Analysis of ground motion caused by propagating air pressure waves

The soil is assumed to be a horizontally layered viscoelastic medium over a rigid base. A semi-analytical method is used to calculate the Green functions for line loads in the frequency domain. The displacements in the soil are then obtained by spatial convolution and Fourier transforms from the frequency to the time domain. The influence of significant parameters is studied for a homogeneous layer. Maximum response is obtained when the air pressure wave velocity equals the Rayleigh wave velocity in the soil. The response spectra computed for an actual soil profile, which is subjected to an air pressure wave from a deflagration-type explosion, show predominant response at frequencies, which are significantly higher than those of earthquake motion.     

Numerical Modeling of Transient Seismic Fields in Viscoelastic Media Based on the Laguerre Spectral Method

--The wave propagation in real media can be described within the theory of linear viscoelasticity. The presence of convolutional integral in Boltzmann's superposition principle poses the main difficulties in implementing the direct numerical methods in time domain. The paper presents a new algorithm, based on the application of the spectral Laguerre method for the approximation of temporal derivatives and convolution as applied to the problem of seismic wave propagation in the heterogeneous viscoelastic medium. Examples of the calculation of synthetic seismograms for different models of viscoelastic media are presented.     

Dynamic interaction numerical models in the time domain based on the high performance scaled boundary finite element method

Consideration of structure-foundation-soil dynamic interaction is a basic requirement in the evaluation of the seismic safety of nuclear power facilities. An efficient and accurate dynamic interaction numerical model in the time domain has become an important topic of current research. In this study, the scaled boundary finite element method （SBFEM） is improved for use as an effective numerical approach with good application prospects. This method has several advantages, including dimensionality reduction, accuracy of the radial analytical solution, and unlike other boundary element methods, it does not require a fundamental solution. This study focuses on establishing a high performance scaled boundary finite element interaction analysis model in the time domain based on the acceleration unit-impulse response matrix, in which several new solution techniques, such as a dimensionless method to solve the interaction force, are applied to improve the numerical stability of the actual soil parameters and reduce the amount of calculation. Finally, the feasibility of the time domain methods are illustrated by the response of the nuclear power structure and the accuracy of the algorithms are dynamically verified by comparison with the refinement of a large-scale viscoelastic soil model.     

Stability and identification for rational approximation of frequency response function of unbounded soil

Exact representation of unbounded soil contains the single output–single input relationship between force and displacement in the physical or transformed space. This relationship is a global convolution integral in the time domain. Rational approximation to its frequency response function (frequency‐domain convolution kernel) in the frequency domain, which is then realized into the time domain as a lumped‐parameter model or recursive formula, is an effective method to obtain the temporally local representation of unbounded soil. Stability and identification for the rational approximation are studied in this paper. A necessary and sufficient stability condition is presented based on the stability theory of linear system. A parameter identification method is further developed by directly solving a nonlinear least‐squares fitting problem using the hybrid genetic‐simplex optimization algorithm, in which the proposed stability condition as constraint is enforced by the penalty function method. The stability is thus guaranteed a priori. The infrequent and undesirable resonance phenomenon in stable system is also discussed. The proposed stability condition and identification method are verified by several dynamic soil–structure‐interaction examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic characteristics and seismic response of adjacent buildings linked by discrete dampers

Coupling adjacent buildings using discrete viscoelastic dampers for control of response to low and moderate seismic events is investigated in this paper. The complex modal superposition method is first used to determine dynamic characteristics, mainly modal damping ratio and modal frequency, of damper-linked linear adjacent buildings for practical use. Random seismic response of linear adjacent buildings linked by dampers is then determined by a combination of the complex modal superposition method and the pseudo-excitation method. This combined method can effectively and accurately determine random seismic response of non-classically damped systems in the frequency domain. Parametric studies are finally performed to identify optimal parameters of viscoelastic dampers for achieving the maximum modal damping ratio or the maximum response reduction of adjacent buildings. It is demonstrated that using discrete viscoelastic dampers of proper parameters to link adjacent buildings can reduce random seismic responses significantly. Copyright © 1999 John Wiley & Sons Ltd.     

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.     

弱形式时域完美匹配层——滞弹性近场波动数值模拟

本文旨在构建适用于滞弹性近场时域波动有限元模拟的高精度人工边界条件：完美匹配层（Perfectly Matched Layer：PML）,其中阻尼介质时域本构基于广义标准线性体建立.与以往研究不同,本文采用复坐标延拓技术变换弱形式波动方程构建了可直接用有限元离散的弱形式时域PML,规避以往独立对无限域内波动方程及界面条件进行延拓可导致的PML场方程和界面条件匹配不合理引发数值失稳、计算精度低下等问题.其次,针对PML中多极点有理分式与频域函数乘积的傅里叶反变换难以计算的问题,利用PML精度对复坐标延拓函数中延拓参数微调不敏感这一特点,明确给出了参数微调准则以规避多重极点,进而利用有理分式分解给出了一种普适、简便的计算方法,极大地简化了PML计算.基于该方法可实现任意高阶PML.最后,将本文构建滞弹性PML与高阶勒让德谱元（高精度集中质量有限元）结合得到滞弹性近场波动谱元离散方案.基于算例验证了滞弹性PML的计算效率、精度及新离散方案的长持时稳定特性.新离散方案可应用于计入实际介质阻尼的地震波动正、反问题数值模拟,提高波形模拟的精度以及地下波速结构反演的精度和可靠性.     

Time‐domain viscoelastic analysis of earth structures

In this paper two causal models that approximate the nearly frequency‐independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time‐domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non‐causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero‐frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non‐causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high‐frequency limit of the Biot model. Although both models have a closed‐form time‐domain representation, only the Biot model is suitable for a time‐domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse‐type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time‐domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems. When explicit time-domain integration algorithms are used, the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary. The lack of a clear and practical stability criterion for this problem, however, affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries. In this study, we investigate the stability conditions of explicit integration algorithms when using three-dimensional (3D) viscoelastic artificial boundaries through an analysis method based on a local subsystem. Several boundary subsystems that can represent localized characteristics of a complete numerical model are established, and their analytical stability conditions are derived from and further compared to one another. The stability of the complete model is controlled by the corner regions, and thus, the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained. Next, by analyzing the impact of different factors on stability conditions, we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

     

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.     

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.     

基于卷积完全匹配层的非规则网格时域有限元探地雷达数值模拟

复频移完全匹配层(Complex Frequency-Shifted PML,CFS-PML)在长时间时域计算中对凋落波、倏失波具有好的吸收效果,并被广泛应用于时域有限差分模拟中.而本文采用卷积方法将CFS-PML应用于时域有限元求解GPR波动方程的数值模拟中.论文以TM波为例,推导了基于CPML(Convolutional PML)边界的时域有限元GPR波动方程求解公式,采用Newmark-β方法对时间导数进行离散,有效改善了时域有限元GPR数值计算程序的稳定性.并以狭长模型为例,开展了CPML边界中关键参数m、R和κ的选取实验,通过对比反射误差大小确定了综合最优参数组合.相同时刻UPML与CPML波场快照、3个检测点的反射误差比较,说明CPML较UPML具有更好的吸收效果.最后,采用非规则四边形网格对1个复杂GPR模型进行剖分,应用加载CPML边界条件的FETD程序对该模型进行了正演,得到了二维剖面法、宽角法正演GPR剖面图,说明非规则四边形对复杂模型的良好适应性,基于CPML边界条件的FETD可有效减少边界反射误差,能实现对任意复杂不规则模型的正演模拟.     

基于delta波包叠加的时间域深度偏移

delta波包可由高斯波束经傅里叶逆变换得到,是高斯波束在时空域的表达.它最早出现在合成理论地震图的研究中,本文将其应用于偏移领域.通过delta波包叠加表达时间域格林函数,可将高斯波束偏移由频率域转换到时间域,再结合Rayleigh积分和激励时间成像条件,本文给出了基于delta波包叠加的深度偏移算法.该偏移算法可在时间域直接计算,但因包含褶积运算,成像时将耗费大量的计算时间.针对这一问题,本文提出了将褶积简化为乘积的近似公式.近似后的偏移算法,不仅保留了高斯波束偏移的优点,而且计算效率得到显著提升.文中通过两个数值算例验证了上述结论.     

Non-linear soil-structure-interaction analysis using dynamic stiffness or flexibility of soil in the time domain

The basic equation of motion to analyse the interaction of a non-linear structure and an irregular soil with the linear unbounded soil is formulated in the time domain. The contribution of the unbounded soil involves convolution integrals of the dynamic-stiffness coefficients in the time domain and the corresponding motions. Alternatively, a flexibility formulation for the contribution of the unbounded soil using the dynamic-flexibility coefficients in the time domain, together with the direct-stiffness method for the structure and the irregular soil can be applied. The dynamic-stiffness or flexibility coefficient in the time domain is calculated as the inverse Fourier transform of the corresponding value in the frequency domain. The dynamic-stiffness coefficient's asymptotic behaviour for high frequencies determines the singular part whose transformation exists only in the sense of a distribution. As the dynamic-flexibility coefficient converges to zero for the frequency approaching infinity, the corresponding coefficient in the time domain is simpler to calculate, as no singular part exists. The salient features of the dynamic-stiffness and flexibility coefficients in the time domain are illustrated using a semi-infinite rod with exponentially increasing area. The dynamic-flexibility coefficients in the time domain are calculated for a rigid circular disc resting on the surface of an elastic halfspace and of a layer built-in at its base. Material damping is also introduced using the three-parameter Kelvin and the Voigt models.     

Mode-based equivalent multi-degree-of-freedom system for one-dimensional viscoelastic response analysis of layered soil deposit

Discrete models such as the lumped parameter model and the finite element model are widely used in the solution of soil amplification of earthquakes. However, neither of the models will accurately estimate the natural frequencies of soil deposit, nor simulate a damping of frequency independence. This research develops a new discrete model for one-dimensional viscoelastic response analysis of layered soil deposit based on the mode equivalence method. The new discrete model is a one-dimensional equivalent multi-degree-of-freedom (MDOF) system characterized by a series of concentrated masses, springs and dashpots with a special configuration. The dynamic response of the equivalent MDOF system is analytically derived and the physical parameters are formulated in terms of modal properties. The equivalent MDOF system is verified through a comparison of amplification functions with the available theoretical solutions. The appropriate number of degrees of freedom (DOFs) in the equivalent MDOF system is estimated. A comparative study of the equivalent MDOF system with the existing discrete models is performed. It is shown that the proposed equivalent MDOF system can exactly present the natural frequencies and the hysteretic damping of soil deposits and provide more accurate results with fewer DOFs.     

A simplified model for lateral response of large diameter caisson foundations—Linear elastic formulation

The transient response of large embedded foundation elements of length-to-diameter aspect ratio D/B=2–6 is characterized by a complex stress distribution at the pier–soil interface that cannot be adequately represented by means of existing models for shallow foundations or flexible piles. On the other hand, while three-dimensional (3D) numerical solutions are feasible, they are infrequently employed in practice due to their associated cost and effort. Prompted by the scarcity of simplified models for design in current practice, we here develop an analytical model that accounts for the multitude of soil resistance mechanisms mobilized at their base and circumference, while retaining the advantages of simplified methodologies for the design of non-critical facilities. The characteristics of soil resistance mechanisms and corresponding complex spring functions are developed on the basis of finite element simulations, by equating the stiffness matrix terms and/or overall numerically computed response to the analytical expressions derived by means of the proposed Winkler model. Sensitivity analyses are performed for the optimization of the truncated numerical domain size, the optimal finite element size and the far-field dynamic boundary conditions to avoid spurious wave reflections. Numerical simulations of the transient system response to vertically propagating shear waves are next successfully compared to the analytically predicted response. Finally, the applicability of the method is assessed for soil profiles with depth-varying properties. The formulation of frequency-dependent complex spring functions including material damping is also described, while extension of the methodology to account for nonlinear soil behavior and soil–foundation interface separation is described in the conclusion and is being currently investigated.     

Numerical viscoelastic modelling by the spectral Laguerre method

Seismic wave propagation in a viscoelastic media can be described by a system of integro-differential equations. The solution of such equations requires special methods when using finite-difference techniques in the time domain. In the frequency domain, the integral terms are represented by complex elastic parameters. This paper presents an efficient algorithm for viscoelastic modelling based on the integral Laguerre transform for the approximation of temporal derivatives and for the calculation of convolution integrals. For the calculation of spatial derivatives, it is possible to use various methods: finite-difference and finite-element techniques, spectral and pseudo-spectral methods. We then obtain a system of algebraic equations with a matrix independent of the parameter m , i.e. the degree of the Laguerre polynomials. In this case, only the right-hand side of the system has recurrent dependence on the parameter m , which is an analogue of the temporal frequency in the frequency domain. The obtained system with a large number of right-hand sides can be solved using fast methods, where the matrix is transformed only once, as opposed to the frequency-domain approach, when the matrix is transformed for each temporal frequency.     

Time-domain identification of dynamic properties of layered soil by using extended Kalman filter and recorded seismic data

A novel time-domain identification technique is developed for the seismic response analysis of soil-structure interaction. A two-degree-of-freedom (2DOF) model with eight lumped parameters is adopted to model the frequency-dependent behavior of soils. For layered soil, the equivalent eight parameters of the 2DOF model arc identified by the extended Kalman filter (EKF) method using recorded seismic data. The polynomial approximations for derivation of state estimators are applied in the EKF procedure. A realistic identification example is given for the layered-soil of a building site in Anchorage, Alaska in the United States. Results of the example demonstrate the feasibility and practicality of the proposed identification technique. The 2DOF soil model and the identification technique can be used for nonlinear response analysis of soil-structure interaction in the time-domain for layered of complex soil conditions. The identified parameters can be stored in a database for use in other similar soil conditions. If a universal database that covers information related to most soil conditions is developed in the future, engineers could conveniently perform time history analyses of soil-structural interaction.     

COMPUTATION OF ZERO-OFFSET VERTICAL SEISMIC PROFILES INCLUDING GEOMETRICAL SPREADING AND ABSORPTION*

Synthetic vertical seismic profiles (VSP) provide a useful tool in the interpretation of VSP data, allowing the interpreter to analyze the propagation of seismic waves in the different layers. A zero-offset VSP modeling program can also be used as part of an inversion program for estimating the parameters in a layered model of the subsurface. Proposed methods for computing synthetic VSP are mostly based on plane waves in a horizontally layered elastic or anelastic medium. In order to compare these synthetic VSP with real data a common method is to scale the data with the spherical spreading factor of the primary reflections. This will in most cases lead to artificial enhancement of multiple reflections. We apply the ray series method to the equations of motion for a linear viscoelastic medium after having done a Fourier transformation with respect to the time variable. This results in a complex eikonal equation which, in general, appears to be difficult to solve. For vertically traveling waves in a horizontally layered viscoelastic medium the solution is easily found to be the integral along the ray of the inverse of the complex propagation velocity. The spherical spreading due to a point source is also complex, and it is equal to the integral along the ray of the complex propagation velocity. Synthetic data examples illustrate the differences between spherical, cylindrical, and plane waves in elastic and viscoelastic layered media.     

System parameters of soil foundations for time domain dynamic analysis

Soil-structure interaction analysis is usually carried out in the frequency domain, because the compliance functions of the half-space are known only in the frequency domain. Since non-linear analysis cannot be carried out in the frequency domain, a system with frequency independent parameters is used to represent the half-space soil medium so that a nonlinear analysis in the time domain becomes possible. The objective of this paper is to propose a system with lumped parameters, which are independent of frequency, to represent the half-space soil medium. The proposed frequency independent system consists of a number of real discrete structure elements; thus the existing dynamic analysis programs may be adoptable with little modification. In this paper, the parameters are found by minimizing the sum of the squares of deviations between the steady-state responses of the theoretical half-space model and those of the lumped parameter system over a specified frequency range. Once the parameters have been found, the lumped parameter system can be used in practical applications for time domain dynamic analysis of either linear or non-linear structures. In comparison with the dynamic response of the theoretical half-space model, the lumped parameter system yields satisfactory results.