Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by incident SH-waves

Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by SH-waves is studied using wave function expansion and complex function method. Firstly, the analytical model is divided into three parts, and the displacement solutions of wave fields are constructed based on boundary conditions in the three regions. Three domains are then conjoined to satisfy the "conjunction" condition at shared boundary. In addition, combined with the zero-stress condition of semi-cylindrical canyon, a series of infinite algebraic equations for the problem are derived. Finally, numerical examples are provided and the influence of different parameters on ground motion is discussed.     

SH波入射时多个半圆形沉积谷地附近浅埋圆形孔洞的动力分析

本文采用辅助函数的思想,利用复变函数和多级坐标的方法给出了SH波入射条件下多个半圆形沉积谷地附近浅埋圆形孔洞动力分析问题的解答。将整个求解区域分割成两部分来处理,区域I为多个半圆形沉积谷地,区域II为浅埋圆形孔洞附近带半圆形凹陷的半无限弹性空间。在区域I和II中分别构造位移解,并在二个区域的“公共边界”上实施位移应力的连续条件,建立求解该问题的无穷代数方程组。最后,本文给出了算例和数值结果,并对其进行了讨论。     

半无限空间中稳态P波在衬砌周围的散射

本文采用多级坐标和复变函数方法针对半无限空间中P波在一衬砌周围的散射问题提出了一种近似求解分析方法。具体做法是利用一个半径很大的圆来逼近半空间的直边界，将待解问题转化为全空间中P波在一圆孔和一衬砌周围的共同散射问题。我们预先写出问题波函数的一般形式解，利用边界条件并借助复数傅立叶级数展开把问题化为求解波函数中未知系数的无穷线性代数方程组，进而讨论了衬砌内外边界处动应力集中系数针对不同条件组合的分布和变化情况。算例结果表明：本文方法对于研究与P波（SV波）有关的散射问题是可行的。     

各向异性半空间中浅埋孔洞对地表 反平面运动的影响

本文采用复变函数方法研究了稳态水平剪切波(SH波)在各向异性弹性半空间中任意形状孔洞上的散射及其对地面运动的影响.在变换空间中构造出自动满足各向异性半空间水平表面上应力为零的散射波函数,并根据孔洞表面应力为零的边界条件,采用最小二乘法求解散射波函数的系数.用介质的各向异性性质来模拟地质条件,给出了SH波作用下含圆孔、椭圆孔和方孔的各向异性半空间表面位移幅值的数值结果,并分析了介质特性、孔洞的形状、埋深、入射波波数及入射角度等因素对地面运动的影响规律.数值结果表明:介质的各向异性对含有孔洞的半空间表面的地表位移具有显著的影响;沿一定角度的入射波在某一频段内所引起的地表位移幅值比各向同性介质的可能要大,且随着孔洞埋深的增加,地表位移的幅值逐渐减小.      

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.     

Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach

In this paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve the exterior Helmholtz problems with alluvial valleys. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals by locating the null-field point exactly on the real boundary. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. Image concept and technique of decomposition are utilized for half-plane problems. After moving the null-field point to the real boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples including single and successive alluvial valleys are given to test our program. Limiting cases of a single canyon and two successive canyons are also addressed. Amplification of soft basin is also observed in this study. The validity of the semi-analytical method is verified. Our advantages, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence and mesh-free, by using the present method are achieved.     

3D seismic response of a limited valley via BEM using 2.5D analytical Green's functions for an infinite free-rigid layer

This paper presents analytical solutions for computing the 3D displacements in a flat solid elastic stratum bounded by a rigid base, when it is subjected to spatially sinusoidal harmonic line loads. These functions are also used as Greens functions in a boundary element method code that simulates the seismic wave propagation in a confined or semi-confined 2D valley, avoiding the discretization of the free and rigid horizontal boundaries.The models developed are then used to simulate wave propagation within a rigid stratum and valleys with different dimensions and geometries, when struck by a spatially sinusoidal harmonic vertical line load. Simulations are performed in the frequency domain, for varying spatial wave numbers in the axial direction of the valley. Time results are obtained by means of inverse Fourier transforms, to help understand how the geometry of the valley may affect the variation of the displacement field.     

Application of a perfectly matched layer in seismic wavefield simulation with an irregular free surface

Recently, an effective and powerful approach for simulating seismic wave propagation in elastic media with an irregular free surface was proposed. However, in previous studies, researchers used the periodic condition and/or sponge boundary condition to attenuate artificial reflections at boundaries of a computational domain. As demonstrated in many literatures, either the periodic condition or sponge boundary condition is simple but much less effective than the well‐known perfectly matched layer boundary condition. In view of this, we intend to introduce a perfectly matched layer to simulate seismic wavefields in unbounded models with an irregular free surface. We first incorporate a perfectly matched layer into wave equations formulated in a frequency domain in Cartesian coordinates. We then transform them back into a time domain through inverse Fourier transformation. Afterwards, we use a boundary‐conforming grid and map a rectangular grid onto a curved one, which allows us to transform the equations and free surface boundary conditions from Cartesian coordinates to curvilinear coordinates. As numerical examples show, if free surface boundary conditions are imposed at the top border of a model, then it should also be incorporated into the perfectly matched layer imposed at the top‐left and top‐ right corners of a 2D model where the free surface boundary conditions and perfectly matched layer encounter; otherwise, reflections will occur at the intersections of the free surface and the perfectly matched layer, which is confirmed in this paper. So, by replacing normal second derivatives in wave equations in curvilinear coordinates with free surface boundary conditions, we successfully implement the free surface boundary conditions into the perfectly matched layer at the top‐left and top‐right corners of a 2D model at the surface. A number of numerical examples show that the perfectly matched layer constructed in this study is effective in simulating wave propagation in unbounded media and the algorithm for implementation of the perfectly matched layer and free surface boundary conditions is stable for long‐time wavefield simulation on models with an irregular free surface.     

SH波在浅埋可移动圆柱形刚性夹杂处的散射与地震动

采用复函数、多极坐标法研究了含有可移动圆柱形刚性夹杂的弹性半空间对SH波的散射问题。构造一个能自动满足含可移动刚性圆柱的弹性半空间自由表面上应力为零的边界条件的散射波,应用可移动圆柱形刚性夹杂的运动条件来确定该散射波。最终则可将求解问题归结为求解一个无穷代数方程组,采用截断有限项的方法对其进行求解。给出了地表位移幅值的数值结果,并讨论了各种参数对它的影响。     

Perfectly matched layer boundary conditions for the second-order acoustic wave equation solved by the rapid expansion method

We derive a governing second-order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite-difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.     

Scattering of SH waves by a scalene triangular hill

<正>The influence of local landforms on ground motion is an important problem.The antiplane response of a scalene triangular hill to incident SH waves is studied in this paper by using a complex function,moving coordinates and auxiliary functions.First,the model is divided into two domains:a scalene triangular hill with a semi-circular bottom;and a half space with a semi-circular canyon.Wave functions that satisfy the zero-stress condition at the triangular wedges and at the horizontal surface are constructed in both domains.Then,considering the displacement continuity and stress equilibrium, algebraic equations are established.Finally,numerical examples are provided to illustrate the influence of the geometry of the hill and the characteristics of the incident waves on the ground motions.     

SH波入射时半圆形凸起地形附近浅埋圆形衬砌结构的动应力分析

本文给出了地下圆形衬砌结构与地面上的半圆形凸起地形对垂直于地面入射的SH波散射问题的解答。方法是将求解区域分割成两部分。其一为包含半圆形凸起地形在内的圆形区域Ⅰ，其二为带有一个半圆形凹陷和一个圆形衬砌结构的弹性半空间Ⅱ，半圆形凹陷部分为其公共边界，在区域Ⅰ和Ⅱ中分别构造其位移解，然后再通过移动坐标，使其满足“公共边界”上的条件和地下衬砌的边界条件，建立起求解该问题的无穷代数方程组。最后，本文给出了算例，并讨论了数值结果，给出了圆形衬砌结构周边上的动应力集中系数变化规律。     

半无限空间界面附近SH波对圆形衬砌的散射

建立了求解半无限空间中SH波对浅埋圆形衬砌结构的散射与动应力集中问题的解析方法。利用SH波散射的对称性和多极坐标的方法，在复平面上构造出了一个可以预先满足半空间自由表面上应力自由的边界条件的浅埋圆形衬砌对稳态SH波散射的波函数，并构造出衬砌内的散射波函数。然后根据衬砌周围的边界条件，将该问题转化为对一组无穷代数方程组的求解。最后给出了具体算例，并讨论了其数值结果。     

半无限空间中圆形孔洞周围SH波的散射

建立了求解在含有圆形孔洞的弹性半空间中SH波散射与圆形孔洞附近动应力集中问题的解析方法。利用SH波散射的对称性和多极坐标的方法，构造了一个可以预先满足半空间自由表面上应力自由边界条件的圆形孔洞对SH波散射的波函数。利用这一波函数，则可将该问题转化成对一个圆形孔洞散射的求解问题。该问题的解答最终又可归结为对一组无究代数方程组的求解问题，并可利用截断有限项的方法对其进行计算，最后给出了有关圆形也洞附近动应力集中问题的算例和数值结果，并讨论了波数与圆孔至自由边界距离变化对动应力集中的影响。     

无破碎带断层场地对SH波的散射

在层状半空间精确动力刚度矩阵和斜线荷载动力格林函数的基础上建立间接边界元方法,在频域内求解无破碎带断层场地对入射平面SH波的散射。为方便求解,将总波场分解为自由波场和散射波场,自由波场由直接刚度法求得,断层两侧的散射波场通过在断层面上分别对两侧施加均布斜线荷载产生的动力响应来模拟,虚拟荷载的密度可通过引入断层表面的边界条件确定,最后叠加自由波场和散射波场求得总波场。以有落差断层和无落差断层模型为例进行数值计算,分析断层落差、断层倾角以及断层两侧介质的刚度比对散射效应的影响。研究表明,断层落差与波长相当时,断层对SH波的放大作用最大;地表位移幅值随着断层倾角的增大逐渐增大;若断层无落差且其两侧刚度不同时,一般刚度较小一侧地表位移幅值较大且振荡更为剧烈,波从刚度较小一侧入射时位移幅值放大尤为显著。     

Dynamic response of a piecewise circular tunnel embedded in a poroelastic medium

Recently, considerable efforts have been devoted to evaluation of seismic dynamic response of a circular tunnel. Conventional approaches have considered integral liners embedded in an elastic medium. In this study, we re-examine the problem with piecewise liners embedded in a porous medium. Surrounding saturated porous medium of tunnels is described by Biot's poroelastic theory, while the liner pieces and the connecting joints are treated as curved beams and characterized by curved beam theories. The scattered wave field in the porous medium is obtained by the wave function expansion method. The differential equations governing the vibration of a curved beam is discretized by the General Differential Quadrature (GDQ) method. The domain decomposition method is used to establish the global discrete dynamic equations for the piecewise tunnel. The surrounding soil and the tunnel are coupled together via the stress and the displacement continuation conditions which are implemented by the boundary collocation method. Numerical results demonstrate that the stiffness difference between the liner piece and the connecting joints has a considerable influence on the internal forces of the liner piece.     

2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves

The problem of harmonic wave diffraction by tunnels in an infinite poroelastic saturated soil obeying Biot's theory is studied numerically under conditions of plane strain and the effect of poroelasticity on the response is assessed through some parametric studies. The method is based on the theory of Mei and Foda, which considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer at the free boundaries. Both problems are solved numerically by the boundary element method in the frequency domain. Results dealing with the response of a circular tunnel with and without an elastic concrete liner in an infinite poroelastic medium to incident harmonic P and SV plane waves are provided and compared against analytical ones as well as to those corresponding to linear elastic soil behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

An unsplit complex frequency-shifted perfectly matched layer for second-order acoustic wave equations

The perfectly matched layer(PML) boundary condition has been proven to be effective for attenuating reflections from model boundaries during wavefield simulation. As such, it has been widely used in time-domain finite-difference wavefield simulations. The conventional PML has poor performance for near grazing incident waves and low-frequency reflections. To overcome these limitations, a more complex frequency-shifted stretch(CSF) function is introduced, which is known as the CFSPML boundary condition and can be implemented in the time domain by a recursive convolution technique(CPML). When implementing the PML technique to second-order wave equations, all the existing methods involve adding auxiliary terms and rewriting the wave equations into new second-order partial differential equations that can be simulated by the finite-difference scheme, which may affect the efficiency of numerical simulation. In this paper, we propose a relatively simple and efficient approach to implement CPML for the second-order equation system, which solves the original wave equations numerically in the stretched coordinate. The spatial derivatives in the stretched coordinate are computed by adding a correction term to the regular derivatives. Once the first-order spatial derivatives are computed, we computed the second-order spatial derivatives in a similar way; therefore, we refer to the method as two-step CPML(TS-CPML). We apply the method to the second-order acoustic wave equation and a coupled second-order pseudo-acoustic TTI wave equation. Our simulations indicate that amplitudes of reflected waves are only about half of those computed with the traditional CPML method, suggesting that the proposed approach has computational advantages and therefore can be widely used for forwarding modeling and seismic imaging.     

用波函数展开法求解界面圆孔的SH波散射问题

研究了ＳＨ波绕界面圆孔散射的波函数展开理论。由入射,反射和透射波组成的自由波场与孔的散射场叠加成总波场。将定义于两个半平面的散射场按一定方式延拓到全平面。通过Ｈａｎｋｅｌ－Ｆｏｕｒｉｅｒ型积分来满足界面与孔缘处的边界条件,得到了确定特定系数的封闭代数方程组。给出了具体算例,计算了孔缘处的动应力系数分布。     

3D seismic response of a 2D hill-valley staggered topography modeled by a 2.5D multi-domain IBEM

A two and a half dimensional(2.5 D) multidomain indirect boundary element method(IBEM) is developed to study the wave scattering of obliquely incident P-, SVand SH-waves by a hill-valley staggered topography in a multi-layered half-space. The IBEM algorithm includes using2.5 D full-space and half-space Green's functions to construct scattered fields in decomposed closed and opened half-space regions, respectively, and using the dynamic stiffness method to solve the free fields. All regions are finally integrated by introducing the compatibility conditions to obtain the total wave fields. The proposed 2.5 D IBEM has the flexibility in dealing with complex boundaries by directly applying the fictitious loads on the regions' boundaries, with a less storage requirement compared to the full 3 D models. Besides, by combining the specific advantages of the two kinds of Green's functions, the method is well suitable for handling coupled topographies with high accuracy. The method is validated by comparison with published results for a single valley as well as a single hill topography. The effects of height-to-width ratio of hill and layering on dynamic responses are further parametrically investigated by numerical implementations in frequency domain. Results show that the interaction between valley and hills can lead to a more significant amplification within the valley region, and dynamic responses are deeply influenced by the height-towidth of hill, simultaneously depending on incident angle and frequency. Besides, the site effects become more complex when the stratification feature is taken into account.