Logarithmic Singular Kernels Research Articles

A simple yet efficient solution for TE scattering from a right trapezoidal groove

AbstractA simple yet efficient solution to the problem of transverse electric (TE) scattering by a two‐dimensional right trapezoidal groove placed in a perfect electric conductor (PEC) surface is presented. Using superposition of plane waves and applying the Neumann boundary condition at the groove walls, the transverse magnetic field is expressed as the sums of infinite series of waveguide modes. Considering the equivalent magnetic current on the aperture, a Fredlholm's integral equation with logarithmic singular kernel is extracted and then discretised by a collocation method based on a piecewise constant approximant. The extracted linear system of equations can be easily solved by the matrix methods for the unknown coefficients. Numerous examples are presented to verify the results and show the ability of the proposed method. Finally, this method is employed to examine the effect of the groove parameters and incidence angle on the scattering signatures.

Singular asymptotic expansion and Legendre collocation method for two-term weakly singular Volterra integral equation of the second kind

This paper develops a singularity-separation Legendre collocation method for solving two-term linear Volterra integral equation with algebraic and logarithmic singular kernels. The psi-series solution at the origin is obtained by an efficient Picard iteration, which is very accurate near the origin. By splitting the interval into a singular subinterval and a regular one, the equation is converted to one with a sufficiently smooth solution on the regular interval, which is discretized by the Legendre collocation method. The efficient computational process is discussed in detail, and the convergence under the square norm is proved. Finally, three numerical examples, including a Caputo fractional differential equation and a Basset equation with initial conditions, are provided to illustrate the efficiency and accuracy of the proposed singularity-separation method.

Application of dual-Chebyshev wavelets for the numerical solution of boundary integral equations with logarithmic singular kernels

In this paper, the discrete Galerkin method based on dual-Chebyshev wavelets has been presented to approximate the solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of a boundary value problem of Laplace's equation with linear Robin boundary conditions. The discrete Galerkin methods for solving logarithmic boundary integral equations with Chebyshev wavelets as a basis encounter difficulties for computing their singular integrals. To overcome this problem, we establish the dual-Chebyshev wavelets, such that they are orthonormal without any weight functions. This property adapts Chebyshev wavelets to discrete Galerkin method for solving logarithmic boundary integral equations. We obtain the error bound for the scheme and find that the convergence rate of the proposed method is of $$O(2^{-Mk})$$O(2-Mk). Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique and confirm the theoretical error analysis.

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions

The main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations result from boundary value problems of Laplace’s equations with linear Robin boundary conditions. The method approximates the solution using the radial basis function (RBF) expansion with polynomial precision in the discrete collocation method. The collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae. To overcome this problem, we utilize the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method. Since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements, we can call it as the meshless discrete collocation method. The new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels. We also provide the error estimate of the proposed method. The efficiency and accuracy of the new approach are illustrated by some numerical examples.

Mode-III Stress Intensity Factors of an Arbitrarily Oriented Crack Crossing Interface in a Layered Structure

ABSTRACTThe problem of a layered structure containing an arbitrarily oriented crack crossing the interface in anti-plane elasticity is considered in this paper. The fundamental solution of displacements and stresses is obtained in a series form via the method of analytical continuation in conjunction with the alternating technique. A dislocation distribution along the prospective site of a crack is used to model a crack crossing the interface and the singular integral equations with logarithmic singular kernels for a line crack are then established. The crack is approximated by several line segments and the linear interpolation equation with undetermined coefficients was applied for the dislocation function along line segments. Once the undetermined dislocation coefficients are solved, the mode-III stress intensity factors KIII at two crack tips can be obtained for various crack inclinations with different material property combinations. All the numerical results are checked to achieve a good approximation that demonstrates the accuracy and the efficiency of the proposed method.

Anti-Plane Interaction between a Crack and an Elliptically Cylindrical Layered Media

AbstractAnti-plane interaction of an elliptically cylindrical layered media with an arbitrarily oriented crack embedded in an infinite matrix, intermediate layer, or inner inclusion under a remote uniform shear load is considered in this paper. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the solution for a screw dislocation in the inclusions and the matrix is first derived in a series form. The integral equations with logarithmic singular kernels for a line crack can easily be obtained by using the screw dislocation solutions as the Green's function together with the principle superposition. The stress intensity factors, which can properly reflect the interaction between a crack and an elliptically cylindrical layered media, are then obtained numerically in terms of the values of the dislocation density functions of the integral equations. The effects of material property combinations and geometric parameters on the normalized mode-III stress intensity factors are discussed in detail and shown in graphic form.

Numerical Verification of Solutions of Nekrasov’s Integral Equation

This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method.

Numerical Verification of Solutions of Periodic Integral Equations with a Singular Kernel

This paper proposes the method of numerical verification of solutions of a periodic integral equation with a logarithmic singular kernel, which is typically found in some two-dimensional potential problems. The verification method utilizes a property of the singular integral for trigonometric polynomials, the periodic Sobolev space and Schauder's fixed point theorem.

High order product integration methods for a Volterra integral equation with logarithmic singular kernel

This work is concerned with the construction and analysis of highorder product integration methods for a class of Volterra integralequations with logarithmic singular kernel. Sufficient conditionsfor the methods to be convergent are derived and it is shown thatoptimal convergence orders are attained if the exact solution issufficiently smooth. The case of non-smooth solutions is dealtwith by making suitable transformations so that the new equation possesses smooth solutions.Two particular methods are considered and their convergenceproved. A sample of numerical examples is included.

A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load

The antiplane elasticity problem for a thin cracked layer bonded to an elastic half-space under an antiplane concentrated load is considered. The fundamental solution is obtained as a rapidly convergent series in terms of the complex potentials via iterations of Mobius transformation. The singular integral equation with a logarithmic singular kernel is derived to model a crack problem that can be solved numerically in a straightforward manner. The dimensionless mode-III stress intensity factors obtained for various crack inclinations and crack lengths are discussed in detail and provided in graphic form. A strip problem with an arbitrarily oriented crack is also considered.

Interaction between a crack and a circular elastic inclusion under remote uniform heat flow

The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix with an arbitrarily oriented crack, located either in the matrix or in the inclusion under remote uniform heat flow, is considered. By using the complex variable theory and the existing solutions for dislocation functions, the thermoelastic problem of a crack in the form of an arbitrary shape in the vicinity of the interface is formulated. The integral equations for a line crack are then obtained as a system of singular integral equations with logarithmic singular kernels. The stress intensity factors, which can properly reflect the interaction between a crack and a circular inclusion, are obtained in terms of the values of the density functions of the integral equations. Several numerical examples are given to demonstrate the effects of geometrical parameters and material property combinations on the strength of the thermal stress singularity.

Stress-intensity factors for curved circular crack in bonded dissimilar materials

Complex potentials are used to solve the curved circular crack problem in two bonded half-planes made of dissimilar materials. The mixed boundary value problem is solved by application of superposition and the dislocation distribution for determining the unknown function. A logarithmic singular kernel is obtained for the governing integral equation. Displayed graphically are stress intensity factors for a variety of the curved crack are length and orientation and their changes due to the dissimilar properties are also exhibited.

Product integration methods for an integral equation with logarithmic singular kernel

Product integration methods for an integral equation with logarithmic singular kernel

Numerical solution of the curved rigid line problem in an infinite plate

This paper deals with the problem of evaluating the stress singularity coefficients at the tips of a rigid line in an infinite plate. If the traction difference between the upper and lower borders of the rigid line is taken as an unknown function and the displacement of the curved rigid line as the right-hand term of the integral equation, then a new integral equation for the curved rigid line problem is obtained which is presented in this paper. The newly obtained integral equation has a logarithmic singular kernel. To solve the integral equation an interpolation equation for the traction difference functions (the undetermined functions in the integral equation) is proposed. A numerical examination is carried out to demonstrate the efficiency of the proposed technique. Also two numerical examples are given in this paper.

THERMAL STRESSES AROUND AN ELASTIC RIBBONLIKE INCLUSION WITH GOOD THERMAL CONDUCTIVITY

Plane-deformation problems of thermal stresses around an elastic ribbonlike conductor in an infinite, homogeneous, isotropic medium under uniform heat flow are analyzed by the method of continuous distribution of singular points. The line heat sources and concentrated forces are distributed continuously along the ribbonlike conductor. The boundary conditions yield a system of singular integral equations for density functions with Cauchy kernels and logarithmic singular kernel. A solution is assumed in the form of the product of a series of Chebyshev polynomials of the first kind and their weight function and is determined by transforming the singular integral equations into linear algebraic equations. One of the advantages of this method is that the strength of the thermal-stress singularity at the tips of the ribbonlike conductor can be easily evaluated. Numerical results for the strength of the thermal-stress singularity are plotted in terms of nondimensional parameters, consisting of the dimensions of the ribbonlike conductor and the material constants of the conductor and matrix.