A posteriori local error estimation and adaptive time-stepping for newmark integration in dynamic analysis

A simple a posteriori local error estimate for Newmark time integration schemes in dynamic analysis is presented, based on the concept of a so called ‘post-processing’ technique. In conjunction with the error estimate, an adaptive time-stepping algorithm is described, which adjusts the time step size so that the local error of each time step is within a prescribed error tolerance. Numerical examples given in the paper indicate that the error estimate is asymptotically convergent, computationally efficient and convenient, and the adaptive time-stepping scheme can predict a nearly optimal step size from time to time, thus making the numerical solution reliable in an efficient manner.     

A post-processing technique and an a posteriori error estimate for the newmark method in dynamic analysis

In this paper, we present a post-processing technique and an a posteriori error estimate for the Newmark method in structural dynamic analysis. By post-processing the Newmark solutions, we derive a simple formulation for linearly varied third-order derivatives. By comparing the Newmark solutions with the exact solutions expanded in the Taylor series, we achieve the local post-processed solutions which are of fifth-order accuracy for displacements and fourth-order accuracy for velocities in one step. Based on the post-processing technique, a posteriori local error estimates for displacements, velocities and, thus, also the total energy norm error estimate are obtained. If the Newmark solutions are corrected at each step, the post-processed solutions are of third-order accuracy in the global sense, i.e. one-order improvement for the original Newmark solutions is achieved. We also discuss a method for estimating the global time integration error. We find that, when the total energy norm is used, the sum of the local error estimates will give a reasonable estimate for the global error. We present numerical studies on a SDOF and a 2-DOF example in order to demonstrate the performance of the proposed technique.     

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.     

Cumulative experimental errors in pseudodynamic tests

In pseudodynamic tests, experimental feedback errors are accumulated in the step-by-step integration procedure. In this paper, the growth of cumulative experimental errors is examined. Approximate cumulative error bounds are derived for linear single- and multi-degree-of-freedom systems, based on realistic models of random and systematic feedback errors. These studies show that the rate of cumulative error growth with respect to the integration time step increases rapidly with the natural frequency of the specimen and the integration time interval used. Hence, the higher modes of a multi-degree-of-freedom system are more sensitive to experimental errors than the lower ones. Furthermore, it is shown that some systematic errors are extremely undesirable. Rational criteria for assessing the reliability of pseudodynamic test results are presented.     

Efficient finite-difference modelling of seismic waves using locally adjustable time steps

Conventional finite-difference modelling algorithms for seismic forward modelling are based on a time-stepping scheme with a constant (global) time step. Large contrasts in the velocity model or in the spatial sampling rate cause oversampling in time for some regions of the model. The use of locally adjustable time steps can save large amounts of computation time for certain modelling configurations. The computation of spatial derivatives across the transition zone between regions of the model with different temporal sampling requires the definition of the wavefield at corresponding time levels on both sides of the transition zone. This condition can be obtained by extrapolation in time, which is inaccurate, or by multiple time integration in the transition zone. The error in the latter solution is of the same order as the conventional time-stepping scheme because both methods are based on the same iteration formula. The technique of multiple time integration simply requires the use of different sizes of time step. It is applicable only for certain factors of variation of the time step.     

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。     

On the Integrability of Limited-Area Numerical Weather Prediction Model ALADIN over Extended Time Periods

This note summarizes results of the first integration of regional numerical weather prediction model ALADIN in a climate mode. The ALADIN model, developed in an international cooperation led by Météo France, is operationally used for weather prediction. The grid step of the model is 12 km; the integration domain covers a major part of Europe. A one-month-long run has been performed with this model on observed boundary conditions (represented by assimilations by the global model ARPEGE). It is demonstrated that no excessive error is generated and accumulated in the model during the integration; hence the model is integrable for extended time periods and may serve a basis for a development towards a regional climate model.     

The discretization error of newmark's method for numerical integration in structural dynamics

The usual assessment of performance of Newmark's method for direct integration in structural dynamics is by reference to amplitude and period error of a single mode of vibration. As an alternative the local and global truncation errors due to the time discretization are introduced. Methods of obtaining norms of the error sequences are presented. The results of numerical experiments confirm the anticipated error order and show how the error formula varies with β. The effect of the presence of physical damping on the error order and formula is also examined.     

Geoid determination by spectral combination of an Earth gravitational model with airborne and terrestrial gravimetry — a theoretical study

Today air-gravimetry is a versatile technique to quickly collect gravity data over large regions, where terrestrial gravity data are sparse and/or of poor quality. The method requires the data to be downward continued to sea level for use in geoid determination, an inverse problem operation that calls for smoothing of the data and/or the kernel function involved (in either spectral or space domain). In this purely theoretical study we avoid this separate computational step by performing direct geoid estimation by so-called spectral combination/filtering of the data, which includes terrestrial gravimetry, airgravimetry, an Earth Gravitational Model (EGM) as well as their signal and error degree variances. Each derived geoid estimator is presented as the sum of one or two integral formulas and the harmonic series of the EGM together with the expected mean square error of the estimator. The article is limited to a theoretical study, leaving its practical tests for future investigation.     

An unconditionally stable hybrid pseudodynamic algorithm

There is a significant motivation to implement an unconditionally stable scheme in the pseudodynamic test method. As more complex experiments with many degrees of freedom are tested, explicit time integration methods limit the size of time step on the basis of the highest natural frequency of the system. This is true even though the response of the structure may be dominated by a few lower frequency modes. The limit on step size is undesirable because it physically increases the duration of a test, but more importantly, because the number of steps to completion increases and error propagation problems increase with the number of steps in a test. In addition, incremental displacements within each step become smaller, introducing the potential for problems associated with stress relaxation. An unconditionally stable algorithm allows the time step to be selected to give accurate response in the modes of interest without regard for higher mode characteristics.     

求解运动方程的一种不等时间步长的显式数值积分方法

在波动有限元模拟中, 若采用传统的显式数值积分方法求解运动方程, 计算时间步长需采用计算区内满足稳定条件要求的最小时间步长. 然而, 对于大部分计算区域, 这一时间步长过小, 是不必要的. 本文提出了一种不等时间步长的显式数值积分方法, 其基本思想是不同的计算区域采用满足各自稳定条件的计算时间步长. 最后, 本文通过数值试验检验了这一方法的可行性及其对数值计算精度的影响.      

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.     

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。     

Local hydraulic gradient estimator analysis of long-term monitoring networks

Three measurements of head at unique locations form a three-point estimator of the local magnitude and orientation of the hydraulic gradient. The relative head measurement error (RHME) is defined here as the measurement error normalized by the head drop across the three-point estimator. Monte Carlo simulation results show that estimators with base to height ratios between 0.5 and 5.0 and that are large enough to keep the RHME below 0.05 create the most accurate gradient estimates and provide criteria for identifying good estimators. These criteria are applied to an example ground water monitoring network design problem in the Culebra dolomite near the Waste Isolation Pilot Plant repository to both analyze temporal changes and modify and expand the current monitoring network. Limiting the three-point estimators to those that meet the shape and RHME criteria reduces the number of possible estimators by >50% and leads to approximately 1 order of magnitude decrease in the average estimated magnitude of the gradient relative to using all estimators. Application of these criteria also reduces the variability in estimated gradient magnitude and orientation between the two time periods of measurements. Redundant wells in the network are identified by removing each existing well in turn and determining which removals yield the smallest decrease in the number of acceptable estimators. Optimal new well locations are identified by mapping the increase in total number of acceptable estimators for a single new well placed in the study domain.     

自适应时间步长法及三维堤坝地震液化数值模拟

三维大模型数值计算因巨大的单元和结点数目而非常耗时,在地震响应分析中受计算时间步长的限值则更加耗时。在饱和砂土动力液化计算平台上开发时域离散误差评估方法和时间步长自适应调整的计算程序,并成功应用于三维堤坝地震液化响应分析。时域离散误差包括土骨架的位移误差和单元孔压误差,通过定义孔压误差影响系数计算出混合误差,根据混合误差和设定的误差允许值进行计算步长的自适应调整。在三维堤坝地震液化数值模拟中,采用自适应时间步长法有效避免小步长精确但耗时、大步长省时而不精确的缺点。在大模型和超大模型计算中,最优调整每一步的计算时间步长,完美实现既节省时间又不失精度的时域离散策略。     

A new integration scheme for application to seismic hybrid simulation

Hybrid simulation is a powerful test method for evaluating the seismic performance of structural systems. This method makes it feasible that only critical components of a structure be experimentally tested. This paper presents a newly proposed integration algorithm for seismic hybrid simulation which is aimed to extend its capabilities to a wide range of systems where existing methods encounter some limitations. In the proposed method, which is termed the variable time step (VTS) integration method, an implicit scheme is employed for hybrid simulation by eliminating the iterative phase on experimental element, the phase which is necessary in regular implicit applications. In order to study the effectiveness of the VTS method, a series of numerical investigations are conducted which show the successfulness of the VTS method in obtaining accurate, stable and converged responses. Then, in a comparative approach, the improved accuracy of the VTS method over commonly used integration methods is demonstrated. The stability of the VTS method is also studied and the results show that it provides conditional stability; however, its stability limit is well beyond the accuracy limit. The effect of time delay on the VTS method results is also investigated and it is shown that the VTS method is quite successful in handling this experimental error.     

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.

     

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.     

Higher order time integration methods for two-phase flow

Time integration methods that adapt in both the order of approximation and time step have been shown to provide efficient solutions to Richards' equation. In this work, we extend the same method of lines approach to solve a set of two-phase flow formulations and address some mass conservation issues from the previous work. We analyze these formulations and the nonlinear systems that result from applying the integration methods, placing particular emphasis on their index, range of applicability, and mass conservation characteristics. We conduct numerical experiments to study the behavior of the numerical models for three test problems. We demonstrate that higher order integration in time is more efficient than standard low-order methods for a variety of practical grids and integration tolerances, that the adaptive scheme successfully varies the step size in response to changing conditions, and that mass balance can be maintained efficiently using variable-order integration and an appropriately chosen numerical model formulation.     

On the accuracy of an implicit algorithm for pseudodynamic tests

The error-propagation characteristics of an implicit time integration algorithm in pseudodynamic testing are examined. It is shown that the implicit algorithm is superior to explicit integration algorithms in terms of experimental error amplification. The influence of systematic experimental errors is studied and methods for controlling these errors are examined. In spite of the fact that the implicit algorithm is unconditionally stable, it is shown that the integration time interval in a pseudodynamic test is limited by the calibration range of the electronic hardware as well as the degree of participation of the higher modes. Furthermore, the tolerance for experimental errors decreases as the integration time interval increases.