Two-dimensional Elastic Wave Propagation Research Articles

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

Spectral-element simulation of two-dimensional elastic wave propagation in fully heterogeneous media on a GPU cluster

We present an implementation of the spectral-element method for simulation of two-dimensional elastic wave propagation in fully heterogeneous media. We have incorporated most of realistic geological features in the model, including surface topography, curved layer interfaces, and 2-D wave-speed heterogeneity. To accommodate such complexity, we use an unstructured quadrilateral meshing technique. Simulation was performed on a GPU cluster, which consists of 24 core processors Intel Xeon CPU and 4 NVIDIA Quadro graphics cards using CUDA and MPI implementation. We speed up the computation by a factor of about 5 compared to MPI only, and by a factor of about 40 compared to Serial implementation.

Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation

In this paper, we carry out a numerical dispersion analysis for the linear two-dimensional elastodynamics equations approximated by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method; specifically, we consider the analysis of harmonic plane waves in an isotropic and homogeneous elastic medium. We compare and discuss the errors associated with the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considering B-spline and NURBS basis functions of different regularity, namely globally C0- and Cp−1-continuous, p being the polynomial degree. We conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface.

Seismic Waveform Inversion Using the Finite-Difference Contrast Source Inversion Method

This paper extends the finite-difference contrast source inversion method to reconstruct the mass density for two-dimensional elastic wave inversion in the framework of the full-waveform inversion. The contrast source inversion method is a nonlinear iterative method that alternatively reconstructs contrast sources and contrast function. One of the most outstanding advantages of this inversion method is the highly computational efficiency, since it does not need to simulate a full forward problem for each inversion iteration. Another attractive feature of the inversion method is that it is of strong capability in dealing with nonlinear inverse problems in an inhomogeneous background medium, because a finite-difference operator is used to represent the differential operator governing the two-dimensional elastic wave propagation. Additionally, the techniques of a multiplicative regularization and a sequential multifrequency inversion are employed to enhance the quality of reconstructions for this inversion method. Numerical reconstruction results show that the inversion method has an excellent performance for reconstructing the objects embedded inside a homogeneous or an inhomogeneous background medium.

Two-dimensional elastic wave propagation analysis in finite length FG thick hollow cylinders with 2D nonlinear grading patterns using MLPG method

Two-dimensional elastic wave propagation analysis in finite length FG thick hollow cylinders with 2D nonlinear grading patterns using MLPG method

Time reverse modeling of low‐frequency microtremors: Application to hydrocarbon reservoir localization

Time reverse modeling is applied to synthetic and real low‐frequency microtremors measured at the Earth surface with synchronized seismometers. In contrast to previous time‐reverse applications, no single event or first arrival time identification is applied for microtremor localization. Synthetic low‐frequency microtremors are numerically generated within a small underground area and modeled with a finite‐difference algorithm simulating two‐dimensional elastic wave propagation. Time reverse modeling using synthetic microtremors shows that small underground source areas can be accurately located. Different source characteristics which emit mainly P‐waves or S‐waves vertically influence the localization accuracy. Time reverse modeling is applied to two real microtremor data sets acquired 16 months apart above known oil reservoirs nearby Voitsdorf, Austria, to investigate whether spectral anomalies observed above the reservoirs originate from the reservoirs. Time reverse modeling indicates that low‐frequency microtremor signals originate from the reservoir locations and provides a possible method for reservoir localization.

Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids

SUMMARY A Triangular Spectral Element Method (TSEM) is presented to simulate elastic wave propagation using an unstructured triangulation of the physical domain. TSEM makes use of a variational formulation of elastodynamics based on unstructured straight-sided triangles that allow enhanced flexibility when dealing with complex geometries and velocity structures. The displacement field is expanded into a high-order polynomial spectral approximation over each triangular subdomain. Continuity between the subdomains of the triangulation is enforced using a multidimensional Lagrangian interpolation built on a set of Fekete points of the triangle. High-order accuracy is achieved by resorting to an analytical computation of the associated internal product and bilinear forms leading to a non-diagonal mass matrix formulation. Therefore, the time stepping involves the solution of a sparse linear algebraic system even in the explicit case. In this paper the accuracy and the geometrical flexibility of the TSEM is explored. Comparison with classical spectral elements on quadrangular grids shows similar results in terms of accuracy and stability even for long simulations. Surface and interface waves are shown to be accurately modelled even in the case of complex topography with the TSEM. Numerical results are presented for 2-D canonical examples as well as more specific problems, such as 2-D elastic wave scattering by a cylinder embedded in an homogeneous half-space. They all illustrate the enhanced geometrical flexibility of the TSEM. This clearly suggests the need of further investigations in computational seismology specifically targeted towards efficient implementations of the TSEM both in the time and the frequency domain.

Estimation of Thermal Contact Resistance Using Ultrasonic Waves

In this paper, numerical simulations of both the three-dimensional heat conduction and two-dimensional elastic wave propagation at the interface of contact solids have been carried out. Numerical results of heat conduction simulations show that both the true contact area and thermal contact conductance increase linearly with an increase in the contact pressure. Numerical results of the ultrasonic wave propagation show that the intensity of a transmitted wave is very weak but depends clearly on the contact pressure. On the other hand, the intensity of reflected wave amounts to more than 99% of the standard reflected wave that results from the case of one cylindrical specimen without contact. However, the intensity of the modified reflected wave defined by the difference between the reflected wave and standard reflected wave shows the same tendency as that of the transmitted wave. The intensities of both transmitted and modified reflected waves could be expressed by the same power function of the contact pressure. By comparing the results of heat conduction with those of ultrasonic propagation calculations, a power functional correlation between the thermal contact conductance and transmitted or modified reflected intensity has been obtained. Using this correlation, it will be possible to estimate the thermal contact conductance between two solids through measuring the intensity of either reflected or transmitted ultrasonic waves.

A wavelet-based method for simulation of two-dimensional elastic wave propagation

Summary A wavelet-based method is introduced for the modelling of elastic wave propagation in 2-D media. The spatial derivative operators in the elastic wave equations are treated through wavelet transforms in a physical domain. The resulting second-order differential equations for time evolution are then solved via a system of first-order differential equations using a displacement-velocity formulation. With the combined aid of a semi-group representation and spatial differentiation using wavelets, a uniform numerical accuracy of spatial differentiation can be maintained across the domain. Absorbing boundary conditions are considered implicitly by including attenuation terms in the governing equations and the traction-free boundary condition at a free surface is implemented by introducing equivalent forces in the semi-group scheme. The method is illustrated by application to SH and P–SV waves for several models and some numerical results are compared with analytical solutions. The wavelet-based method achieves a good numerical simulation and shows an applicability for an elastic-wave study.

Finite-difference modelling of two-dimensional elastic wave propagation in cracked media

Summary We present a finite-difference modelling technique for 2-D elastic wave propagation in a medium containing a large number of small cracks. The cracks are characterized by an explicit boundary condition. The embedding medium can be heterogeneous. The boundaries of the cracks are not represented in the finite-difference mesh, but the cracks are incorporated as distributed point sources. This enables one to use grid cells that are considerably larger than the crack sizes. We compare our method with an accurate integral representation of the solution and conclude that the finite-difference technique is accurate and computationally fast.

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++ (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.

Closure to “Discussion of ‘Numerical Solution for Two-Dimensional Elastic Wave Propagation in Finite Bars’” (1968, ASME J. Appl. Mech., 35, p. 199)

Closure to “Discussion of ‘Numerical Solution for Two-Dimensional Elastic Wave Propagation in Finite Bars’” (1968, ASME J. Appl. Mech., 35, p. 199)

Numerical Solution for Two-Dimensional Elastic Wave Propagation in Finite Bars

Numerical solutions of the exact equations for axisymmetric wave propagation are obtained with continuous and discontinuous loadings at the impact end of an elastic bar. The solution for a step change in stress agrees with experimental data near the end of the bar and exhibits a region that agrees with the one-dimensional strain approximation. The solution for an applied harmonic displacement closely approaches the Pochhammer-Chree solution at distances removed from the point of application. Reflections from free and rigid-lubricated ends are studied. The solutions after reflection are compared with the elementary one-dimensional stress approximation.

Homogeneous solutions in elastic wave propagation

Busemann’s method of conical flows is formulated for two-dimensional elastic wave propagation. The equations of motion are reduced to either Laplace’s equation in two dimensions or the wave equation in one dimension, and solutions then are obtained with the aid of complex variable or characteristics theory, respectively. Special attention is paid to that class of problems in which the hyperbolic domains (of the two-dimensional wave equation) are simple wave zones, in consequence of which the solutions may be continued into the elliptic domain (of Laplace’s equation) without explicitly posing the boundary conditions on the boundary separating the two domains. The method is applied to the diffraction of P P - and S V SV -pulses by a perfectly weak half-plane.