求解加速度反应的显式积分格式研究

目前波动显式有限元分析多以位移或速度作为输入,而加速度记录更直接地保留了地震动的原始信息。为此,本文发展了一种直接以地震加速度作为输入的显式算法,该算法以中心差分和New-m ark-β法结合,并通过平衡方程约简得到。以水塔及框架结构为例,与现有振型分解联合Duham el积分方法、Newm ark-β隐式解法及五种显式算法进行了对比分析。结果表明:该算法与文献[3]的方法,可以在保证计算效率的前提下,得到与振型分解联合Duham el积分方法、Newm ark-β隐式解法相吻合的加速度反应。     

结构地震响应分析新方法

本文通过对传统动力反应分析方法的总结,阐明了建立隐式和显式方法的一般思路及数学本质,提出了使用系统位移反应向量三阶导数的实用显式高阶积分方法,分析了该显式方法的精度和稳定性,并对建立更高阶隐式和显式方法以及方法的精度和稳定性作了初步讨论。结果表明,本文方法具有明显的优点。     

结构动力反应分析的三阶显式方法

本通过对传统动力反应分析方法的总结，阐明了建立隐式和显式方法的一般思路及数学本质，提出了使用系统位移反向向量三阶导数的隐工和实用显式积分方法－3阶显式方法，分析了该显式方法的精度和稳定性，并对建立更高阶隐式和显式方法以及方法的精度和稳定性作了初步讨论。最后，通过算例对本方法、献[1]方法和经典的常平均加速度法（隐式方法、视为精确解）的精度和稳定性进行了比较分析。结果表明，本方法具有明显的优点。     

结构动力方程的显式级数积分格式

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.     

基于一种新显式时间积分算法的场地非线性地震反应分析

针对线弹性结构动力学方程,作者已提出一种具有良好稳定性的二阶精度单步显式时间积分算法。本文将该方法推广到求解材料非线性结构动力学方程中,采用带误差控制的修正欧拉算法计算单元应力,提高显式时间积分算法的精度。将求解非线性问题的显式算法应用于地震波垂直入射时非线性地震反应分析中,使用黏性边界模拟场地土层底部半空间基岩的辐射阻尼,并考虑地震动输入。与中心差分法计算结果进行对比,以表明新显式算法的有效性。     

有阻尼振动方程常用显式积分格式稳定性分析

本文主要介绍了用于求解有阻尼振动方程的四种常用显式积分方法，并针对其稳定性和精度进行了分析对比，讨论了其适用范围及阻尼对稳定性的影响。     

Newmark-精细积分方法的选择及稳定性

针对Newmark-精细直接积分法,对其非齐次项的积分方法进行了讨论,并分析了该逐步积分方法的稳定性。通过理论推导和数值验证,此方法的非齐次项采用高斯积分公式,其计算误差均比采用柯特斯积分公式和辛浦生积分公式的误差小,且其计算工作量比原方法的少,因此Newmark-精细直接积分法得到了改进。通过稳定性的分析得知,改进的Newmark-精细直接积分法虽是条件稳定的,但是其稳定性条件极易满足。综合分析,此方法可推广应用于实际结构的动力反应分析中。     

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。     

求解振动方程的一种显式积分格式及其精度与稳定性

介绍了一种求解有阻尼体系振动方程的显式积分格式及其逐步求解的过程，并对其计算精度和稳定性进行了分析。该方法不但能同时求得体系的位移、速度和加速度反应，而且所得到的加速度反应的精度能满足工程需要。     

显式分析方法在高层建筑弹塑性地震反应分析中的适用性研究

目前隐式方法是动力弹塑性时程分析最常用的分析方法,然而隐式方法在强非线性分析中常常存在迭代不收敛的问题,并且刚度矩阵求解消耗的存储空间随结构自由度增加呈几何级数增长。因此,在求解高层建筑这种大规模问题中,极易遭遇计算瓶颈。显式分析方法直接求解解耦的方程组,不需要迭代。本文对隐式方法和显式方法进行了对比分析,研究了显式分析方法在高层建筑弹塑性地震反应分析中适用性。实例分析表明,从计算精度来讲,隐式方法和显式方法在稳定条件下都能得到较好的精度。从计算效率来讲,对于自由度较少的结构,隐式方法计算效率较高;对于自由度庞大的结构,显式方法计算效率较高。建议在进行自由度多的高层建筑弹塑性地震反应分析时,采用显式分析方法。     

Time domain dynamic analysis of piles under impact loading

In this paper, time domain dynamic analysis of piles under impact loading is presented. For this purpose a hybrid boundary element technique is implemented. Linear beam column finite elements are used to model the piles and resulting governing equations are solved using an implicit integration scheme. The continuum is assumed to be elastic and an efficient step-by-step time integration scheme is implemented by using an approximate half space integral formulation. By enforcing displacement equilibrium conditions at each time step, a system of equations is generated which yields the solution. Results of this time domain formulation under linear material behavior are compared with Laplace domain results to validate the methods.     

Evaluation of numerical time‐integration schemes for real‐time hybrid testing

We present a comparison of methods for the analysis of the numerical substructure in a real‐time hybrid test. A multi‐tasking strategy is described, which satisfies the various control and numerical requirements. Within this strategy a variety of explicit and implicit time‐integration algorithms have been evaluated. Fully implicit schemes can be used in fast hybrid testing via a digital sub‐step feedback technique, but it is shown that this approach requires a large amount of computation at each sub‐step, making real‐time execution difficult for all but the simplest models. In cases where the numerical substructure poses no harsh stability condition, it is shown that the Newmark explicit method offers advantages of speed and accuracy. Where the stability limit of an explicit method cannot be met, one of the several alternatives may be used, such as Chang's modified Newmark scheme or the α‐operator splitting method. Appropriate methods of actuator delay compensation are also discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Combined implicit or explicit integration steps for hybrid simulation

Explicit integration procedures have been widely adapted and applied to hybrid simulations of the seismic response of structures due to their simplicity. However, these procedures are only conditionally stable and have limited recent applications of hybrid simulations to simple structural models with few degrees of freedom. A novel integration procedure is proposed herein, in which a fully implicit formulation is applied to solve the equation of motion for the hybrid model, but defaults to an explicit or noniterative formulation in steps that fail to converge. The advantages to this approach are the ensured continuity of the simulation and the reduced accumulation of errors that occur during consecutive explicit steps that may lead to instability. The implicit procedure is applied by loading the experimental substructures beyond the expected displacement for the current step, then using the displacements and forces measured through the load path in the iterative implicit scheme. This approach captures the instantaneous behaviour of experimental substructures without physically imposing iterations. Numerical and experimental simulations demonstrate the effectiveness of the proposed integration scheme for multi‐degree‐of‐freedom models, especially in utilization of longer time steps that exceed stability limits of explicit methods, prevention of excitation of higher modes, and testing of stiff systems. Copyright © 2007 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.

     

Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step

Calculation of actual crop evapotranspiration under soil water stress conditions is crucial for hydrological modeling and irrigation water management. Results of actual evapotranspiration depend on the estimation of water stress coefficient from soil water storage in the root zone, which varies with numerical methods and time step used. During soil water depletion periods without irrigation or precipitation, the actual crop evapotranspiration can be calculated by an analytical method and various numerical methods. We compared the results from several commonly used numerical methods, including the explicit, implicit and modified Euler methods, the midpoint method, and the Heun's third‐order method, with results of the analytical method as the bench mark. Results indicate that relative errors of actual crop evapotranspiration calculated with numerical methods in one time step are independent of the initial soil water storage in the range of soil water stress. Absolute values of relative error decrease with the order of numerical methods. They also decrease with the number of time step, which can ensure the numerical stability of successive simulation of soil water balance. Considering the calculation complexity and calculation errors caused by numerical approximation for different time step and maximum crop evapotranspiration, the explicit Euler method is recommended for the time step of 1 day (d) or 2 d for maximum crop evapotranspiration less than 5 mm/d, the midpoint method or the modified Euler method for the time step of up to one week or 10 d for maximum crop evapotranspiration less than 5 mm/d, and the Heun's third‐order method for the time step of up to 15 d. Copyright © 2011 John Wiley & Sons, Ltd.     

Extensions of nonlinear error propagation analysis for explicit pseudodynamic testing

Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm (Chang, 2003) are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom (SDOF) system, can be applied to a nonlinear multiple degree of freedom (MDOF) system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.     

Implicit/explicit multi-time step co-computations for predicting reinforced concrete structure response under earthquake loading

The paper explores the GC (Gravouil and Combescure) partitioning strategy recently adopted in real-time testing (RTDS) and pseudo-dynamic testing (PsD) with dynamic substructuring. The GC method is a multi-time step subdomain algorithm able to couple any time integration schemes from the Newmark family with the appropriate time step size dependent on the subdomain. The partitioning method is numerically tested by developing an external software able to couple finite element codes based on implicit and explicit time integration schemes. A complex Finite Element mesh partitioning, exhibiting a large number of interface points, has been considered for a full-size reinforced concrete frame structure subjected to an earthquake loading: the well-known SPEAR structure pseudo-dynamically tested at the ELSA laboratory, in Ispra, Italy. Implicit and explicit parts of the structure are modelled using multi-fibre beam elements whose cross-section is divided into steel and concrete fibres associated with cyclic and nonlinear behaviours. The accuracy of the results from the Explicit/Implicit multi-time step co-computation has been proved by comparing with the results from full explicit and full implicit computations. Despite the very large duration of the earthquake excitation and the number of interface nodes involved into the mesh partitioning, the Explicit/Implicit multi-time step co-computations provide very accurate global (displacements, forces at the base) and local (maximum strains in concrete) results while reducing by a factor 2 the computation times. Finally, it has been observed that the dissipated energy at the interface coming from the GC coupling algorithm remains very low at the end of the earthquake loading, less than 2% of the external energy, for time step ratios between the large and fine time steps ranging from 20 to 200.     

一种显隐式积分方法及其应用

在实际工程结构动力反应分析中,往往由于结构型式十分复杂,常用的两种直接积分方法,即显式积分方法和隐式积分方法,在使用中都存在着一定的局限性,如何将这两种积分方法合理有效地结合起来,是一个十分有意义的研究课题。针对实际工程问题中整体结构计算时间步长的选择往往受局部区域的材料特性、尺寸大小等因素影响的这一现象,提出了一种对结构局部区域进行隐式积分、对其余区域进行显式积分的显隐式积分方法,这种积分格式相对于显式积分格式而言,能显著提高整体结构的计算速度。最后采用两个数值计算实例对这一方法进行验证。