Solution Of Singular Integral Equations Research Articles

Singular Integral Equation Method for Interaction Between Elliptical Inclusions

This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0≦φk≦π/2 is equivalent to determining an original unknown density in the range 0≦φk≦2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.

Inversion Formulas for the Discretized Hilbert Transform on the Unit Circle

A discrete version of the Hilbert transform on the unit circle is considered. Its Moore--Penrose inverse with respect to suitable scalar products is derived for different side conditions. Furthermore, stability of the pseudo-inverse is studied. These results allow the efficient computation of approximate solutions of singular integral equations with Hilbert kernel. Furthermore, the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions.

Numerical Solution of Singular Integral Equations of the Body Force Method in Stress Concentration Problems under In-Plane Bending.

This paper delas with numerical solutions of singular integral equations in an interaction of elliptical holes under in-plane bending. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknowns are the densities of body forces distributed in an infinite plate. In order to satisfy the boundary conditions along elliptical boundaries, the unknown body force densities are approximated by a linear combination of fundamental densities functions and polynomials. The calculations are carried out for several arrangement of elliptical holes, and it is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of present solutions.

Singular Integral Equation Method in the Analysis of Interaction between Rectangular Inclusions.

This paper deals with numerical solutions of singular integral equations in interaction problems of rectangular inclusions under various loading conditions. The body force method is used to formulate the problems as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns 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 are expressed as piecewize smooth functions using two types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity of 1/γ<1-λ1> and the skew-symmetric stress singularity of 1/γ<1-λ2>. Then, newly defined stress intensity factors at the end of inclusions are systematically calculated for various shapes and spacings of two rectangular inclusions in a plate subjected to longitudinal tension, transverse tension, and in-plane shear. The present method is found to be effective for accurate and efficient analysis of rectangular inclusions.

Interaction Effect between Ellipsoidal Inclusions is an Infinite Body under Asymmetric Uniaxial Tension.

This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of two ellipsoidal inclusions under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads ; (i) biaxial tension and (ii) plane state of pure shear. These 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 γ, θ, z directions are unknown functions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynominals. The present method is found to yield rapidly converging numerical results for stress distributions along the boundaries.

Numerical solution of singular integral equations in stress concentration problems

In this paper, the numerical solution of singular integral equations in stress concentration problems is considered. The idea of the body force method stress field induced by a point force in an infinite body is used as a fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r−1. In solving the integral equations, the boundary conditions satisfied by two types of numerical procedure are examined. Then, it is found that the unknown functions of body force densities should be approximated by the product of a polynomial and several types of fundamental density functions. The calculation shows that this latter method gives a smooth variation of stresses along the elliptical boundary for various geometrical and loading conditions. In addition, this method gives rapidly converging numerical results and highly satisfied boundary conditions along the entire boundary.

Numerical Solution of Singular Integral Equations of the Body Force Method in Stress Concentration Problems for Longitudinal Shear Loading.

This paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problems as a system of singular integral equations with Cauchy-type singularities, where unknowns are the densities of body forces distributed in the longitudinal direction of an infinite body. In order to satisfy the boundary conditions along elliptical boundaries, four kinds of fundamental density functions are newly introduced. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. The calculations are carried out for several arrangements of elliptical holes, and it is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions.

Analysis of Intensity of Singular Stress at the End of a Cylindrical Inclusion.

This paper deals with numerical solutions of singular integral equations in the problem of an elastic cylindrical inclusion with ends in an infinite body under tension. The problem is formulated as a system of singular integral equations with Cauchy type or logarithmic type singularities, where unknown functions are densities of body forces distributed in infinite bodies having the same elastic constants as those of the matrix and inclusion. In the numerical analysis, 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 express the symmetric stress singularity of the form 1/r1-λ1 and the skew-symmetrics stress singularity of the form 1/r1-λ2. Then, the singular stress fields at one end of a cylindrical inclusion are discussed for various fiber lengths and elastic ratios. The results are also compared with ones for a rectangular inclusion.

On Projective Methods of Approximate Solution of Singular Integral Equations

Abstract The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov–Galerkin and collocation methods are investigated.

On projective methods of approximate solution of singular integral equations

The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov-Galerkin and collocation methods are investigated.

Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cages are considered, in one of which the range of integration is a Single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.

A probabilistic-informational algorithm for approximate solution of singular integral equations. The method of reduction to Riemann-Hilbert problems

We propose a numerical-analytic method of solving singular integral equations with a singular kernel of Cauchy type on an interval. The method relies on the construction of a regular operator of special form, whose action on the original singular equation leads to an integral equation that admits an explicit solution.

Singular Integral Equation Method in the Analysis of Interaction between Diamond-Shaped Inclusions.

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.

任意分布だ円形介在物の干渉効果の解析法

This paper deals with the numerical solutions of singular integral equations of the body force method in an interaction problem of arbitrarily distributed elliptical inclusions under general loading conditions. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the densities of body forces distributed in the x- and y-directions of infinite plates having the same elastic constants as those of the matrix and the inclutions are unknown functions. In order to satisfy the boundary conditions along the inclusions, eight kinds of 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 calculations are carried out for several arrangement of the inclusions, and it is found that the present method yields rapidly conversing numerical results for arbitrarily distributed elliptical inclusions. The numerical results of weight functions and stress distributions along the boundaries are shown in figures to demonstrate the present solution.

Interaction Effect between Ellipsoidal Cavities in an Infinite Body under Uniaxial Tension.

This paper deals with the numerical solution of singular integral equations of the body force method in the interaction problem of ellipsoidal cavities under uniaxial tension. The problem is solved by the superposition of two auxiliary loads : (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the densities of body forces distributed in the γ, θand z directions are unknown functions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous paper are applied. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries.

Antenna-diffraction problems and singular solutions of singular integral equations

The theory for solving of singular integral equations (s.i.e) in the class of singular functions is given briefly. The antenna-diffraction problem (a.d.p.) is presented as an example. The solution of this problem can be reduced to the considered integral equation type. It is shown that the theory can be applied to solution for the problem of removing "luminescence" of contour angle points by location of current sources on a contour near these points. Amplitudes of these sources are selected in a special manner. One can control the scattering diagram with the help of known amplitudes of other sources. The numerical methods for the solution of the considered integral equations are elaborated. The examples of calculations for the mentioned above physical effects are given.

Interaction Effect among a Row of Ellipsoidal Cavities in an Infinite Body under Uniaxial Tension.

This paper deals with the numerical solution of singular integral equations of the body force method in the interaction problem for a row of ellipsoidal cavities under uniaxial tension. The problem is solved by the superposition of two auxiliary loads: (i)biaxial tension and (ii)plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the densities of body forces distributed in the r, θ and z directions are unknown functions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous paper are applied. In the analysis, the number, shape, and distance of cavities are varied systematically; then the magunitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of the squared number of cavities.

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

This paper describes a numerical approach for obtaining the solution of singular integral equations using an L p-approximation method. The basic elements of this approach are a minimization with respect to a set of parameters in a function chosen to represent the solution of the integral equation and a numerical integration to evaluate the L p-norm . Attention is given to different values of p ≥ 1. The performance of the numerical procedures is investigated by solving a number of test problems.

Finite sections for singular integral operators by weighted Chebyshev polynomials

A finite section method for the approximate solution of singular integral equations with piecewise continuous coefficients on intervals is considered. The problem is transformed in such a way that results which were previously obtained for singular integral equations on the unit circle using localization methods in Banach algebras are applicable to it. Thus, necessary and sufficient conditions for the stability of the approximation method can be proved.

On the solution of singular integral equations with automorphic kernels by mechanical quadrature

On the solution of singular integral equations with automorphic kernels by mechanical quadrature