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1.
The manner in which boundary conditions are approximated and introduced into finite difference schemes may have an important influence on the stability and accuracy of the results. The standard von Neumann condition for stability applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. Working on elastic media with free stress boundary conditions we found that the boundary approximation gives rise to serious stability problems especially for regions with high Poisson's ratio. In order to detect these effects apriori and to analyse them, we have used a more elaborate procedure for checking the stability of the scheme which takes into consideration the boundary conditions. It is based on studying a locally spaced time propagating matrix which governs the time-space behavior of a small region of the grid which includes free surface points. By using this procedure a better insight into the nature of instability caused by the approximations to the boundary conditions was gained which led us to a new stable approximation for the free surface boundary conditions. 相似文献
2.
— 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
3.
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
4.
5.
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
6.
Based on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non‐welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite‐difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non‐welded contact boundary conditions. In this paper, finite‐difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite‐difference grid. The new finite‐difference schemes explicitly have an additional part that is different from the conventional second‐order finite‐difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite‐difference scheme is accurate and stable and agrees well with the results of previously published finite‐difference schemes (the Coates and Schoenberg method). The results of the new finite‐difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium. 相似文献
7.
The staggered grid finite-difference method is a powerful tool in seismology and is commonly used to study earthquake source
dynamics. In the staggered grid finite-difference method stress and particle velocity components are calculated at different
grid points, and a faulting problem is a mixed boundary problem, therefore different implementations of fault boundary conditions
have been proposed. Viriuex and Madariaga (1982) chose the shear stress grid as the fault surface, however, this method has
several problems: (1) Fault slip leakage outside the fault, and (2) the stress bump beyond the crack tip caused by S waves
is not well resolved. Madariaga et al. (1998) solved the latter problem via thick fault implementation, but the former problem remains and causes a new issue;
displacement discontinuity across the slip is not well modeled because of the artificial thickness of the fault. In the present
study we improve the implementation of the fault boundary conditions in the staggered grid finite-difference method by using
a fictitious surface to satisfy the fault boundary conditions. In our implementation, velocity (or displacement) grids are
set on the fault plane, stress grids are shifted half grid spacing from the fault and stress on the fictitious surface in
the rupture zone is given such that the interpolated stress on the fault is equal to the frictional stress. Within the area
which does not rupture, stress on the fictitious surface is given a condition of no discontinuity of the velocity (or displacement).
Fault normal displacement (or velocity) is given such that the normal stress on the fault is continuous across the fault.
Artificial viscous damping is introduced on the fault to avoid vibration caused by onset of the slip. Our implementation has
five advantages over previous versions: (1) No leakage of the slip prior to rupture and (2) a zero thickness fault, (3) stress
on the fault is reliably calculated, (4) our implementation is suitable for the study of fault constitutive laws, as slip
is defined as the difference between displacement on the plane of z = + 0 and that of z = − 0, and (5) cessation of slip is achieved correctly. 相似文献
8.
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
9.
采用规则网格有限差分方法对二维平面弹性波动方程进行差分离散,得到相应的弹性波动方程的有限差分方程,再将弹性波动方程的差分格式与吸收边界、自由边界的离散形式结合形成弹性波动方程有限差分方程解决问题的主体,将其应用于含方形凹陷半无限非均匀介质的模型中进行数值模拟,得到此离散化模型中不同时刻不同节点的位移值。针对具体算例,运用上述方法结合科学计算软件MATLAB和结果后处理软件DIFEM ISOLINE PLOTER得到不同时刻的水平方向位移等值线图与接收器测量点处的合成位移记录,讨论非均匀介质、吸收边界、方形凹陷等对波动特性的影响。 相似文献
10.
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
11.
矩形网格有限差分法在地震波传播数值模拟方面具有计算速度快的显著优势,但该方法在处理复杂边界问题上存在着效率低的严重缺陷.本文针对分段光滑曲线边界定义了尖点处的一种正则导数,给出了矩形网格情形分段光滑曲线网格边界点法向导数的一种插值计算方法.采用矩形网格有限差分法对复杂边界地球介质模型进行地震波场数值模拟,并采用波场系列快照技术揭示地震波在起伏地表和复杂介质中的传播规律.模拟结果表明：法向导数插值计算方法为矩形网格有限差分法处理复杂边界提供了有效途径,采用波场系列快照技术可以清晰地展现地震波在反射界面的反射和透射规律、在尖点的绕射规律以及在自由表面的直达波和多次反射规律. 相似文献
12.
I.D. James 《Continental Shelf Research》1988,8(12)
A three-dimensional numerical model is used to simulate the development of disturbances on shelf-sea coastal currents and fronts. The model, which has a free surface, uses a finite difference grid ☐ scheme based on sigma coordinates. It has a semi-implicit scheme for the barotropic flow and a hydrid advection scheme to retain sharp fronts. The results demonstrate that (i) eddy formation follows changes at the inflow of a coastal current, (ii) a simple radiation boundary condition at the outflow produces nearly identical results for different outflow boundary positions, (iii) eddy growth, with matching behaviour of surface and bottom fronts, follows a small displacement on a tidal mixing front and (iv) effects of friction and mixing can significantly alter the behaviour of the front and the relative strength of the cyclonic and anticyclonic eddies formed. 相似文献
13.
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
14.
The modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case. 相似文献
15.
This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response. 相似文献
16.
实际工程场地中的断层通常具有三维尺寸,应进行三维分析。本文考虑均匀弹性半空间内存在不同长度、深度的三维断层,由下向上垂直入射一个脉冲波,主要采用三维时域显式有限元结合透射人工边界的方法及相应编写的三维显示有限元波动程序DSI3,计算断层附近地震地面运动的时程反应,研究含有三维断层场地地面运动的一些特点,并与自由场反应对比,以说明三维竖向断层对地震波传播的特点和对断层附近地面运动的影响。 相似文献
17.
A method of coupling finite elements (FE), boundary elements (BE) and infinite boundary elements (IBE) in the frequency domain is presented for wave propagation and soil-structure interaction problems. The procedure has the advantage of being suitable for considering non-homogeneous materials in the near region and the radiation condition at the far field. At the same time, the displacement field on the traction free boundary of the semi-plane is simulated by the IBE rather than by additional discretization of the ground surface as is required in the normal BE procedure. Verification examples show that excellent accuracy is achieved compared with the results by Wong18 who used the generalized inverse method. Finally, the proposed procedure is employed to obtain the response of a semi-elliptical rock canyon-alluvial deposit under seismic P and SV waves. Amplification behaviour due to the alluvial soil is observed to be significant for the given geometric dimensions of the canyon and the mechanical properties of the media under the excitation of the El Centro and Parkfield earthquakes. 相似文献
18.
An adaptive free‐surface expression for three‐dimensional finite‐difference frequency‐domain modelling of elastic wave 
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下载免费PDF全文 Finite‐difference frequency‐domain modelling of seismic wave propagation is attractive for its efficient solution of multisource problems, and this is crucial for full‐waveform inversion and seismic imaging, especially in the three‐dimensional seismic problem. However, implementing the free surface in the finite‐difference method is nontrivial. Based on an average medium method and the limit theorem, we present an adaptive free‐surface expression to describe the behaviour of wavefields at the free surface, and no extra work for the free‐surface boundary condition is needed. Essentially, the proposed free‐surface expression is a modification of density and constitutive relation at the free surface. In comparison with a direct difference approximate method of the free‐surface boundary condition, this adaptive free‐surface expression can produce more accurate and stable results for a broad range of Poisson's ratio. In addition, this expression has a good performance in handling the lateral variation of Poisson's ratio adaptively and without instability. 相似文献
19.
Modified image algorithm to simulate seismic channel waves in 3D tunnel model with rugged free surfaces 总被引：1，自引：0，他引：1 下载免费PDF全文
下载免费PDF全文 The existence of rugged free‐surface three‐dimensional tunnel conditions in the coal seams, caused either by geological or mining processes, will inevitably influence wave propagation characteristics when the seismic waves go through the coal mines. Thus, a modified image algorithm has been developed to account for seismic channel waves propagating through this complicated topography with irregular free surfaces. Moreover, the seismic channel waves commonly exhibit damped and dispersive signatures, which is not only because of their own unique sandwich geometry of rock–coal–rock but also because of the viscoelastic behavior of coal. Considering the complexity of programming in three‐dimensional tunnel models with rugged free surfaces, an optimized vacuum grid search algorithm, enabling to model highly irregular topography and to compute efficiently, is also proposed when using high‐order staggered finite‐difference scheme to simulate seismic channel wave propagations in viscoelastic media. The numerical simulations are implemented to investigate the accuracy and stability of the method and the impact of coal's viscoelastic behavior on seismic channel wave propagation characteristics. The results indicate that the automatic vacuum grid search algorithm can be easily merged into high‐order staggered finite‐difference scheme, which can efficiently be applied to calculate three‐dimensional tunnel models with rugged free surfaces in the viscoelastic media. The simulation also suggests that the occurrence of a three‐dimensional tunnel with free surfaces has a remarkable influence on the seismic channel wave propagation characteristics and elastic energy distribution. 相似文献
20.
We recently proposed an efficient hybrid scheme to absorb boundary reflections for acoustic wave modelling that could attain nearly perfect absorptions. This scheme uses weighted averaging of wavefields in a transition area, between the inner area and the model boundaries. In this paper we report on the extension of this scheme to 2D elastic wave modelling with displacement‐stress formulations on staggered grids using explicit finite‐difference, pseudo‐implicit finite‐difference and pseudo‐spectral methods. Numerical modelling results of elastic wave equations with hybrid absorbing boundary conditions show great improvement for modelling stability and significant absorption for boundary reflections, compared with the conventional Higdon absorbing boundary conditions, demonstrating the effectiveness of this scheme for elastic wave modelling. The modelling results also show that the hybrid scheme works well in 2D rotated staggered‐grid modelling for isotropic medium, 2D staggered‐grid modelling for vertically transversely isotropic medium and 2D rotated staggered‐grid modelling for tilted transversely isotropic medium. 相似文献