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1.
The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil-structure-interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic-stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.
- (i) The impulse-invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement.
- (ii) In the segment approach, the dynamic-stiffness coefficients in the time domain are interpolated piecewise. Applying the z-transformation analytically then results in an explicit recursive equation without solving a system of equations.
2.
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. 相似文献
3.
To calculate the hydrodynamic interaction forces of the reservoir directly in the time-domain, the dynamic stiffness of each mode of the semi-infinite uniform fluid channel is either represented by a lumped-parameter model with frequency-independent real coefficients of the springs, dashpots and masses and with only a few additional internal degrees of freedom, or the interaction forces are calculated recursively. For each mode characterized by its eigenvalue, the coefficients of the lumped-parameter model and the recursive coefficients are specified, which can be used directly in a practical application. The procedures exhibit many advantages: the only approximation (replacing the rigorous dynamic stiffness by a ratio of two polynomials) can be evaluated visibly. No unfamiliar discrete-time manipulations such as the z-transformation are used. The stiffness, damping and mass matrices corresponding to the lumped-parameter model are automatically symmetrical. Stability of the procedures is also guaranteed. Combining the lumped-parameter model of the semi-infinite uniform channel with the finite-element discretization of the irregular fluid region or calculating the interaction forces recursively allows a reservoir of arbitrary shape to be analysed directly in the time domain. Non-linearities in the dam can, thus, be taken into consideration in a seismic analysis. 相似文献
4.
We have developed a novel method for missing seismic data interpolation using f‐x‐domain regularised nonstationary autoregression. f‐x regularised nonstationary autoregression interpolation can deal with the events that have space‐varying dips. We assume that the coefficients of f‐x regularised nonstationary autoregression are smoothly varying along the space axis. This method includes two steps: the estimation of the coefficients and the interpolation of missing traces using estimated coefficients. We estimate the f‐x regularised nonstationary autoregression coefficients for the completed data using weighted nonstationary autoregression equations with smoothing constraints. For regularly missing data, similar to Spitz f‐x interpolation, we use autoregression coefficients estimated from low‐frequency components without aliasing to obtain autoregression coefficients of high‐frequency components with aliasing. For irregularly missing or gapped data, we use known traces to establish nonstationary autoregression equations with regularisation to estimate the f‐x autoregression coefficients of the complete data. We implement the algorithm by iterated scheme using a frequency‐domain conjugate gradient method with shaping regularisation. The proposed method improves the calculation efficiency by applying shaping regularisation and implementation in the frequency domain. The applicability and effectiveness of the proposed method are examined by synthetic and field data examples. 相似文献
5.
A discrete model to represent the unbounded soil (halfspace) in a soil–structure interaction analysis in the time domain is developed. For each dynamic degree of freedom of the foundation node, the discrete model consists of a mass M0 which is attached to a rigid support with a spring K and with a damper C0. In addition, a free node with the mass M1 is introduced, which is connected to the foundation node with a damper C1. All coefficients are frequency-independent. The discrete model is semi-empirical. It is based on a semi-infinite truncated cone, whereby, after enforcing the static stiffness, the remaining parameters are modified to achieve an optimal fit of the dynamic-stiffness coefficient in the frequency domain. The spring K is equal to the static stiffness. The coefficients appearing in the equations for the dampers C0, C1 and the masses M0, M1 are specified (assuming a homogeneous halfspace) for the disc, the embedded cylinder, the rectangle (also embedded) and the strip. A square on a layer whose stiffness increases with depth resting on a homogeneous halfspace is also treated. For an embedded foundation, eccentricities arise. Material damping increases the damper C0 and the mass M0. 相似文献
6.
The problem of the propagation of acoustic waves in a two-dimensional layered medium can be easily solved in the frequency domain if the Dix approximation is used, i.e. when only the primary reflections are considered. The migrated data at a depth z are obtained by convolving the time section with a proper two-dimensional operator dependent on z. The same result can be obtained by multiplying their two-dimensional spectra and summing for all the values of the temporal frequency. The aspect of the operator in the time-space domain has the classic hyperbolic structure together with the prescribed temporal and spatial decay. The main advantages of the frequency domain approach consist in the noticeable computer time savings and in the better approximation. On the other hand lateral velocity variations are very difficult to be taken into account. This can be done if a space variant filter is used in the time-space domain. To reduce computer time, this filter has to be recursive; the problem has been solved by Claerbout by transforming the hyperbolic partial differential equation into a parabolic one, and using the latter to generate the recursion operator. In the presentation a method is given for the generation of recursive filters with a better phase characteristics that have a pulse response with the requested hyperbolic shape instead of the parabocli one. This allows a better migration of steeper dips. 相似文献
7.
John P. Wolf 《地震工程与结构动力学》1991,20(1):11-32
A systematic procedure to develop a consistent lumped-parameter model with real frequency-independent coefficients to represent the unbounded soil is developed. Each (modelled) dynamic-stiffness coefficient in the frequency domain is approximated as a ratio of two polynomials, which is then formulated as a partial-fraction expansion. Each of these terms is represented by a discrete model, which is the building block of the lumped-parameter model. A second-order term, for example, leads to a discrete model with springs and dampers with two internal degrees of freedom, corresponding to two first-order differential equations, or, alternatively, results in a discrete model with springs, dampers and a mass with one internal degree of freedom, corresponding to one second-order differential equation. The lumped-parameter model can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear. A thorough evaluation shows that highly accurate results are achieved, even for dynamic systems with a cutoff frequency. 相似文献
8.
To calculate the dynamic-stiffness matrix in the time domain (unit-impulse response functions) of the unbounded medium, the infinitesimal finite element cell method based solely on the finite element formulation and working exclusively in the time domain is developed. As in the cloning algorithm, the approach is based on similarity of the unbounded media corresponding to the interior and exterior boundaries of the infinitesimal finite element cell. The derivation can be performed exclusively in the time domain, or alternatively in the frequency domain. At each time station a linear system of equations is solved. The consistent-boundary method to analyse a layered medium in the frequency domain and the viscous-dashpot boundary method are special cases of the infinitesimal finite element cell method. The error is governed by the finite element discretization in the circumferential direction, as the width of the finite-element cell in the radial direction is infinitesimal. The infinitesimal finite element cell method is thus ‘exact in the finite-element sense’. This method leads to highly accurate results for a vast class of problems, ranging from a one-dimensional spherical cavity to a rectangular foundation embedded in a half-plane. 相似文献
9.
Acoustic plane wave scattering at a vertical fault structure represents the simplest two-dimensional model of geophysical exploration that can be investigated by analytical techniques. The exact and complete solution, in the time domain, for the scattering of the pressure field of an acoustic plane wave normally incident on a vertical fault structure is determined adapting previous results given for the frequency domain. The wave form of the pressure field of the incident plane wave is expressed by a causal time function that decays exponentially with time at every point above the fault (z<0). The zero-order term of the scattered pressure field has been computed above the fault. This zero-order term consists of an inverse Fourier transform which reduces to a closed expression forx=0, and contains an integral of a Hankel function forx#0. The high frequency part of the inverse Fourier transform forx#0 is computed employing asymptotic expressions for the Hankel function. The integral of the asymptotic expression of the Hankel function reduces to: (i) a Fresnel integral which contains a plane wave term for |x||z|; and (ii) a stationary point plane wave term plus an upper limit term for |x|=O(|z|). For the latter case the plane wave term cancels, leaving a cylindrical wave emanated from the edge of the fault. The wave front is well defined in shape, in phase and in amplitude. The amplitude of the scattered field is discontinuous atx=0, presents a jump and is well defined for |x| small and is rather smooth for |x| large. 相似文献
10.
Approximate dynamic-stiffness coefficients of a disk on the surface of a single layer on a half-space may be calculated using cone models. This concept is generalized to the case of a horizontally stratified site consisting of many layers on a homogeneous half-space. After constructing the so-called ‘backbone cone’ determining the radii of the disks at all interfaces, the dynamic-stiffness matrices of the layers (modelled as cone frustums) and the dynamic-stiffness coefficient of the underlying half-space (modelled as a cone) are assembled to that of the site. The dynamic-stiffness matrix of a layer is a complex-valued function of frequency because radiation of energy in the horizontal direction is considered. In this model of the layered half-space the properties of the cone reproduce themselves (cloning). The advantages of using cone models are also present for the layered half-space; in particular, no transformation to the wave-number domain is performed. 相似文献
11.
W. G. CLEMENT 《Geophysical Prospecting》1973,21(1):125-145
The geophysicist involved in the analysis of two-dimensional data should have an understanding of the two-dimensional finite Fourier transform and the mechanics of two-dimensional filtering. Frequency aliasing must be considered when working with sampled data. In two dimensions it is advantageous to consider aliasing in terms of the overlap of the repeating spectra inherent in the finite Fourier transform. Two-dimensional filtering can be performed as a transient convolution in the space domain, as cyclic convolution utilizing the frequency domain or as the multiplication of polynomials using the z-transform. If the “edge” effects are removed, the results of the three methods are identical. 相似文献
12.
The least squares estimation procedures used in different disciplines can be classified in four categories:
- a. Wiener filtering,
- b. b. Autoregressive estimation,
- c. c. Kalman filtering,
- d. d. Recursive least squares estimation.
13.
14.
A systematic procedure to construct the (symmetric) static-stiffness, damping and mass matrices representing the unbounded medium is presented addressing the unit-impulse response matrix corresponding to the degrees of freedom on the structure–medium interface. The unit-impulse response matrix is first diagonalized which then permits each term to be modelled independently from the others using expansions in a series of Legendre polynomials in the time domain. This leads to a rational approximation in the frequency domain of the dynamic-stiffness coefficient. Using a lumped-parameter model which provides physical insight the property matrices are constructed. 相似文献
15.
In the damping-solvent extraction method, to calculate the dynamic-stiffness matrix of an unbounded medium, a finite region of the medium, adjacent to the structure is analysed in the first step, whereby hysteretic material damping is introduced artificially as a solvent. This leads to the dynamic-stiffness matrix of the damped bounded medium, which is assumed in the second step to be equal to that of the damped unbounded medium. In the third step, the effect of the material damping on the dynamic-stiffness matrix is eliminated, i.e. the damping solvent is extracted, resulting in the dynamic-stiffness matrix of the unbounded medium. The damping-solvent extraction method permits an efficient calculation of the dynamic-stiffness matrix of an unbounded medium by analysing the adjacent bounded medium only, which exhibits the same dynamic characteristics as the (bounded) structure. The familiar standard finite-element method is sufficient for the analysis and the hysteretic damping is introduced by multiplying the elastic moduli by 1 + 2i£. The introduced hysteretic material damping, the solvent, is extracted at the end of the analysis for each coefficient of the dynamic-stiffness matrix and for each frequency independently of the others by a very concise equation based on a Taylor expansion. The method is evaluated thoroughly for dynamic soil-structure interaction and for seismic reservoir-dam interaction using stringent simple cases with analytical solutions available and is also applied to practical examples, by calculating the dynamic-stiffness matrix of a semi-infinite wedge and an embedded foundation. 相似文献
16.
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. 相似文献
17.
A technique of interpolation based on a stochastic approach and referred to as ‘kriging’ technique has recently been contributed by the French School. A primary feature of the algorithm is its ability to provide an assessment of the predictive reliability. The accuracy of estimate depends on the evaluation of two stochastic quantities: the variogram γ and the main trend m of the hydrologic event z to be reconstructed.For an effective use of the method a correct understanding of the actual role played by m is required. With some ad hoc examples it is shown that using a polynomial trend with unspecified coefficients as suggested by the general theory may lead to paradoxical results whose behaviour is hard to predict a priori. It turns out that increasing the degree of m may yield an increase of the estimation error where one would expect to obtain the opposite.An alternative formulation is suggested which assumes m to be fully known in advance. Its expression is supposed to be derived from both the general behaviour of z as is recognizable from the available records and some extra-amount of information related to the general physical knowledge of the hydrological context. If this extra-amount of information is missing, the use of a constant trend should be recommended. 相似文献
18.
The contribution of the (linear) unbounded soil to the basic equation of motion of a non-linear analysis of soil-structure interaction consists of convolution integrals of the displacement-force relationship in the time domain and the history of the interaction forces. The former is calculated using the indirect boundary-element method, which is based on a weighted-residual technique and involves Green's functions. As an example of a non-linear soil-structure-interaction analysis, the partial uplift of the basemat of a structure is examined. As the convolution integrals have to be recalculated for each time step, the computational effort in this rigorous procedure is substantial. A reduction can be achieved by simplifying the Green's function by ‘concentrating’ the region of influence. Alternatively, assuming a specified wave pattern, a coupled system of springs and dashpots with frequency-independent coefficients can be used as an approximation. 相似文献
19.
The authors present a method for estimation of interval velocities using the downward continuation of the wavefield to perform
layer-stripping migration velocity analysis. The generalized, phase-shift migration MG(F-K) in wavenumber-frequency domain
was used for fulltime downward extrapolation of the wavefield. Such downward depth extrapolation accounts for strong changes
of velocity in lateral and vertical directions and helps in correct positioning of the wavefield image in complex structures.
Determination of velocity is the recursive process which means that the wavefield on depth level z
n−1 (n = 0, 1, ...) is an input data-set for determination of velocity on level z
n
. The velocity ν [x, z
n
− z
n−1] can be thus treated as interval velocity in Δz
n
= z
n
− z
n−1 step. This method was tested on synthetic Marmousi data-set and showed satisfactory results for complex, inhomogeneous media. 相似文献
20.
A time domain boundary element in a cylindrical coordinate system is developed for the analysis of wave propagation in a half space. The integral formulation is based on Graffi's dynamic reciprocal theorem and Stokes' fundamental solutions. The field quantities (displacements and tractions) are expressed as products of Fourier series in the tangential direction and linear polynomials in the other spatial directions. Gaussian integration is used to integrate the non-singular parts of the integral equations, whereas the integration of the singular components, which are either of order 1/r or 1/r2, is handled by special numerical schemes. In the time marching aspect, the field quantities are assumed to vary linearly in the temporal direction as well. Examples for wave propagation due to various forms of surface excitations are reported to demonstrate the accuracy of the method. 相似文献