An elastic hollow cylinder under axial tension containing a crack and two rigid inclusions of ring shape

This paper is concerned with the fracture of an axisymmetric hollow cylindrical bar containing rigid inclusions. The cylinder is under the action of uniformly distributed axial tension applied at infinity. The bar contains a ring-shaped crack at the symmetry plane whose surfaces are free of tractions and two ring-shaped rigid inclusions with negligible thickness symmetrically located on both sides of the crack. It is assumed that the material of the cylinder is linearly elastic and isotropic. The mixed boundary conditions of the problem lead the analysis to a system of three singular integral equations for crack surface displacement derivative and normal and shearing stress jumps on rigid inclusions. These integral equations are solved numerically and the stress intensity factors are calculated.     

A numerical scheme for mixed boundary value problems in elasticity

A numerical scheme, based upon the Kobayashi-Tranter method with certain modifications, is given for axisymmetric punch and crack problems in elasticity. The problems are reduced to solving a system of linear algebraic equations instead of a Fredholm integral equation of the second kind. A standard program thus allows the treatment of a range of different cases.The indentation of a rigid punch on an elastic layer overlying an elastic foundation is formulated in this fashion and numerical results for various cases are presented.     

Planar crack occupying a finite doubly connected region inside an infinite elastic solid

A method is proposed for the approximate solution of the problem of an embedded pressurized planar crack occupying a finite doubly connected region inside an infinite elastic solid. The formulation of the problem produces a system of two integral equations for determining the unknown normal stresses on the plane of the crack outside the crack region, which can be solved using numerical procedures. The proposed method has been applied to obtain the opening mode stress intensity factors K_I, along the boundary lines of an annular crack subjected to a uniform internal pressure.     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.     

Structured matrix algorithms for inverse scattering on the line

Abstract In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schr?dinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation     

Solution of discontinuous interior Helmholtz problems by the boundary and shell element method

The Helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method. In this paper it is shown how the Helmholtz equation can be recast as an integral equation known as the boundary and shell integral equation. The application of collocation to the integral equation gives rise to a method termed the boundary and shell element method.

The associated problem of finding the eigenvalues and eigenfunctions of the Helmholtz equation in a discontinuous domain via the same method is also considered. This leads to a non-linear eigenvalue problem. Such a problem may be solved through polynomial interpolation of the matrix components. In this paper methods for solving the Helmholtz equation and the associated eigenvalue problem are implemented and applied to a test problem.     

Boundary integral method for multispecies 1-D convection-diffusion-reaction problems

Boundary integral method has been proposed to solve a system of differential equations with a convection-diffusion operator and a nonlinear rate term for each equation. The operator part can be eliminated by using an inverse formulation by a proper choice of the weighting function for each differential equation. This reduces the problem to an integral formulation which is then solved by approximating the dependent variables by cubic osculation. The method has been demonstrated to a packed bed reactor with heat and mass dispersion and for a countercurrent heat exchanger.     

Axisymmetric Finite Element Method for Analysis of Plate on Layered Soil

To obtain the fundamental solution of soil has become the key problem for the semi-analytical and semi-numerical (SASN) method in analyzing plate on layered soil. By applying axisymmetric finite element method (FEM),an expression relating the surface settlement and the reaction of the layered soil can be obtained. Such a reaction can be treated as load acting on the applied external load. Having the plate modelled by four-node elements,the governing equation of the plate can be formed and solved. In this ca...     

Numerical solutions in axisymmetric elasticity

Paper presents the formulation of the axisymmetric elasticity problem with thermal and rotational loading using the boundary-integral equation method. The resulting one dimensional numerical model is evaluated for a series of problems, including a problem solved by a finite element model. Results show the validity of the formulation for non-trivial problems, as well as the advantage of significant modeling efficiency relative to the finite element method for certain classes of problems.     

Numerical solution of laplace's equation on non-simply connected regions on R 2

We consider the interior Dirichlet problem for Laplace's equation on a non-simply connected two-dimensional regions with smooth boundaries.The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral.The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind.Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “Modified Dirichlet problem”.The Nystrom's method coupled with the trapezoidal rule is used as numerical integration scheme.The linear system derived from the integral equation is solved using the conjugate gradient applied to the normal equation.Theoretical and computational details of the method are presented.     

Saulyev'S Techniques For Solving A Parabolic Equation With A Non Linear Boundary Specification

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\hskip1pty, t) = \vint u(x, y, t)\,\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.     

Linear transport theory for particles moving on a spherical surface

The paper deals with a linear transport model for particles moving on a spherical surface in the presence of absorption and scattering events. Starting from the analytical expression for the particle density in the case of a pure absorbing sphere, an integral equation is derived for situations in which isotropic scattering is present. The equation has been solved numerically in order to obtain the Green function of the problem. Analytical and numerical results are compared with those obtained resorting to the Monte Carlo method.     

Collocation method using quintic B-spline and sinc functions for solving a model of squeezing flow between two infinite plates

An axisymmetric viscous flow, generated by two large parallel plates slowly approaching each other is investigated. The steady nonlinear governing equations are converted into a fourth-order nonlinear differential equation using integrability condition. The resulting nonlinear boundary value problem is solved using quintic B-spline collocation and sinc-collocation methods. The approach consists of reducing the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the techniques and a comparison is made with existing results in the literature.     

An integral-equation function-space method for solving inequality-constrained optimal control problems†

A function-space method using an integral equation representation of the second variation is applied to inequality-constrained optimal control problems. Numerical solutions for a bang-bang control problem and a singular control problem are solved using the computational algorithm.     

求解Cauchy型奇异积分方程的数值方法

1.引 言 断裂力学中许多裂纹问题的数学模型都可归结为奇异积分方程(SIE)[1,2].由于这些奇异积分方程的封闭解一般情况下都难以得到,因而数值方法受到广泛的注意.Muskhelishvili[3]对奇异积分方程的一般理论进行了深入的研究.这些研究成果为奇异积分方程的求解,不论     

Numerical Modeling of Plasma Devices by the Particle-In-Cell Method on Unstructured Grids

The paper considers methods and algorithms providing the basis for a computer program implementing an axial-symmetric electrostatic version of the particle-in-cell method on unstructured triangular grids. In the presented implementation, the Poisson equation is approximated using the finite volume method. A discrete analog of the Poisson equation is solved by the multigrid method. Charged particle trajectories are calculated using the Boris method. Methods for interpolating electrostatic fields on unstructured grids and obtaining the charge density in the computational domain are considered. Special attention is paid to the specifics of implementing these methods in axisymmetric geometry. The developed computer code is tested on the problem of a flat diode operating in the space charge mode.     

Shape optimization of quasi-brittle axisymmetric shells by genetic algorithm

This paper describes the application of genetic algorithm to the shape optimization of axisymmetric shells. The primary problem of axisymmetric shell optimization under fracture mechanics constraint is formulated as the weight (volume of shell material) minimization under stress intensity constraints. It is assumed that the shells are made from quasi-brittle materials and through-thickness crack presence is admitted. Taking into account the fact of incomplete information concerning crack arising (size, location and orientation) this paper presents some numerical results based on a guaranteed approach.     

Method for determining the projection of an arrhythmogenic focus on the heart surface, based on solving the inverse electrocardiography problem

The problem of determining the point on the heart surface (projection) nearest to the arrhythmogenic focus, which is located inside the heart, is considered. Localization of this point is crucial for a successful cardiac ablation procedure. The sought projection is calculated on the basis of solving the inverse electrocardiography problem, which is a generalization of the Cauchy problem for the Laplace equation. The inverse electrocardiography problem is solved by the boundary integral equation and Tikhonov regularization methods. Examples of test computations are demonstrated, and the results of processing real electrophysiological data are presented and compared with the medical observation data.     

A finite element-integral equation solution to torsion problem

The torsion problem can be formulated as a Fredholm integral equation of the second kind and then solved by a finite element method based on a weighted residual technique. This procedure effectively reduces a two dimensional problem to a one dimensional problem, and thereby significantly simplified the computer implementation. The procedure is illustrated by an example and results are compared with the exact solution.