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1.
A boundary integral formulation is presented and applied to model the ground motion on alluvial valleys under incident P, S and Rayleigh waves. It is based on integral representations for the diffracted and the refracted elastic waves using single-layer boundary sources. This approach is called indirect BEM in the literature as the sources' strengths should be obtained as an intermediate step. Boundary conditions lead to a system of integral equations for boundary sources. A discretization scheme based on the numerical and analytical integration of exact Green's functions for displacements and tractions is used. Various examples are given for two-dimensional problems of diffraction of elastic waves by soft elastic inclusion models of alluvial deposits in an elastic half-space. Results are displayed in both frequency and time domains. These results show the significant influence of locally generated surface waves in seismic response and suggest approximations of practical interest. For shallow alluvial valleys the response and its resonant frequencies are controlled by a coupling mechanism that involves both the simple one-dimensional shear beam model and the propagation of surface waves.  相似文献   

2.
We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.  相似文献   

3.
Elastic wave propagation in an irregularly layered medium   总被引：1，自引：0，他引：1  
The indirect boundary element method (IBEM) is used to simulate wave propagation in two-dimensional irregularly layered elastic media for internal line sources. The method is based on the integral representation for scattered elastic waves using single layer boundary sources. Fulfillment of the boundary conditions leads to a system of integral equations. Results are obtained in the frequency domain and seismograins are computed through Fourier synthesis. In order to test and validate the method we present various comparisons between our results and the time series obtained analytically for a buried line source in a half-space and by using the recently developed spectral element method (SEM).  相似文献   

4.
Scattering of SH waves by an embedded rigid elliptic cylinder of finite length, which is partially debonded from elastic soil is studied. The debonding regions are modeled as multiple elliptic arc-shaped interface cracks with non-contacting faces. The scattered wave field is expressed as a Mathieu function expansion with unknown coefficients. The mixed boundary conditions of the problem lead to a set of singular integral equations of the first type in terms of the dislocation density functions of the cracks. A quadrature method is used to solve these integral equations numerically. The results for dynamic stress intensity factors, far-field pattern of the displacement and scattering cross sections are presented. We particularly discuss the effects of the ratio of the short radius to long radius of the cylinder.  相似文献   

5.
In this work, a hybrid boundary integral equation method (BIEM) is developed, based on both displacement and hypersingular traction formulations, for the analysis of time-harmonic seismic waves propagating through cracked, multi-layered geological regions with surface topography and under plane strain conditions. Specifically, the displacement-based BIEM is used for a multi-layered deposit with interface cracks, while the regularized, traction-based BIEM is used when internal cracks are present within the layers. The standard uni-dimensional boundary element with parabolic shape functions is employed for discretizing the free surface and the layer interfaces, while special discontinuous boundary elements are placed near the crack tips to model the asymptotic behaviour of both displacements and tractions. This formulation yields displacement amplitudes and phase angles on the free surface of a geological deposit, as well as stress intensity factors near the tips of the cracks. Finally, in the companion paper, numerical results are presented which show that both scattered wave and stress concentration fields are sensitive to the incidence seismic wave parameters and to specific site conditions such as surface topography, layering, the presence of cracks and crack interaction.  相似文献   

6.
In this paper scattering of elastic waves in fluid–solid interfaces is investigated. We use the Indirect Boundary Element Method to study this wave propagation phenomenon in 2D models. Three models are analyzed: a first one with an interface between two half-spaces, one fluid on the top part and the other solid in the bottom; a second model including a fluid half-space above a layered solid; and finally, a third model with a fluid layer bounded by two solid half-spaces. The source, represented by Hankel's function of the second kind, is always applied in the fluid. This indirect formulation can give to the analyst a deep physical insight on the generated diffracted waves because it is closer to the physical reality and can be regarded as a realization of Huygens' principle. In any event, mathematically it is fully equivalent to the classical Somigliana's representation theorem. In order to gauge accuracy we test our method by comparing with an analytical solution known as Discrete Wave Number. A near interface pulse generates scattered waves that can be registered by receivers located in the fluid and it is possible to infer wave velocities of solids. Results are presented in both time and frequency domain, where several aspects related to the different wave types that emerge from this kind of problems are pointed out.  相似文献   

7.
Scattering of elastic waves by two dimensional multilayered dipping sediments of arbitrary shape embedded in an elastic half-sapce is investigated by using a bondary method. The displancement field is evaluated throughout the elastic media for both steady state and transient incident SH waves. The unknown scattered field is expressed in terms of wave functions which satisfy the equation of motion, traction-free boundary condition and appropariate radiation conditions. The transient response is constructed from the steady state solution by using the fast Fourier transform technique. The numerical results presented demonstrate that scattering of waves by subsurface irregularities may cause locally very large amplification of surface ground motion. The motion can be affected greatly by the scattered surface waves in the sediments. The results clearly indicate that the surface ground motion depends upon a number of parameters present in the problem, such as frequency and the angle of incidence of the incoming wave, impedance contrast between the layers and location of the observation point.  相似文献   

8.
In earthquake engineering and seismology it is of interest to know the surface motion at a given site due to the incoming and scattered seismic waves by surface geology. This can be formulated in terms of diffraction of elastic waves and then the indirect boundary element method (IBEM) for dynamic elasticity is used. It is based on the explicit construction of diffracted waves at the boundaries from which they radiate. This provides the analyst with insight on the physics of diffraction. The IBEM has been applied to study the amplification of elastic waves in irregular soil profiles. From the strong or weak satisfaction of boundary conditions and a simple analytical discretization scheme a linear system of equations for the boundary sources is obtained. Here, we explore the use of a weak discretization strategy with more collocation points than force densities. The least squares enforcement of boundary conditions leads to a system with reduced number of unknowns. This approach naturally allows one to use both coarser and finer boundary discretizations for smooth and rapidly varying profiles, respectively. A well studied semicircular canyon under incident P or SV in-plane waves is used to calibrate this method. Several benefits are obtained using mixed meshing that leads to the least squares condensation of the IBEM.  相似文献   

9.
One of the many important contributions that Aki has made to seismology pertains to the origin of coda waves (Aki, 1969; Aki and Chouet, 1975). In this paper, I revisit Aki's original idea of the role of scattered surface waves in the seismic coda. Based on the radiative transfer theory, I developed a new set of scattered wave energy equations by including scattered surface waves and body wave to surface wave scattering conversions. The work is an extended study of Zeng et al. (1991), Zeng (1993) and Sato (1994a) on multiple isotropic-scattering, and may shed new insight into the seismic coda wave interpretation. The scattering equations are solved numerically by first discretizing the model at regular grids and then solving the linear integral equations iteratively. The results show that scattered wave energy can be well approximated by body-wave to body wave scattering at earlier arrival times and short distances. At long distances from the source, scattered surface waves dominate scattered body waves at surface stations. Since surface waves are 2-D propagating waves, their scattered energies should in theory follow a common decay curve. The observed common decay trends on seismic coda of local earthquake recordings particular at long lapse times suggest that perhaps later seismic codas are dominated by scattered surface waves. When efficient body wave to surface wave conversion mechanisms are present in the shallow crustal layers, such as soft sediment layers, the scattered surface waves dominate the seismic coda at even early arrival times for shallow sources and at later arrival times for deeper events.  相似文献   

10.
Scattering of plane harmonic waves by a three‐dimensional basin of arbitrary shape embedded within elastic half‐space is investigated by using an indirect boundary integral equation approach. The materials of the basin and the half‐space are assumed to be the most general anisotropic, homogeneous, linearly elastic solids without any material symmetry (i.e. triclinic). The unknown scattered waves are expressed in terms of three‐dimensional triclinic time harmonic full‐space Green's functions. The results have been tested by comparing the surface response of semi spherical isotropic and transversely isotropic basins for which the numerical solutions are available. Surface displacements are presented for a semicircular basin subjected to a vertical incident plane harmonic pseudo‐P‐, SV‐, or SH‐wave. These results are compared with the motion obtained for the corresponding equivalent isotropic models. The results show that presence of the basin may cause significant amplification of ground motion when compared to the free‐field displacements. The peak amplitude of the predominant component of surface motion is smaller for the anisotropic basin than for the corresponding isotropic one. Anisotropic response may be asymmetric even for symmetric geometry and incidence. Anisotropic surface displacement generally includes all three components of motion which may not be the case for the isotropic results. Furthermore, anisotropic response strongly depends upon the nature of the incident wave, degree of material anisotropy and the azimuthal orientation of the observation station. These results clearly demonstrate the importance of anisotropy in amplification of surface ground motion. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

11.
This paper presents a simple, flexible way of introducing stress-free boundary conditions for including cracks and cavities in 2D elastic media by a finite difference method (FDM). The surfaces of cracks and cavities are discretized in a staircase on a rectangular grid scheme. When zero-stress is applied to free surfaces, the resulting finite difference schemes require a set of adjacent fictitious points. These points are classified based on the geometry of the free surface and their displacement is computed as a prior step to later calculation of motion on the crack surface. The use of this extra line of points does not involve a significant drain on computational resources. However, it does provide explicit finite difference schemes and the construction of displacement on the free surfaces by using the correct physical boundary conditions. An accuracy analysis compares the results to an analytical solution. This quantitative analysis uses envelope and phase misfits. It estimates the minimum number of points per wavelength necessary to achieve suitable results. Finally, the method is employed to compute displacement in various models with cavities in the P-SV formulation. The results show suitable construction of the reflected P and S waves from the free surface as well as diffraction produced by these cavities.  相似文献   

12.
Numerical modelling techniques are now becoming common for understanding the complicated nature of seismic wave propagation in fractured rock. Here the Indirect Boundary Element Method (IBEM) is applied to study scattering of elastic waves by cracks. The problem addressed in this paper is the diffraction of P and S waves by open 3-D cracks of arbitrary shape embedded in a homogeneous isotropic medium. The IBEM yields the value of the jump of displacements between opposite surfaces of the crack, often called Crack Opening Displacement (COD). This is used to evaluate the solution away from the crack. We use a multi-regional approach which consists of splitting a surface S into two identical surfaces S+ and S− chosen such that the crack lies at the interface. The resulting integral equations are not hyper-singular and wave propagation within media that contain open cracks can be rigorously solved. In order to validate the method, we compare results of displacements of a penny-shaped crack for a vertical incident P-wave with the classic results by Mal (1970) obtaining excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P or S waves upon cracks with various shapes are depicted and the salient aspects of the method are also discussed. Both frequency and time-domain results are included.  相似文献   

13.
An approximate method is proposed for the scattering of SH-waves by foundations of irregular shape and the resulting soil-structure interaction problems. The scattering of elastic waves by the rigid foundation embedded in half-space is solved approximately by using integral representation of the wave equation. The procedure is the Born approximation which has been widely used in quantum mechanics for collision and scattering theory though not well-known in elastodynamics. This paper extends the previous work of the authors on the scattering of waves to account for soil-structure interaction. The motion of the foundation is evaluated by the balance of momentum under stresses due to the incident waves as well as the waves generated by its own motion and the forces coming from the superstructure. The model investigated consists of an infinitely long elastic shear wall of height H and thickness h erected on a rigid infinitely long foundation. Results are presented for the cases with circular, elliptical and rectangular foundations. For a circular foundation, excellent agreement is found with the exact solutions for the foundation displacement and the relative displacement between the top and bottom of the structure for the entire range of wave numbers. For an elliptical foundation, accuracy decreases with increasing wave numbers. Foundation displacements are compared for foundation shapes that are shallow elliptical, deep elliptical, rectangular and circular. It is observed that foundation displacements are dependent on the angle of incidence except for a semi-circle. The results on the details of the scattered field are, however, not as accurate.  相似文献   

14.
A hybrid indirect boundary element – discrete wavenumber method is presented and applied to model the ground motion on stratified alluvial valleys under incident plane SH waves from an elastic half-space. The method is based on the single-layer integral representation for diffracted waves. Refracted waves in the horizontally stratified region can be expressed as a linear superposition of solutions for a set of discrete wavenumbers. These solutions are obtained in terms of the Thomson–Haskell propagators formalism. Boundary conditions of continuity of displacements and tractions along the common boundary between the half-space and the stratified region lead to a system of equations for the sources strengths and the coefficients of the plane wave expansion. Although the regions share the boundary, the discretization schemes are different for both sides: for the exterior region, it is based on the numerical and analytical integration of exact Green's functions for displacements and tractions whereas for the layered part, a collocation approach is used. In order to validate this approach results are compared for well-known cases studied in the literature. A homogeneous trapezoidal valley and a parabolic stratified valley were studied and excellent agreement with previous computations was found. An example is given for a stratified inclusion model of an alluvial deposit with an irregular interface with the half-space. Results are displayed in both frequency and time domains. These results show the significant influence of lateral heterogeneity and the emergence of locally generated surface waves in the seismic response of alluvial valleys.  相似文献   

15.
Statistical properties of small-scale inhomogeneities (wavelengths between 20 and 70 km) near the core-mantle boundary are inferred from scattered core waves. Observations of scattered core waves at large seismic arrays and worldwide networks indicate that the inhomogeneities have a global nature with similar characteristics. However, there may exist a few regions having markedly stronger or weaker strengths. Scattering by volumetric inhomogeneities of about 1% inP-wave velocity in the lower mantle or by about 300 m of topographic relief of the core-mantle boundary can explain the observations. At present it is not possible to rule out either of these two alternatives, or a combination of both.  相似文献   

16.
Scattering of Seismic Waves by Cracks with the Boundary Integral Method   总被引：3，自引：0，他引：3  
— We develop a new scheme to compute 2-D SH seismograms for media with many flat cracks, based on the boundary integral method. A dry or traction-free boundary condition is applied to crack surfaces although other kinds of cracks such as wet or fluid-saturated cracks can be treated simply by assigning different boundary conditions. While body forces are distributed for cavities or inclusions to express scattered wave, dislocations (or displacement discontinuities between the top and the bottom surfaces of each crack) are used as fictitious sources along crack surfaces. With these dislocations as unknown coefficients, the scattered wave is expressed by the normal derivative of Green's function along the crack surface, which is called “double-layer potentials” in the boundary integral method, while we used “single-layer potentials” for cavities or inclusions. These unknowns are determined so that boundary conditions or crack surfaces are satisfied in the least-squared sense, for example, traction-free for dry cracks. Seismograms with plane-wave incidence are synthesized for homogeneous media with many cracks. First, we check the accuracy of our scheme for a medium with one long crack. All the predicted phases such as reflected wave, diffraction from a crack tip and shadow behind the crack are simulated quite accurately, under the same criterion as in the case for cavities or inclusions. Next, we compute seismograms for 50 randomly distributed cracks and compare them with those for circular cavities. When cracks are randomly oriented, waveforms and the strength of scattering attenuation are similar to the cavity case in a frequency range higher than k d $\simeq$ 2 where the size of scatterers d (i.e., crack length or cavity diameter) is comparable with the wavelength considered (k is the wavenumber). On the other hand, the scattering attenuation for cracks becomes much smaller in a lower frequency range (k d<2) because only the volume but not detail geometry of scatterers becomes important with wavelength much longer than each scatterer. When all the cracks are oriented in a fixed direction, the scattering attenuation depends strongly on the incident angle to the crack surface as frequency increases (k d>2): scattering becomes weak for cracks oriented parallel to the direction of the incident wave, while it gets close to the cavity case for cracks aligned perpendicular to the incident wave.  相似文献   

17.
Reflection and transmission of elastic wave motion by a layer of compact inhomogeneities has been analyzed. For identical inhomogeneities whose geometrical centers are periodically spaced, the problem has been formulated and solved rigorously. The reflected and transmitted longitudinal and transverse wave motions have been expressed as superpositions of wavemodes, where each wavemode has its own cut-off frequency. At its cut-off frequency a mode converts from a standing into a propagating wavemode. The standing wavemodes decay exponentially with distance to the plane of the centers of the inhomogeneities. At small frequencies only the lowest order modes of longitudinal and transverse wave motion are propagating. Reflection and transmission coefficients have been defined in terms of the coefficients of the zeroth-order scattered wavemodes. These coefficients have been computed by a novel application of the Betti-Rayleigh reciprocal theorem. They are expressed as integrals over the surface of a single inhomogeneity, in terms of the displacements and tractions on the surface of the inhomogeneity. The system of singular integral equations for the surface fields has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted longitudinal and transverse waves as functions of the frequency. Some results are also presented for planar distributions of cracks whose spacing and size are random variables. Finally, dispersion relations are discussed for solids which are completely filled with periodically spaced inhomogeneities.  相似文献   

18.
The standard free-surface boundary conditions for in-plane crack dynamics are shown to be identical to the conditions for crack dynamics on a liquefied crack. The surfaces of both the free and liquefied cracks do not separate during faulting and hence the static normal stress is not relaxed by the faulting. A crack with either free or liquid boundary conditions deforms in the transverse direction during slip. It follows that both the free and liquefied cracks may represent solutions to the heat-flow paradox. As an application of the proof, we derive a physical understanding of the properties of harmonic Rayleigh waves on a uniform elastic half-space without solving a cubic equation.  相似文献   

19.
Scattering of elastic waves by a three‐dimensional transversely isotropic basin of arbitrary shape embedded in a half‐space is considered using an indirect boundary integral equation approach. The unknown scattered waves are expressed in terms of point sources distributed on the so‐called auxiliary surfaces. The sources are expressed in terms of the full‐space Green's functions with their intensities determined from the requirement that the boundary and the continuity conditions are to be satisfied in the least‐squares sense. Steady‐state results were obtained for incident plane pseudo‐P‐, SH‐, SV‐, and Rayleigh waves. Using the Radon transform the Green's functions are obtained in the form of finite integrals over a unit sphere or a unit circle which can be numerically evaluated very efficiently. Detailed analysis of the method includes the discussion on the shape of the auxiliary surfaces and the distribution of the collocation points and sources. The convergence criteria is defined in terms of transparency tests, isotropic limit test, and minimization of a certain norm. The isotropic limit tests show excellent agreement with the isotropic results available in literature. For anisotropic materials the numerical results are given for a semispherical basin. The results show that presence of an anisotropic basin may result in significant amplification of surface motion atop the basin. While the amplitude of peak surface motion may be similar to the corresponding isotropic results, the difference in the displacement patterns may be quite different between the two. Therefore, this study clearly demonstrates that material anisotropy may be very important for accurate assessment of surface ground motion amplification atop basins. Copyright © 2000 John Wiley & Sons, Ltd.  相似文献   

20.
This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.  相似文献   

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