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.     

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.     

Acoustic VTI modeling and pre-stack reverse-time migration based on the time–space domain staggered-grid finite-difference method

Reverse-time migration (RTM) is based on seismic numerical modeling algorithms, and the accuracy and efficiency of RTM strongly depend on the algorithm used for numerical solution of wave equations. Finite-difference (FD) methods have been widely used to solve the wave equation in seismic numerical modeling and RTM. In this paper, we derive a series of time–space domain staggered-grid FD coefficients for acoustic vertical transversely isotropic (VTI) equations, and adopt these difference coefficients to solve the equations, then analyze the numerical dispersion and stability, and compare the time–space domain staggered-grid FD method with the conventional method. The numerical analysis results demonstrate that the time–space domain staggered-grid FD method has greater accuracy and better stability than the conventional method under the same discretizations. Moreover, we implement the pre-stack acoustic VTI RTM by the conventional and time–space domain high-order staggered-grid FD methods, respectively. The migration results reveal that the time–space domain staggered-grid FD method can provide clearer and more accurate image with little influence on computational efficiency, and the new FD method can adopt a larger time step to reduce the computation time and preserve the imaging accuracy as well in RTM. Meanwhile, when considering the anisotropy in RTM for the VTI model, the imaging quality of the acoustic VTI RTM is better than that of the acoustic isotropic RTM.     

The ρ-family of algorithms for time-step integration with improved numerical dissipation

The dynamic analysis of complex non-linear structural systems by the finite element approach requires the use of time-step algorithms for solving the equations of motion in the time domain. Both an implicit and an explicit version of such a time-step algorithm, called the ρ-method, the parameter ρ being used for controlling numerical damping in the higher modes, are presented in this paper. For the implicit family of algorithms unconditional stability, consistency, convergence, accuracy and overshoot properties are first discussed and proved. On the basis of the algorithmic damping ratio (dissipation) and period elongation (dispersion) the ρ-method is then compared with the well-known implicit algorithms of Hilber, Newmark, Wilson, Park and Houbolt. An explicit version of the algorithm is also derived and briefly discussed. This shows numerical properties similar to the central difference method. Both versions of the algorithm have been implemented in a general purpose computer program which has been often used for both numerical tests and practical applications.     

Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm

Real‐time hybrid testing combines experimental testing and numerical simulation, and provides a viable alternative for the dynamic testing of structural systems. An integration algorithm is used in real‐time hybrid testing to compute the structural response based on feedback restoring forces from experimental and analytical substructures. Explicit integration algorithms are usually preferred over implicit algorithms as they do not require iteration and are therefore computationally efficient. The time step size for explicit integration algorithms, which are typically conditionally stable, can be extremely small in order to avoid numerical stability when the number of degree‐of‐freedom of the structure becomes large. This paper presents the implementation and application of a newly developed unconditionally stable explicit integration algorithm for real‐time hybrid testing. The development of the integration algorithm is briefly reviewed. An extrapolation procedure is introduced in the implementation of the algorithm for real‐time testing to ensure the continuous movement of the servo‐hydraulic actuator. The stability of the implemented integration algorithm is investigated using control theory. Real‐time hybrid test results of single‐degree‐of‐freedom and multi‐degree‐of‐freedom structures with a passive elastomeric damper subjected to earthquake ground motion are presented. The explicit integration algorithm is shown to enable the exceptional real‐time hybrid test results to be achieved. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Novel coupling Rosenbrock‐based algorithms for real‐time dynamic substructure testing

Real‐time testing with dynamic substructuring is a novel experimental technique capable of assessing the behaviour of structures subjected to dynamic loadings including earthquakes. The technique involves recreating the dynamics of the entire structure by combining an experimental test piece consisting of part of the structure with a numerical model simulating the remainder of the structure. These substructures interact in real time to emulate the behaviour of the entire structure. Time integration is the most versatile method for analysing the general case of linear and non‐linear semi‐discretized equations of motion. In this paper we propose for substructure testing, L‐stable real‐time (LSRT) compatible integrators with two and three stages derived from the Rosenbrock methods. These algorithms are unconditionally stable for uncoupled problems and entail a moderate computational cost for real‐time performance. They can also effectively deal with stiff problems, i.e. complex emulated structures for which solutions can change on a time scale that is very short compared with the interval of time integration, but where the solution of interest changes on a much longer time scale. Stability conditions of the coupled substructures are analysed by means of the zero‐stability approach, and the accuracy of the novel algorithms in the coupled case is assessed in both the unforced and forced conditions. LSRT algorithms are shown to be more competitive than popular Runge–Kutta methods in terms of stability, accuracy and ease of implementation. Numerical simulations and real‐time substructure tests are used to demonstrate the favourable properties of the proposed algorithms. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and identification for rational approximation of frequency response function of unbounded soil

Exact representation of unbounded soil contains the single output–single input relationship between force and displacement in the physical or transformed space. This relationship is a global convolution integral in the time domain. Rational approximation to its frequency response function (frequency‐domain convolution kernel) in the frequency domain, which is then realized into the time domain as a lumped‐parameter model or recursive formula, is an effective method to obtain the temporally local representation of unbounded soil. Stability and identification for the rational approximation are studied in this paper. A necessary and sufficient stability condition is presented based on the stability theory of linear system. A parameter identification method is further developed by directly solving a nonlinear least‐squares fitting problem using the hybrid genetic‐simplex optimization algorithm, in which the proposed stability condition as constraint is enforced by the penalty function method. The stability is thus guaranteed a priori. The infrequent and undesirable resonance phenomenon in stable system is also discussed. The proposed stability condition and identification method are verified by several dynamic soil–structure‐interaction examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation

The implicit dissipative generalized‐ α method is analyzed using discrete control theory. Based on this analysis, a one‐parameter family of explicit direct integration algorithms with controllable numerical energy dissipation, referred to as the explicit KR‐α method, is developed for linear and nonlinear structural dynamic numerical analysis applications. Stability, numerical dispersion, and energy dissipation characteristics of the proposed algorithms are studied. It is shown that the algorithms are unconditionally stable for linear elastic and stiffness softening‐type nonlinear systems, where the latter indicates a reduction in post yield stiffness in the force–deformation response. The amount of numerical damping is controlled by a single parameter, which provides a measure of the numerical energy dissipation at higher frequencies. Thus, for a specific value of this parameter, the resulting algorithm is shown to produce no numerical energy dissipation. Furthermore, it is shown that the influence of the numerical damping on the lower mode response is negligible. It is further shown that the numerical dispersion and energy dissipation characteristics of the proposed explicit algorithms are the same as that of the implicit generalized‐ α method. A numerical example is presented to demonstrate the potential of the proposed algorithms in reducing participation of undesired higher modes by using numerical energy dissipation to damp out these modes. Copyright © 2014 John Wiley & Sons, Ltd.     

A two-step unconditionally stable explicit method with controllable numerical dissipations

A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α≥-0.5, β≤ 0.5 and γ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.     

Stability analysis of SDOF real‐time hybrid testing systems with explicit integration algorithms and actuator delay

Real‐time hybrid testing is a method that combines experimental substructure(s) representing component(s) of a structure with a numerical model of the remaining part of the structure. These substructures are combined with the integration algorithm for the test and the servo‐hydraulic actuator to form the real‐time hybrid testing system. The inherent dynamics of the servo‐hydraulic actuator used in real‐time hybrid testing will give rise to a time delay, which may result in a degradation of accuracy of the test, and possibly render the system to become unstable. To acquire a better understanding of the stability of a real‐time hybrid test with actuator delay, a stability analysis procedure for single‐degree‐of‐freedom structures is presented that includes both the actuator delay and an explicit integration algorithm. The actuator delay is modeled by a discrete transfer function and combined with a discrete transfer function representing the integration algorithm to form a closed‐loop transfer function for the real‐time hybrid testing system. The stability of the system is investigated by examining the poles of the closed‐loop transfer function. The effect of actuator delay on the stability of a real‐time hybrid test is shown to be dependent on the structural parameters as well as the form of the integration algorithm. The stability analysis results can have a significant difference compared with the solution from the delay differential equation, thereby illustrating the need to include the integration algorithm in the stability analysis of a real‐time hybrid testing system. Copyright © 2007 John Wiley & Sons, Ltd.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized‐α methods for seismic structural testing

This paper presents novel predictor–corrector time‐integration algorithms based on the Generalized‐α method to perform pseudo‐dynamic tests with substructuring. The implicit Generalized‐α algorithm was implemented in a predictor–one corrector form giving rise to the implicit IPC–ρ∞ method, able to avoid expensive iterative corrections in view of high‐speed applications. Moreover, the scheme embodies a secant stiffness formula that can closely approximate the actual stiffness of a structure. Also an explicit algorithm endowed with user‐controlled dissipation properties, the EPC–ρb method, was implemented. The resulting schemes were tested experimentally both on a two‐ and on a six‐degrees‐of‐freedom system, using substructuring. The tests indicated that the numerical strategies enhance the fidelity of the pseudo‐dynamic test results even in an environment characterized by considerable experimental errors. Moreover, the schemes were tested numerically on severe non‐linear substructured multiple‐degrees‐of‐freedom systems reproduced with the Bouc–Wen model, showing the reliability of the seismic tests under these conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

A FE–BE method for dynamic analysis of dam–foundation–reservoir systems in the time domain

By coupling FEM and BEM, a numerical method was developed for dynamic response analyses of dam–foundation–reservoir systems in the time domain. During formulation, the weighted residual procedure was applied to the coupling of several equations of motion for solid and fluid in the FE and BE regions, and an algorithm similar to the Newmark beta procedure was finally obtained. The algorithm is advantageous in that it takes into account all the effects of dam–foundation, dam–reservoir and reservoir–foundation interactions, as well as of the absorption of both elastodynamic and hydrodynamic waves at the boundaries of the foundation and the reservoir. To demonstrate the validity of the present method, the impulsive response of a dam–foundation–reservoir system was calculated using the algorithm, and showed a good agreement with the existing results obtained by other researchers.     

考虑阻尼影响的修正CD-Newmark组合算法数值特性研究

显-隐式组合数值积分算法结合了显式算法无需迭代和隐式算法无条件稳定的各自优点,是结构抗震拟动力试验顺利运行的关键.在对传统显式中央差分法和隐式Newmark β组合算法进行参数修正的基础上,建立了修正CD-Newmark算法,考虑阻尼的影响分析了组合算法的稳定性条件、周期失真率和数值阻尼比,分别得到了试验子结构的稳定性条件和计算子结构无条件稳定的参数合理取值范围,并对计算精度进行了分析.通过算例分析验证了算法的数值特性,从而初步解决了CD-Newmark算法存在稳定性界限过严的问题,为结构抗震拟动力混合试验提供了研究参考.     

A novel predictor–corrector algorithm for sub‐structure pseudo‐dynamic testing

A new predictor–corrector (P–C) method for multi‐site sub‐structure pseudo‐dynamic (PSD) test is proposed. This method is a mixed time integration method in which computational components separable from experimental components are solved by implicit time integration method (Newmark β method). The experiments are performed quasi‐statically based on explicit prediction of displacement. The proposed P–C method has an important advantage as it does not require the determination of the initial stiffness values of experimental components and is thus suitable for representing elastic and inelastic systems. A parameter relating to quality of displacement prediction at boundaries nodes is introduced. This parameter is determined such that P–C method can be applicable to many practical problems. Error‐propagation characteristics of P–C method are also presented. A series of examples including linear and non‐linear soil–foundation–structure interaction problem demonstrate the performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Convergence and stability of step-by-step integration for model with negative-stiffness

The convergence and stability of step-by-step integration schemes used in the inelastic dynamic analysis of structures and their corresponding criteria were studied for a restoring force model with negative-stiffness. Convergence conditions and stability conditions 1, 2 or 3 were established. The numerical stability of the integration under negative-stiffness belongs to the category of relative stability; consequently, the concepts and the conclusions concerning numerical stability in the case of positive-stiffness (which belongs to absolute stability) cannot be used. Research into several step-by-step integration methods usually employed in inelastic dynamic analysis has shown great differences in numerical stability for models with negative-stiffness as compared with positive-stiffness models. The central difference method is convergent and unconditionally stable in the case of negative-stiffness, though it is only conditionally stable in the case of positive-stiffness. The Houbolt method satisfies the requirement for convergence; its stability, however, depends not only on the integration step size Δt but also on the stiffness ratio β for the model with negative-stiffness, unlike the unconditional stability for the model with positive-stiffness. The Newmark constant acceleration method is convergent and unconditionally stable in the case of negative-stiffness just like it is in the case of positive-stiffness.     

Non‐iterative time integration scheme for non‐linear dynamic FEM analysis

This paper proposes a non‐iterative time integration (NITI) scheme for non‐linear dynamic FEM analysis. The NITI scheme is constructed by combining explicit and implicit schemes, taking advantage of their merits, and enables stable computation without an iteration process for convergence even when used for non‐linear dynamic problems. Formulation of the NITI scheme is presented and its stability is studied. Although the NITI scheme is not unconditionally stable when applied to non‐linear problems, it is stable in most cases unless stiffness hardening occurs or the problem has a large velocity‐dependent term. The NITI scheme is applied to dynamic analysis of the non‐linear soil–structure system and computation results are compared with those by the central difference method (CDM). Comparison shows that the stability of the NITI scheme is superior to that of the CDM. Accuracy of the NITI scheme is verified because its results are identical with those by the CDM in which the time step is set as 1/10 of that for the NITI scheme. The application of the NITI scheme to the mesh‐partitioned FEM is also proposed. It is applied to dynamic analysis of the linear soil–structure system. It yields the same results as a conventional single‐domain FEM analysis using the Newmark β method. This result verifies the usability of mesh‐partitioned FEM analysis using the NITI scheme. Copyright © 2003 John Wiley& Sons, Ltd.     

Hybrid formulation of operator splitting (OS) and Newmark‐β methods for collaborative structural analysis (CSA)

A collaborative structural analysis (CSA) system is developed, which is capable of performing highly sophisticated structural analyses utilizing beneficial features of existing individual structural analysis programs. In the system, the global equations of motion for the overall structural system are formulated in the host program. Some substructures, whose behaviors are relatively simple, are directly solved in the host program, whereas those having complex behavior are analyzed by the station programs. A time‐consuming static condensation procedure is needed for the substructures analyzed by the station programs if adopting an implicit integration scheme. The operator splitting (OS) method, which does not require tangential stiffness, can be used to improve the system efficiency. To this end, a hybrid formulation of the Newmark‐β and OS methods is proposed, and a CSA scheme based on the hybrid formulation is developed. In the CSA system adopting the hybrid formulation, the degrees of freedom whose tangential stiffness are unavailable are formulated by the OS method, whereas the rest are still formulated by the commonly used Newmark‐β method. Using the system, analyses of a three‐story‐braced steel moment‐resisting frame are conducted. In the analyses, the column bases are analyzed using the commercial finite element method software ABAQUS, and the remaining structural elements are analyzed using a frame analysis program called NETLYS. Results suggest that the hybrid formulation is very effective for the CSA system. Copyright © 2008 John Wiley & Sons, Ltd.     

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.