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1.
Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling 
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下载免费PDF全文 We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion. 相似文献
2.
We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method. 相似文献
3.
A finite element viscoplastic computation is proposed where the strain dependent nonlinear stiffness matrix represents plasticity effects and a general nonlinear strain-rate dependent damping matrix accounts for the overall viscous loss. This model can assimilate any conventional plasticity data measured in the laboratory, where the loss coefficient is characterized by Q−1. In field testing the same is estimated from seismological observations, usually stated as a strain, strain-rate (or frequency) independent loss factor. It is demonstrated herein that the solution of any auxiliary differential equation even for the constant Q model can be avoided when a Laurent series expansion is sought where the coefficients are calculated by a least square fit of the experimental Q-data. Therein the causality condition is satisfied exactly. Since the procedure yields an integro-differential equation the required time steps are considerably small as compared with those in standard explicit and implicit schemes. 相似文献
4.
Iterative implicit integration procedure for hybrid simulation of large nonlinear structures 总被引：1，自引：0，他引：1
A fully implicit iterative integration procedure is presented for local and geographically distributed hybrid simulation of the seismic response of complex structural systems with distributed nonlinear behavior. The purpose of this procedure is to seamlessly incorporate experimental elements in simulations using existing fully implicit integration algorithms designed for pure numerical simulations. The difficulties of implementing implicit integrators in a hybrid simulation are addressed at the element level by introducing a safe iteration strategy and using an efficient procedure for online estimation of the experimental tangent stiffness matrix. In order to avoid physical application of iterative displacements, the required experimental restoring force at each iteration is estimated from polynomial curve fitting of recent experimental measurements. The experimental tangent stiffness matrix is estimated by using readily available experimental measurements and by a classical diagonalization approach that reduces the number of unknowns in the matrix. Numerical and hybrid simulations are used to demonstrate that the proposed procedure provides an efficient method for implementation of fully implicit numerical integration in hybrid simulations of complex nonlinear structures. The hybrid simulations presented include distributed nonlinear behavior in both the numerical and experimental substructures. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
5.
Zbigniew Sienkiewicz 《地震工程与结构动力学》1993,22(11):1009-1014
Approximate formulas are derived to evaluate in the frequency domain the dynamic impedances of a weakly anelastic medium based on its pure elastic behaviour. The correspondence principle is applied to the elastic solution of a boundary-value problem followed by the expansion of the anelastic solution in a Taylor series about the elastic state. Taking the magnitude of material damping into account (small damping) only the first two terms of the Taylor series have been used. The derivatives of first order in the expansion can be determined by the central difference approximation; this requires only the evaluation of differences between neighbouring elastic solutions. 相似文献
6.
在深度偏移方法中，把二维隐式方法推广到三维，就会面对一个分块对角矩阵求逆问题. 通常，这种矩阵的求逆将耗费大量计算时间，严重制约了三维隐式方法偏移在实际资料处理中的广泛应用. 在螺旋边界条件下，该矩阵H具有Toeplitz结构的正定厄密矩阵，其快速求逆可由谱法LU分解或直解法快速实现. 本文结合谱法LU分解和直接解法方法的优点，提出了一种混合算法. 文中采用谱分解方法建立起矩阵列元素的谱分解表，并采用直解法的递推公式，可以快速给出矩阵的分解. 通过与谱法分解和直解法在分解精度和分解速度两方面的比较表明，本文方法与谱法相比，在非均匀介质中亥姆霍兹算子矩阵分解时的精度提高10倍；在计算速度方面，混合方法比简化后的直解法快. 因此，该方法的提出，在计算精度许可的条件下，最大限度地减少三维隐式差分偏移中矩阵求逆占用的时间，从而使得该方法能真正用于实际地震资料的处理. 相似文献
7.
位场向下与向上延拓之间存在固有的内在联系,向上延拓解算具有稳定可靠的优良特性,本文据此提出了借助向上延拓信息实现航空重力向下延拓稳定解算的两种方法,分别建立了点对点向下解析延拓模型和最小二乘向下解析延拓模型.其核心思想是,依据泰勒级数展开模型,将位场向下延拓解算过程转换为向上延拓计算和垂向偏导数解算两个步骤,通过第一步的处理有效抑制数据观测噪声对解算结果的干扰,通过第二步的处理成功实现向下延拓反问题的稳定解算,较好地解决了向下延拓解算固有的不适定性问题.分析研究了两种解析延拓模型的计算精度及适用条件,利用超高阶位模型EGM2008建立的模拟标准场数据对两种模型解算结果的合理性和有效性进行了数值验证,证明本文新方法实用易行,具有较高的应用价值. 相似文献
8.
将Newmark-β法中常平均加速度法的基本假定与精细指数算法结合,根据指数矩阵的Taylor级数展开式,提出了动力方程的显式级数解,并设计了相应的时程积分算法.该算法的精度可根据Taylor级数展开式的项数进行灵活控制.算例的结果表明:在满足稳定性条件的前提下,随着时间步长的增加,其精度优于传统的时程积分法.通过稳定性的分析,指出其稳定性条件是显然满足的. 相似文献
9.
The magnetic interface forward and inversion method is realized using the Taylor series expansion to linearize the Fourier transform of the exponential function. With a large expansion step and unbounded neighborhood, the Taylor series is not convergent, and therefore, this paper presents the magnetic interface forward and inversion method based on Padé approximation instead of the Taylor series expansion. Compared with the Taylor series, Padé’s expansion’s convergence is more stable and its approximation more accurate. Model tests show the validity of the magnetic forward modeling and inversion of Padé approximation proposed in the paper, and when this inversion method is applied to the measured data of the Matagami area in Canada, a stable and reasonable distribution of underground interface is obtained. 相似文献
10.
土层参数的随机性对场地传递函数的影响 总被引：6，自引：0，他引：6
本文采用秦勒级数展开估计土层参数随机性对场地传递函数的影响。对单层土模型在土层参数服从独立高斯分布假定下进行了公式推导和实例计算,比较了不同的泰勒级数阶次、土层参数变异系数以及阻尼比对传递系数随机性的综合效应。结果表明,即使在小参数摄动范围内采用一阶泰勒级数也是不够的;一般情况下采用二阶泰勒级数即可给出满意的估计。 相似文献
11.
IntroductionThe analysis of dynamic soil-structure interaction for important engineering project is still based on linear model (including equivalent linear model) with complex damping, and traditional frequency domain method (Lysmer, et al, 1975, 1981; DING, et al, 1999). Namely, first calculating frequency domain solution by Fourier transform, and then calculating time domain solution by Fourier inverse transform. The motion equation of a system in frequency domain is usually written as (… 相似文献
12.
13.
《Advances in water resources》2005,28(8):793-806
In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution. 相似文献
14.
B. URSIN 《Geophysical Prospecting》1982,30(5):569-579
The principles for ray-tracing and wavefront curvature calculations in a three-dimensional medium are reviewed. A new derivation of the transformation of the wavefront curvature matrix at an interface between two inhomogeneous media is given. The derivation is based on a Taylor series expansion of the ray refraction equation at the interface between two inhomogeneous media, and only elementary geometric arguments are used. The wave-front curvature transformation at the interface is obtained by neglecting all terms in the direction of the surface normal. With proper definition of the variables, the derivation is also valid for a reflected wave-front. A simplified transformation rule is derived for a reflected wave of the same type as the incident wave. 相似文献
15.
The time integration method proposed by Kolay and Ricles, which was claimed to be both explicit and unconditionally stable, is shown to be implicit in the sense of requiring the factorization of an effective stiffness matrix where an explicit method needs no solver. Its original derivation procedure employed discrete control theory concepts, which are in fact, equivalent to conventional recurrence relation concepts aiming to match its spectral properties with those of the three-parameter optimal/generalized-α method, thus giving rise to an implicit method within the class of linear multistep methods. It is shown that the resulting method possesses several added computational drawbacks due to its derivation procedure, such as additional effective stiffness inversions and a degraded order of accuracy in general. 相似文献
16.
An accurate algorithm for the integration of the equations of motion arising in structural dynamics is presented. The algorithm is an unconditionally stable single-step implicit algorithm incorporating algorithmic damping. The displacement for a Single-Degree-of-Freedom system is approximated within a time step by a function which is cubic in time. The four coefficients of the cubic are chosen to satisfy the two initial conditions and two weighted integral equations. By considering general weight functions, eight additional coefficients arise. These coefficients are selected to (i) minimize the difference between exact and approximate solutions for small time steps, (ii) incorporate specified algorithmic damping for large time steps, (iii) ensure unconditional stability and (iv) minimize numerical operations in forming the amplification matrix. The accuracy of the procedure is discussed, and the solution time is compared with a widely used algorithm. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
17.
A. C. LIAKOPOULOS Dr. Eng. 《水文科学杂志》2013,58(1):69-110
ABSTRACT The one-dimensional transient downward entry of water in unsaturated soils is investigated theoretically. The mathematical equation describing the infiltration process is derived by combining Darcy's dynamic equation of motion with the continuity and thermodynamic state equations adjusted for the unsaturated flow conditions. The resulting equation together with the corresponding initial and boundary conditions constitues a mathematical initial boundary value problem requiring the solution of a nonlinear partial differential equation of the parabolic type. The volumetric water content is taken as the dependent variable and the time and the position along the vertical direction are taken as the independent variables. The governing equation is of such nature that a solution exists for t > 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0 and at the two boundaries. The two required relations are those of pressure versus permeability and pressure versus volumetric water content. Since the partial differential equation has strong non-linear terms, a discrete solution is obtained by approximating the derivatives with finite-differences at discrete mesh points in the solution domain and integrated for the corresponding initial and boundary conditions. The use of an implicit difference scheme is employed in order to generate a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with a back-substitution technique. The FORTRAN statements used for obtaining the solution with an electronic digital computer (IBM 704) are presented together with the input data. Analysis of the errors involved in the numerical solution is made and the stability and convergence of the solution of the approximate difference equation to that of the differential equation is investigated. The method applied is that of making a Fourier series expansion of a whole line of errors and then following the progress of the general term of the series expansion and also the behavior of each constituent harmonic. The errors (forming a continuous function of points in an abstract Banach space) are represented by vectors with the Fourier coefficients constituting a second Banach space. The amplification factor of the difference equation is shown to be always less than unity which guarantees the stability of the employed implicit recurrence scheme. Experiments conducted on a vertical column packed uniformly with very fine sand, show a satisfactory agreement between the theoretically and experimentally obtained values. Many experimental results are shown in an attempt to explain the infiltration phenomenon with emphasis on the shape and movement of the wet front, and the effects of the degree of compaction, initial water content and deaired water on the infiltration rate. 相似文献
18.
Glenn R. Ierley 《地球物理与天体物理流体动力学》2013,107(1-4):143-173
Abstract Two distributions of the α-effect in a sphere are considered. The inviscid limit is approached both by direct numerical solution and by solution of a simpler nonlinear eigenvalue problem deriving from asymptotic boundary layer analysis for the case of stress-free boundaries. The inviscid limit in both cases is dominated by the need to satisfy the Taylor constraint which states that the integral of the Lorentz force over cylindrical (geostrophic) contours in a homogeneous fluid must tend to zero. For a small supercritical range in α, this condition can only be met by magnetic fields which vanish as the viscosity goes to zero. In this range, the agreement of the two approaches is excellent. In a portion of this range, the method of finite amplitude perturbation expansion is useful, and serves as a guide for understanding the numerical results. For larger α, evidence from the nonlinear eigenvalue problem suggests both that the Taylor state exists, and that the transition from small to large amplitude can require a finite amplitude (oscillatory) instability in accord with the findings of Soward and Jones (1983). However, solutions of the full equations have not been found which are independent of viscosity at larger values of α. 相似文献
19.
E. J. DOUZE 《Geophysical Prospecting》1985,33(8):1093-1102
The linear filter is used extensively in exploration geophysics, and is usually computed using the least squares normal equations. In the general field of time series, the inverse problem is often solved through eigenvalue expansion solutions to integral equations. The normal equations can be solved in terms of the eigenvalues and eigenvectors of the autocorrelation matrix. It has been suggested that a spectral expansion technique should be used which computes the inverse directly without explicit use of the normal equations. If all possible spiking positions are calculated using the normal equations, the spiking operator matrix is obtained. The matrix operator obtained from the spectral expansion is closely related to the spiking operator matrix. Thus, it is possible to compute the spectral expansion filter using the normal equations. Therefore, it is possible to use the best features of both methods, i.e. obtaining the optimum filter with the normal equations, and discarding the poorly determined parts of the solution based on spectral expansion. 相似文献