Point force excitation of a thick-walled elastic infinite pipe with an embedded inhomogeneity

The propagation of elastic waves in thick-walled pipe with an embedded inhomogeneity is considered. The pipe is excited by a point force applied on its surface and the time harmonic problem is solved using the null field approach, a method whose main characteristics are surface integral representations and expansions in spherical and cylindrical vector wave functions. Entering in the expression for the scattered field are the transition matrix for the cavity, the reflection matrices for the inner and outer surfaces of the pipe, the transformation functions between the spherical and cylindrical vector wave functions and also the translation for the cylindrical waves. Numerical examples, both in the frequency and time domain, are presented for a spherical cavity and an open circular crack.     

Scattering of anti-plane shear waves by a partially debonded piezoelectric circular cylindrical inclusion

Summary This paper presents an analysis of the scattering of anti-plane shear waves by a single piezo-electric cylindrical inclusion partially bonded to an unbounded matrix. The anti-plane governing equations for piezoelectric materials are reduced to Helmholtz and Laplacian equations. The fields of scattered waves are obtained by means of the wave function expansion method when the bonded interface is perfect. When the interface is partially debonded, the region of the debonding is modeled as an interface crack with non-contacting faces. The electric permeable boundary conditions are adopted, i.e. the normal electric displacement and electric potential are continuous across the crack faces. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients.     

Transient scattering of Rayleigh-Lamb waves by a surface-breaking crack: Comparison of numerical simulation and experiment

The transient scattering of Rayleigh-Lamb waves by a surface-breaking crack in a plate is investigated in both time and frequency domains by using the hybrid numerical method which combines the finite element discretization of the vicinity of the crack with the Green's function integral representation of the exterior scattered field. The frequency domain response is obtained by solving a set of large sparse unsymmetric complex matrix equations, elements of which are stored in a compacted data structure, by the biconjugate gradient method. The time domain solution is then obtained by using FFT. The source function generated by a steel ball impact is extracted by a direct integration technique. It is then used to simulate some available experimental results. Good agreement has been obtained. Numerical and experimental results show the effect of the crack more in the near field than far away.     

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.     

The linear sampling method for a mixed scattering problem

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by a mixed scatterer, which is a combination of an open crack and an impenetrable obstacle. Firstly, the well-posedness of the solution to the direct scattering problem is established using the variational method. Then, a uniqueness result for the inverse scattering problem is proved, that is, both of the crack and the impenetrable obstacle can be uniquely determined simultaneously by the knowledge of the far-field pattern. Furthermore, a mathematical basis is given to reconstruct the shape of the crack and the impenetrable obstacle using the linear sampling method, and some numerical examples are given to establish the viability of our approach.     

Multiple Scattering of Thermal Waves from a Subsurface Cylindrical Inclusion in Semi-infinite Functionally Graded Materials Using Non-Fourier Model

In this study, a theoretical method is applied to investigate the multiple scattering of thermal waves and temperature field resulting from a subsurface cylindrical inclusion in a semi-infinite functionally graded material (FGM). The adiabatic boundary condition at the semi-infinite surface is considered. The thermal waves are excited at the surface of semi-infinite functionally graded materials by modulated optical beams. The model includes the multiple scattering effects of the cylindrical thermal wave generated by the line heat source. According to the wave equation of heat conduction, a general solution of scattered thermal waves is presented. Numerical calculations illustrate the effect of subsurface inclusion on the temperature and phase change at the sample surface under different physical and geometrical parameters. It is found that the temperature above the conducting cylindrical inclusion decreases because of the existence of the inclusion. The effect of the inclusion on the temperature and phase change at the surface is also related to the non-homogeneous parameter of FGMs, the wave frequency of thermal waves, and the distance between the inclusion and the semi-infinite surface. Finally, the effect of the relaxation time of buried inclusion on the temperature and phase change at the surface is examined.     

Quasi-three-dimensional method of moments for analyzing electromagnetic wave scattering in microwave tomography systems

A new quasi-three-dimensional method of moments (Quasi-3-D MoM) for analyzing electromagnetic wave scattering from a cylindrical dielectric object surrounded by a dipole array in microwave tomography systems is presented in this paper. A wire-volumetric electric field integral equation is derived for the electromagnetic wave scattering phenomena in microwave tomography systems. The new method is based on the MoM and involves rectangular cylindrical cells modeling the cylindrical object. The distribution of electric flux densities along the axial direction of cylindrical cells is expanded as a Fourier series multiplied by an attenuation factor, which is one part of basis functions. Therefore, the Quasi-3-D MoM is performed in a two-dimensional discretization, and the computational complexity is reduced. Detailed mathematical steps along with some numerical results are presented to illustrate the efficacy and accuracy of this approach.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

Effects of sensor locations on air-coupled surface wave transmission measurements across a surface-breaking crack

Previous studies show that the surface wave transmission (SWT) method is effective to determine the depth of a surface-breaking crack in solid materials. However, nearfield wave scattering caused by the crack affects the reliability and consistency of surface wave transmission measurements. Prior studies on near-field scattering have focused on the case where crack depth h is greater than wavelength λ of surface waves (i.e., h/λ > 1). Near-field scattering of surface waves remains not completely understood in the range of h/λ for the SWT method (i.e., 0 ≤ h/λ ≤ 1/3), where the transmission coefficient is sensitive to crack depth change and monotonically decreases with increasing h/λ. In this study, the authors thoroughly investigated the near-field scattering of surface waves caused by a surface-breaking crack using experimental tests and numerical simulations for 0 ≤ h/λ ≤ 1/3. First, the effects of sensor locations on surface wave transmission coefficients across a surface-breaking crack are studied experimentally. Data are collected from Plexiglas and concrete specimens using air-coupled sensors. As a result, the variation of transmission coefficients is expressed in terms of the normalized crack depth (h/λ) as well as the normalized sensor location (x/λ). The validity of finite element models is also verified by comparing experimental results with numerical simulations (finite element method). Second, a series of parametric studies is performed using the verified finite element model to obtain more complete understanding of near-field scattering of surface waves propagating in various solid materials with different mechanical properties and geometric conditions. Finally, a guideline for selecting appropriate sensor arrangements to reliably obtain the crack depth using the SWT method is suggested.     

Application of a novel formulation to the MFIE using RWG basis functions in method of moments

We investigate the inaccuracy of the traditional magnetic field integral equation (MFIE) in the analysis of scattering of small conducting objects, with the use of Rao–Wilton–Glisson (RWG) basis functions. As a remedy to such problems, we propose to use a novel impedance matrix elements (IME) formulation of the MFIE, which is suitable to use RWG basis functions. Techniques to calculate the IME of the new formulation are outlined. It is shown that similar technique to compute the IME of the MFIE resulted from the use of curl-conforming basis functions can also be used here. Based on the examples of several relatively small conducting objects, it is demonstrated that significant improvement in the accuracy of the MFIE can be obtained for this new IME formulation with the use of RWG basis functions.     

Effects of sinusoidal crack-face perturbations on scattering of elastic waves

The scattering of a plane longitudinal wave from a two-dimensional crack, with a sinusoidal surface perturbation whose amplitude and wavelength are much smaller than the length of the crack, is investigated. The amplitude of the cylindrical body waves in the far field are calculated from a Kirchhoff approximation that utilizes the solution to the reflection from the sinusoidal surface profile of a semiinfinite solid. The results are compared to those for a flat crack, and conditions for significant differences of the amplitude as a function of the angle of observation are discussed. Characteristic changes in the scattered field produced by profiles with different amplitudes and periods are explained.     

Spectral-domain analysis of multilayer cylindrical–rectangular microstrip antennas

In this paper, the resonance and radiation characteristics of multilayered cylindrical–rectangular microstrip antennas are analysed. The problem is formulated, in the spectral-domain, using an electric field integral equation and the spectral-domain Green's function. An efficient compact method is developed to obtain the Green's function of the electric field due to the current distribution of a patch located in an arbitrary layer of a cylindrical stratified medium. To accomplish this, efficient matrix representations are used to describe the electric and magnetic fields in each layer of the geometry in terms of ABCD matrices characteristic of its material properties. The boundary integral equation for the unknown patch current is solved numerically by applying the boundary element method using three kinds of basis functions. Taking into account the symmetry properties of the elements of the moment method matrix, the CPU time is reduced significantly. The convergence of the method is proven by performing the resonant frequencies for a single layer cylindrical–rectangular microstrip patch vs. the substrate thickness. The computed data are found to be in very good agreement with those found in the literature, using only two basis functions. Once the validity of the method is checked, further results for various antennas structures are presented proving the generality of the method. Finally, the effect of an air gap between the substrate layer and the ground conducting cylinder on far zone radiation field is investigated using the steepest-descent method.     

Green's-function method for the analysis of propagation in holey fibers

A Green's-function method is employed to provide a rigorous analysis to the propagation and coupling phenomena in holey fibers. The analysis is carried out for an arbitrary grid of circular air holes of the fiber guide, while the electromagnetic field is taken to be a vector quantity. Application of the Green's-function concept leads to a coupled system of equations incorporating as unknowns the field expansion coefficients to cylindrical wave functions within the air holes. The propagation constants of the guided waves are computed accurately by determining the singular points of the corresponding system's matrix. Field distribution and dispersion properties of guided modes as well as coupling phenomena between parallel-running holey fibers are investigated, and numerical results are presented.     

Scattering of antiplane shear wave by a kinked crack

In this paper the scattering of antiplane shear waves by a kinked crack for a linearly elastic medium is considered. In order to solve the proposed problem, at first the broken crack problem is reduced to two coupled single cracks. Fourier integral transform method is employed to calculate the scattered field of a single crack. In order to derive the Cauchy type integral equations of a broken crack and analyze the singular stresses at the breakpoint, the scattered field of a single crack is separated into a singular part and a bounded part. The single crack solution is applied to derive the generalized Cauchy type integral equations of a broken crack. The singular stress and singular stress order are analyzed in the paper and the dynamic stress intensity factor (DSIF) at breakpoint is defined. Numerical solution of the obtained Cauchy type integral equations gives the DSIF at the crack tips and at the breakpoint. Comparison of the present results in some special cases with the known results confirms the proposed method. Some typical numerical results and corresponding analysis are presented at the end of the paper.     

A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions

Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by linear Fredholm integral equations, the main focus of this paper is to present a fast numerical method for solving them. The method uses Whittaker cardinal functions as a set of basis functions. By using the properties of these cardinal functions together with an appropriate quadrature rule, the computational cost of the method becomes low and the calculations are done quickly. This is due to the fact that the method uses just sampling of functions instead of integration. To show computational efficiency of this approach, some practical one- and two-dimensional scatterers are analyzed by it and the results are compared with those of two other numerical methods.     

Numerical simulation bistatic scattering from three-dimensional conducting rough surface with forward?backward method-based vector basic functions

To more really simulate distribution of vector-induced currents along a three-dimensional (3-D) conducting rough surface, it is better for vector basic functions to be used to describe the vector-induced currents. The forward-backward method (FBM) has been further developed to be combined with the vector basic functions to numerically calculate 3-D conducting rough surface scattering problem in this paper, meanwhile, the FBM is first well explained through some matrix forms. The FBM has been used to numerically simulate the bistatic scattering from 3-D conducting rough surface. Some numerical results have been discussed in the paper.     

Ultrasonic 2D SH Crack Detection in a Layered Anisotropic Plate

The 2-D scattering problem of an internal crack in a layered anisotropic plate is considered in this paper. In the model, two ultrasonic SH probes are attached on the upper surface of the plate and the incoming displacement field is generated by one of the probes and the other probe is acting as a receiver. The transmitting and the receiving probe may be the same. The problem is solved by deriving the Green function for the layered plate and then using the integral representation for the total field to obtain an integral equation for the crack opening displacement. The integral equation is solved by expanding the crack opening displacement (COD) in Chebyshev functions. A crucial part of the method is the expansion of the Green function in a free space part, expressed in the crack coordinate system, and a reflection part, expressed in the plate coordinate system. The electrical signal response is calculated by an electromechanical reciprocity relation. Numerical examples are given for a transversally isotropic graphite-epoxy plate, where the symmetry axes are mutually perpendicular in the layers. The results are presented as A-scans, i.e. the electrical response as a function of time.     

Numerical resolution of the boundary integral equations for elastic scattering by a plane crack

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross-section, the back-scattered amplitude and far-field patterns.     

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.     

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。