Guided elastic waves and perfectly matched layers

Elastic waveguides support propagating modes that have two possible features, negative group velocities and long wavelengths that, for some frequencies, degrade the accuracy or otherwise poison existing numerical schemes that utilise perfectly matched layers (PMLs) to mimic infinite domains. We illustrate why negative group velocities and long waves are potentially an issue and describe how these problems are overcome. Detailed numerical simulations confirm the accuracy of the modified scheme and provide both theoretical and pragmatic estimates for the parameters within the PML model, in particular for the damping function. We also contrast and compare different implementations of the PML model using spectral and finite difference methods.     

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

An efficient method to compute the scattering of a guided wave by a localized defect, in an elastic waveguide of infinite extent and bounded cross section, is considered. It relies on the use of perfectly matched layers (PML) to reduce the problem to a bounded portion of the guide, allowing for a classical finite element discretization. The difficulty here comes from the existence of backward propagating modes, which are not correctly handled by the PML. We propose a simple strategy, based on finite-dimensional linear algebra arguments and using the knowledge of the modes, to recover a correct approximation to the solution with a low additional cost compared to the standard PML approach. Numerical experiments are presented in the two-dimensional case involving Rayleigh–Lamb modes.     

Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides

Perfectly matched layers (PMLs) are now a standard approach to simulate the absorption of waves in open domains. Wave propagation in elastic waveguides has the possibility to support back-propagating modes (propagating modes with oppositely directed group and phase velocities) with long wavelengths. Back-propagating modes can lead to temporally growing solutions in the PML. In this paper, we demonstrate that back-propagating modes in a two space dimensional isotropic elastic waveguide are not harmful to a discrete and finite width PML. Analysis and numerical experiments confirm the accuracy and stability of the PML.     

A perfectly matched layer for finite-element calculations of diffraction by metallic surface-relief gratings

We introduce a perfectly matched layer approach for finite element calculations of diffraction by metallic surface-relief gratings. We use a non-integrable absorbing function which allows us to use thin absorbing layers, which reduces the computational time when simulating this type of structure. In addition, we numerically determine the best choice of the absorbing layer parameters and show that they are independent of the wavelength.     

Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.     

Propagation and interaction of non-linear elastic plane waves in soft solids

Using the perturbation method of weakly non-linear asymptotics we analyze the propagation and interaction of elastic plane waves in a model of a soft solid proposed by Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44]. We derive the evolution equations for the wave amplitudes and find analytical formulas for all interaction coefficients of quadratically non-linear interacting waves. We show that in spite of the assumption of almost incompressibility used in Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44], the model behaves essentially like that of a compressible isotropic material. Both the structure of the equations and the interaction patterns are similar. The models differ, however, in the elastic constants that characterize them, and hence the values of the coefficients in the evolution equations and the values of the interaction coefficients differ.     

A knife-edge load traveling on the surface of an elastic halfspace

A load moving on the surface of an elastic halfspace forms a basic problem that is related to different fields of engineering, such as the subsoil response due to vehicle motion or the ultrasound field due to an angle beam transducer. Many numerical techniques have been developed to solve this problem, but these do not provide the fundamental physical insights that are offered by closed form solutions, which are very rare in comparison. This paper describes the development and analysis of the closed form space-time domain solution for a knife-edge load, i.e. a line segment of normal traction, moving at a constant speed on the surface of an elastic halfspace. The various contributions making up the exact solution, obtained with the Cagniard-De Hoop method, produce all the complicated wave patterns from this distributed type of loading. Examples are the transient wave field at the starting position of the load, angled conical and plane waves propagating into the solid, Rayleigh waves propagating along the surface, and head waves spreading and attenuating in specific directions from the loading path. The influence of the load speed on the wave field is investigated by considering the singularities in the relevant complex domains, for each sonic range relative to the bulk wave velocities. The characteristic wave fronts and wave patterns as exhibited by the particle displacements are evaluated for subsonic, transonic and supersonic load speeds.     

Complex frequency shifted convolution PML for FDTD modelling of elastic waves

The perfectly matched layer (PML) is nowadays considered as the best optimum absorbing boundary condition available. However, the PML with the classical stretching tensor has certain limitations. Strangely, these limitations have rarely been addressed in elastic wave modelling. For example, substantial reflections occur when strong evanescent waves are propagating parallel to the interface. To circumvent problems like this, the complex frequency shifted stretching tensor has been introduced in electromagnetic modelling. In this paper we show that the convolution PML with this stretching tensor as used in electromagnetic modelling can be adapted for elastic wave modelling. Numerical results of a model where the presence of evanescent waves is predominant show that the PML based on the complex frequency shifted stretching tensor can improve the performance of the absorbing boundary layer considerably.     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

Scattering attenuation of elastic waves due to low-contrast inclusions

Seismic scattering attenuation due to random lithospheric heterogeneity has been theoretically modeled using two approaches. One approach is the Born approximation theory (BAT), which is primarily used to treat weak continuous heterogeneity, and the other approach is the Foldy approximation theory (FAT), which deals with sparsely distributed discrete inclusions. We apply the BAT to elastic wave scattering due to inclusions having low contrast with the matrix, and compare the results with those predicted by the FAT. We thus investigate the valid wavenumber range of the BAT based on a reasonable assumption that the inclusions are distributed so sparsely that the FAT is effectively correct for any wavenumber. For simplicity, we consider a specific type of round inclusion, which is either two- or three-dimensional and has a two-valued wave velocity and/or mass density. Both theories are confirmed to yield essentially equivalent results below a certain wavenumber limit, depending on the contrast. This is known as the Rayleigh-Gans scattering regime. Beyond the wavenumber limit, the BAT overestimates the attenuation for common-mode scattering due to wave-velocity contrast, but remains valid with respect to the attenuation for scattering due to mass-density contrast and/or conversion scattering. These conclusions are independent of the spatial dimensions of the media as well as the modes of the elastic waves (P or S). Some advantages of the BAT over the FAT for application to low-contrast inclusions are discussed.     

Precise method to control elastic waves by conformal mapping

The transformation method to control waves has received widespread attention in electromagnetism and acoustics. However, this machinery is not directly applicable to the control of elastic waves, because it has been shown that the Navier's equation does not usually retain its form under coordinate transformation. In this letter, we prove the form invariance of the Navier's equation under the conformal mapping based on the Helmholtz decomposition method. The needed material parameters are provided to manipulate elastic waves. The validity of this approach is confirmed by an active stealth device which can disguise the signal source by changing its position. Experimental verifications and potential applications may be expected in nondestructive testing, structural seismic design and other fields.     

A novel method for investigation of acoustic and elastic wave phenomena using numerical experiments

The emergence of new types of composite materials, the depletion of existing hydrocarbon deposits, and the increase in the speed of trains require the development of new research methods based on wave scattering. Therefore, it is necessary to determine the laws of wave scattering in inhomogeneous media. We propose a method that combines the advantages of a numerical simulation with an analytical study of the boundary value problem of elastic and acoustic wave equations. In this letter we present the results of the study using the proposed method: the formation of a response from a shear wave in an acoustic medium and the formation of shear waves when a vertically incident longitudinal wave is scattered by a vertical gas-filled fracture. We have obtained a number of analytical expressions characterising the scattering of these wave types.     

Dynamics of structural interfaces: Filtering and focussing effects for elastic waves

Dynamics of thick interfaces separating different regions of elastic materials is investigated. The interfaces are made up of elastic layers or inertial truss structures. The study of evanescent mode propagation and transmission properties reveals that the discrete nature of structural interfaces introduces unusual filtering characteristics in the system, which cannot be obtained with multilayered interfaces. An example of metamaterial is presented, namely, a planar structural interface, which acts as a flat lens, therefore evidencing the negative refraction and focussing of elastic waves.     

弹性波散射与多重散射的T矩阵方法

研究弹性波散射与多重散射的T矩阵方法。首先,基于Helmholtz体内和体外公式推导了对应于圆柱型散射体的T矩阵元素的具体表达式;接着分析了在含多个随机分布圆柱型散射体的随机非均匀介质中弹性波的多重散射并给出在统计平均意义下的相干波的定义以及波速和衰减系数计算公式;最后,针对Ge/Al、Sic/Al复合材料用Matlab进行了编程和数值计算;计算单个柱型散射体的散射截面以及随机非均匀介质中相干波的速度和衰减系数,分析了这种介质的频散特性。     

Interface damage modeled by spring boundary conditions for in-plane elastic waves

In-plane elastic wave propagation in the presence of a damaged interface is investigated. The damage is modeled as a distribution of small cracks and this is transformed into a spring boundary condition. First the scattering by a single interface crack is determined explicitly in the low frequency limit for the case of a plane wave normally incident to the interface. The transmission at an interface with a random distribution of small cracks is then determined and is compared to periodically distributed cracks. The cracked interface is then described by a distributed spring boundary condition. As an illustration the dispersion relation of the first modes in a thick plate with a damaged interface in the middle is given.     

A method to study interactions between narrow-banded random waves and multi-chamber perforated structures

A time-domain method, based on linear velocity potential theory, is presented to study the interaction between narrow-banded random waves and perforated structures. A simple relation is derived to estimate the jet length of flows through the perforated wall. The reflection coefficient of narrow banded random waves from perforated structures is calculated by assuming a Rayleigh distribution of the heights of incident random waves. For reflection of narrow-banded waves from a single-chamber perforated breakwater, a comparison of the predicted and measured reflection coefficients shows that the method presented in this paper can provide a prediction better than that of regular waves. Numerical results are also reported on the reflection of narrow-banded waves from multi-chamber perforated breakwaters.The project partially supported by the Hong Kong Research Grant Council (DAG03/04.EG39, DAG04/05.EG32).     

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.     

A new version of smoothed point method to simulate linear elastic wave propagation

By using the kernel function of the smoothed particle hydrodynamics (SPH) and modification of statistical volumes of the boundary points and their kernel functions, a new version of smoothed point method is established for simulating elastic waves in solid. With the simplicity of SPH kept, the method is easy to handle stress boundary conditions, especially for the transmitting boundary condition. A result improving by de-convolution is also proposed to achieve high accuracy under a relatively large smooth length. A numerical example is given and compared favorably with the analytical solution.     

A semi-analytical elastic solution for stress field of lined non-circular tunnels at great depth using complex variable method

In this paper, a semi-analytical elastic plane strain solution was provided for stress field around a lined non-circular tunnel subjected to uniform ground load. Concrete lining and the surrounding rock mass were assumed as linearly elastic materials. Due to complexity of the problem for non-circular geometric configurations, complex variable method introduced by Muskhelishvili and conformal mapping functions were used to determine stress components within concrete lining and the surrounding rock mass. Finally, the solution was validated by ABAQUS finite element software through an example. Very good agreement was demonstrated between semi-analytical and numerical solution although some discrepancies were found at tunnel corners where large curvature existed. It was demonstrated that the solution predicted stress components more accurately around the tunnels, especially the corners with large stress concentration. Practical significance of the solution was placed in the fact that it could be used as a quick-solver with high accuracy.