the Somigliana-identity (hypersingular identity) in 3D-elastostatics. Through this method it is possible to evaluate the stress tensor on the boundary of a complex 3D structure in a very accurate manner by employing the direct boundary element method (BEM). The geometry of the elements and their arrangements over the boundary of the structure are not subjected to any restrictions. Numerical examples illustrate the accuracy of the proposed method.     

Non-linear co-ordinate transformation of finite part integrals in two-dimensional boundary element analysis

This paper describes a method for the numerical computation of hypersingular integrals as they appear in the boundary element analysis. The proposed method is based on the finite part concept and allows for arbitrary curved boundary elements. Owing to the unknown transformation properties of finite part integrals undergoing a non-linear co-ordinate transformation, the definition formula of finite part integrals is applied prior to the transformation into the usual element co-ordinate system. The resulting integrals are regular and may be evaluated by standard Gaussian quadrature rules. The method is described in detail for the boundary integrals of two-dimensional linear elastostatics. Numerical examples are inclcded for this type of problem, but the method may easily be adapted to other two-dimensional problems.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

On the calculation of boundary stresses with the Somigliana stress identity

The computation of boundary stresses by Boundary Element Method (BEM) is usually performed either by expressing the boundary tractions in a local co-ordinate system, calculating the remaining stresses by shape function differentiation and inserting into Hooke's law or recently also by solving the hypersingular integral equation for the stresses. While direct solution of the hypersingular integral equation, the so-called Somigliana stress identity, has been shown to be more reliable, the interpretation and numerical treatment of the hypersingularity causes a number of problems. In this paper, the limiting procedure in taking the load point to the boundary is carried out by leaving the boundary smooth and the contributions of all different types of singularities to the boundary integral equation are studied in detail. The hypersingular integral in the arising boundary integral equation is then reduced to a strongly singular one by considering a traction free rigid body motion. For the numerical treatment, an algorithm for multidimensional Cauchy Principal Value (CPV) integrals is extended that is applicable for the calculation of boundary stresses. Moreover, the shape of the surrounding of the singular point is studied in detail. A numerical example of elastostatics confirms the validity of the proposed method.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: Numerical analysis

In this paper, elliptical cracks and rectangular cracks embedded in a three-dimensional infinite transversely isotropic piezoelectric solid are analyzed under combined mechanical tension and electric fields. The hypersingular integral equation method is used to solve the mentioned problems. The unknown function in the hypersingular integral equations is approximated with a product of the fundamental density function and polynomials. The hypersingular integrals can be numerically evaluated by using a method of Taylor series expansion. Therefore, the hypersingular integral equations for the crack problems can be solved immediately. Finally, numerical examples of the stress and electric displacement intensity factors as well as the energy release rates for these crack configurations are presented. The numerical results demonstrate the present approach to be very efficient.     

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.     

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.     

Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals

The typical Boundary Element Method (BEM) for fourth‐order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second‐order problems. Actually, the direct method for the evaluation of singular integrals is completely revised and presented in an alternative way. All aspects are dealt with in detail and full generality, including the evaluation of free‐term coefficients. Numerical tests and comparisons with other regularization techniques show that the direct evaluation of singular integrals is easy to implement and leads to very accurate results. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

THE EVALUATIONS OF CAUCHY PRINCIPAL VALUE INTEGRALS AND WEAKLY SINGULAR INTEGRALS IN BEM AND THEIR APPLICATIONS

A new technique is developed to evaluate the Cauchy principal value integrals and weakly singular integrals involved in the boundary integral equations. The boundary element method is then applied to analyse scattering of waves by cracks in a laminated composite plate. The Green's functions are obtained in discrete form through the thickness of the plate using a stiffness method. To circumvent the difficulties associated with the evaluation of hypersingular integrals due to the presence of cracks, the multidomain technique is applied. Numerical computations have shown the accuracy and reliability of the proposed technique. Scattered wave fields for a composite plate with a horizontal crack are computed. The numerical results show that the applications of the technique in non-destructive evaluation of defects is very promising.     

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.     

Hypersingular quarter-point boundary elements for crack problems

The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.     

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.     

Complex hypersingular integrals and integral equations in plane elasticity

Summary Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.     

Analytical integrations for two-dimensional elastic analysis by the symmetric Galerkin boundary element method

In the context of linear elasticity, this paper presents a way to perform hypersingular integrals arising from the symmetric Galerkin Boundary Element Method (BEM). In contrast to the existing integration techniques, the one here proposed does not need any regularization or limit processes: in fact it works directly on the final form of the hypersingular double integrals without any previous manipulation. The present method is applied to 2D linear elastic problems, using straight elements with continuous piecewise-linear displacements and piecewise-constant tractions. Numerical tests are presented for the validation of the obtained analytic results.     

Evaluations of certain hypersingular integrals on interval

In this paper, we investigate a hypersingular integral on an interval. The definition of Hadamard's finite‐part integrals and some of its properties are given. Some numerical methods on approximate computation of the finite‐part integrals are constructed. The new method is very simple, easy to implement, reliable, and above all, not affected by the location of singular point. Some numerical experiments are carried out using the current formulae, and numerical results show that the current methods are feasible and effective. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical implementation of the symmetric Galerkin boundary element method in 2D elastodynamics

The symmetric Galerkin boundary element method (SGBEM) employs both the displacement integral equation and the traction integral equation which lead to a symmetric system of equations. A two‐dimensional SGBEM is implemented in this paper, with emphasis on the special treatments of singular integrals. The integrals in the time domain are carried out by an analytical method. In order to evaluate the strong singular double integrals and the hypersingular double integrals in the space domain which are associated with the fundamental solutions G ^pu and G ^pp, artificial body forces are introduced which can be used to indirectly derive the singular terms. Thus, those singular integrals which behave like 1/r and 1/r² are all avoided in the proposed SGEBM implementation. An artificial body force scheme is proposed to evaluate the body force term effectively. Two numerical examples are presented to assess the accuracy of the numerical implementation, and show similar accuracy when compared with the FEM and the analytical solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Nearly singular approximations of CPV integrals with end- and corner-singularities for the numerical solution of hypersingular boundary integral equations

A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming C⁰ quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment.With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.     

Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE,and piecewise-constant and piecewise-linear approximations

In this article the hypersingular integrals that arise when boundary integral equation (BIE) methods are used to solve fracture mechanics problems are considered. An approach for hypersingular integral regularization is based on the theory of distribution and Green's theorems. This approach is applied for regularization of the hypersingular integrals over triangular boundary elements (BEs) for the case of piecewise-constant and piecewise-linear approximations. The hypersingular integrals are transformed into regular contour integrals that can be easily calculated analytically.