Full operator algorithm for hybrid simulation

One of the weaknesses of the operator splitting method (OSM) is that its corrector step employs the approximation that incremental forces are linearly related to the tested structure's initial stiffness matrix. This paper presents a new predictor–corrector technique in which the assumptions about the tested structure's response are shifted to the predictor step, which results in an enhancement in overall simulation accuracy, especially for nonlinear structures. Unlike OSM, which splits the displacement and velocity operators into explicit and implicit terms, the new method uses predicted accelerations to compute fully explicit displacement and velocity values in the predictor step. Another advantage of the proposed technique, termed the full operator method (FOM) is that its formulation makes it suitable for both quasi‐static and real‐time hybrid simulation. The effectiveness of FOM is first evaluated by investigating error propagation in an undamped single degree‐of‐freedom model. It is shown that the corrector step in FOM is able to significantly suppress aberrant simulation results caused by incorrect estimation of the structure's stiffness matrix. The performance of FOM is demonstrated by exercising two additional models, which exhibit significant inelastic behavior under the prescribed excitation. The simulation results show that the proposed FOM algorithm is capable of producing accurate solutions and that the corrector step is influential in effectively reducing simulation errors. It is also shown that FOM suppresses actuator displacement control errors because of its reliance on measured quantities in the corrector step. Copyright © 2009 John Wiley & Sons, Ltd.     

错格傅里叶伪谱微分算子在波场模拟中的应用

本文在传统傅里叶微分矩阵的基础上,对原始微分算子进行改进,引入了错格微分算子的傅里叶伪谱方法．尽管该方法增加了一些计算量,但却极大地提高了计算精度和稳定性．而且,该方法将微分计算过程由传统的傅里叶变换转换为一般的矩阵矢量乘积,大大降低了微分求解过程的复杂程度．在均匀介质中,将错格伪谱微分算子计算的结果和解析解进行比较,结果表明本文算子几乎达到了解析解的精度．而在分层均匀介质中的实验结果同时显示,该方法精度高、稳定性好,是一种研究层状介质中地震波传播的有效数值方法．     

A Rosenbrock‐W method for real‐time dynamic substructuring and pseudo‐dynamic testing

A variant of the Rosenbrock‐W integration method is proposed for real‐time dynamic substructuring and pseudo‐dynamic testing. In this variant, an approximation of the Jacobian matrix that accounts for the properties of both the physical and numerical substructures is used throughout the analysis process. Only an initial estimate of the stiffness and damping properties of the physical components is required. It is demonstrated that the method is unconditionally stable provided that specific conditions are fulfilled and that the order accuracy can be maintained in the nonlinear regime without involving any matrix inversion while testing. The method also features controllable numerical energy dissipation characteristics and explicit expression of the target displacement and velocity vectors. The stability and accuracy of the proposed integration scheme are examined in the paper. The method has also been verified through hybrid testing performed of SDOF and MDOF structures with linear and highly nonlinear physical substructures. The results are compared with those obtained from the operator splitting method. An approach based on the modal decomposition principle is presented to predict the potential effect of experimental errors on the overall response during testing. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element techniques for convective-diffusive transport in porous media

Operator-splitting techniques are applied to convective-diffusive transport problems in porous media. The convection is treated by applying a modified method of characteristics to time-step along the characteristics of the convective part of the flow. The nonsymmetry in the spatial operator is addressed via a Petrov-Galerkin method which uses a test function to achieve stability through a balancing of the remaining convection, the diffusion, and any possible reaction terms. The use of time-stepping along characteristics allows the use of large time-steps in a stable but accurate fashion. If local phenomena are important, self-adaptive local grid refinement techniques can be coupled with the operator splitting.     

错格实数傅里叶变换微分算子及其在非均匀介质波动传播研究中的应用

提出一种新的数值微分运算方法，即错格实数傅里叶变换微 分法. 该方法的运算速度 比错格复数傅里叶变换数值微分解法快0.33倍；因为该微分算法在整个微分运算过程中保留 了奈奎斯特分量，使得它比普通分格的实数傅里叶变换数值微分算法的精度高，稳定性好. 将该方法和Cagniard De Hoop解析法在求解半无限空间地震波动的问题中进行比较，结果 表明，新微分法的精度和解析方法的精度相同. 在非均匀介质中的地震波传播数值模拟的结 果表明，该方法是一种研究非均匀介质中地震波传播问题的有效的数值微分方法.     

Modified state–space procedures for pseudodynamic testing

The existing on‐line numerical integration algorithms are derived from the Newmark method, which is based on an approximation of derivatives in the differential equation. The state–space procedure (SSP), based on an interpolation of the discrete excitation signals for piecewise convolution integral, has been confirmed as more reliable than the Newmark method in terms of numerical accuracy and stability. In an attempt to enhance the pseudodynamic test, this study presents an on‐line integration algorithm (referred to as the OS–SSP method) via an integration of the state–space procedure with Nakashima's operator‐splitting concept. Numerical stability and accuracy assessment of the proposed algorithm in addition to the explicit Newmark method and the OS method were investigated via an eigenvalue, frequency‐domain and time‐domain analysis. Of the on‐line integration algorithms investigated, the OS–SSP method is demonstrated as the most accurate method with an acceptable stability (although not unconditionally stable) characteristic. Therefore, the OS–SSP method is the most desirable method for pseudodynamic testing if the numerical stability criterion (Δt/T⩽0.5) is ensured for every vibration mode involved. Copyright © 2001 John Wiley & Sons, Ltd.     

A method for estimating specimen tangent stiffness for hybrid simulation

Researchers have long recognized the importance and potential benefits of utilizing the tangent stiffness matrix of a test specimen in hybrid simulations employing implicit and mixed‐integration schemes. However, the computation of the tangent stiffness matrix during testing has proved to be challenging, particularly for test specimens with more than one degree of freedom (dof). This paper presents a new methodology that is more straightforward and simpler than existing techniques for computing the tangent stiffness matrix of a multi‐dof test specimen. The proposed method is combined with the operator‐splitting method (OSM), and the capabilities, advantages and limitations of the new formulation are demonstrated through several examples. The accuracy, stability, and error propagation characteristics of the modified OSM are also studied theoretically as well as numerically. The research results show that the proposed algorithm provides results that are better than those produced via the regular OSM alone, especially for damped structures undergoing highly inelastic behavior during testing. Copyright © 2008 John Wiley & Sons, Ltd.     

补偿向下延拓方法研究及应用

位场向下延拓是重、磁处理和解释的常用方法,但其不稳定性限制了其在资料处理及反演中的应用.本文基于补偿圆滑滤波思想以及空间域向下延拓迭代法,通过逐次补偿的办法实现位场的稳定向下延拓.同时,在频率域空间给出了该下延方法的频率域响应因子,并讨论了其低通滤波特性,理论模型和实际位场资料试验表明该方法向下延拓稳定性具有较高的延拓精度.将其应用于重力密度界面反演中,改进反演的稳定性,实际莫霍界面反演表明下延因子具备实用性.     

Lattice Boltzmann method with two relaxation times for advection–diffusion equation: Third order analysis and stability analysis

The objectives of this study are to investigate the third order accuracy and linear stability of the lattice Boltzmann method (LBM) with the two-relaxation-time collision operator (LTRT) for the advection–diffusion equation (ADE) and compare the LTRT model with the single-relaxation-time (LBGK) model. While the LBGK has been used extensively, the LTRT appears to be a more flexible model because it uses two relaxation times. The extra relaxation time can be used to improve solution accuracy and/or stability. This study conducts a third order Chapman–Enskog expansion on the LTRT to recover the macroscopic differential equations up to the third order. The dependency of third order terms on the relaxation times is obtained for different types of equilibrium distribution functions (EDFs) and lattices. By selecting proper relaxation times, the numerical dispersion can be significantly reduced. Furthermore, to improve solution accuracy, this study introduces pseudo-velocities to develop new EDFs to reduce the second order numerical diffusion. This study also derives stability domains based on the lattice Peclet number and Courant number for different types of lattices, EDFs and different values of relaxation times, while conducting linear stability analysis on the LTRT. Numerical examples demonstrate the improvement of the LTRT solution accuracy and stability by selecting proper relaxation times, lattice Peclet number and Courant number.     

一种全局优化的隐式交错网格有限差分算法及其在弹性波数值模拟中的应用

在数值模拟中,隐式有限差分具有较高的精度和稳定性.然而,传统隐式有限差分算法大多由于需要求解大型矩阵方程而存在计算效率偏低的局限性.本文针对一阶速度-应力弹性波方程,构建了一种优化隐式交错网格有限差分格式,然后将改进格式由时间-空间域转换为时间-波数域,利用二范数原理建立目标函数,再利用模拟退火法求取优化系数.通过对均匀模型以及复杂介质模型进行一阶速度-应力弹性波方程数值模拟所得单炮记录、波场快照分析表明:这种优化隐式交错网格差分算法与传统的几种显式和隐式交错网格有限差分算法相比不但降低了计算量,而且能有效的压制网格频散,使弹性波数值模拟的精度得到有效的提高.     

非稳态相移法叠前深度偏移

介绍一种能够适应介质速度横向变化的非稳态相移算子及其叠前深度偏移方法．为了克服常规相移偏移算法中要求速度横向不变的缺点,出现了基于非稳态滤波器理论的非稳态相移算子,即PSPI算子、NSPS算子和SNPS算子,其中SNPS算子是将前二者结合起来的一种对称的非稳态相移算子,它比前二者具有更高的精度和稳定性．为了提高运算速度,基于非稳态相移算子的叠前深度偏移算法采取了分片均匀近似的策略,Marmousi模型的叠前深度偏移结果证明了该算法的可行性和有效性。     

双变参数标量纵波方程正演模拟方法

常见弹性波动理论的建立是基于介质均匀这一基本假设,实际介质的非均匀性非常普遍.为研究连续介质中波的传播特征,本文从弹性力学中建立弹性波动方程的三个基本方程出发,考虑连续介质弹性参数的空变特征,建立非均匀介质的弹性波动方程,利用Alkhalifah声学近似思想建立位移表征的纵波波动方程,利用本征值问题求解方法建立标量波频率-波数域传播算子,从而建立描述纵波传播的标量波方程,其中波函数为纵波位移的散度,不同于均匀介质标量波方程的波函数为位移势.随后推导含PML边界波动方程差分格式并建立不同模型数值模拟进行数值试算,与均匀假设标量波方程和变密度方程对比证明本方法的准确性和稳定性.     

地震偏移的最优可分近似算法实现

针对地震偏移算法中单程波算子的特征函数近似，采用最优可分表示法将该特征函数展开为空间变量(g)和水平波数(k)的可分表达式，以此可分近似表达式为基础，运用正反Fourier变换重新构造单程波算子. 为了克服特征函数最优可分近似计算在奇点及其领域产生的数值振荡，引入等价黏性技巧以增强算法的数值计算稳定性，并采用分频最优可分及对空间变量(g)线性插值的方法，不仅提高了计算精度，也节约计算机时. 文中具体研究了单程波算子的脉冲响应和二维叠前深度偏移. 结果表明，在不甚大步长情况下，本文构造的算子具有适应横向强变速的能力，运用Marmousi模型验证了本文方法适合复杂构造的成像.     

弹性波数值模拟的非规则网格差分法

基于应力、速度混合变量弹性波方程及任意四边形网格差分算子,给出了交错计算应力及速度的非规则网格弹性波应力一速度差分法该方法融合了有限元法能适应复杂形状边界及差分法无需计算刚度阵的特点,具有较高的计算精度,所需计算机存储空间较少,计算效率也很高．基于积分平衡方程引入了任意形状自由表面的边界条件,且通过局部滤波改善了自由表面边界条件的稳定性,使得该方法可应用于考虑地表形状影响的地震波数值模拟     

NUMERICAL SEISMOGRAM COMPUTATIONS FOR INHOMOGENEOUS MEDIA USING A SHORT,VARIABLE LENGTH CONVOLUTIONAL DIFFERENTIATOR1

A short convolutional differentiator (CD) for computing second spatial derivatives in the acoustic wave equation is presented. This differentiator is obtained by tapering the inverse Fourier transform of the band-limited Fourier spectrum of the second-derivative operator. This new filter has been applied to seismogram computations for inhomogeneous media and results are compared with the conventional high-order finite-difference (FD) and Fourier schemes. The operator can be progressively shortened at the model edges to reduce boundary artefacts. The CD method is superior to the conventional FD operator and comparable with the Fourier method in accuracy but faster to run. A strategy to reduce computation time by 20%, which exploits the localized nature of the operator, is given. The method is illustrated using simple 2D models.     

Stability and accuracy of waterhammer analysis

The first-order accurate explicit finite-difference scheme based on the method of characteristics (MOC) has been widely used for the analysis of waterhammer in pipelines. For the stability of this scheme, the Courant condition has been developed by neglecting the nonlinear terms. When the friction loss term is large, this scheme becomes unstable even if the Courant stability is satisfied. In this paper, the stability and accuracy including friction are investigated by systematically analyzing typical piping systems over a wide range of parameters and empirical stability limits are presented for practical applications. In addition, the suitability of the implicit finite difference method for transient analysis is investigated.     

基于短算子设计的黏性介质叠前时间偏移

为了提高频域黏性介质叠前时间偏移的计算效率,本文采用加权最小平方方法设计高精度的、最优时域褶积短算子,发展了一套表驱动的时域黏性介质叠前时间偏移方法.该方法将大量的逐频率补偿运算转化为少量的时域褶积运算,并将走时,振幅表和补偿褶积短算子系数表的计算过程与补偿成像过程相剥离,提高了时域算法的计算效率;通过控制最大的补偿因...     

Operator‐splitting method for real‐time substructure testing

It has been shown that the operator‐splitting method (OSM) provides explicit and unconditionally stable solutions for quasi‐static pseudo‐dynamic substructure testing. However, the OSM provides only an explicit target displacement but not an explicit target velocity, so that it is essentially an implicit method for real‐time substructure testing (RST) when the velocity‐dependent restoring force is considered. This paper proposes a target velocity formulation based on the forward difference of the predicted displacements so as to render the OSM explicit for RST. The stability and accuracy of the resulting OSM‐RST algorithm are investigated. It is shown that the OSM‐RST is unconditionally stable so long as the non‐linear stiffness and damping are of the softening type (i.e. the tangent stiffness and damping never exceed the initial values). The stability of the OSM‐RST for structures with infinite tangent damping coefficient or stiffness is also proved, and the stability of the method for MDOF structures with a non‐classical damping matrix is demonstrated by an energy criterion. The effects of actuator delay and compensation are analysed based on the bilinear approximation of the actuator step response. Experiments on damped SDOF and MDOF structures verify that the stability of the OSM‐RST is preserved when the experimental substructure generates velocity‐dependent reaction forces, whereas the stability of real‐time substructure tests based on the central difference method is worsened by the damping of the specimen. Copyright © 2005 John Wiley & Sons, Ltd.     

Acoustic finite-difference modeling beyond conventi-onal Courant-Friedrichs-Lewy stability limit: App-roach based on variable-length temporal and spatial operators

Conventional finite-difference (FD) methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy (CFL) numbers 0.707 and 0.577 for two-dimensional (2D) and three-dimensional (3D) equal spacing cases, respectively, thereby limiting time step selection. Based on the definition of temporal and spatial FD operators, we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy. First, to simulate wave propagation beyond the conventional CFL stability limit, the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones. Second, to preserve the modeling accuracy, the velocity-dependent lengths of the temporal and spatial operators are adaptively varied. The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples, respec-tively. We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.     

TTI介质隐式有限差分平面波偏移

文章研究了TTI介质隐式有限差分(IFD)波场外推算子和TTI介质平面波偏移.与各向同性和VTI介质相比,TTI介质频散关系要复杂的多,很难得到频散关系的显性表达,也不能再只用对称的偶函数表示,需要加入一个奇函数.文中设计TTI介质IFD波场外推算子并用非线性优化方法求解算子系数.将各向同性介质下平面波偏移理论与TTl...