Wave penetration in elastic solids with periodic array of rectangular defects: oblique incidence

Summary. In the context of wave propagation in damaged (elastic) solids, an analytical approach for anti-plane oblique penetration of a plane wave through a periodic array of rectangular defects is developed. Reducing the problem to integral equations holding over the openings, an approximation of low-frequency type leads directly to explicit analytical formulas for the scattering parameters. Numerical solution of the governing equations provides graphs which are compared with an exact full-numerical solution and previous analytical results.     

In-plane problem for wave propagation through elastic solids with a periodic array of cracks

Summary In the context of wave propagation in damaged (elastic) solids, an analytical method previously introduced for scalar problems, is now applied to study the (vector) problem for normal penetration of a longitudinal plane wave into a periodic array of collinear cracks. Reduced the problem to an integral equation holding over the openings, an approximation of one-mode type leads to analytical solutions and then to explicit representations for the wave fields and the scattering parameters. Some graphs will finally compare our results with the numerical ones by other authors.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

Boundary integral equation method to solve embedded planar crack problems under shear loading

The solution of three-dimensional planar cracks under shear loading are investigated by the boundary integral equation method. A system of two hypersingular integral equations of a three-dimensional elastic solid with an embedded planar crack are given. The solution of the boundary integral equations is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack without a polygonal contour shape and permit the fast convergence for the results. The stress intensity factors at the crack front are calculated in the case of a circular and an elliptic crack and are compared with the analytical solution.     

Effect of a submerged vertical barrier on flexural gravity waves

An explicit solution is provided for the scattering of flexural gravity waves by a rigid vertical barrier submerged in an infinite depth of water. By applying recently developed mode-coupling relation for eigenfunctions, the mixed boundary value problem has been converted to solve dual integral equations with kernel consisting of trigonometric functions. And then complete analytical solutions are derived with an aid of singular integral equations whose solutions are bounded at the end points. The important hydrodynamical scattering quantities such as reflection and transmission coefficients associated with the flexural gravity wave scattering have been obtained analytically in terms of modified Bessel functions and Struve functions. It is observed that these quantities are sensitive to both combined as well as individual effect of plate thickness and barrier depth of submergence. Numerical results are computed and explained graphically for different parameters such as time period and non-dimensional wave length. Further, the effect of compressive force and plate thickness on the flexural gravity waves against a submerged vertical barrier is studied.     

Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains

This paper presents an analytical approach to solving the plane non-axisymmetric elasticity and thermoelasticity problems in terms of stresses for isotropic, homogeneous or inhomogeneous annular domains. The key feature of this approach is integration of the equilibrium equations in order to: a) express all the stress-tensor components in terms of a governing stress; b) deduce the integral equilibrium conditions, which are vital for the solution. Because the equilibrium equations are insensitive of material properties, the obtained expressions and integral conditions fit both homogeneous and inhomogeneous cases. The governing stress is derived out of the compatibility equation. Regarding complete construction of the solution, the integral compatibility conditions are deduced by integrating the strain-displacement relations. In the case of inhomogeneous material, the governing compatibility equation is reduced to Volterra type integral equation which then is solved by simple iteration method. The rapid convergence of the iterative procedure is established.     

Acoustic scattering by a circular semi-transparent conical surface

The scattering of a plane acoustic wave by a circular semi-transparent conical surface with impedance-type boundary conditions is studied. The analytic solution is constructed on the basis of the incomplete separation of variables and the reduction of the problem to a functional difference equation of the second order. Although the latter is equivalent to a Carleman boundary-value problem for analytic vectors, the solution is studied by means of the direct reduction method, that is, converting the functional difference equations to a Fredholm-type integral equation. Its unique solvability is then studied and the expression for the scattering amplitude of the spherical wave from the vertex is discussed. Some numerical results for axial incidence are also presented.     

Boundary element solution of 3-D wave scatter problems in a poroelastic medium

This study details the development of boundary integral equations suitable for treating problems involving the scatter of a plane harmonic wave by an inclusion embedded in an infinite poroelastic medium. The pore pressure-solid displacement form of the harmonic equations of motion are developed from the form of the equations originally presented by Biot. Fundamental solutions and a generalized reciprocal work relation are developed, and these are used to formulate a solution representation in terms of an integral over the inclusion surface. The corresponding boundary integral equations are developed in a form that is integrable in the usual sense, eliminating the need to evaluate Cauchy principal value integrals. These boundary integral equations are discretized and implemented into a boundary element computer program. The so-called forbidden frequency problem which causes computational difficulties in boundary integral treatments of wave scatter in elastic and acoustic media is shown to be absent in the poroelastic case. Numerical results obtained from the boundary element program are compared with analytical results for some test problems, and the program appears to produce accurate results at moderate frequencies.     

半圆柱型沉积盆地对SH波散射的数值分析

该文实现了一种半无限域SH波散射问题的数值分析方法。采用传递矩阵法得到SH波斜入射时的自由场,将其作为输入;采用集中质量显式有限元方法计算区域内节点的位移;采用透射人工边界计算人工边界点的位移;通过编写的FORTRAN程序实现计算过程。运用该方法对均匀半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析,与Trifunac M D的解析解进行了对比,验证了该文方法的有效性,分析了不同入射角对地表位移和位移谱放大系数的影响。最后,对成层半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析。相对于解析方法而言,该方法可以考虑更为复杂地形情况。     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

The scattering of oblique flexural waves of a cracked conducting plate in a uniform magnetic field

Summary Following a classical plate bending theory for magneto-elastic interactions under quasistatic electromagnetic field, we consider the scattering of time harmonic flexural waves by a through crack in a conducting plate under a uniform magnetic field normal to the crack surface. It is assumed that the plate has the finite electric conductivity, and the electric and magnetic permeabilities of the free space. An incident wave giving rise to moments symmetric about the crack plane is applied in an arbitrary direction. Fourier transform method is used to solve the mixed boundary value problem which reduces to a pair of dual integral equations. These dual integral equations are further reduced to a Fredholm integral equation of the second kind. The dynamic moment intensity factor versus frequency for several values of incident angle is computed and the influence of the magnetic field on the normalized values is displayed graphically.     

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.     

Fracture of a surface layer bonded to a half space

The plane problem of a cracked elastic surface layer bonded to an elastic half space is considered. The surface layer is assumed to contain a transverse crack whose surface is subjected to uniform compression. The problem is formulated in terms of a singular integral equation, the derivative of the crack surface displacement being the density function. By using appropriate quadrature formulas, the integral equation reduces to a system of linear algebraic equations. This system is solved; the stress intensity factors and the crack surface displacement for various crack geometries, namely for internal crack, edge crack, crack touching the interface, and completely broken layer cases, are obtained.     

Integral equation model of light scattering by an oriented monodisperse system of triaxial dielectric ellipsoids: application in ectacytometry

A novel mathematical model of light scattering by an oriented monodisperse system of triaxial dielectric ellipsoids of complex index of refraction is presented. It is based on an integral equation solution to the scattering of a plane electromagnetic wave by a single triaxial dielectric ellipsoid. Both the position and the orientation of a single representative scatterer in a given coordinate system are considered arbitrary. A Monte Carlo simulation is developed to reproduce the diffraction pattern of a population of aligned ellipsoids. As an example of practical importance, light scattering by a population of erythrocytes subjected to intense shear stress is modeled. Agreement with experimental observations and the anomalous diffraction theory is illustrated. Thus a novel check of the electromagnetic basis of ektacytometry is provided. Furthermore, the versatility of the integral equation method, particularly in the advent of parallel processing systems, is demonstrated.     

Mathematical boundary integral equation analysis of an embedded shell under dynamic excitations

A boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half-space under axisymmetric excitations. By virtue of a set of ring-load Green's functions for the shell and a group of dynamic fundamental solutions for the semi-infinite medium, the structure–medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations. Through the analysis of an auxiliary pair of Cauchy integral equations, the singularities of the contact stress distributions arc rendered explicit. With a direct incorporation of such analytical features into the formulation, an effective computational procedure is developed which involves an interpolation of regular functions only. Typical results for the dynamic contact load distributions, displacements, and complex compliance functions are included as illustrations. In addition to furnishing quantities of direct engineering interest, this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods.     

Interaction of antiplane cracks with elastic waves in transversely isotropic materials

Summary The interaction of plane time-harmonic SH-waves with micro-cracks in transversely isotropic materials is investigated. Elastic wave scattering by a single micro-crack is first analyzed. The scattered displacement is expressed as a Fourier integral containing the crack opening displacement. By using this representation formula and by invoking the traction-free boundary condition on the faces of the crack, a boundary integral equation for the unknown crack opening displacement is obtained. Expanding the crack opening displacement into a series of Chebyshev polynomials and adopting a Galerkin method, the boundary integral equation is converted into an infinite system of inear algebraic equations for the expansion coefficients which is solved numerically. Numerical results are presented for the elastodynamic stress intensity factors, the scattered far-field and the scattering cross section of a single crack. Then, propagation of plane time-harmonic SH-waves in a transversely isotropicmaterial permeated by a random and dilute distribution of micro-cracks is investigated. The effects of the micro-crack density on the attenuation coefficient and the phase velocity are analyzed by appealing to a simple energy consideration and by using Kramers-Kronig relations.     

Improved beam propagation method equations

Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff-Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens.     

Diffraction of elastic waves by an elliptic crack—II

A recently developed integral equation technique is used to obtain a low frequency solution for the diffraction of a plane compressional or shear wave by an elliptic crack embedded in an elastic medium. The mixed boundary value problem is reduced to a coupled system of integro-differential equations. A formal power series solution for the coupled system of integro-differential equations is developed. Attention is focussed on the farfield scattered amplitudes and the dynamic stress intensity factor. The limiting values when the ellipse degenerates into a circle agree with those of a circular crack.     

The scattering of a scalar wave by a thin oblate body of revolution

The scattering of a scalar incident wave by a thin oblate body of revolution is studied. The scattering wave is represented as a distribution of ring singularities in a disk inside the body. Boundary conditions result in one-dimensional integral equations for singularity distributions. The asymptotic solution of these equations for thin bodies furnishes the singularity distribution as well as the radius α of the disk containing ring singularities. Example of the plane wave axially incident upon an oblate ellipsoid is presented. The total scattering cross-section is computed.     

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.