Application of a second-order paraxial boundary condition to problems of dynamics of circular foundations on a porous layered half-space

A half-space finite element and a consistent transmitting boundary in a cylindrical coordinate system are developed for analysis of rigid circular (or cylindrical) foundations in a water-saturated porous layered half-space. By means of second-order paraxial approximations of the exact dynamic stiffness for a half-space in plane-strain and antiplane-shear conditions, the corresponding approximation for general three-dimensional wave motion in a Cartesian coordinate system is obtained and transformed in terms of cylindrical coordinates. Using the paraxial approximations, the half-space finite element and consistent transmitting boundary are formulated in a cylindrical coordinate system. The development is verified by comparison of dynamic compliances of rigid circular foundations with available published results. Examination of the advantage of the paraxial condition vis-á-vis the fixed condition shows that the former achieves substantial gain in computational effort. The developed half-space finite element and transmitting boundary can be employed for accurate and effective analysis of foundation dynamics and soil–structure interaction in a porous layered half-space.     

Response of unbounded soil in scaled boundary finite‐element method

The scaled boundary finite‐element method is a powerful semi‐analytical computational procedure to calculate the dynamic stiffness of the unbounded soil at the structure–soil interface. This permits the analysis of dynamic soil–structure interaction using the substructure method. The response in the neighbouring soil can also be determined analytically. The method is extended to calculate numerically the response throughout the unbounded soil including the far field. The three‐dimensional vector‐wave equation of elasto‐dynamics is addressed. The radiation condition at infinity is satisfied exactly. By solving an eigenvalue problem, the high‐frequency limit of the dynamic stiffness is constructed to be positive definite. However, a direct determination using impedances is also possible. Solving two first‐order ordinary differential equations numerically permits the radiation condition and the boundary condition of the structure–soil interface to be satisfied sequentially, leading to the displacements in the unbounded soil. A generalization to viscoelastic material using the correspondence principle is straightforward. Alternatively, the displacements can also be calculated analytically in the far field. Good agreement of displacements along the free surface and below a prism foundation embedded in a half‐space with the results of the boundary‐element method is observed. Copyright © 2001 John Wiley & Sons, Ltd.     

A three‐dimensional transmitting boundary formulated in Cartesian co‐ordinate system for the dynamics of non‐axisymmetric foundations

A three‐dimensional transmitting boundary is formulated in the Cartesian co‐ordinate system. It is developed for the dynamic soil–structure interaction problems of arbitrary shape foundations in laterally heterogeneous strata overlying rigid bedrock. Dynamics of a rectangular rigid surface foundation on a homogeneous stratum is analysed by a hybrid approach in which the finite region including foundation is modelled by the conventional finite element method and the surrounding infinite region by the newly developed transmitting boundary. To demonstrate its strength, the present method is applied to a rectangular foundation in a horizontally heterogeneous ground consisting of two distinct regions divided by and welded along a vertical plane. Copyright © 2000 John Wiley & Sons, Ltd.     

Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method

The scaled boundary finite‐element method is extended to simulate time‐harmonic responses of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. The unbounded domains and the elasticity matrices are transformed to the scaled boundary coordinates. The scaled boundary finite‐element equation in displacement amplitudes are derived directly from the governing equations of elastodynamics. To enforce the radiation condition at infinity, an asymptotic expansion of the dynamic‐stiffness matrix for high frequency is developed. The dynamic‐stiffness matrix at lower frequency is obtained by numerical integration of ordinary differential equations. Only the boundary is discretized yielding a reduction of the spatial dimension by one. No fundamental solution is required. Material anisotropy is modelled without additional efforts. Examples of two‐ and three‐dimensional non‐homogeneous isotropic and transversely isotropic unbounded domains are presented. The results demonstrate the accuracy and simplicity of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Transmitting boundary for water-saturated transversely isotropic strata based on the u–U formulation

A new transmitting boundary in a cylindrical coordinate system has been developed for modeling the elastic waves radiating out to an infinite boundary in water-saturated transversely isotropic soil strata over a rigid bedrock. The saturated soil strata are assumed to consist of a porous material and modeled as a transversely isotropic two-phase medium, based on the u−U formulation. The newly developed transmitting boundary is combined with the finite-elements model of the near-field region, using the same u−U formulation, and applied to the study of the dynamics of a rigid circular foundation in porous isotropic or transversely isotropic layered strata, either fully or partly saturated with water. The verification and application examples give valuable insights into new and interesting aspects of the dynamic behavior of rigid circular foundations in fully or partly saturated two-phase ground in terms of permeability, transverse anisotropy, and ground-water table level.     

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.     

CALCULATING THE DYNAMIC STIFFNESS MATRIX OF 2-D FOUNDATIONS BY DISCRETE WAVE NUMBER INDIRECT BOUNDARY ELEMENT METHODS

Apart from some special cases, calculating the dynamic stiffness matrix of foundations on a layered half-space, especially for embedded foundations, is computationally expensive. An efficient method for two-dimensional foundations in a horizontally layered soil media is presented in this paper. This method is based on indirect boundary element methods and uses discrete wave number solution methods for calculating Green's functions for displacements and analytical methods for the integrations over the boundary. For surface foundations, the present method applies at all frequencies. For embedded foundations or for constructing energy transmitting boundaries, because the free-field part is modelled by boundary elements and the excavated part is modelled by finite elements, the present method applies only at low frequencies for the spring coefficients (the real parts of the dynamic stiffness matrix) but applies at all frequencies for the damping coefficients (the imaginary part of the dynamic stiffness matrix) for undamped sites. The novelty of the method can be used for three-dimensional foundations. © 1997 by John Wiley & Sons, Ltd.     

基于混合边界条件的有限单元法GPR正演模拟

从Maxwell方程组出发,推导了探地雷达(GPR)有限元波动方程.阐述了透射边界条件和Sarma边界条件的原理,推导了这两种边界条件的理论公式;通过在衰减层内加入过渡带优化了Sarma边界条件的加载方法,压制了介质区和衰减层交界面处的人为反射.考虑到透射边界条件与Sarma边界条件不同的理论机制,提出了一种结合透射边界条件和Sarma边界条件的混合边界条件,它利用Sarma边界条件对到达边界区域的GPR波能量衰减功能和透射边界对GPR波能量的透射功能,使GPR波经过Sarma边界条件的衰减吸收后,再通过透射边界条件将剩余能量透射出去,集成了二者的优势.并以二维均匀模型中的中心脉冲激励源方式为例,通过Matlab程序实现,以GPR的全波场快照的直观方式,对比了有、无边界条件及不同边界条件对人工截断边界的处理效果,说明了该混合边界条件对到达截断边界处的GPR波的处理优于单一边界条件.最后,以基于混合边界条件的有限单元法对两个典型的GPR地电模型进行了正演模拟,指导了GPR数据处理与工程实践.     

Transient analysis of wave propagation in non‐homogeneous elastic unbounded domains by using the scaled boundary finite‐element method

The scaled boundary finite‐element method has been developed for the dynamic analysis of unbounded domains. In this method only the boundary is discretized resulting in a reduction of the spatial dimension by one. Like the finite‐element method no fundamental solution is required. This paper extends the scaled boundary finite‐element method to simulate the transient response of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. To reduce the number of degrees of freedom and the computational cost, the technique of reduced set of base functions is applied. The scaled boundary finite‐element equation for an unbounded domain is reformulated in generalized coordinates. The resulting acceleration unit‐impulse response matrix is obtained and assembled with the equation of motion of standard finite elements. Numerical examples of non‐homogeneous isotropic and transversely isotropic unbounded domains demonstrate the accuracy of the scaled boundary finite‐element method. Copyright © 2006 John Wiley & Sons, Ltd.     

饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法

本文基于Biot的饱和多孔介质本构方程,考察具有辐射阻尼的外行球面波,推导了饱和多孔介质三维黏弹性人工边界的法向和切向边界方程;在已有的饱和多孔介质二维显式有限元数值计算方法基础上,提出该理论的三维方法,并开发了实现该三维方法的有限元程序.算例表明饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法具有较好的精度和稳定性.     

Three‐dimensional controlled‐source electromagnetic modelling with a well casing as a grounded source: a hybrid method of moments and finite element scheme

Steel well casings in or near a hydrocarbon reservoir can be used as source electrodes in time‐lapse monitoring using grounded line electromagnetic methods. A requisite component of carrying out such monitoring is the capability to numerically model the electromagnetic response of a set of source electrodes of finite length. We present a modelling algorithm using the finite‐element method for calculating the electromagnetic response of a three‐dimensional conductivity model excited using a vertical steel‐cased borehole as a source. The method is based on a combination of the method of moments and the Coulomb‐gauged primary–secondary potential formulation. Using the method of moments, we obtain the primary field in a half‐space due to an energized vertical steel casing by dividing the casing into a set of segments, each assumed to carry a piecewise constant alternating current density. The primary field is then substituted into the primary–secondary potential finite‐element formulation of the three‐dimensional problem to obtain the secondary field. To validate the algorithm, we compare our numerical results with: (i) the analytical solution for an infinite length casing in a whole space, excited by a line source, and (ii) a three‐layered Earth model without a casing. The agreement between the numerical and analytical solutions demonstrates the effectiveness of our algorithm. As an illustration, we also present the time‐lapse electromagnetic response of a synthetic model representing a gas reservoir undergoing water flooding.     

饱水介质波动分析的谐波输入初始间断的处理方法

本文以饱水两相介质的土力学模型为研究对象,在假定两相介质为弹性介质条件下,采用了显式有限元法和透射边界进行了饱和弹性半空间动力响应问题的研究。为避免谐波输入初始间断的影响,文中提出了一个处理函数,并以弹性半空间为算例,对饱水介质和单相介质分别进行了在底边界P波垂直入射时的动力响应分析,验证了该处理函数的有效性和实用性。     

基于吸收边界条件的瞬变电磁法三维矢量有限元快速正演

瞬变电磁法的三维有限元正演通常采用齐次边界条件,为满足该边界条件,需要构建较大尺寸的模型,这降低了正演问题的求解速度.针对该问题,本文采用吸收边界条件代替齐次边界条件,以缩小模型体积,加快正演速度:首先,从时间域麦克斯韦方程组出发,推导了基于库伦规范的矢量势的微分控制方程,结合一阶吸收边界条件推导了相应的的弱形式方程;在此基础上采用一阶四面体矢量单元进行单元分析、Newmark法进行时间离散,实现了瞬变电磁法的快速三维正演.通过均匀半空间模型的解析解,H型地电断面的CR1Dmod解和相应模型有限元解的对比,验证了本文算法的正确性.均匀半空间模型分别采用吸收边界条件和齐次边界条件的正演结果对比表明:吸收边界条件确实可以提高三维正演的精度或者缩小模型尺寸、加快计算速度.     

斜入射条件下地下结构时域地震反应分析初探

通过采用平面波和远场散射波混合透射的应力人工边界条件,得到了地震波斜入射的解析方式,以此为基础建立了地震波斜入射条件下,土体与地下结构动力相互作用的时域计算分析模型。以实际建设的南京地铁某车站结构为研究对象,应用大型通用有限元分析软件ANSYS,进行了地震波斜入射条件下地下结构时域地震反应的计算和分析。初步结果表明:在地震波斜入射的情况下,地下结构的动力反应与地震波垂直入射时有较为明显的差异。     

A practical method for estimating dynamic soil stiffness on surface of multi‐layered soil

It is important to estimate the influence of layered soil in soil–structure interaction analyses. Although a great number of investigations have been carried out on this subject, there are very few practical methods that do not require complex calculations. In this paper, a simple and practical method for estimating the horizontal dynamic stiffness of a rigid foundation on the surface of multi‐layered soil is proposed. In this method, waves propagating in the soil are traced using the conception of the cone model, and the impulse response function can be calculated directly and easily in the time domain with a good degree of accuracy. The characteristics of the impedance, that is the transformed value to the frequency domain of the obtained impulse response, are studied using two‐ to four‐layered soil models. The cause of the fluctuation of impedance is expressed clearly from its relation to reflected waves from the lower layer boundary in the model. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic finite element analysis of three-dimensional soil models with a transmitting element

A method for the dynamic finite element analysis of a non-axisymmetric soil model with an axisymmetric boundary is presented. In the non-axisymmetric soil domain an arbitrary discretization with three-dimensional isoparametric solid elements is used. At the boundary a transmitting element is arranged. It is based on the semi-analytical element of Waas and Kausel. The transformation of the stiffness matrix of the Waas/Kausel element with cyclic symmetric displacements to general displacement fields is presented. For earthquake excitation the forces acting on the discretized domain are given. The method is illustrated by the dynamic analysis of an embedded box-type building. The distribution and magnitude of significant section forces are discussed.     

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。     

Finite element simulation of steady state SH wave motion

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An absorbing boundary condition for wave propagation in saturated poroelastic media — Part II: Finite element formulation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation towards infinity. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. In Part I (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 411-21), a local absorbing boundary condition for wave propagation in saturated poroelastic media has been developed. In the present paper, this boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid. Spurious reflections for oblique incident waves on the absorbing boundary contribute to the solution errors. Therefore, a spectral element method, based on classical analytical solution techniques, is used to assess the accuracy of the finite element formulation.     

3D scattering of obliquely incident plane SV waves by an alluvial valley embedded in a fluid-saturated,poroelastic layered half-space

The indirect boundary element method is used to study the 3D dynamic response of an infinitely long alluvial valley embedded in a saturated layered half-space for obliquely incident SV waves. A wave-number transform is first applied along the valley’s axis to reduce a 3D problem to a 2D plane strain problem. The problem is then solved in the section perpendicular to the axis of the valley. Finally, the 3D dynamic responses of the valley are obtained by an inverse wave-number transform. The validity of the method is confirmed by comparison with relevant results. The differences between the responses around the valley embedded in dry and in saturated poroelastic medium are studied, and the effects of drainage conditions, porosity, soil layer stiffness, and soil layer thickness on the dynamic response are discussed in detail resulting in some conclusions.