Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.     

Acoustic diffraction by a rigid annular disk

Summary The subject of this paper is the problem of acoustic diffraction by a perfectly rigid annular disk. The method of solution rests on formulating the problem in terms of an integral equation which embodies the steady state wave equation as well as the boundary conditions. This Fredholm integral equation of the first kind is converted into four simultaneous integral equations of the second kind by using Williams' integral equation technique. These four integral equations are subsequently solved by the standard iterative procedure when the frequency of the incident wave is low and the inner radius of the annulus is small.     

Mutually constrained partial differential and integral equations for an exterior field problem

A new implementation of the mutually constrained partial differential and integral equation method for the exterior 2-dimensional field problem is described. It is shown, that the method is applicable to exterior problems in an inhomogeneous medium. The inhomogeneity is considered in the finite element procedure and in boundary element method, where an adequate Green's function is applied. The temperature distribution around a three-cable system is then computed as an illustration. The eddy-current losses in the cable sheaths are calculated using the Fredholm integral equation of the second kind.     

Diffraction of transient horizontal shear waves by a finite crack at the interface of two bonded dissimilar elastic solids

Scattering of transient horizontal shear waves by a finite crack located at the interface of two bonded dissimilar elastic solids is investigated in this study. Laplace and Fourier transform technique is used to reduce the problem to a pair of dual integral equations. The solution of the dual integral equation is expressed in terms of the Fredholm integral equation of the second kind having the kernel of a finite integration. Dynamic stress intensity factor is obtained as a function of the material and geometric properties and time.     

Solution of a plane,axisymmetric and three-dimensional single-phase stefan problem

Using a superposition method we construct a solution of the multidimensional problem of the steady-state fusion regime of semibounded solids. The solution of the problem is reduced to a generalized Fredholm integral equation of the first kind. A method is given for solving the integral equation for plane problems by converting to a linear system of algebraic equations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 27, No. 2, pp. 341–350, August, 1974.     

An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data

We present here the theoretical results and numerical analysis of a regularization method for the inverse problem of determining the rate constant distribution from biosensor data. The rate constant distribution method is a modern technique to study binding equilibrium and kinetics for chemical reactions. Finding a rate constant distribution from biosensor data can be described as a multidimensional Fredholm integral equation of the first kind, which is a typical ill-posed problem in the sense of J. Hadamard. By combining regularization theory and the goal-oriented adaptive discretization technique, we develop an Adaptive Interaction Distribution Algorithm (AIDA) for the reconstruction of rate constant distributions. The mesh refinement criteria are proposed based on the a posteriori error estimation of the finite element approximation. The stability of the obtained approximate solution with respect to data noise is proven. Finally, numerical tests for both synthetic and real data are given to show the robustness of the AIDA.     

Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

Computational methods for solving static field and eddy current problems via Fredholm integral equations

Two-dimensional static field problems can be solved by a method based on Fredholm integral equations (equations of the second kind). This has numerical advantages over the mote commonly used integral equation of the first kind. The method is applicable to both magnetostatic and electrostatic problems formulated in terms of either vector or scalar potentials. It has been extended to the solution of eddy current problems with sinusoidal driving functions. The application of the classical Fredholm equation has been extended to problems containing boundary conditions: 1) potential value, 2) normal derivative value, and 3) an interface condition, all in the same problem. The solutions to the Fredholm equations are single or double (dipole) layers of sources on the problem boundaries and interfaces. This method has been developed into computer codes which use piecewise quadratic approximations to the solutions to the integral equations. Exact integrations are used to replace the integral equations by a matrix equation. The solution to this matrix equation can then be used to directly calculate the field anywhere.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem

Summary This is the first paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. A novel solution procedure has been developed to systematically solve a type of Fredholm integral equations based on symmetry, self-similarity, and invariant group arguments. In this paper, we consider a two-dimensional (2D) circular inclusion within a finite, circular representative volume element (RVE). The RVE is considered isotropic, linear elastic and is subjected to a displacement (Dirichlet) boundary condition. Starting from the 2D plane strain Navier equation and by using our new solution technique, we obtain the exact disturbance displacement and strain fields due to a prescribed constant eigenstrain field within the inclusion. The solution is characterized by the so-called Dirichlet-Eshelby tensor, which is provided in closed form for both the exterior and interior region of the inclusion. Some immediate applications of the Dirichlet-Eshelby tensor are discussed briefly.     

Synthesis of magnetic fields

The paper deals with some problems of magnetic fields synthesis, depending on determination of the current density distribution, which generates the required magnetic field in the investigated region. Such problems can be reduced to the linear, or nonlinear Fredholm integral equations of the first kind, or to the set of these equations. Fredholm integral equation of the first kind belongs to the class of the ill-posed problems, and for its solving the method of regularisation has been used. In the paper there are given some useful results of synthesis of magnetic fields in few practical configurations.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

The stress intensity factor of an edge crack in a finite elastic disc

In this paper, a Mellin transform technique is used to express the stress intensity factor and the crack energy of an edge crack in a finite elastic disc directly in terms of the solution of a Fredholm integral equation of the second kind. The constant loading case is considered in detail and the results given in graphical form.     

Axisymmetric deformation of a short magneto-strictive current-carrying cylinder

The title-problem has been reduced to that of solving a Fredholm integral equation of the second kind. One end of the cylinder is assumed to be fixed, while the cylinder is deformed by an axial current. The vertical displacement on the upper flat end of the cylinder has been determined from an iterative solution of the Fredholm equation valid for large values of the length. The radial displacement of the curved boundary has also been determined at the middle of the cylinder, by using the iterative solution.     

A central crack normal to the interface between two different materials in a rectangular sheet under an anti-plane shear stress

In this paper, a Fourier transform and Fourier series technique is used to express the stress intensity factor of a central crack in a finite rectangular sheet with two different materials whose interface normal to the crack in terms of the solution of a Fredholm integral equation of the second kind. The constant loading case is considered in detail and the results given in numerical form.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

Stresses in an orthotropic strip containing A Griffith Crack

The problem of determining the stress and displacement fields in an orthotropic elastic strip containing a Griffith crack situated symmetrically and oriented in a direction normal to the edges of the strip is considered. A general solution in terms of two potential functions is presented. The mixed boundary conditions lead to dual integral equations, which are reduced to Fredholm integral equation of second kind and are solved by the use of Gaussian quadrature formula. Numerical solutions for a fiber-reinforced composite material and some isotropic materials are carried out and the effect of orthotropy on various quantities of physical interest, in fracture mechanics, is discussed.     

A problem of Reissner-Sagoci type for a semi-infinite composite elastic cylinder

Summary The problem considered is that of the torsion of a semi-infinite elastic cylinder which is embedded in a semi-infinite elastic cylindrical shell of different material. By the use of integral transforms and the theory of dual integral equations, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Numerical solution of the integral equation is obtained and the numerical values of the torque are displayed graphically.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

Contact between an elastically supported circular plate and a rigid indenter

The axially symmetric problem of a finite circular plate loaded at its center by a smooth, rigid punch is solved by superposing an infinite layer elasticity solution with a pure bending plate theory solution. The problem is reduced to dual integral equations, which are further reduced to a single Fredholm integral equation of the second kind. The Fredholm equation is numerically solved and the results are used to compute contact stresses under the indenter as well as the overall load-deflection behavior. The problem is formulated to model a partially fixed edge around the plate's perimeter, and calculations are carried out for the limiting cases of simple supports and complete fixity. Various ratios of plate diameter to plate thickness are studied, and the results are compared to both Hertzian contact theory and standard plate theory.