Numerical solution of Cauchy type singular integral equations with logarithmic weight,based on arbitrary collocation points

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points x _k. Until now these x _k have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of x _k without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.     

A numerical solution of singular integral equations without using special collocation points

A new technique for the solution of singular integral equations is proposed, where the unknown function may have a particular singular behaviour, different from the one defined by the dominant part of the singular integral equation. In this case the integral equation may be discretized by two different quadratures defined in such a way that the collocation points of the one correspond to the integration points of the other. In this manner the system is reduced to a n × n system of discrete equations and the method preserves, for the same number of equations, the same polynomial accuracy. The main advantage of the method is that it can proceed without using special collocation points. This new technique was tested in a series of typical examples and yielded results which are in good agreement with already existing solutions.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

Numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points

An integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points (ACP). As a first step a Gaussian quadrature of order n (hence of polynomial accuracy 2n? 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x?_k, determined as roots of appropriate transcedental functions, in order to retain the 2n ? 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that x_k may be arbitrary and yet the high (2n ? 1) accuracy of the Gaussian quadrature is retained.     

Reducing the boundary value problems of elasticity theory for cracked bodies to boundary integral equations

A general method has been proposed for constructing integral representations of general solutions and boundary integral equations of multidimensional boundary value problems of mathematical physics for regions with cuts. It involves the use of the theory of generalized functions, and in particular of the surface delta function. At first, the boundary value problems of Dirichlet and Neumann were studied for n-dimensional Poisson and Helmholtz equations in a space with cuts along piecewise-smooth surfaces. After that the method is extended to the case of a system of differential equations. In this way the basic spatial and plane problems of elasticity theory were considered for an anisotropic infinite body with cracks under static and dynamic loading. The corresponding axisymmetric problems were also studied.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 6, pp. 61–71, November–December, 1990.     

The tanh transformation for the solution of singular integral equations

Using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained. The formula has the same step length h as the formula for regular integrals derived by F. Stenger. These quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity. Using these formulae the solution of singular integral equations does not depend on the order of the end point singularities. Furthermore the collocation points are given by a very simple equation and, in the case of constant coefficients, by a closed-form formula.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

A weakly singular integral formulation for displacement prescribed problems of elasticity

Summary. This paper introduces a new integral formulation for displacement prescribed problems in linear elasticity. The formulation uses a weakly singular kernel and extends to the case of linear elasticity the integral formulation introduced by Mikhlin [1] to solve Dirichlet problems for Laplaces equation in multiply connected domains. A detailed proof of the proposed formulation is given for displacement prescribed problems in two-dimensional multiply-connected domains. The proposed method can also be readily extended to solve three-dimensional problems.     

On some numerical methods for the solution of the plane elasticity problem for bodies with cracks by means of singular integral equations

Comparison is given of the accuracy of calculation of stress intensity factors at the crack tips by various methods when solving plane elasticity problems for bodies with cruciform and edge cracks. It is shown that, within the range of quadrature formulas for singular integrals discussed, the type of the formula chosen for the solution of an equation, if used correctly, affects negligibly the accuracy of the stress intensity factor evaluation at the crack tip, and in view of this a method is proposed based on simple relationships.

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Résumé On compare l'exactitude des calculs des facteurs d'intensité de contrainte aux extrémités d'une fissure à l'aide de diverses méthodes à l'occasion de la solution de problèmes d'élasticité plane dans des corps présentant des fissures cruciformes et des fissures de bord. On montre que dans les limites des formules de quadrature des intégrales singulières qui sont discutées, le type de formule choisie pour la solution d'une équation n'a qu'une influence négligeable si elle est utilisée correctement sur l'exactitude du facteur d'intensité de contrainte évalué à l'extrémité de la fissure. Dans cette optique, on propose une méthode basée sur des relations simples.

     

Numerical solutions of the singular integral equations in the crack analysis using the body force method

In this paper, numerical solutions of the singular integral equations of the body force method in the crack problems are discussed. The stress fields induced by two kinds of displacement discontinuity are used as fundamental solutions. Then, the problem is formulated as a hypersingular integral equation with the singularity of the form r ². In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density function and the Chebyshev polynomials. As examples, the stress intensity factors of the oblique edge crack, kinked crack, branched crack and zig-zag crack are analyzed. The calculation shows that the present method gives accurate results even for the extremely oblique edge crack and kinked crack with extremely short bend which has been difficult to analyze by the previous method using the approximation by the products of the fundamental density function and the stepped functions etc.