Elastic Wave Modeling Research Articles

Improved detection of rough defects for ultrasonic nondestructive evaluation inspections based on finite element modeling of elastic wave scattering.

Defects which possess rough surfaces greatly affect ultrasonic wave scattering behavior, usually reducing the magnitude of reflected signals. Understanding and accurately predicting the influence of roughness on signal amplitudes is crucial, especially in nondestructive evaluation (NDE) for the inspection of safety-critical components. An extension of Kirchhoff theory has formed the basis for many practical applications; however, it is widely recognized that these predictions are pessimistic because of analytical approximations. A numerical full-field modeling approach does not fall victim to such limitations. Here, a finite element (FE) modeling approach is used to develop a realistic methodology for the prediction of expected backscattering from rough defects. The ultrasonic backscatter from multiple rough surfaces defined by the same statistical class is calculated for normal and oblique incidence. Results from FE models are compared with Kirchhoff theory predictions and experimental measurements to establish confidence in the new approach. At lower levels of roughness, excellent agreement is observed between Kirchhoff theory, FE, and experimental data, whereas at higher values, the pessimism of Kirchhoff theory is confirmed. An important distinction is made between the total, coherent, and diffuse signals and it is observed, significantly, that the total signal amplitude is representative of the information obtained during an inspection. This analysis provides a robust basis for a less sensitive, yet safe, threshold for inspection of rough defects.

Optimal rotated staggered-grid finite-difference schemes for elastic wave modeling in TTI media

The rotated staggered-grid finite-difference (RSFD) is an effective approach for numerical modeling to study the wavefield characteristics in tilted transversely isotropic (TTI) media. But it surfaces from serious numerical dispersion, which directly affects the modeling accuracy. In this paper, we propose two different optimal RSFD schemes based on the sampling approximation (SA) method and the least-squares (LS) method respectively to overcome this problem. We first briefly introduce the RSFD theory, based on which we respectively derive the SA-based RSFD scheme and the LS-based RSFD scheme. Then different forms of analysis are used to compare the SA-based RSFD scheme and the LS-based RSFD scheme with the conventional RSFD scheme, which is based on the Taylor-series expansion (TE) method. The contrast in numerical accuracy analysis verifies the greater accuracy of the two proposed optimal schemes, and indicates that these schemes can effectively widen the wavenumber range with great accuracy compared with the TE-based RSFD scheme. Further comparisons between these two optimal schemes show that at small wavenumbers, the SA-based RSFD scheme performs better, while at large wavenumbers, the LS-based RSFD scheme leads to a smaller error. Finally, the modeling results demonstrate that for the same operator length, the SA-based RSFD scheme and the LS-based RSFD scheme can achieve greater accuracy than the TE-based RSFD scheme, while for the same accuracy, the optimal schemes can adopt shorter difference operators to save computing time.

A staggered-grid convolutional differentiator for elastic wave modelling

The computation of derivatives in governing partial differential equations is one of the most investigated subjects in the numerical simulation of physical wave propagation. An analytical staggered-grid convolutional differentiator (CD) for first-order velocity-stress elastic wave equations is derived in this paper by inverse Fourier transformation of the band-limited spectrum of a first derivative operator. A taper window function is used to truncate the infinite staggered-grid CD stencil. The truncated CD operator is almost as accurate as the analytical solution, and as efficient as the finite-difference (FD) method. The selection of window functions will influence the accuracy of the CD operator in wave simulation. We search for the optimal Gaussian windows for different order CDs by minimizing the spectral error of the derivative and comparing the windows with the normal Hanning window function for tapering the CD operators. It is found that the optimal Gaussian window appears to be similar to the Hanning window function for tapering the same CD operator. We investigate the accuracy of the windowed CD operator and the staggered-grid FD method with different orders. Compared to the conventional staggered-grid FD method, a short staggered-grid CD operator achieves an accuracy equivalent to that of a long FD operator, with lower computational costs. For example, an 8th order staggered-grid CD operator can achieve the same accuracy of a 16th order staggered-grid FD algorithm but with half of the computational resources and time required. Numerical examples from a homogeneous model and a crustal waveguide model are used to illustrate the superiority of the CD operators over the conventional staggered-grid FD operators for the simulation of wave propagations.

Numerical Approximation of an Elastic Wave Model

Wave operators play an important role modeling various physical, engineering and biological problems. We consider an initial boundary value problem modeling an elastic wave propagation. We approximate the model using a Legendre spectral Galerkin method for the spatial approximation. We analyze the accuracy of such a spatial approximation. We perform some numerical experiments to demonstrate the scheme.

Identification of co-seismic ground motion due to fracturing and impact of the Tsaoling landslide, Taiwan

Earthquakes can generate seismic disturbances that propagate vast distances and trigger landslides that can achieve high-speeds. It remains difficult to identify the co-seismic ground motion of these landslides and their triggering earthquakes. In this paper, we report on the analysis of co-seismic ground motions generated by the initiating fracture and deposition impact of the Tsaoling landslide. The landslide, with a source volume of 125×106m3, was triggered by the 1999 Chi-Chi earthquake in Taiwan. The ground motion was recorded by a strong ground motion station, CHY080, near the scar area. The polarization of the seismic waves indicates that the peak acceleration was parallel to the dip direction. Modified ensemble empirical mode decomposition (EEMD), with additional clustering analysis, was applied to decompose the seismic signals. Two instances were found in the seismic records of a series of peculiar wave packets, with the first being associated with the landslide initiation and the second the landslide impact on the deposit valley. To confirm the first landslide breakage, the decomposed signals were compared with the predictions of the analytic elastic wave model and Newmark analysis. The landslide impact was verified with a computational fluid dynamic simulation. Comparison between the EEMD decomposed signals, elastic wave theory, Newmark rigid body analysis, and numerical simulation demonstrates the claimed landslide motion.

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

2D and 3D frequency-domain elastic wave modeling in complex media with a parallel iterative solver

Full-waveform inversion and reverse time migration rely on an efficient forward-modeling approach. Current 3D large-scale frequency-domain implementations of these techniques mostly extract the desired frequency component from the time-domain wavefields through discrete Fourier transform. However, instead of conducting the time-marching steps for each seismic source, in which the time step is limited by the stability condition, performing the wave modeling directly in the frequency domain using an iterative linear solver may reduce the entire computational complexity. For 2D and 3D frequency-domain elastic wave modeling, a parallel iterative solver based on a conjugate gradient acceleration of the symmetric Kaczmarz row-projection method, named the conjugate-gradient-accelerated component-averaged row projections (CARP-CG) method, shows interesting convergence properties. The parallelization is realized through row-block division and component averaging operations. Convergence is achieved systematically even when different physical factors such as the space-dependent Poisson’s ratio, free-surface condition, and seismic attenuation are incorporated in the wave modeling. We determined that the scalability of CARP-CG was satisfactory, especially for large-scale applications, using up to several hundred computational cores. We found a potential improvement in computational complexity compared to time-domain modeling through numerical experiments. Finally, we achieved a convergence at 5 Hz in a 3D heterogeneous model, involving fast-slow-fast layers resembling waveguide geometries, with up to several hundred million unknowns, in fewer than 10 h on fewer than 200 cores. All of these results make CARP-CG a potential candidate of the forward modeling engine for seismic imaging on challenging models.

A 2-D enlarged cell technique (ECT) for elastic wave modelling on a curved free surface

The conventional finite-difference time-domain (FDTD) method for elastic waves suffers from the staircasing error when applied to model a curved free surface because of the structured grid. This is similar to the situation for the FDTD method in electromagnetics when it is applied to model a curved perfect conductor surface, where the conformal FDTD methods have been recently developed to avoid this error. In this work a stable and second-order accurate 2-D FDTD method for elastic wave modelling on a curved free surface is presented based on the finite volume method and enlarged cell technique (ECT). To achieve a sufficiently accurate implementation, a finite volume scheme is applied to the curved free surface to remove the staircasing error; in the meantime, to achieve the same stability as the FDTD method without reducing the time step increment, the ECT is introduced to preserve the solution stability even for small irregular cells. This method is verified by several 2-D numerical examples. Results show that the method is second-order accurate and stable at the Courant stability limit for a regular FDTD grid.

The multi-domain FMM-IBEM to model elastic wave scattering by three-dimensional inclusions in infinite domain

The scattering of elastic wave by three-dimensional (3-D) inclusions embedded in solid infinite domain is solved by a Fast Multipole accelerated multi-domain Indirect Boundary Element Method (FMM-IBEM). Based on the single-layer potential theory, the scattered field is constructed by virtual uniform loads acted on circular elements covering the boundary surface, and then the analytic integration of the exact Green’s function for infinite space can be applied. By using the Taylor series expansion of compressional and shear waves potential function, the multipole expansion and local expansion coefficients of Green’s function are deduced. The detailed procedure of multi-domain FMM-IBEM for such problem is presented. It is verified that this method substantially improves the computing efficiency and reduces the memory storage, then the elastic wave scattering problems involving millions of degrees of freedom (DOFs) can be solved accurately and efficiently on an ordinary workstations. Finally, the scattering problems of P, SV waves by spherical inclusions group and 3-D random inclusions in elastic infinite domain are solved, and some important characteristics of plane P and SV waves scattering by inclusions group are discussed through typical numerical results.

Steady and unsteady shear flows of a viscoplastic fluid in a cylindrical Couette cell

Yield stress fluid flows in Couette cells have been widely studied in the last decades for their intriguingly exhibiting phenomena. In this paper, we use a PIV technique to investigate the axisymmetric flow and rheological properties of a Carbopol gel in a relatively wide cylindrical Couette device. Carbopol gel is known to exhibit viscoplastic behavior and is often described using a Herschel–Bulkley law, which is characterized by a plastic yield stress τy and a shear-dependent nonlinear viscosity. In some cases, the elasticity of the material has to be accounted for to understand the whole dynamics of the system, in particular for unsteady flows as observed in the present study. Two set of experiments are conducted here in order to highlight these different rheological behaviors and the resulting dynamics: (i) a steady shear configuration and (ii) an unsteady shear configuration, in which the angular velocity of the inner cylinder is either constant or time dependent (sin profile), respectively. In the steady configuration, a simple optimization model, based on the Herschel–Bulkley law, is developed to extract the rheological parameters of the viscoplastic contribution of the gel from the steady velocity fields. Results are shown to be in good agreements with rheological parameters obtained from a standard rheometer. On the other hand, the elastic contribution of the material is highlighted in the unsteady shear configuration, for which a spatio-temporal transition between solid-elastic and fluid behaviors is observed. Different models are proposed to describe the dynamics of the unsteady flow. First, quasi-steady state models allow to predict both the fluid shear zone close to the inner cylinder and the elastic deformation of the material as long as their contributions can be decoupled in space and in time. For more complex dynamics, i.e. when the flow becomes strongly unsteady, an elasto-viscoplastic model is developed to describe the flow dynamics. It is shown to quantitatively reproduce the experimental measurements. Finally, an elastic wave model is derived to describe an elastic front propagating from the inner cylinder to the outer one, and observed at every half forcing period. The front velocity is thus shown to scale on the phase velocity of an elastic wave in a deformable solid.

Least squares staggered-grid finite-difference for elastic wave modelling

Staggered-grid finite-difference (SFD) methods have been used widely in seismic wave numerical modelling and migration. The conventional way to calculate the high-order SFD coefficients on spatial derivatives is the Taylor-series expansion method, which generally leads to great accuracy on just a small frequency zone. In this paper, we first derive the SFD coefficients of arbitrary even-order accuracy for the first-order spatial derivatives by the dispersion relation and the least-squares method, which can satisfy the specific numerical solution accuracy of the derivative on a wide frequency zone. Then we use the SFD coefficients based on the least-squares to solve the first-order spatial derivatives and analyse the accuracy of the numerical solution. Finally, we perform elastic wave numerical modelling with the least-squares staggered-grid finite-difference (LSSFD) method. Meanwhile, the numerical dispersion, the modelling accuracy and the computing costs of the new method are compared with that of the Taylor-series expansion staggered-grid finite-difference (TESFD) method. The numerical dispersion analysis and elastic wavefield modelling results demonstrate that the LSSFD method can efficiently suppress the numerical dispersion and has greater modelling accuracy than the conventional TESFD method under the same discretisations and without extra computing costs.

Modeling of elastic waves in a blocky medium based on equations of the Cosserat continuum

Based on the equations of the dynamics of piecewise-homogeneous elastic material, parallel computational algorithms are developed to simulate the process of stress and strain wave propagation in a medium consisting of a large number of blocks interacting through compliant interlayers. Computations of waves caused by localized impulse perturbations show that such a medium can be considered as isotropic only in the case of sufficiently thin interlayers. In the case of relatively thick interlayers, the anisotropy effects are observed which consist in the appearance of elongated wave fronts along the coordinate directions and characteristic oscillations of velocities and stresses because of the rotational motion of blocks. For the description of waves in a blocky medium with thin interlayers, the equations of the isotropic Cosserat continuum are applicable. As the thickness of interlayers increases, the orthotropic Cosserat continuum theory which takes into account the symmetry of elastic properties relative to the coordinate planes can be applied. By comparing the elastic wave velocities in the framework of piecewise-homogeneous model and continuum model, a simple method is obtained to estimate the mechanical parameters of the orthotropic Cosserat continuum modeling a blocky medium. In two-dimensional formulation of the orthotropic model the computational algorithm and the program system are worked out for the analysis of propagation of elastic waves. The comparison showed good qualitative agreement between the results of computations of waves caused by localized impulses, in the framework of the model of a blocky medium with compliant interlayers and the model of orthotropic Cosserat continuum.

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities—the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position × wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal (2005). The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.

Investigation of model based beamforming and Bayesian inversion signal processing methods for seismic localization of underground sources.

Techniques have been studied for the localization of an underground source with seismic interrogation signals. Much of the work has involved defining either a P-wave acoustic model or a dispersive surface wave model to the received signal and applying the time-delay processing technique and frequency-wavenumber processing to determine the location of the underground tunnel. Considering the case of determining the location of an underground tunnel, this paper proposed two physical models, the acoustic approximation ray tracing model and the finite difference time domain three-dimensional (3D) elastic wave model to represent the received seismic signal. Two localization algorithms, beamforming and Bayesian inversion, are developed for each physical model. The beam-forming algorithms implemented are the modified time-and-delay beamformer and the F-K beamformer. Inversion is posed as an optimization problem to estimate the unknown position variable using the described physical forward models. The proposed four methodologies are demonstrated and compared using seismic signals recorded by geophones set up on ground surface generated by a surface seismic excitation. The examples show that for field data, inversion for localization is most advantageous when the forward model completely describe all the elastic wave components as is the case of the FDTD 3D elastic model.

Complex frequency-shifted multi-axial perfectly matched layer for elastic wave modelling on curvilinear grids

SUMMARY Perfectly matched layer (PML) is an efficient absorbing technique for numerical wave simulations.Sinceitappeared,variousimprovementshavebeenmade.Thecomplexfrequency-shifted PML (CFS-PML) improves the absorbing performance for near-grazing incident waves and evanescent waves. The auxiliary differential equation (ADE) formulation of the PML provides a convenient unsplit-field PML implementation that can be directly used with high order time marching schemes. The multi-axial PML (MPML) stabilizes the PML on anisotropic media. However, these improvements were generally developed for Cartesian grids. In this paper, we extend the ADE CFS-PML to general curvilinear (non-orthogonal) grids for elastic wave modelling. Unlike the common implementations to absorb the waves in the computational space, we apply the damping along the perpendicular direction of the PML layer in the local Cartesian coordinates. Further, we relate the perpendicular and parallel components of the gradient operator in the local Cartesian coordinates to the derivatives in the curvilinear coordinates, to avoid mapping the wavefield to the local Cartesian coordinates. It is thus easy to be incorporated with numerical schemes on curvilinear grids. We derive the PML equations for the interior region and for the free surface separately because the free surface boundary condition modifies the elastic wave equations. We show that the elastic wave modelling on curvilinear grids exhibits anisotropic effects in the computational space, which may lead to unstable simulations. To stabilize the simulation, we adapt the MPML strategy to also absorb the wavefield along the two parallel directions of the PML. We illustrate the stability of this ADE CFS-MPML for finite-difference elastic wave simulations on curvilinear grids by two numerical experiments.

Moulding and shielding flexural waves in elastic plates

Platonic crystals (PlCs) are the elastic plate analogue of the photonic crystals widely used in optics, and are thin structured elastic plates along which flexural waves cannot propagate within certain stop band frequency intervals. The practical importance of PlCs is twofold: These can be used either in the design of microstructured acoustic metamaterials or as an approximate model for surface elastic waves propagating in meter scale seismic metamaterials. Here, we make use of the band spectrum of PlCs created by an array of either very small or densely packed clamped circles to achieve surface wave reflectors at very large wavelengths, a flat lens, a waveguide effect, a directive antenna near the stop band frequencies. The limit in which the circles reduce to points is particularly appealing as there is an exact dispersion relation available so the origin of these phenomena can be explained and interpreted using Fourier series and high-frequency homogenization (HFH). We then enlarge the radius of clamped circles, which both makes the zero-frequency stop band up to five times wider and flattens the dispersion curves. Here, HFH notably captures the essence of localized modes, one of which appears in the zero-frequency stop band and is used in the design of a highly directive waveguide.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However, the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap, combined with variable grid-size and time-step, this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

Modeling of elastic waves in a block medium based on equations of the Cosserat continuum

Modeling of elastic waves in a block medium based on equations of the Cosserat continuum

Study of thermal stress behavior of sheet glass during laser irradiation using one-dimensional elastic wave model

The thermoelastic behavior of sheet glass during laser irradiation was clarified using a one-dimensional model. A thermoelastic equation based on an equation of motion with the damping force proportional to the deformation velocity was applied. The thermal and thermoelastic equations were numerically solved by the finite difference method based on an implicit method for the time evolution. A proportional damping coefficient (PDC) was adopted for the calculation. In the case of a stationary heat source, compressive stress on the heated region changed into tensile stress 0.1 s after the end of heating. In the case of a moving heat source, the highest tensile stress was induced when the PDC was less than 0.1. An optimal combination of velocity and the PDC that induced the maximum tensile stress was obtained.

Numerical Study of Elastic Wave Propagation Characteristics in Cracked Rock

Numerical methods can provide extremely powerful tools for analysis and design of engineering systems with complex factors that are not possible or very difficult with the use of the conventional methods. In this paper, we use the 2-D boundary element method (BEM) program to model elastic wave excited by a point explosive source propagating in cracked rocks. As an example, we consider the typical crack distributions in rocks, both models for the real crack structure are also talked about. The elastic wave propagating in rocks with aligned cracks and parallel fractures is assumed. Effects of different crack parameters, such as crack scale length and crack density are analyzed. Numerical results show that the BEM is a powerful interpretive tool for understanding the complicated wave propagation and interaction in cracked solids.