Numerical solution of singular integral equations in stress concentration problems

The Numerical Solution of Integral Equations on the Half-Line

Several methods for the numerical solution of Cauchy-type singular integral equations, frequently encountered in two-dimensional elasticity, are established. The well known numerical integration rules are properly combined in order to handle numerically Cauchy-type singular integral equations along the integration interval [-1, 1] and for a large variety of weight functions. In this way, such an equation (or a system of such equations) can be numerically sloved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected points of the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation.

In this section, we obtain theorems on the approximation of functions of the complex variable dened on closed contours bounding a simply connected domain in the complex plane; the approximations are constructed in the space of continuous functions and in Lebesgue spaces with the use of Lozinskii interpolation polynomials. These results are used to obtain a theoretical justication of the subdomain method for the approximate solution of singular integral equations dened on closed contours of various smoothness. The solvability of the computational scheme of this method is proved by the technique in [1]. Note that issues related to the approximate solution of singular integral equations have been considered in numerous papers (see [2{10] and the bibliography therein). However, the subdomain method for singular integral equations on closed contours other than the unit circle has not been studied. Let be a closed smooth contour bounding a connected domain F + containing the origin, and let F = CnfF + [ g ,w hereC is the entire complex plane. We say that belongs to the class (m) ,w herem is a positive integer, if the Riemann function t = (w), w2 0 =f : jj =1 g, (1 )= 1, 0 (1 )= a1 (a1 > 0), of this contour has continuous :

This paper deals with numerical solutions of singular integral equations in interaction problems of diamond-shaped inclusions with angular corners under various loading conditions. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. In order to analyze the problems accurately, the unknown functions of the body force densities are expressed as a linear combination of two types of fundamental density functions and power series, where the fundamental density functions are chosen to represent the symmetric stress singularity of 1/r1-λ1 and the skew-symmetric stress singularity of 1/r1-λ2. Then, newly defined stress intensity factors of angular corners are systematically calculated for various shapes and spacings of two diamond-shaped inclusions in a plate subjected to uniaxial tension, biaxial tension and in-plane shear. The present method is found to yield rapidly converging numerical results for the interaction of diamond-shaped inclusions.

L p-approximation method for the numerical solution of singular integral equations

This paper deals with numerical solutions of singular integral equations of the body force method in interaction problems of notches and holes under general loading conditions. The problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the densities of body forces distributed in the x-and y-directions are to be unknown functions. In order to satisfy the boundary conditions along the notches, several types of fundamental density functions are proposed ; then, the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The accuracy of the present analysis is verified by comparing with the results obtained by the previous method. The present method is found to give rapidly converging numerical results for stress distribution along the notch boundaries.

Numerial Methods for Volterra Integral Equations with Singular Kernels

The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type

This paper describes numerical solutions of singular integral equations of the body force method in interaction problems of ellipsoidal inclusions under uniform tension. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the densities of body forces distributed in the r-and z-directions of infinite bodies having the same elastic constants of the matrix and inclusions are unknown functions. In order to satisfy the boundary conditions along the inclusions, eight fundamental density functions proposed in our previous paper are used. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The present method is found to give rapidly converging numerical results for stress distribution along the boundaries of both the matrix and inclusions.

We present a new method for the numerical solution of singular integral equations on the real axis. The method’s value stems from a new formula for the Cauchy integral of a rational function with an oscillatory exponential factor. The inner product of such functions is also computed explicitly. With these tools in hand, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. In specific cases, ideas from Fredholm theory and Riemann–Hilbert problems are used to motivate preconditioners for these singular integral equations. A significant acceleration in convergence is realized for these examples. This presents a useful link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann–Hilbert problem.

A new approach for investigating the collocation method for singular integral equations is given. Using Banach algebra techniques and a local principle established by I. Gohberg and N. Krupnik under suitable conditions the convergence of the Lagrange- and Mul thopp-collocation method is proved for one-dimensional singular integral equations and systems of such equations with piecewise continuous coefficients having a countable infinite number of discontinuities.

Existing Green functions for a conducting wedge in the form of infinite series of orthogonal eigenfunctions are known to suffer from slow and conditional convergence 1) near the source point, due to the singularity of the Green functions at source points, and 2) when the source point is near the surface of the wedge, due to the presence of a nearby image source. As a result, such expansions are unsuited for the exact solution of singular integral equations wherein values of the Green functions at the source point do appear inside the integral. In this paper, alternative expansions are proposed which converge rapidly for any combination of the source and observation points. The main idea is to extract in closed form both the singular term and the nearby image term together with certain simple asymptotic terms. This helps isolate the inherent singularities and, thus, transform the remaining part of the Green function into a rapidly converging series of elementary terms. Numerical examples and case studies illustrate the stability and high accuracy of the new expansions. Application to the solution of an integral equation associated with wave diffraction by perfectly conducting curved strips in the vicinity of a wedge is exemplified.

Stability of the numerical procedure for solution of singular integral equations on semi-infinite interval. Application to fracture mechanics

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss‐Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto‐Chebyshev method for the numerical solution of the same class of equations.

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].