不规则形状垂直井孔中波传播的2.5维边界元模拟

本文给出了求解声波测井工作中具有任意形状的垂直井孔中波传播的波数域边界元公式。我们将问题看成是垂向均匀的二维介质,并沿垂直方向变换到离散波数域。通过与有限差分计算的频散曲线的比较验证了算法和程序的正确性。边界元方法可以处理任意形状的边界,作为初步应用,我们研究了垂向裂纹对井孔中波传播的影响。     

Boundary element simulation of scattering of elastic waves by 3-D cracks

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.     

改进的二维数值波浪水槽在波浪破碎中的应用

Song和Banner（2002,简称SB02）利用二维数值波浪水槽（Drimer和Agnon开发,并被Segre改进,简称DAS）研究了深水及中等水深下波群破碎,并依据波群内部能量调制和演变特性提出一个新的波浪破碎阈值.本文利用两个DAS改进模型对SB02的波浪破碎结果进行检验和比较,并研究底面斜坡坡度对SB02破碎判据的影响,其中第一个改进模型（简称MDAS1）修正了DAS中某些积分的计算错误,第二个改进模型（简称MDAS2）在自由表面上用三阶元取代原来的线性元.研究表明： MDAS1和DAS的结果非常一致;而MDAS2和MDAS1相比,波浪临界破碎时的造波板振幅、破碎时刻和局部能量极大值的最大平均变化率δmax都会发生变化,但MDAS2仍然符合SB02提出的破碎阈值.由MDAS1模拟中等水深下波群在坡度为1∶500、1∶300、1∶150和1∶100斜坡上的破碎结果表明：随着斜坡变陡,波群的局部能量极大值μ增大,δmax在临界非破碎情况下微弱变小,在临界破碎情况下急剧增大,SB02提出的破碎阈值在中等水深下坡度小于1∶100的斜坡上仍然有效.     

A viscous-spring transmitting boundary for cylindrical wave propagation in saturated poroelastic media

Based on the u–U formulation of Biot equation and the assumption of zero permeability coefficient, a viscous-spring transmitting boundary which is frequency independent is derived to simulate the cylindrical elastic wave propagation in unbounded saturated porous media. By this viscous-spring boundary the effective stress and pore fluid pressure on the truncated boundary of the numerical model are replaced by a set of spring, dashpot and mass elements, and its simplified form is also given. A u–U formulation FEA program is compiled and the proposed transmitting boundaries are incorporated therein. Numerical examples show that the proposed viscous-spring boundary and its simplified form can provide accurate results for cylindrical elastic wave propagation problems with low or intermediate values of permeability or frequency content. For general two dimensional wave propagation problems, spuriously reflected waves can be greatly suppressed and acceptable accuracy can still be achieved by placing the simplified boundary at relatively large distance from the wave source.     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

一种模拟非均匀介质中弹性波传播新的k-space方法

传统的伪谱（PS）方法,采用傅里叶变换（FT）计算空间导数具有很高的精度,每个波长仅需要两个采样点,而时间导数采用有限差分（FD）近似因而精度较低.当采用大时间步长时,由于时空精度不平衡,PS法存在不稳定性问题.原始的k-space方法可以有效地克服这些问题但是却无法适用于非均匀介质.为了提高原始k-space方法模拟非均匀介质波动方程的精度,我们提出了一种新的k-space算子族.它是用非均匀介质的变速度代替原k-space算子中的常数补偿速度构造得到,引入低秩近似可以高效求解.我们将构造的新的k-space算子应用于耦合的二阶位移波动方程,而不是交错网格一阶速度应力波动方程,使模拟弹性波的计算存储量减少.我们从数学上证明了基于二阶波动方程的k-space方法与基于一阶波动方程的k-space方法是等价的.数值模拟实验表明,与传统的PS、交错网格PS和原始的k-space方法相比,我们的新方法可以在时间和空间步长较大的均匀和非均匀介质中,为弹性波的传播提供更精确的数值解.在保持稳定性和精度的同时,采用较大的时空采样间隔,可以大大降低数值模拟的计算成本.

     

点源两层大地三维地电模型视电阻率边界元解

本文用两层均匀大地人工点源作用下的空间电位分布作为基本解，导出了下伏基岩中陷含三维地质体情况下，关于电位的边界积分方程组，用边界单元法解积分方程求得地表电位，从而求得了一定装置下的视电阻率。本文方法的独特优点是积分只须在异常体边界面上进行了测线可以沿任意方向。用联剖法计算了均匀半空间下含球形异常体的ρｓ曲线，所得结果与其他学者用解析方法所求得的结果一致。用中间梯度法计算下层基岩中含有一长方体，测线     

Spectral-element simulations of elastic wave propagation in exploration and geotechnical applications

We apply the spectral-element method (SEM), a high-order finite-element method (FEM) to simulate seismic wave propagation in complex media for exploration and geotechnical problems. The SEM accurately treats geometrical complexities through its flexible FEM mesh and accurately interpolates wavefields through high-order Lagrange polynomials. It has been a numerical solver used extensively in earthquake seismology. We demonstrate the applicability of SEM for selected 2D exploration and geotechnical velocity models with an open-source SEM software package SPECFEM2D. The first scenario involves a marine survey for a salt dome with the presence of major internal discontinuities, and the second example simulates seismic wave propagation for an open-pit mine with complex surface topography. Wavefield snapshots, synthetic seismograms, and peak particle velocity maps are presented to illustrate the promising use of SEM for industrial problems.     

Combined paraxial-consistent boundary conditions finite element model for simulating wave propagation in elastic half-space media

In this paper, a finite element model of a soil island is coupled to both a consistent transmitting boundary and a paraxial boundary, which are then used to model the propagation of waves in semi-infinite elastic layered media. The formulation is carried out in the frequency domain while assuming plane strain conditions. It is known that a discrete model of this type, while providing excellent results for a wide range of physical parameters in the context of a half-space problem, may deteriorate rapidly at low frequencies of excitation. This is so because at low frequencies the various waves in the model eventually attain characteristic wavelengths which exceed the distance of the bottom boundary, which then causes that boundary to fail. Also, the paraxial boundaries themselves break down at very low frequencies. In this paper, this difficulty is overcome and the model׳s performance is improved upon dramatically by incorporating an artificial buffer layer sandwiched between the bottom of the soil medium and the underlying elastic half-space. Applications dealing with rigid foundations resting on homogenous or layered half-space media are shown to exhibit significant improvement. Following extensive simulations, clear guidelines are provided on the performance of the coupled model and an interpretation is given on the engineering significance of the findings. Finally, clear recommendations are provided for the practical use of the proposed modelling strategy.     

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.     

Finite element equations and numerical simulation of elastic wave propagation in two—phase anisotropic media

Based on Biot theory of two-phase anisotropic media and Hamilton theory about dynamic problem,finite element equations of elastic wave propagation in two-phase anisotopic media are derived in this paper.Numerical solution of finite element equations is given.Finally,properties of elastic wave propagation are observed and analyzed through FEM modeling.     

Condensed hyperelements method of non-vertical consistent boundaries for wave propagation analysis in irregular media

The study of wave propagation in finite/infinite media has many applications in geotechnical and structural earthquake engineering and has been a focus of research for the past few decades. This paper presents an analysis of 2D anti- plane problems （Love waves） and 2D in-plane problems （Rayleigh waves） in the frequency domain in media consisting of a near-field irregular and a far-field regular part. The near field part may contain structures and its boundaries with the far-field can be of any shape. In this study, the irregular boundaries of the near-field are treated as consistent boundaries, extending the concept of Lysmer＇s vertical consistent boundaries. The presented technique is called the Condensed Hyperelements Method （CHM）. In this method, the irregular boundary is limited to a vertical boundary at each end that is a consistent boundary at the far-field side. Between the two ends, the medium is discretized with hyperelements. Using static condensation, the stiffness matrix of the far-field is derived for the nodes on the irregular boundary. Examples of the application of the CHM illustrate its excellent accuracy and efficiency.     

Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity‡

This paper presents a Lebedev finite difference scheme on staggered grids for the numerical simulation of wave propagation in an arbitrary 3D anisotropic elastic media. The main concept of the scheme is the definition of all the components of each tensor (vector) appearing in the elastic wave equation at the corresponding grid points, i.e., all of the stresses are stored in one set of nodes while all of the velocity components are stored in another. Meanwhile, the derivatives with respect to the spatial directions are approximated to the second order on two‐point stencils. The second‐order scheme is presented for the sake of simplicity and it is easy to expand to a higher order. Another approach, widely‐known as the rotated staggered grid scheme, is based on the same concept; therefore, this paper contains a detailed comparative analysis of the two schemes. It is shown that the dispersion condition of the Lebedev scheme is less restrictive than that of the rotated staggered grid scheme, while the stability criteria lead to approximately equal time stepping for the two approaches. The main advantage of the proposed scheme is its reduced computational memory requirements. Due to a less restrictive dispersion condition and the way the media parameters are stored, the Lebedev scheme requires only one‐third to two‐thirds of the computer memory required by the rotated staggered grid scheme. At the same time, the number of floating point operations performed by the Lebedev scheme is higher than that for the rotated staggered grid scheme.     

Transient mechanical wave propagation in semi-infinite porous media using a finite element approach

In this study, we propose a numerical investigation in the time domain of the mechanical wave propagation of an impulsional load on semi-infinite soil. The ground is modelled as a porous saturated viscoelastic medium involving complete Biot theory. All the couplings and a hysteretic Rayleigh damping are taken into consideration. An accurate and efficient finite element method using a matrix-free technique and an expert multigrid system are applied. Our results present the displacements of the fluid and solid particles over the surface and in depth. The arrival times of body and surface waves are studied. Particularly, the compressional wave of the second kind is highlighted. The influence of the different couplings and more specifically, the influence of the permeability on the response of the soil are analyzed.     

Numerical modelling of wave propagation in anisotropic soil using a displacement unit-impulse-response-based formulation of the scaled boundary finite element method

An efficient method for modelling the propagation of elastic waves in unbounded domains is developed. It is applicable to soil–structure interaction problems involving scalar and vector waves, unbounded domains of arbitrary geometry and anisotropic soil. The scaled boundary finite element method is employed to derive a novel equation for the displacement unit-impulse response matrix on the soil–structure interface. The proposed method is based on a piecewise linear approximation of the first derivative of the displacement unit-impulse response matrix and on the introduction of an extrapolation parameter in order to improve the numerical stability. In combination, these two ideas allow for the choice of significantly larger time steps compared to conventional methods, and thus lead to increased efficiency. As the displacement unit-impulse response approaches zero, the convolution integral representing the force–displacement relationship can be truncated. After the truncation the computational effort only increases linearly with time. Thus, a considerable reduction of computational effort is achieved in a time domain analysis. Numerical examples demonstrate the accuracy and high efficiency of the new method for two-dimensional soil–structure interaction problems.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

TTI介质弹性波近似解耦波动方程

借助Christoffel方程可求解出各向异性介质弹性波精确频散关系.利用近似方法进行处理,再通过傅里叶逆变换将频率波数域算子变换为时空域算子,可导出解耦的qP波或qS波波动方程.本文在TTI介质弹性波精确频散关系的基础上,利用近似配方法推导了qP波和qSV波近似频散关系,通过傅里叶逆变换推导了TTI介质qP波和qSV波解耦的波动方程.为了验证近似频散关系的有效性,利用两组模型参数对其进行数值计算,分析了相对误差在不同传播方向上的分布.随后使用有限差分方法分别对均匀、层状及复杂TTI介质弹性波近似解耦波动方程进行数值模拟,结果显示qP波和qSV波完全解耦,并且在各向异性参数η < 0以及介质对称轴倾角变化较大的情况下,纯qP波和纯qSV波近似波动方程依然可以保持稳定.

     

Modelling of pile wave barriers by effective trenches and their screening effectiveness

The three-dimensional problem of isolation of vibration by a row of piles is studied numerically on the basis of a model replacing the row of piles by an effective trench in order to reduce the modelling complexity. The analysis is accomplished with the aid of an advanced frequency domain boundary element method, which is used for both the infilled trench and the soil medium in conjunction with a coupling procedure based on enforcement of equilibrium and compatibility at the trench–soil interface. Linear elastic or viscoelastic material behaviour is assumed for both the piles and the soil. The piles can be tubular or solid and have circular or square cross-section. The vibration source is a vertical force, harmonically varying with time, and the row of piles acts as a passive wave barrier. The effective trench model is constructed by invoking well known homogenization techniques used in the mechanics of fibre-reinforced composite materials, and its accuracy is compared against a rigorous boundary element analysis modelling each pile separately in full contact with the soil medium. On the basis of the effective trench model, the screening effectiveness of a row of piles is studied through parametric studies.     

Numerical analysis of seismic wave amplification in Nice (France) and comparisons with experiments

The analysis of site effects is very important since the amplification of seismic motion in some specific areas can be very strong. In this paper, the site considered is located in the centre of Nice on the French Riviera. Site effects are investigated considering a numerical approach (Boundary Element Method) and are compared with experimental results. The experimental results are obtained thanks to real earthquakes (weak motion) and microtremor measurements. The investigation of seismic site effects through numerical approaches is interesting because it shows the dependency of the amplification level on such parameters as wave velocity in surface soil layers, velocity contrast with deep layers, seismic wave type, incidence and damping.In this specific area of Nice, experimental measurements obtained for weak motion lead to strong site effects. A one-dimensional (1D) analytical analysis of amplification does not give a satisfactory estimation of the maximum reached levels. A boundary element model is then proposed considering different wave types (SH, P, SV) as the seismic loading. The alluvial basin is successively assumed as an isotropic linear elastic medium and an isotropic linear viscoelastic solid with Zener type behaviour (standard solid). The influence of frequency and incidence is analysed. The thickness of the surface layer, its mechanical properties, its general shape as well as the seismic wave type involved have a great influence on the maximum amplification and the frequency for which it occurs. For real earthquakes, the numerical results are in very good agreement with experimental measurements for each motion component. The boundary element method leads to amplification values very close to the actual ones and much larger than those obtained in the 1D case. Two-dimensional basin effects are then very strong and are well reproduced numerically.     

Analysis of Rayleigh waves in saturated porous elastic media by finite element method

A numerical procedure for the analysis of Rayleigh waves in saturated porous elastic media is proposed by use of the finite element method. The layer stiffness matrix, the layer mass matrix and the layer damping matrix in a layered system are presented for the discretized form of the solid-fluid equilibrium equation proposed by Biot. In order to consider the influence of the permeability coefficient on the behavior of Rayleigh waves, attention is focused on the following states: ‘drained’ state, ‘undrained’ state and the states between two extremes of ‘drained’ and ‘undrained’ states. It is found from computed results that the permeability coefficient exerts a significant effect on dispersion curves and displacement distributions of Rayleigh waves in saturated porous media.