基于Chebyshev多项式的弯曲射线Kirchhoff叠前时间偏移

弯曲射线Kirchhoff叠前时间偏移被认为是一种综合了叠前时间偏移效率和叠前深度偏移精度的方法.本文以保精度地减少高阶Kirchhoff叠前时间偏移走时计算量为目标,在分析了Chebyshev正交多项式性质的基础上,建立了Chebyshev多项式约简系数表,进而用模拟退火法对转换系数进行分段优化,从而实现了在大炮检距...     

NUMERICAL SOLUTION OF THE ACOUSTIC AND ELASTIC WAVE EQUATIONS BY A NEW RAPID EXPANSION METHOD1

We present a new rapid expansion method (REM) for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions. The method is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method. It is based on a Chebyshev expansion of the formal solution to the appropriate wave equation written in operator form. The method yields machine accuracy yet it is faster than methods based on temporal differencing. Its disadvantages are that it does not apply to all types of material rheology, and it can also require much storage when many snapshots and time sections are desired. Comparisons between numerical and analytical solutions for simple acoustic and elastic problems demonstrate the high accuracy of the REM.     

动力反应分析的显式积分方法及其稳定性

Newm ark-更新精细积分法是动力方程求解的隐式的时域逐步积分法,其稳定性条件非常容易满足。与隐式方法相比较,显式积分方法不需要求解耦联的方程组,可以有效地减少内存占用和机时耗费。因此,根据显式积分方法的特点和优点,基于Newm ark-更新精细积分法的基本思想,提出其显式积分格式。对显式积分方法的精度与稳定性进行了初步的分析,指出该显式积分方法具有极好的稳定性,其精度比隐式积分方法的精度稍低。随着时间步长的增加,其精度优于传统的方法。     

均匀半空间瞬变电磁场直接时域响应数值分析

近源时域电磁场具有信号强、探测深度大和精度高等优点,但传统瞬变电磁场理论中偶极子近似在近源区会引起较大误差,推导瞬变电磁场直接时域解析式是解决这一问题的关键.本文在点电荷微元假设下通过时域格林函数,采用分离变量等方法推导出了上半空间一次有源波动场和反射波的时域解析式和下半空间二次无源波动场的时域解析式,结合均匀半空间瞬变电磁场的边界条件给出了均匀半空间瞬变电磁场的直接时域解析式,进而利用第一型曲线积分,通过沿回线源叠加推导出圆回线源在瞬变电磁场中的直接时域解析式.然后在半空间表面上,与传统的电偶极源假设下的表达式作了比较.数值结果表明两者在远源区的计算结果相差甚微,而近源区则存在很大误差.本文利用真正点元（点电荷）严密推导给出的均匀半空间表面上瞬变电磁场的直接时域解析式适用于全场区探测,克服了偶极子假设下只适用远场区的不足,为瞬变电磁法的进一步发展和实际勘探提供了新的理论基础.     

地震波时域数值优化研究及应用

将最小二乘拟合法与单自由度地震动力反应递归法相结合来解决地震观测和人工模拟地震波过程中速度、位移的基线漂移问题。首先以3次多项式来拟合加速度的均值线(一次优化);再对一次优化后的积分速度、位移时程仍存在的长周期基线波动问题,运用单自由度地震动力反应递归法进行二次优化。文中通过3个数值算例体现出优化算法的优越性,并将优化后的地震波加速度时程应用到江坪河水电站溢洪道控制段三维有限元动力计算分析中。结果表明,该算法消除了积分基线漂移影响,具有较好的可行性、数值稳定性和易操作性。     

频率波数域内层状地基动力柔度的精细求解

精细积分法既可得到在计算机精度意义下的精确解,又能够保持哈密顿体系的辛结构。其是求解一阶线性常微分方程组的精确数值方法,既可以用于时间域的初值问题,又可以应用于空间域的两点边值问题。运用精细积分法求解微层区段矩阵,并对微层区段矩阵合并得到整个层状地基的区段矩阵,最终得到层状地基的动力柔度值。运用数值算例验证了本文方法的计算精度。     

Dynamic stiffness of rigid rectangular foundations on the half-space

The dynamic soil–structure interaction of a rigid rectangular foundation with the subsoil represents a mixed-boundary value problem. This problem is formulated in terms of a system of coupled Fredholm integral equations of the first kind. The subsoil is modelled by a homogeneous, linear-elastic and isotropic half-space which is perfectly bonded to the rigid, rectangular foundation. An approximate solution for the resultant loads between the foundation and the half-space due to a unit forced displacement or rotation is obtained using the Bubnov–Galerkin method. Using this method the displacement boundary value conditions are exactly satisfied and the contact stress distributions between the foundation and the half-space are approximated by series expansions of Chebyshev polynomials. This method provides a simple means of studying the soil-structure interaction of rectangular foundations with different inertia properties.     

切比雪夫伪谱法模拟地震波场

介绍了切比雪夫伪谱法以及快速傅立叶算法在其中的应用,并用切比雪夫伪谱法模拟二维有限区域弹性介质地震波场。分别计算了兰姆问题,均匀介质中心爆破源问题,介质内部的速度异常体问题以及各种分界面情况下地震首波传播问题。     

AN ACCURATE SCHEME FOR SEISMIC FORWARD MODELLING*

A new time integration technique for use in forward modelling programmes is introduced. The technique presents an alternative to second-order temporal differencing. It is based on a Chebyshev expansion of the formal evolution operator to the spatially discretized wave equation. The computational effort in forward modelling based on the new technique is about the same as in methods based on temporal differencing. However, machine accuracy can be obtained. The implementation of the technique to solve the acoustic wave equation in two spatial dimensions is described. Finally, an example of using the technique to solve a problem of wave propagation in a single layer is presented.     

长偏移距瞬变电磁测深甚晚期响应及视电阻率的数值计算

文中讨论了提高长偏移距瞬变电磁测深甚晚期响应计算精度及相应的全区视电阻率的计算方法。在频率域 ,利用直接数值积分结合连分式展开提高低频段响应的计算精度 ,从而提高时间域甚晚期响应的计算精度。同时 ,利用连分式渐进展开 ,将均匀半空间电阻率表示为该模型长偏移距瞬变电磁测深响应的反函数 ,利用迭代方法求出适合全时间段的全区视电阻率。数值结果表明 ,文中的方法可有效地提高长偏移距瞬变电磁测深 (甚 )晚期数据的模拟与解释精度     

Extracting the Virtual Reflected Wavelet from TEM Data Based on Regularizing Method

A pseudo-seismic interpretation method is an alternative way to process and explain transient electromagnetic (TEM) data, and has become a popular research field in recent years. TEM signals which satisfy the diffusion equation can be converted by means of a mathematical transformation into ones which obey the wave equation. For an ill-posed problem of this kind of transformation, a sub-regularization algorithm is developed in this paper to extract a virtual wavelet of the TEM field. According to the conventional designation of TEM recordings, the entire integration period is divided into seven time intervals. In order to avoid low accuracy in the calculations, high-density wavefield data has been calculated based on the former sub-division. Therefore, the virtual wavelet can be extracted successfully by using an optimized algorithm to obtain high-density integral coefficients for all time windows, and a satisfactory condition number of the coefficient matrix while taking a different channel number in each time period. The Tikhonov regularization inversion scheme is used to determine the optimal parameters based on minimizing a least squares misfit, and the Newton iterative formula is used to obtain optimal regularization parameters. Both synthetic model simulations and a real data interpretation example indicate that the proposed pseudo-seismic wavefield method is a suitable alternative way to interpret TEM data.     

采用复合样条有限元方法求解板的动力初值问题

结构动力分析是工程设计中的重要组成部分，传统结构动力分析不能全面反映结构动力的初值特征，而Gurtin变分原理被认为是目前唯一能全面反映结构动力初值特征的变分原理。本文应用以位移为参变量的Gurtin变分原理，采用复合样条有限元的方法，即在时间域及空间域的y方向采用三次B样条函数，而在空间域的x方向采用多项式逼近广义位移，从而建立了精度较高的计算板动力初值问题的样条有限元模型。数值计算结果表明，本文所建立的方法能有效地求解板的动力初值问题，且计算精度高。     

Modified precise time step integration method of structural dynamic analysis

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method （e.g., an algorithm with an arbitrary order of accuracy） is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix H. If the matrix H is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.     

Wave propagation in semi-infinite media with topographical irregularities using Decoupled Equations Method

In this paper, a novel semi-analytical method, called Decoupled Equations Method (DEM), is presented for modeling of elastic wave propagation in the semi-infinite two-dimensional (2D) media which are involved surface topography. In the DEM, only the boundaries of the problem are discretized by specific subparametric elements, in which special shape functions as well as higher-order Chebyshev mapping functions are implemented. For the shape functions, Kronecker Delta property is satisfied for displacement function. Moreover, the first derivatives of displacement function with respect to the tangential coordinates on the boundaries are assigned to zero at any given node. Employing the weighted residual method and using Clenshaw–Curtis numerical integration, coefficient matrices of the system of equations are transformed into diagonal ones, which leads to a set of decoupled partial differential equations. To evaluate the accuracy of the DEM in the solution of scattering problem of plane waves, cylindrical topographical features of arbitrary shapes are solved. The obtained results present excellent agreement with the analytical solutions and the results from other numerical methods.     

Modified predictor–corrector numerical scheme for real‐time pseudo dynamic tests using state‐space formulation

This paper deals with an explicit numerical integration method for real‐time pseudo dynamic tests. The proposed method, termed the MPC‐SSP method, is suited to use in real‐time pseudo dynamic tests as no iteration steps are involved in each step of computation. A procedure for implementing the proposed method in real‐time pseudo dynamic tests is described in the paper. A state‐space approach is employed in this study to formulate the equations of motion of the system, which is advantageous in real‐time pseudo dynamic testing of structures with active control devices since most structural control problems are formulated in state space. A stability and accuracy analysis of the proposed method was performed based on linear elastic systems. Owing to an extrapolation scheme employed to predict the system's future response, the MPC‐SSP method is conditionally stable. To demonstrate the effectiveness of the MPC‐SSP method, a series of numerical simulations were performed and the performance of the MPC‐SSP method was compared with other pseudo dynamic testing methods including Explicit Newmark, Central Difference, Operator Splitting, and OS‐SSP methods based on both linear and non‐linear single‐degree‐of‐freedom systems. Copyright © 2004 John Wiley & Sons, Ltd.     

An explicit finite element－finite difference method for analyzing the effect of visco－elastic local topography on the earthquake motion

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Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

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.     

Transient elastodynamic analysis by time domain bem in cylindrical coordinates

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.     

On enhancement of accuracy in direct integration dynamic response problems

Enhancement of accuracy of the direct integration methods used in structural dynamic response analysis is presented. The accuracy achieved is obtained at the same time as the cost of the problem solution is reduced. The method presented is based on Richardson's extrapolation technique.     

Newmark精细积分格式及其在地震反应分析中的应用

在Newmark精细直接积分法的基础上,应用高斯积分与精细指数运算,提出该方法的两种逐步积分格式。文中对两种积分格式的稳定性和精度进行了分析。经过分析比较,第1种逐步积分格式计算精度较高,其稳定性明显地满足算法稳定性分析的条件;而第2种积分格式计算精度相对较差,且是不稳定的。因此本文将第1种积分格式应用于结构的地震反应分析中。算例表明,该逐步积分格式对地震作用有很好的适应性。