Modified fundamental solutions for the scattering of elastic waves by a cavity: Numerical results

The boundary integral equation method is very often used to solve exterior problems of scattering of waves (elastic waves, acoustic waves, water waves and electromagnetic waves). It is known, however, that this method fails to provide a unique solution at the so-called irregular frequencies. This difficulty is inherent to the method used rather than the nature of the problem. In the context of elastodynamics. we proposed, in a recent work¹, two methods for eliminating these irregular frequencies. Both are based on modifying the fundamental solution. Here we present numerical results pertaining to the solutions of the modified and unmodified integral equations.     

Virtual boundary element-integral collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element—integral collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the basic idea of the virtual boundary element method for elasticity and the fundamental solutions of the plane magnetoelectroelastic solids. Besides sharing all the advantages of the conventional boundary element method (BEM) over domain discretization methods, it avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. The method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.     

Numerical modelling for scattering of waves in three-dimensional cracked material

In this paper, a mathematical model is proposed for the problem of the scattering of plane waves in a three-dimensional cracked materials. Instead of obtaining closed-form solutions as in conventional theoretical analysis methods, this approach, called the Equivalent Nodal Force (ENF) method formulates the mechanical effects of cracks as an equivalent nodal force in a numerical procedure, and physically translates cracked material into an equivalent continuous one. Several mechanical relations between waves and cracks are evident from this method. Also the results of several numerical calculations are presented and these are compared with those obtained by the conventional methods.     

平面电磁弹性固体的虚边界元——最小二乘配点法

从电磁弹性固体平面问题的基本方程出发,依据弹性力学虚边界元法的基本思想,利用电磁弹性固体平面问题的基本解,提出了电磁弹性固体平面问题的虚边界元——最小二乘配点法。电磁弹性固体的虚边界元法在继承传统边界元法优点的同时,有效地避免了传统边界元法的边界积分奇异性的问题。由于仅在虚实边界选取配点,此方法不需要网格剖分,并且不用进行积分计算。最后给出了一些具体算例,并和已有的解析解进行了对比,结果表明提出的虚边界元方法有很高的精度。     

各向异性材料平面问题基本解析解的特征方程解法

该文提出了一种利用特征方程解法构造基本解析解的新方法,并将其应用到各向异性材料平面问题,成功构造了完备且独立的系列基本解析解.构造各向异性材料平面问题控制微分方程的算子矩阵,通过其行列式计算可得到问题特征通解所需满足的特征方程,将所求得特征通解代入到微分方程算子矩阵所对应的伴随矩阵,可推导得出各向异性材料平面问题的基本解析解.根据基本解析解独立性的论证,可得到系列独立且完备各向异性材料平面问题基本解析解.利用特征方程解法求解基本解析解思路简单、并且容易找到独立且完备的解析解,其结果可以成为相关数值计算方法的基础.     

The Almansi method of fundamental solutions for solving biharmonic problems

A new fundamental solutions method for the numerical solution of two-dimensional biharmonic problems is described. In this method, which is based on the Almansi representation of a biharmonic function in the plane, the approximate solution is expressed in terms of fundamental solutions of Laplace's equation, and is determined by a least squares fit of the boundary conditions. The results of numerical experiments which demonstrate the efficacy of the method are presented.     

基于拓展单位分解有限元法分析声波在多域场内的响应

基于拓展单位分解有限元方法，将平面波函数和贝塞尔函数作为基函数进行拓展。将亥姆霍兹方程离散，求解时不变情况下多域场内声波的响应，并分析基函数对求解精度的影响。将波动方程的时间导数利用二阶中心差分方法离散，得到方程的隐式表达式，划分时间步迭代求解时变情况下声波在多域场内的响应，分析迭代时间间隔对计算精度的影响，与典型算例的精确解进行比较，验证精确性。结果表明，平面波函数作为拓展基函数，利用二阶中心差分法离散时间导数，分析时不变以及时变情况下多域场内高波数声波的响应问题，具有较高的计算精度和计算效率。     

A scattering problem by means of the spectral representation of Green's function for a layered acoustic half-space

 The problem in this paper is for scattering waves caused by an object and a plane wave in a layered acoustic half space. The boundary integral equation method as well as the spectral representation of Green's function for a layered acoustic half space are introduced to the present analyses. The spectral form of Green's function developed here is expressed in terms of the eigenfunctions for the point and the continuous spectra, that is the extension form of Green's function expressed by Ewing, Jardetsky and Press (1957). The advantage of the spectral representation of Green's function is that it enables us to decompose the scattering waves into eigenfunctions for the layered medium. Several numerical calculations are carried out to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the spectral form of Green's function provides accurate values and is applicable to the boundary element analysis for a layered medium. The spectral structures of the scattering waves are also found to be able to explain their properties. Received 2 November 1999     

Helical localized wave solutions of the scalar wave equation

A right-handed helical nonorthogonal coordinate system is used to determine helical localized wave solutions of the homogeneous scalar wave equation. Introducing the characteristic variables in the helical system, i.e., u = zeta - ct and v = zeta + ct, where zeta is the coordinate along the helical axis, we can use the bidirectional traveling plane wave representation and obtain sets of elementary bidirectional helical solutions to the wave equation. Not only are these sets bidirectional, i.e., based on a product of plane waves, but they may also be broken up into right-handed and left-handed solutions. The elementary helical solutions may in turn be used to create general superpositions, both Fourier and bidirectional, from which new solutions to the wave equation may be synthesized. These new solutions, based on the helical bidirectional superposition, are members of the class of localized waves. Examples of these new solutions are a helical fundamental Gaussian focus wave mode, a helical Bessel-Gauss pulse, and a helical acoustic directed energy pulse train. Some of these solutions have the interesting feature that their shape and localization properties depend not only on the wave number governing propagation along the longitudinal axis but also on the normalized helical pitch.     

Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams

Based on the metamaterial cloaking technique, we propose the use of a new photon funnel to compress a plane electromagnetic (EM) wave. The theoretical analysis and numerical simulations indicate that the compression ratio can be designed optionally and the compressed wave beam remains the original wave shape without any distortions. Here we apply the method to EM waves but it can be applied to acoustic waves and other fields as well.     

Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.     

The method of fundamental solutions for acoustic wave scattering by a single and a periodic array of poroelastic scatterers

The method of fundamental solutions (MFS) is now a well-established technique that has proved to be reliable for a specific range of wave problems such as the scattering of acoustic and elastic waves by obstacles and inclusions of regular shapes. The goal of this study is to show that the technique can be extended to solve transmission problems whereby an incident acoustic pressure wave impinges on a poroelastic material of finite dimension. For homogeneous and isotropic materials, the wave equations for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. This allows for a simple and systematic way to construct fundamental solutions for describing the wave displacement field in the material. The efficiency of the technique relies on choosing an appropriate set of fundamental solutions as well as properly imposing the transmission conditions at the air–porous interface. In this paper, we address this issue showing results involving bidimensional scatterers of various shapes. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Comparisons with results computed using a mixed pressure–displacement finite element formulation illustrate the great advantages of the MFS both in terms of computational resources and mesh preparation. The extension of the method for dealing with the scattering by an infinite array of periodic scatterers is also presented.     

Numerical simulation and visualization of elastic waves using mass-spring lattice model

A computer program package has been developed for simulation and visualization of two-dimensional elastic wave propagation and scattering using the mass-spring lattice model (MSLM) and, for comparison, a finite difference model. To assess the reliability of the numerical schemes, their convergence and accuracy have been analysed using the Taylor series expansion and the von Neumann analysis methods. As a result, the grid spacing-time increment combinations previously adopted in the literature have proved to be non-optimal. The optimal combinations have been found and shown to yield the most accurate results with the least computation time, particularly in the high frequency regime. Using these algorithms, a program package has been developed in Visual C++(R) (Microsoft, Redmond, WA) with graphic user interfaces for convenient exploration of visualized results. Numerical results have been obtained for some fundamental problems in ultrasonic testing such as plane waves incident on cracks. All numerical results have shown excellent qualitative agreements with the analytical results of the wave physics, as the reflected, diffracted, head, and Rayleigh waves have been observed. Also, for numerical results with anisotropic media, the cusps on the shear wavefronts have been observed. Finally, slight modification of the modeling method for free surfaces has led to more accurate prediction of Rayleigh waves.     

A new hybrid finite element approach for plane piezoelectricity with defects

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.     

一种声发射源的新型平面定位方法研究

声发射现象中产生的弹性波在传播时声压遵从指数衰减规律,文中提出一种利用声衰减特性对声发射源进行平面定位的能量定位新方法。从理论上导出了该方法的定位计算公式,并用AE21C型声发射仪在高分子合成纤维平板上以铅笔芯折断产生信号为模拟源进行了实验测量,证明这种新的源定位方法可行,并具有无须测量传声媒质的声衰减和声速等优点。     

Application of the MFS to inverse obstacle scattering problems

In this paper, the method of fundamental solutions (MFS) is used to detect the shape, size and location of a scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered acoustic pressure in the vicinity of the obstacle. A nonlinear constrained minimization regularized MFS technique is proposed for the numerical solution of the inverse problem in question. The stability of the technique is investigated by inverting measurements contaminated by random noise. The results of several numerical experiments are presented.     

The mode‐III fracture analysis of an inclined crack embedded in thin layer systems by analytical alternating methods

Abstract

This paper analyzes the mode‐III stress intensity factor of an inclined crack, embedded in a thin layer, bonded to a half plane, subjected to arbitrary distributed anti‐plane loads. Special alternating procedures are presented to evaluate the mode‐ III S.I.F. and the numerical results confirm the validity of the proposed alternating procedure. The solution of a bi‐material problem in an infinite plane with an inclined crack and the analytical solution of a thin layer, without crack, bonded to a half plane, subjected to an anti‐plane point force applied on the boundary are referred to as fundamental solutions. By using these fundamental solutions and alternating procedures, the stress intensity factors of a crack in a thin layer bonded to a half plane are evaluated. The numerical results of some reduced problems are computed and excellent agreements with existing solutions are obtained.     

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.     

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.     

The effect of acoustic diffraction and acoustic attenuation on the dynamic range of acoustooptic Bragg cells

A three-dimensional model for the propagation of finite acoustic waves in nonlinear media is developed. This model implicitly includes the effects of acoustic attenuation and divergence due to diffraction. The generation of intermodulation products in the case of a two-tone input signal is numerically analyzed. It is found that acoustic diffraction can have a significant effect on the dynamic range of a Bragg cell if the acoustic field extends well into the Fraunhofer region. Inclusion of the effect of diffraction in the model predicts a dynamic range that can be considerably larger than the value obtained by using the infinite plane wave assumption. It is shown that acoustic attenuation significantly reduces the level of the acoustic intermodulation products relative to the level of the fundamental modes. This also increases the dynamic range. The influence of these effects on design considerations for Bragg cells is discussed.