Analysis of implicit HHT‐α integration algorithm for real‐time hybrid simulation

Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

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 of average acceleration method for structures with nonlinear damping

The energy approach is used to theoretically verify that the average acceleration method （AAM）, which is unconditionally stable for linear dynamic systems, is also unconditionally stable for structures with typical nonlinear damping, including the special case of velocity power type damping with a bilinear restoring force model. Based on the energy approach, the stability of the AAM is proven for SDOF structures using the mathematical features of the velocity power function and for MDOF structures by applying the virtual displacement theorem. Finally, numerical examples are given to demonstrate the accuracy of the theoretical analysis.     

A semi-analytical time integration for numerical solution of Boussinesq equation

A semi-analytical time integration method is proposed for the numerical simulation of transient groundwater flow in unconfined aquifers by the nonlinear Boussinesq equation. The method is based on the analytical solution of the system of ordinary differential equations with constant coefficients. While it is unconditionally stable and more accurate than the finite difference methods, the computational cost is much more expensive than (can be more than 10 times) that of the finite difference methods for a single time step. However, by partitioning the nonlinear parameters into linear and nonlinear parts, the costly computation can be performed only once. With larger and less variable time step sizes, the total computational cost can be significantly reduced. Three examples are included to illustrate the advantages and limitations of the proposed method.     

Discussion of “development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation” by Chinmoy Kolay and James M. Ricles

The time integration method proposed by Kolay and Ricles, which was claimed to be both explicit and unconditionally stable, is shown to be implicit in the sense of requiring the factorization of an effective stiffness matrix where an explicit method needs no solver. Its original derivation procedure employed discrete control theory concepts, which are in fact, equivalent to conventional recurrence relation concepts aiming to match its spectral properties with those of the three-parameter optimal/generalized-α method, thus giving rise to an implicit method within the class of linear multistep methods. It is shown that the resulting method possesses several added computational drawbacks due to its derivation procedure, such as additional effective stiffness inversions and a degraded order of accuracy in general.     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

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.     

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.     

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.     

A generalized least-squares family of algorithms for transient dynamic analysis

By use of the generalized least-squares procedure, in conjunction with a finite element approximation in time, a simple three-time-level family of time integration schemes is derived. This results in fourth-order accurate unconditionally stable algorithms and stable eighth-order accurate non-dissipative algorithms. Numerical examples show the accuracy of the proposed schemes in comparison with the Fox-Goodwin formula and Newmark's average acceleration method.     

Efficient mode superposition algorithm for seismic analysis of non-linear structures

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes are needed to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the non-linear terms in the equations of motion for non-linear systems may be considered as additional external forces, and the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system's response in one of its modes of vibration. To increase the efficiency of the method and account properly for the participation of higher modes, these independent equations are solved using the Nigam-Jennings technique in conjunction with the so-called mode acceleration method. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, including only a few modes, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on an average, about 47 per cent less computer time than when the direct integration method is used.     

Dynamic response by large step integration

A family of unconditionally stable algorithms for the economical computation of large linear dynamic systems is described and applied. Possible application to divergent systems is considered and some of the difficulties of extending the use of the algorithms to non-linear systems are discussed. In the Appendix a previously developed conditionally stable algorithm is applied to the non-linear gust response of a prestressed cable roof structure over the Munich Olympic Stadium. The idealization involves 1,164 degrees of freedom.     

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 modal procedure for seismic analysis of non-linear base-isolated multistorey structures

A modal procedure for non-linear analysis of multistorey structures with high-damping base-isolation systems was proposed. Two different isolation devices were considered in the analysis: an high-damping laminated rubber bearing and a lead-rubber bearing. Starting from deformational properties verified by tests, the isolation systems were characterized using three different analytical models (an Elastic Viscous, a Bilinear Hysteretic and a Wen's Model) with parameters depending from maximum lateral strain. After non-linear modelling of isolation and lateral-force-resisting systems, the effects of material non-linearities were considered as pseudo-forces applied to the equivalent linear system (Pseudo-Force Method) and the formally linearized equations of motion were uncoupled by the transformation defined by the complex mode shapes. The modal responses were finally obtained with an extension of Nigam–Jennings technique to non-linear and non-classically damped systems, in conjunction with an iterative technique searching for non-linear contributions satisfying equations of motion and constitutive laws. Since the properties of the isolated structure usually change with maximun lateral strain of isolation bearings, the integration of a new set of governing equations was required for each design-displacement value. The procedure proposed was described in detail and then applied for the determination of modal and total seismic responses in some real cases. At first, a very good agreement between non-linear responses obtained with the proposed mode superposition and with a direct integration method was observed. Then a comparison of results obtained with the three different analytical models of the isolation bearings was carried out. At last, the exact modal response obtained with analytical models depending from the design displacement of the isolation bearings was compared with two different approximated solutions, evaluated using mode shapes and isolation properties, respectively, calculated under simplified hypothesis.© 1998 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.     

Stability of an explicit time-integration algorithm for hybrid tests,considering stiffness hardening behavior

An explicit unconditionally stable algorithm for hybrid tests,which is developed from the traditional HHT-α algorithm,is proposed.The unconditional stability is first proven by the spectral radius method for a linear system.If the value of α is selected within [-0.5,0],then the algorithm is shown to be unconditionally stable.Next,the root locus method for a discrete dynamic system is applied to analyze the stability of a nonlinear system.The results show that the proposed method is conditionally stable for dynamic systems with stiffness hardening.To improve the stability of the proposed method,the structure stiffness is then identified and updated.Both numerical and pseudo-dynamic tests on a structure with the collision effect prove that the stiffness updating method can effectively improve stability.     

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.     

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。