Singular integral equation for an electric dipole taking into account the finite metal conductivity from which it is made

A singular integral equation for an electric dipole has been obtained, which makes it possible to take into account the finite conductivity of the metal from which it is made. The derivation of the singular integral equation is based on the application of the Greens function for free space, written in a cylindrical coordinate system, taking into account the absence of the dependence of the field on the azimuthal coordinate, on a point source located on the surface of an electric dipole. Methods for its solution are proposed. In contrast to the well-known mathematical models of an electric dipole, built in the approximation of an ideal conductor, the use of the singular integral equation obtained in this work makes it possible to take into account heat losses and calculate the efficiency.

Computational contact mechanics for a medium consisting of functionally graded material coating and orthotropic substrate

This paper presents a semi-analytical method to investigate the frictionless contact mechanics between a functionally graded material (FGM) coating and an orthotropic substrate when the system is indented by a rigid flat punch. From the bottom, the orthotropic substrate is completely bonded to the rigid foundation. The body force of the orthotropic substrate is ignored in the solution, while the body force of the FGM coating is considered. An exponential function is used to define the smooth variation of the shear modulus and density of the FGM coating, and the variation of Poisson’s ratio is assumed to be negligible. The partial differential equation system for the FGM coating and the orthotropic substrate is solved analytically through Fourier transformations. After applying boundary and interface continuity conditions to the mixed boundary value problem, the contact problem is reduced to a singular integral equation. The Gauss–Chebyshev integration method is then used to convert the singular integral equation into a system of linear equations, which are solved using an appropriate iterative algorithm to calculate the contact stress under the rigid flat punch. The parametric analyses presented here demonstrate the effects of normalized punch length, material inhomogeneity, dimensionless press force, and orthotropic material type on contact stresses at interfaces, critical load factor, and initial separation distance between FGM coating and orthotropic substrate. The developed solution procedures are verified through the comparisons made to the results available in the literature. The solution methodology and numerical results presented in this paper can provide some useful guidelines for improving the design of multibody indentation systems using FGMs and anisotropic materials.

Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

Solution for the degenerate scale for a rigid curve in antiplane elasticity by using a weakly singular integral equation

This paper provides a numerical solution for the degenerate scale for a rigid curve in antiplane elasticity. The degenerate scale problem for the rigid curve is formulated on the usage of the logarithmic potential. After assuming the displacement to be a vanishing value along the rigid curve, the boundary integral equation (BIE) is formulated. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant BIE in the ordinary scale. Finally, a numerical solution is achieved. Several numerical examples are provided. In addition, the degenerate scale problem for the multiple rigid curves is also solved.

On the solvability of some operators with several moved and fixed

В работе рассматриваются двумерные сингулярные интегральные операторы по ограниченной области с коэффициентами при интегралах, содержащими в нескольких точках существенный разрыв и операторы с ядрами, имеющими в нескольких точках фиксированные особенности типа однородных функций порядка -2. Такие операторы широко применяются при изучении различных краевых задач для эллиптических систем уравнений первого и второго порядка с сингулярными коэффициентами на плоскости (см. напр. [1]-[4]). Одно из таких приложений приведено в конце настоящей работы. Сначала излагаются результаты исследования разрешимости (нетеровости) двумерного сингулярного интегрального уравнения с коэффициентом при интеграле, содержащим в одной точке существенный разрыв. In this paper we consider two-dimensional singular operators over a bounded domain with coefficients of the integrals, containing an essential discontinuity at several points and operators with kernels having fixed singularities at several points of the type of homogeneous functions order -2. Such operators are widely used in various boundary value problems for elliptic systems of equations of the first and second order with singular coefficients on the plane (see eg. [1]-[4]). One such application is given at the end of this work. First of all set out the results of studying the solvability (Noethericity) of a two-dimensional singular integral equation with a coefficient of the integral containing an essential discontinuity at one point.