A fast volume integral equation method for the direct/inverse problem in elastic wave scattering phenomena

A fast method for solving the volume integral equation is introduced for the solution of forward and inverse multiple scattering problems in an elastic 3-D full space. For both forward and inverse scattering analysis, the volume integral equation in the wavenumber domain is used. By means of the discrete Fourier transform, the volume integral equation in the wavenumber domain can be dealt with as a Fredholm equation of the 2nd kind with respect to a non-Hermitian operator on a finite dimensional vector space. The Bi-CGSTAB method is employed to construct the Krylov subspace in the wavenumber domain. The current procedure establishes a fast and simplified method without requiring the derivation of a coefficient matrix. Several numerical results validate the accuracy and effectiveness of the current method for both forward and inverse scattering analysis. According to the numerical results, the reconstruction of inhomogeneities of the wave field is successful, even for multiple scattering of several cubes.     

Application of the boundary integral equation (BIE) method to transient response analysis of inclusions in a half space

Transient scattering of elastic waves by inclusions in a half space is investigated by the boundary integral equation (BIE) method. The formulation of BIE presented here is based on the Fourier transform method, and involves the analysis of transformed problems and the reconstitution of transient solutions by Fourier inversion. After the BIE has been solved numerically in the transformed domain, the transient wave fields are obtained with the help of the fast Fourier transform (FFT) algorithm. After confirmation of the accuracy of the present method, some numerical examples are shown for various inclusions in a half space, such as a cavity, an elastic inclusion, and a fluid inclusion.     

Time-dependent motion of a two-dimensional floating elastic plate

Three methods are presented to determine the motion of a two-dimensional finite elastic plate floating on the water surface, which is released from rest and allowed to evolve freely. The first method is based on a generalized eigenfunction expansion and it is valid for all water depths. The second method is based on an integral equation derived from the Fourier transform, and it is valid for all water depths, although computations are made only for water of infinite depth. These two methods are both based on the frequency-domain solution—however no obvious connection exists between the two methods. The third method is valid only for shallow water, and it expresses the solution as the sum over decaying modes. We present a new derivation of the integral equation for a floating plate based on the Fourier transform of the equations of motion in the time domain. The solution obtained by each method is compared in the appropriate regime, and excellent agreement is found, thereby providing benchmark solutions. We also investigate the regime of validity of the infinite and shallow-depth solutions, and show that both give good results for a quite wide range of depths.     

移动荷载作用下饱和土地基中的波动特性分析

基于Biot波动方程,经过Fourier变换和逆Fourier变换后可获得波数-频率域以及时间-空间域的解析解。通过数值分析的手段研究了移动荷载作用下饱和多孔弹性地基中波的传播特性。重点就弥散曲线、多谱勒效应、波的成分和动力响应频率等几个特性进行了分析,发现饱和土地基由于比弹性地基多了一项流体介质,波动特性明显差异于弹性介质。     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

SH波在压电材料条中垂直界面裂纹处的散射

研究了SH波在压电材料条中裂纹处的散射.压电材料条两侧涂有相同梯度参数的两个半无限大功能梯度材料,裂纹垂直于界面.通过Fourier变换,利用边界条件把问题转化为柯西核奇异积分方程,然后利用Chebyshev多项式对奇异积分方程进行数值求解.通过数值计算,分析讨论了压电条的几何参数和SH波频率对标准动应力强度因子的影响.     

Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

介质参数反演的广义射线近似方法

在对无粘性介质参数反演问题进行的研究中，引入一种全波场广义射线近似形式，提出一种新的反演参数的方法，文中，首先对由弹性波动方程演变成的声波方程进行分析，引入背景场量和扰动量，并结合Ｇｒｅｅｎ函数理论，得到了介质参数的积分方程；然后结合前人对非均匀介质中波函数局部理论的定性分析，引入一种全波场广度射线近似形式，把问题归结为一个第一类Ｆｒｅｄｈｏｌｍ积分方程；最后对半空间问题层状介质模型进行了反演，算     

The Problem of an External Circular Crack Under Asymmetric Loadings

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Simulation of elastic wave diffraction by multiple strip-like cracks in a layered periodic composite

The problem of numerical simulation of the steady-state harmonic vibrations of a layered phononic crystal (elastic periodic composite) with a set of strip-like cracks parallel to the layer boundaries is solved, and the accompanying wave phenomena are considered. The transfer matrix method (propagator matrix method) is used to describe the incident wave field. It allows one not only to construct the wave fields but also to calculate the pass bands and band gaps and to find the localization factor. The wave field scattered by multiple defects is represented by means of an integral approach as a superposition of the fields scattered by all cracks. An integral representation in the form of a convolution of the Fourier symbols of Green’s matrices for the corresponding layered structures and a Fourier transform of the crack opening displacement vector is constructed for each of the scattered fields. The crack opening displacements are determined by the boundary integral equation method using the Bubnov-Galerkin scheme, where Chebyshev polynomials of the second kind, which take into account the behavior of the solution near the crack edges, are chosen as the projection and basis systems. The system of linear algebraic equations with a diagonal predominance of components arising when the system of integral equations is discretized has a block structure. The characteristics describing qualitatively and quantitatively the wave processes that take place under the diffraction of plane elastic waves by multiple cracks in a phononic crystal are analyzed. The resonant properties of a system of defects and the influence of the relative positions and sizes of defects in a layered phononic crystal on the resonant properties are studied. To obtain clearer results and to explain them, the energy flux vector is calculated and the energy surfaces and streamlines corresponding to them are constructed.     

Interaction of one-periodic disk-shaped cracks under an incident elastic harmonic wave

We study a symmetric problem of harmonic wave propagation in an elastic space with a one-periodic array of interacting disk-shaped cracks. Using the Green function obtained by the Fourier transform, we reduce the problem to a boundary integral equation (BIE) for the function characterizing the displacement discontinuity on one of the cracks and numerically determine the desired function by solving the BIE. We present graphs of the dynamic stress intensity factors near a circular crack versus the wave number for various distances between the defects.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer’s free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. The case of circular domain of contact is considered in detail. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

周期界面裂纹的弹性波散射问题研究

本文研究了分布于两个关元限空间的周期界面对垂直入射Ｐ波及ＳＨ波的散射问题，文中利用有限Ｆｏｕｒｉｅｒ变换将一个周期带内散射场的边值问题转化为求解一个带周期核的奇异积分方程，并对ＳＨ波入射的情形进行了详细的分析，求解了相应的异积分方程，最后给出裂纹尖端的应力强度因子的计算公式及远离裂纹时散射位移场的渐进形式，并对散场的动态特性进行了数值分析。     

非均匀弹性材料中反平面的运动裂纹

用解析方法研究了非均匀弹性材料中反平面运动裂纹问题。首先采用余弦变换求解非均匀材料的基本方程，然后根据混合边值条件建立裂纹运动的对偶积分方程，再把对偶积分方程化为第二类Fredholm积分方程。给出了数值算例，计算结果表明材料的非均匀性对动应力强度因子有较大的影响。     

Fast numerical method for crack problem in the porous elastic material

The present paper discusses the crack problem in the linear porous elastic plane using the model developed by Nunziato and Cowin. With the help of Fourier transform the problem is reduced to an integral equation over the boundary of the crack. Some analytical transformations are applied to calculate the kernel of the integral equation in its explicit form. We perform a numerical collocation technique to solve the derived hyper-singular integral equation. Due to convolution type of the kernel, we apply, at each iteration step, the classical iterative conjugate gradient method in combination with the Fast Fourier technique to solve the problem in almost linear time. There are presented some numerical examples for materials of various values of porosity.     

Boundary element analysis for dynamic response of vibration foundation

The boundary elament method (BEM) for numerical solution to dynamic response of vibration fundation in plane, elastic domains are presented. The dynamic boundary integral equation is derived from the Laplace integral transform of the elestodynamic differential equation. Numerical solution can then be completed by the discrete boundary element in the transform space. Finally, dynamic responsed in time domain will be inverted back from the transform space with the numerical method. Excited harmonic load responses of dynamic rigid foundation are calculated and discussed for different frequencies, Layer depths and foundation embedments. Again, screening of exciting wave is also studied.The support of the ressarch project part in this work by Dr. O. Tullberg, Goteborg Universities' Computing Centre, Sweden, is gratefully acknwledged.     

Integral transform solution of multi-material structure with finite crack perpendicular to the interface

The plane elasticity problem for layered elastic systems containing a finite crack perpendicular to the interface is considered. To derive the singular integral equations. Fourier transform in conjunction with dislocation is used. The singular integral equation is solved with the Lobatto-Chebyshev method commonly applied to such problems. In order to have an idea about the usefulness of the method described, a two-layer structure which contains a cut parallel toh is considered.     

GENERALIZED RAY APPROXIMATE METHOD TO INVERSE MEDIUM PARAMETERS

In this paper,the inverse problem of the medium parameters in an inhomogeneousmedium is studied and a generalized ray approximate form of the total wave field is described.First,the acoustic wave equation derived from the elastic wave equation is studied,the referential variablesand perturbational variables are introduced,and the integral equation of the medium perturbational pa-rameters is obtained.Then from the point of view of the local principles of the wave function in an in-homogeneous medium,a generalized ray approximate form of the total wave field in an inhomoge-neous medium is described,and attention is focused on the Fredholm integral equation of the firstkind.Finally,the medium parameters in half-plane are inversed.Numerical examples show when theperturbations of the medium parameters are about 0.5,this method can effectively inverse its varia-tion.Apparently,this method is better than the conventional Born weak scattering approximation.     

State space approach to the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources

A two-dimensional equation of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with the elastic modulus dependent on temperature and with an internal heat source is established using a Laplace transform in time and a Fourier transform in the space variable. The problem for the transforms is solved in the space of states. The problem of heating of the upper and the lower surface of a plate of great thickness by an exponential time law is considered. Expressions for displacements, temperature, and stresses are obtained in the transform domain. The inverse transform is obtained using a numerical method. Results of solving the problem are presented in graphical form. Comparisons are made with the results predicted by the coupled theory and with the case of temperature independence of the elastic modulus.