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Experimental evaluation of CV‐Voronoi based adaptive sampling for Kriging meta‐modeling of multiple responses through real‐time hybrid simulation
Abstract
Real‐time hybrid simulation (RTHS) integrates numerical modeling of analytical substructures with physical testing of experimental sub‐structures thus enabling system global responses to be efficiently evaluated through component testing in size‐limited laboratories. Traditional practice of RTHS focuses on responses evaluation of structures without considering their uncertainties. A cross‐validation (CV)‐Voronoi based adaptive sampling strategy is explored in this study for global meta‐modeling of multiple response quantities of interests through RTHS for engineering systems with uncertainties. Based on the Kriging meta‐model from initial samples, the CV‐Voronoi based adaptive sampling sequentially identifies the sample points for RTHS tests in laboratory and observed responses of interests are used to update the Kriging meta‐model. Multiple response distributions under structural uncertainties are thus acquired through limited number of experiments. RTHS tests of a two‐degree‐of‐freedom system with self‐centering viscous dampers (SC‐VDs) are conducted in this study to experimentally evaluate the effectiveness of the CV‐Voronoi based adaptive sampling for multiple response estimation. The accuracy of multi‐response meta‐models are further evaluated through comparison with validation tests. A stopping criterion is finally proposed for more efficient implementation of the adaptive sampling strategy. It is demonstrated that the CV‐Voronoi based adaptive sampling strategy provides a viable technique to enable accurate global meta‐modeling and estimation for multiple responses with limited number of RTHS tests in laboratory.
... To get the optimal parameters at a low cost and with high accuracy, the metamodeling approach with Kriging interpolation is used and combined with an intelligent sampling and model management strategy. Here, the metamodel is explicitly expressed in accordance with the optimization variables using the Kriging interpolation [17,20,[41][42][43][44][45]. The model management and sampling strategies used throughout the optimization process have a substantial impact on the accuracy and efficacy of metamodels. ...
Reliability‐based design optimization (RBDO) traditionally relies primarily on high‐fidelity and computationally expensive simulations to search for and evaluate design solutions. However, significant disparities could emerge for complex nonlinear behavior that are challenging for numerical modeling. In contrast to mitigating the impact of inaccurate numerical modeling through optimization algorithms, laboratory experiments realistically capture the complex nonlinear behavior of structures or their components. Real‐time hybrid simulation (RTHS) is widely considered as an efficient and cost‐effective technique for integrating numerical modeling with experimental testing for structural response evaluation. This study proposes an innovative framework that utilizes RTHS for the performance assessment of candidate designs to enable RBDO of structures subjected to pulse‐like ground motions. RTHS tests are conducted to physically evaluate structural responses through realistically replicating complex nonlinear behavior of experimental substructures. This study introduces a novel penalty function‐based efficient global optimization (P‐EGO) method to minimize the required number of laboratory tests through surrogating the response quantities of interest derived from RTHS. The proposed framework is experimentally evaluated for design optimization of a two‐story four‐bay steel moment‐resisting frame with self‐centering viscous dampers subjected to pulse‐like ground motions. The results demonstrate innovative application of RTHS in dynamic optimal design to account for uncertainties. It offers an effective and efficient alternative for traditional RBDO through pure computational simulation, particularly when structural components pose challenges for numerical modeling.
Integration algorithm is an important mean to solve the discrete time equations of structural dynamics in earthquake engineering. In this paper, a new model-based explicit integration algorithm, featured by optimization-based integration parameter updating, is developed for a more accurate and efficient solution of structural dynamic problems. The new integration algorithm is designed to yield minimum error and controllable numerical dispersion under the premise of stability by defining objective functions and constraints. Numerical properties of the new integration algorithm are investigated in details using the amplification matrix method for a linear elastic single degree-of-freedom system. Meanwhile, representative numerical examples and complex structure examples are analyzed and compared with other representative algorithms. As a result, the proposed algorithm yields comparable numerical characteristics with other methods, but exhibits pronounced superiority in controllable algorithmic dissipation, higher precision, and better efficiency. Therefore, the new integration algorithm possesses extensive application prospect in solving complicated linear and nonlinear structural dynamic problems.
Seismic response evaluation often poses challenges for computational simulation when parts of structures are difficult for accurate modeling due to their complexity or nonlinearity. Failure to accurately replicate their behavior might lead to errors in response estimation for traditional computational simulation and reliability analysis when uncertainties need to be considered for the structure under investigation. Real-time hybrid simulation (RTHS) provides an efficient experimental technique for performance evaluation of large- or full-scale structures in size limited laboratories. Components that are difficult for accurate modeling are physically tested in the laboratory to enable system responses to be evaluated while the remaining parts of the structure are represented by numerical modeling. Traditional applications of RTHS often considers deterministic structural properties which however often vary with time. In this study, an innovative AK-TL-RTHS approach is proposed to integrate adaptive Kriging (AK) with transfer learning (TL) for RTHS to account for both time-dependent and independent uncertainties in response evaluation. The expansion optimal linear estimation (EOLE) is applied to discretize the stochastic process for time dependent structural deterioration. Kriging meta-model is utilized to surrogate measured structural responses from RTHS tests and a modified U-function from active learning is applied to sequentially select sample points for further RTHS tests. Selected samples and associated experimental results are transferred between random variables of discretized stochastic process to further reduce the number of tests in laboratory. RTHS of a single-degree-of-freedom (SDOF) structure with a self-centering viscous damper (SC-VD) is conducted as proof of concept to experimentally demonstrate the effectiveness of proposed AK-TL-RTHS method. It is shown that the proposed method provides an efficient and effective approach for time variant performance evaluation of structures in laboratory especially when pure numerical analysis is not feasible. The proposed approach also significantly enhances the capacity of RTHS to account for structural uncertainties and deterioration in seismic reliability analysis.
Intensive and complex engineering models drive the need for computational affordability and accuracy. To obtain more accurate prediction, an effective sampling method is proposed and evaluated in this study by integrating CV (cross validation)‐Voronoi (CV‐V) and entropy (EP) strategies. The new CVV (CV‐V) ‐DEP (Distance and EP) method uses CV‐V to divide the sample space into Voronoi units and selects the unit with largest error. The candidate sample point with the largest EP and distance values in the selected unit is then selected. To verify the efficacy of the proposed method, CVV‐DEP is first compared with traditional CV‐V and CVV‐EP (CV‐V and EP) strategies for six nonlinear functions. Computational simulations are further conducted using the proposed method for uncertainty quantification through hybrid simulation of two different nonlinear systems. The proposed method is demonstrated to provide great potential for uncertainty quantification of civil engineering infrastructures through hybrid simulation. An effective sampling method is proposed and evaluated in this study by integrating cross validation‐Voronoi (CV‐V) and entropy (EP) strategies. Computational simulations of hybrid tests of two different nonlinear systems are further conducted using the proposed method for uncertainty quantification. The proposed method is demonstrated to provide great potential for uncertainty quantification through hybrid simulation.
Hybrid simulation is an experimental method used to investigate the dynamic response of a reference prototype structure by decomposing it to physically-tested and numerically-simulated substructures. The latter substructures interact with each other in a real-time feedback loop and their coupling forms the hybrid model. In this study, we extend our previous work on metamodel-based sensitivity analysis of deterministic hybrid models to the practically more relevant case of stochastic hybrid models. The aim is to cover a more realistic situation where the physical substructure response is not deterministic, as nominally identical specimens are, in practice, never actually identical. A generalized lambda surrogate model recently developed by some of the authors is proposed to surrogate the hybrid model response, and Sobol’ sensitivity indices are computed for substructure quantity of interest response quantiles. Normally, several repetitions of every single sample of the inputs parameters would be required to replicate the response of a stochastic hybrid model. In this regard, a great advantage of the proposed framework is that the generalized lambda surrogate model does not require repeated evaluations of the same sample. The effectiveness of the proposed hybrid simulation global sensitivity analysis framework is demonstrated using an experiment.
Time-dependent reliability analysis using surrogate model has drawn much attention for avoiding the high computational burden. But the surrogate training strategies of existing methods do not directly consider the estimation error of failure probability, leading to the limitations that some computationally expensive samples are wasted or some algorithms tend to terminate prematurely. To address the challenges, this work proposes a real-time estimation error-guided sampling method. As the classification of random points may be not completely accurate, the wrong-classification probability is calculated by using the mean and variance of Kriging prediction. With this probability, the total numbers of wrongly classified points are obtained. Furthermore, the confidence intervals of the numbers are computed based on the probability theory, and the estimation error of failure probability is calculated through the confidence intervals. Subsequently, the point maximizing the probability is identified as the new sample for decreasing the estimation error, and the maximum error is used to guide when to stop the refinement of Kriging model. Results of three cases demonstrate the performance of the proposed method.
Finite element (FE) is a powerful tool and has been applied by investigators to real-time hybrid simulations (RTHSs). This study focuses on the computational efficiency, including the computational time and accuracy, of numerical integrations in solving FE numerical substructure in RTHSs. First, sparse matrix storage schemes are adopted to decrease the computational time of FE numerical substructure. In this way, the task execution time (TET) decreases such that the scale of the numerical substructure model increases. Subsequently, several commonly used explicit numerical integration algorithms, including the central difference method (CDM), the Newmark explicit method, the Chang method and the Gui-λ method, are comprehensively compared to evaluate their computational time in solving FE numerical substructure. CDM is better than the other explicit integration algorithms when the damping matrix is diagonal, while the Gui-λ (λ = 4) method is advantageous when the damping matrix is non-diagonal. Finally, the effect of time delay on the computational accuracy of RTHSs is investigated by simulating structure-foundation systems. Simulation results show that the influences of time delay on the displacement response become obvious with the mass ratio increasing, and delay compensation methods may reduce the relative error of the displacement peak value to less than 5% even under the large time-step and large time delay.
Some adaptive sampling approaches have been developed to efficiently and accurately build global metamodels for the deterministic single-response problems. Most complex engineering problems, however, yield multiple responses during one simulation. This article adjusts the framework of the CV-Voronoi adaptive sampling approach for a multi-response system. In the proposed multi-response CV-Voronoi (mCV-Voronoi) sampling approach, a new strategy that combines a weighted-sum term and an extreme term is presented to properly estimate the cell errors by simultaneously considering the characteristics of multiple responses. The performance of this approach is investigated on 57 multi-response systems and two engineering design problems. The results show that mCV-Voronoi is very promising for global metamodeling of deterministic multi-response systems.
Hybrid simulation is a very effective testing strategy for simulating the dynamic response of structural systems whose dimensions and complexities exceed the capacity of conventional testing facilities. The hybrid model of the prototype structure, which combines numerical and physical substructures, accurately reproduces the overall dynamic response of the structure with reduced costs and effort. In current practice, the parameters that characterize the numerical subdomain are deterministic. The values of these parameters are often determined through deliberate simplifications, ignoring the associated uncertainties. However, the effect of uncertainties may be significant. The objective of the work, presented in this paper, is to use uncertainty quantification and global sensitivity analysis to evaluate the effect of uncertainties on the response of a hybrid model in simulations. Sobol indices are selected to explain the total variance of output parameters, e.g. maximum absolute displacement peaks, with respect to the variance of input parameters, e.g. viscous damping. In detail, a generalized polynomial chaos expansion of the hybrid system response is derived and Sobol indices are post-processed from the entailing polynomial coefficients. The proposed approach is illustrated using a nonlinear two-degrees-of-freedom benchmark case study that covers a class of prototype typical for hybrid simulation tests.
Accurate actuator tracking is critical to achieve reliable real-time hybrid simulation results for earthquake engineering research. The frequency-domain evaluation approach provides an innovative way for more quantitative post-simulation evaluation of actuator tracking errors compared with existing time domain based techniques. Utilizing the Fast Fourier Transfonn the approach analyzes the actuator en-or in terms of amplitude and phrase errors. Existing application of the approach requires using the complete length of the experimental data. To improve the computational efficiency, two techniques including data decimation and frequency decimation are analyzed to reduce the amount of data involved in the frequency-domain evaluation. The presented study aims to enhance the computational efficiency of the approach in order to utilize it for future on-line actuator tracking evaluation. Both computational simulation and laboratory experimental results are analyzed and recommendations on the two decimation factors are provided based on the fmdings from this study.
Surrogate models are widely used in simulation-based engineering design and optimization to save the computing cost. The choice of sampling approach has a great impact on the metamodel accuracy. This article presents a robust error-pursuing sequential sampling approach called cross-validation (CV)-Voronoi for global metamodeling. During the sampling process, CV-Voronoi uses Voronoi diagram to partition the design space into a set of Voronoi cells according to existing points. The error behavior of each cell is estimated by leave-one-out (LOO) cross-validation approach. Large prediction error indicates that the constructed metamodel in this Voronoi cell has not been fitted well and, thus, new points should be sampled in this cell. In order to rapidly improve the metamodel accuracy, the proposed approach samples a Voronoi cell with the largest error value, which is marked as a sensitive region. The sampling approach exploits locally by the identification of sensitive region and explores globally with the shift of sensitive region. Comparative results with several sequential sampling approaches have demonstrated that the proposed approach is simple, robust, and achieves the desired metamodel accuracy with fewer samples, that is needed in simulation-based engineering design problems.
Real-time hybrid simulation combines experimental testing and numerical simulation by dividing a structural system into experimental and analytical substructures. Servohydraulic actuators are typically used in a real-time hybrid simulation to apply command displacements to the experimental substructure(s). Servohydraulic actuators may develop a time delay due to inherent actuator dynamics that results in a desynchronization between the measured restoring force(s) and the integration algorithm in a real-time hybrid simulation. Inaccuracy or even instability will occur in a hybrid simulation if actuator delay is not compensated properly. This paper presents an adaptive compensation method for actuator delay. An adaptive control law is developed using an error tracking indicator to adapt a compensation parameter used in the proposed compensation method. Laboratory tests involving large-scale real-time hybrid simulations of a single degree of freedom moment resisting frame with an elastomeric damper are conducted to experimentally demonstrate the effectiveness of the proposed adaptive compensation method. The actuator tracking capability is shown to be greatly improved and exceptional experimental results are still achieved when a good estimate of actuator delay is not available.
Real-time testing is a useful technique to evaluate the performance of structural systems subjected to seismic loading. Servo-hydraulic actuators can introduce an inevitable delay when applying command displacements to a structure during a real-time test due to their inherent dynamics. Various compensation methods have been developed and applied in real-time testing to minimize the effect of actuator delay. This paper proposes an approach to analyze actuator delay compensation methods using an equivalent discrete transfer function. Discrete control theory is introduced and used to formulate the discrete transfer function in the frequency domain and the difference equation in the time domain for an actuator delay compensation method. The discrete transfer function is used to conduct frequency response analysis of the compensation method to provide insight into the expected performance of the compensation method by graphically illustrating the characteristics of the compensation method as a function of frequency of command displacement and the parameters for the compensator. The difference equation gives the form of the mathematical extrapolation in the time domain for the compensation method. Three popular compensation methods are selected to illustrate the proposed approach. Experiments with predefined sine sweep displacement and real-time hybrid simulations of a single-degree-of-freedom moment resisting frame with an elastomeric damper are conducted to experimentally validate the analysis of the selected compensation methods using the proposed approach.
Surrogate models are data-driven models used to accurately mimic the complex behavior of a system. They are often used to approximate computationally expensive simulation code in order to speed up the exploration of design spaces. A crucial step in the building of surrogate models is finding a good set of hyperparameters, which determine the behavior of the model. This is especially important when dealing with sparse data, as the models are in that case more prone to overfitting. Cross-validation is often used to optimize the hyperparameters of surrogate models, however it is computationally expensive and can still lead to overfitting or other erratic model behavior. This paper introduces a new auxiliary measure for the optimization of the hyperparameters of surrogate models which, when used in conjunction with a cheap accuracy measure, is fast and effective at avoiding unexplained model behavior.
In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.
In this article, a study is performed on the accuracy of radial basis functions (RBFs) in creating global metamodels for both low- and high-order nonlinear responses. The response surface methodology (RSM), which typically uses linear or quadratic polynomials, is inappropriate for creating global models for highly nonlinear responses. The RBF, on the other hand, has been shown to be accurate for highly nonlinear responses when the sample size is large. However, for most complex engineering applications only limited numbers of samples can be afforded; it is desirable to know whether the RBF is appropriate in this situation, especially when the augmented RBF has to be used. Because the types of true responses are typically unknown a priori, it is essential for high-fidelity metamodeling to have an RBF or RBFs that are appropriate for linear, quadratic, and higher-order responses. To this end, this study compares a variety of existing basis functions in both non-augmented and augmented forms with various types of responses and limited numbers of samples. This article shows that the augmented RBF models created by Wu’s compactly supported functions are the most accurate for the various test functions used in this study.
Real-time hybrid substructuring (RTHS) has previously been shown to be an effective tool to quantify the effect of parametric uncertainties on the response of a structural system. Proposed and implemented in this paper is a method that combines RTHS, Principal Component Analysis, and Kriging to metamodel the frequency response functions of a structure. The proposed method can be used to account for parametric variation in both the numerical and physical substructures. This approach is demonstrated using a series of bench-scale RTHS tests of a magnetorheological (MR) fluid damper used to control a 2 degree-of-freedom mass-spring system. The numerical system spring stiffnesses and the physical current supplied to the MR damper are each treated as uniformly distributed random variables. The RTHS test data is used to train computationally fast metamodels, which can then be used to conduct Monte Carlo simulations to determine distributions of the system response. The proposed methodology is shown to be both efficient and accurate.
Uncertainties in real-time hybrid simulation include structural parameters and ground motion. Uncertain parameters often do not follow common distribution types. Data-driven arbitrary polynomial chaos constructs optimal orthogonal polynomial basis based on the sample data without distribution assumption. In this study, the data-driven polynomial chaos is compared with other generalized polynomial chaos from the aspects of the rate of error convergence when applied for uncertainty quantification of real-time hybrid simulation. Moreover, uncertainties of ground motion are considered in the RTHS problem to represent the scenarios with more complex input variables. Different statistical indicators are utilized to evaluate the accuracy of the alternative model in comparison with the Monte Carlo simulation results. Compared with generalized polynomial chaos, the data-driven arbitrary polynomial chaos presents potential for uncertainty quantification of real-time hybrid simulation with approximate or better accuracy. Actuator delay in RTHS could change the sensitivity of model output to the random variables.
This research presents a novel self-centering fluidic viscous damper that incorporates preloaded ring springs to offer self-centering capability and a fluidic viscous damper for energy dissipation. A full-scale self-centering fluidic viscous damper was developed and subjected to low-cyclic reversed loading tests. The test results show the self-centering fluidic viscous damper has both displacement-dependent and velocity-dependent hysteric responses with self-centering capability. Fatigue tests further show that the self-centering fluidic viscous damper maintains a stable hysteretic response under reversed loading. An analytical model and a numerical model are developed for the proposed self-centering fluidic viscous damper and analyzed. Comparisons of test results and the numerical and analytical models show similar hysteric responses, thereby validating the accuracy of the numerical and analytical models to simulate the behavior of the proposed damper.
Real-time hybrid simulation (RTHS) is a reliable and cost-effective testing technique to evaluate the dynamic response of a structural system, especially when the system includes rate-dependent components. Numerous studies suggest that the stability and accuracy of a RTHS are governed by the tracking errors between the calculated displacements and measured displacements during the test. In this study, frequency domain-based error indicators were designed to quantify the tracking errors in real-time. Consequently, an adaptive two degree-of-freedom phase-lead compensator was introduced to cancel out the identified tracking errors. Additionally, a proportional-integral -derivative controller was included as a supplement to further improve the tracking performance. The details of how to design the combined outer-loop controller were provided. A benchmark control problem in RTHS was used to assess the performance and robustness of the designed controller by carrying out a series of virtual RTHS (vRTHS), where different force excitations, partitioning configurations, and plant uncertainties were considered in MATLAB Simulink. Nine evaluation criteria were used to quantify the results of the vRTHS in this study. As a result, the proposed combined outer-loop controller was shown to be effective and robust to provide good stability and accuracy in RTHS.
This historical note reports on the early days of the development of an experimental method called “hybrid simulation.” As background, the seeds of this concept, initiated in the early 1970s by Japanese researchers, are presented first, followed by initial efforts (regarded as Stage I) to realize the concept of hybrid simulation and its first applications to explore the seismic performance of structures. The initial research in this now‐seminal field of earthquake engineering began in the early 1970s by Koichi Takanashi and his coworkers at the Institute of Industrial Science, the University of Tokyo. Their highly notable efforts in laying the groundwork for hybrid simulation occurred in the mid‐1970s through the early 1980s by Takanashi (for steel structures) and Tsuneo Okada (for RC structures). These two men and their coworkers first applied hybrid simulation to explore the seismic behavior, performance, and design of various types of building structures. In Stage I, this method was called “the on‐line computer‐controlled test” or “pseudo dynamic test” because the unique feature of the method was the combined test and simulation and the intentional slow loading in the test. Extension of the scope and application of hybrid simulation occurred largely between the early 1980s and the early 1990s (regarded as Stage II) in conjunction with the United States–Japan joint research project. A few notable efforts made around that period are touched upon briefly, including error propagation and suppression in multi‐degree‐of‐freedom hybrid simulation, application of the substructure methodology to hybrid simulation, and real‐time hybrid simulation.
Real-time hybrid simulation is an advanced testing methodology to evaluate structural responses under realistic operating conditions. Typically, the real-time hybrid system uses actuation and control system to apply load and motion boundary conditions on the experimental substructure. The inherent system dynamics present phase lags, in addition to the communication delays, that both cause negative damping in the real-time hybrid system. In this study, the dynamic equation of motion is derived for a generalized multiple-degree-of-freedom hybrid system. The negative damping effect is quantified, which depends not only on the actuator motion control performance, but also more importantly on the partition between the numerical and experimental substructures (i.e. how the stiffness and mass are assumed in both substructures). It is demonstrated the worst-case hybrid substructure partition may have very narrow stability margin in its tolerance of any system delay. The proposed equation of motion gains system level understanding of any arbitrary hybrid substructure partition; thus allow both frequency domain system analysis and time domain evaluation. The benchmark problem is studied to validate the proposed equation of motion compared with the virtual testing method, both approaches show excellent correlation. These pre-testing assessments could establish quantitative predictive measures about the system stability limit and performance criteria. Thus they are very important in the early design stage of a feasible real-time hybrid implementation, help reduce the risks of unintended physical testing responses.
This paper presents the problem definition and guidelines for a benchmark control problem in real-time hybrid simulation for a seismically excited building, to appear in a Special Issue of Mechanical Systems and Signal Processing. Benchmark problems have been especially useful in enabling a community of researchers to leap forward on a given topic, distill the lessons learned, and identify the capabilities and limitations of various approaches. The focus here is on the design of an effective transfer system displacement tracking controller which is a commonly used approach for ensuring that interface conditions between numerical and experimental substructures are satisfied. In this study, a laboratory model of a three-story steel frame is considered as the reference structure. Realistic numerical models are developed and provided to represent the numerical and experimental substructures and the transfer system, which is comprised of hydraulic actuation, sensing instrumentation, and control implementation hardware. Experimental components are identified and provided as Simulink models, which are executed in real-time using Simulink Desktop Real-Time capability to enable realistic virtual real-time hybrid simulation. The task of each participant of the Special Issue is to design, evaluate, and report on their proposed controller approaches using the numerical models and computational codes provided. Such approaches will be assessed for robustness and performance using the provided tools. This benchmark problem is expected to further the understanding of the relative merits, as well as provide a clear basis for evaluating the performance of various control approaches and algorithms for RTHS. To illustrate some of the design challenges, a sample control strategy employing a proportional-integral (PI) controller is included, in addition to the built-in control loop of the transfer system.
Real-time hybrid simulation (RTHS) is an innovative and versatile technique for evaluating the dynamic responses of structural and mechanical systems. This technique separates the emulated system into numerical and physical substructures, which are analyzed by computers and loaded in laboratories, respectively. Ensuring the boundary conditions between the two substructures through a transfer system plays a significant role in obtaining reliable and accurate testing results. However, measurement noise and the delay between commands and responses due to the dynamic performance of the transfer system are inevitable in RTHS. To address these issues and to achieve outstanding tracking performance and excellent robustness, this paper proposes an adaptive Kalman-based noise filter and an adaptive two-stage delay compensation method. In particular, in the novel noise filter strategy, adaptive inverse compensation with parameters updated by the least squares method is adopted to accommodate the amplitude and phase errors induced by a traditional Kalman filter. In the proposed delay compensation method, classic polynomial extrapolation and an adaptive inverse strategy are employed for coarse and fine compensation, respectively. Virtual RTHS on a benchmark problem reveals the satisfactory tracking performance and robustness of the proposed methods. Comparisons with polynomial extrapolation and single-stage adaptive compensation indicate the superiority of the proposed two-stage delay compensation method.
Real-time hybrid simulation (RTHS) is a practical, cost-effective, and versatile experimental technique to evaluate structural performance under dynamic excitation. The simulated structure is commonly split into a physically tested rate-dependent substructure (PS) and a numerically simulated substructure (NS). A transfer system such as a servo-hydraulic actuator is used to impose boundary conditions on the PS. Consequently, efficient actuator control is necessary to guarantee reliable simulation results. However, time delay and uncertainties exist due to the dynamics of the actuator, which adversely influence the accuracy and stability of RTHS. Therefore, an innovative robust actuator dynamics compensation method is proposed in this study comprising three components, namely a mixed sensitivity-based robust H∞ controller to stabilize the actuator–specimen dynamics, a polynomial extrapolation module to further cancel the actuator delay, and an adaptive filter for displacement reconstruction of the actuator–specimen system. A detailed design procedure of the proposed strategy is presented. The efficacy of the proposed strategy is validated through a series of virtual tests on the benchmark problem for RTHS. Results show that the proposed method exhibits excellent tracking performance and robustness. In particular, the maximum values of the calculated time delay (TD), root mean square of the tracking error (RMSE), and peak tracking error (PE) are 0 ms, 2.56%, and 2.12%, respectively, whereas the maximum values of the standard deviation of TD, RMSE, and PE are 0 ms, 0.25%, and 0.33%, respectively.
Actuator control plays an essential role to achieve stable and accurate real-time hybrid simulation (RTHS) results. Delay compensation is often used to minimize the desynchronization at the interface between numerical and experimental substructures. In this study, a new delay compensation method is proposed for RTHS, which integrates the inverse compensation method (IC) and frequency-domain evaluation index (FEI). Window technique is utilized to enable FEI for calculation of almost instantaneous time delay and the IC parameter is then adjusted accordingly for optimal compensation. The performance of this windowed FEI compensation (WFEI) is evaluated and compared with that of the IC and the adaptive inverse compensation (AIC) through computational simulations of a benchmark model with different initial estimates of time delay. It is demonstrated that the WFEI compensation not only provides accurate actuator control when initial estimated time delay deviates from actual values but also have good robustness under unpredicted uncertainties of the servo-hydraulic system.
A backstepping adaptive control method is proposed for on-line estimation of unknown servo-hydraulic dynamics and the compensation of time-varying lags in real-time hybrid simulation tests. The response tracking problem becomes a critical challenge when realistic experimental conditions are taken into consideration, such as control-structure interaction effects and sensor measurement noise. Unlike a conventional time-lag compensator, the proposed adaptive controller generates a command trajectory for the actuated system according to adaptive laws. Besides bringing response tracking error to zeros, the estimation of a first-principle actuator dynamic model is also facilitated in the proposed approach. Lyapunov stability analysis is systematically presented for designing the adaptive control law. Illustratively, a three-story seismically excited structure with different control strategies is utilized to demonstrate the efficiency and robustness of the proposed controller. A benchmark problem is then utilized for the verification of controller's advancement. Four simulation cases with different damping/mass conditions and four ground excitation scenarios are selected for the application. As stated, favorable tracking performance has been observed with a remarkable improvement in performance evaluation.
In a real-time hybrid simulation (RTHS), the actuator delay in experimental results might deviate from actual structural responses and even destabilize the real-time test. Although the assumption of a constant actuator delay helps simplify the stability analysis of RTHS, experimental results often show that the actuator delay varies throughout the test. However, research on the effect of time-varying delay on RTHS system stability is very limited. In this study, the Lyapunov-Krasovskii functional is introduced for the stability analysis of RTHS system. Two stability criteria are proposed for a linear system with a single constant delay and a time-varying delay. It is demonstrated that (1) the stable region of a time-varying delay system shrinks with the increase of the first derivative of time-varying delay; and (2) the stable region of the time-varying delay system is smaller than that of constant-time-delay system. Computational simulations were conducted for RTHS systems with a single degree of freedom to evaluate the proposed criteria. When the experimental specimen is an ideal elastic spring, the stability region of RTHS system with time-varying delay is shown to depend on the stiffness partition, structural natural period, and damping ratio. Significant differences in stability regions indicate that the time-varying characteristics of actuator delay should be considered for stability analysis of RTHS systems.
Uncertainties in structure properties can result in different responses in hybrid simulations. Quantification of the effect of these uncertainties would enable researchers to estimate the variances of structural responses observed from experiments. This poses challenges for real-time hybrid simulation (RTHS) due to the existence of actuator delay. Polynomial chaos expansion (PCE) projects the model outputs on a basis of orthogonal stochastic polynomials to account for influences of model uncertainties. In this paper, PCE is utilized to evaluate effect of actuator delay on the maximum displacement from real-time hybrid simulation of a single degree of freedom (SDOF) structure when accounting for uncertainties in structural properties. The PCE is first applied for RTHS without delay to determine the order of PCE, the number of sample points as well as the method for coefficients calculation. The PCE is then applied to RTHS with actuator delay. The mean, variance and Sobol indices are compared and discussed to evaluate the effects of actuator delay on uncertainty quantification for RTHS. Results show that the mean and the variance of the maximum displacement increase linearly and exponentially with respect to actuator delay, respectively. Sensitivity analysis through Sobol indices also indicates the influence of the single random variable decreases while the coupling effect increases with the increase of actuator delay. © 2017, Institute of Engineering Mechanics, China Earthquake Administration and Springer-Verlag GmbH Germany.
Real-time hybrid simulation is an efficient and cost-effective dynamic testing technique for performance evaluation of structural systems subjected to earthquake loading with rate-dependent behavior. A loading assembly with multiple actuators is required to impose realistic boundary conditions on physical specimens. However, such a testing system is expected to exhibit significant dynamic coupling of the actuators and suffer from time lags that are associated with the dynamics of the servo-hydraulic system, as well as control-structure interaction (CSI). One approach to reducing experimental errors considers a multi-input, multi-output (MIMO) controller design, yielding accurate reference tracking and noise rejection. In this paper, a framework for multi-axial real-time hybrid simulation (maRTHS) testing is presented. The methodology employs a real-time feedback-feedforward controller for multiple actuators commanded in Cartesian coordinates. Kinematic transformations between actuator space and Cartesian space are derived for all six-degrees-offreedom of the moving platform. Then, a frequency domain identification technique is used to develop an accurate MIMO transfer function of the system. Further, a Cartesian-domain model-based feedforward-feedback controller is implemented for time lag compensation and to increase the robustness of the reference tracking for given model uncertainty. The framework is implemented using the 1/5th-scale Load and Boundary Condition Box (LBCB) located at the University of Illinois at Urbana- Champaign. To demonstrate the efficacy of the proposed methodology, a single-story frame subjected to earthquake loading is tested. One of the columns in the frame is represented physically in the laboratory as a cantilevered steel column. For realtime execution, the numerical substructure, kinematic transformations, and controllers are implemented on a digital signal processor. Results show excellent performance of the maRTHS framework when six-degrees-of-freedom are controlled at the interface between substructures. © 2017, Institute of Engineering Mechanics, China Earthquake Administration and Springer-Verlag GmbH Germany.
This article reviews Kriging (also called spatial correlation modeling). It presents the basic Kriging assumptions and formulas, contrasting Kriging and classic linear regression metamodels. Furthermore, it extends Kriging to random simulation, and discusses bootstrapping to estimate the variance of the Kriging predictor. Besides classic one-shot statistical designs such as Latin Hypercube Sampling, it reviews sequentialized and customized designs. It ends with topics for future research.
Real-time hybrid simulation (RTHS) is a powerful cyber-physical technique that is a relatively cost-effective method to perform global/local system evaluation of structural systems. A major factor that determines the ability of an RTHS to represent true system-level behavior is the fidelity of the numerical substructure. While the use of higher-order models increases fidelity of the simulation, it also increases the demand for computational resources. Because RTHS is executed at real-time, in a conventional RTHS configuration, this increase in computational resources may limit the achievable sampling frequencies and/or introduce delays that can degrade its stability and performance. In this study, the Adaptive Multi-rate Interface rate-transitioning and compensation technique is developed to enable the use of more complex numerical models. Such a multi-rate RTHS is strictly executed at real-time, although it employs different time steps in the numerical and the physical substructures while including rate-transitioning to link the components appropriately. Typically, a higher-order numerical substructure model is solved at larger time intervals, and is coupled with a physical substructure that is driven at smaller time intervals for actuator control purposes. Through a series of simulations, the performance of the AMRI and several existing approaches for multi-rate RTHS is compared. It is noted that compared with existing methods, AMRI leads to a smaller error, especially at higher ratios of sampling frequency between the numerical and physical substructures and for input signals with high-frequency content. Further, it does not induce signal chattering at the coupling frequency. The effectiveness of AMRI is also verified experimentally.
Magnetorheological dampers (MR) have the promising ability to mitigate seismic hazard for structures because of their adaptive energy dissipation characteristics and low power requirements that can be met using standby batteries. These attractive characterstics of advanced damping devices, such as MR dampers, are important for achieving the goals of performance-based infrastucture designs. This paper validates the performances of four semiactive control algorithms for the control of a large-scale realistic moment-resisting frame using a large-scale 200-kN MR damper. To conduct this test, a large-scale damper-braced steel frame was designed and fabricated. Four semiactive controllers, namely (1) passive on, (2) clipped optimal controller, (3) decentralized output feedback polynomial controller, and (4) Lyapunov stability based controller, were designed for this frame. Real-time hybrid simulations (RTHS) were carried out for these controllers using three recorded earthquakes. The comparative performance of these controllers was investigated using both RTHS and numerical simulations in terms of reductions in the maximum interstory drifts, displacements, absolute accelerations, and control forces, and comparisons between test and numerical results.
As magnetorheological (MR) control devices increase in scale for use in real-world civil engineering applications, sophisticated modeling and control techniques may be needed to exploit their unique characteristics. Here, a control algorithm that utilizes overdriving and backdriving current control to increase the efficacy of the control device is experimentally verified and evaluated at large scale. Real-time hybrid simulation (RTHS) is conducted to perform the verification experiments using the nees@Lehigh facility. The physical substructure of the RTHS is a 10-m tall planar steel frame equipped with a large-scale MR damper. Through RTHS, the test configuration is used to represent two code-compliant structures, and is evaluated under seismic excitation. The results from numerical simulation and RTHS are compared to verify the RTHS methodology. The global responses of the full system are used to assess the performance of each control algorithm. In each case, the reduction in peak and root mean square (RMS) responses (displacement, drift, acceleration, damper force, etc.) is examined. Beyond the verification tests, the robust performance of the damper controllers is also demonstrated using RTHS.
This paper presents real-time hybrid earthquake simulation (RTHS) on a large-scale steel structure with nonlinear viscous dampers. The test structure includes a three-story, single-bay moment-resisting frame (MRF), a three-story, single-bay frame with a nonlinear viscous damper and associated bracing in each story (called damped braced frame (DBF)), and gravity load system with associated seismic mass and gravity loads. To achieve the accurate RTHS results presented in this paper, several factors were considered comprehensively: (1) different arrangements of substructures for the RTHS; (2) dynamic characteristics of the test setup; (3) accurate integration of the equations of motion; (4) continuous movement of the servo-controlled hydraulic actuators; (5) appropriate feedback signals to control the RTHS; and (6) adaptive compensation for potential control errors. Unlike most previous RTHS studies, where the actuator stroke was used as the feedback to control the RTHS, the present study uses the measured displacements of the experimental substructure as the feedback for the RTHS, to enable accurate displacements to be imposed on the experimental substructure. This improvement in approach was needed because of compliance and other dynamic characteristics of the test setup, which will be present in most large-scale RTHS. RTHS with ground motions at the design basis earthquake and maximum considered earthquake levels were successfully performed, resulting in significant nonlinear response of the test structure, which makes accurate RTHS more challenging. Two phases of RTHS were conducted: in the first phase, the DBF is the experimental substructure, and in the second phase, the DBF together with the MRF is the experimental substructure. The results from the two phases of RTHS are presented and compared with numerical simulation results. An evaluation of the results shows that the RTHS approach used in this study provides a realistic and accurate simulation of the seismic response of a large-scale structure with rate-dependent energy dissipating devices. Copyright © 2015 John Wiley & Sons, Ltd.
Real-time hybrid simulation combines physical testing (experimental substructuring) and numerical simulation (analytical substructuring) such that the dynamic performance of the entire structural system can be considered during the simulation. A grid-based real-time hybrid simulation technique is introduced as a means to perform real-time hybrid simulations of complex structural systems where the analytical substructure poses a large computational demand. Real-time hybrid simulations of the 9-story ASCE benchmark structure with large-scale magnetorheological (MR) dampers are performed at the Lehigh NEES Equipment Site to demonstrate the multi-grid real-time hybrid simulation procedure and illustrate the ability to significantly reduce the time to perform the state determination of the analytical substructure. The 9-story building structure is modeled as the analytical substructure, with the experimental substructure consisting of large-scale MR dampers that are located in the structure. The analytical substructure is divided into two parts and implemented onto a computational grid consisting of two parallel xPCs that run MathWorks real-time Target PC software package. The restoring force data from these two xPCs are synchronized together along with the measured damper forces from the experimental substructure, and processed in a real-time manner for each time step of the hybrid simulation. The results of real-time hybrid simulation are compared to those of numerical simulations to validate the new test methodology.
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.
Hydraulic actuators are typically used in a real‐time hybrid simulation to impose displacements to a test structure (also known as the experimental substructure). It is imperative that good actuator control is achieved in the real‐time hybrid simulation to minimize actuator delay that leads to incorrect simulation results. The inherent nonlinearity of an actuator as well as any nonlinear response of the experimental substructure can result in an amplitude‐dependent behavior of the servo‐hydraulic system, making it challenging to accurately control the actuator. To achieve improved control of a servo‐hydraulic system with nonlinearities, an adaptive actuator compensation scheme called the adaptive time series (ATS) compensator is developed. The ATS compensator continuously updates the coefficients of the system transfer function during a real‐time hybrid simulation using online real‐time linear regression analysis. Unlike most existing adaptive methods, the system identification procedure of the ATS compensator does not involve user‐defined adaptive gains. Through the online updating of the coefficients of the system transfer function, the ATS compensator can effectively account for the nonlinearity of the combined system, resulting in improved accuracy in actuator control. A comparison of the performance of the ATS compensator with existing linearized compensation methods shows superior results for the ATS compensator for cases involving actuator motions with predefined actuator displacement histories as well as real‐time hybrid simulations. Copyright © 2013 John Wiley & Sons, Ltd.
In order to reduce the time and resources devoted to design-space exploration during simulation-based design and optimization, the use of surrogate models, or metamodels, has been proposed in the literature. Key to the success of metamodeling efforts are the experimental design techniques used to generate the combinations of input variables at which the computer experiments are conducted. Several adaptive sampling techniques have been proposed to tailor the experimental designs to the specific application at hand, using the already-acquired data to guide further exploration of the input space, instead of using a fixed sampling scheme defined a priori. Though mixed results have been reported, it has been argued that adaptive sampling techniques can be more efficient, yielding better surrogate models with less sampling points. In this paper, we address the problem of adaptive sampling for single and multi-response metamodels, with a focus on Multi-stage Multi-response Bayesian Surrogate Models (MMBSM). We compare distance-optimal latin hypercube sampling, an entropy-based criterion and the maximum cross-validation variance criterion, originally proposed for one-dimensional output spaces and implemented in this paper for multi-dimensional output spaces. Our results indicate that, both for single and multi-response surrogate models, the entropy-based adaptive sampling approach leads to models that are more robust to the initial experimental design and at least as accurate (or better) when compared with other sampling techniques using the same number of sampling points.
An important challenge in structural reliability is to keep to a minimum the number of calls to the numerical models. Engineering problems involve more and more complex computer codes and the evaluation of the probability of failure may require very time-consuming computations. Metamodels are used to reduce these computation times. To assess reliability, the most popular approach remains the numerous variants of response surfaces. Polynomial Chaos [1] and Support Vector Machine [2] are also possibilities and have gained considerations among researchers in the last decades. However, recently, Kriging, originated from geostatistics, have emerged in reliability analysis. Widespread in optimisation, Kriging has just started to appear in uncertainty propagation [3] and reliability and studies. It presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used in active learning methods. The aim of this paper is to propose an iterative approach based on Monte Carlo Simulation and Kriging metamodel to assess the reliability of structures in a more efficient way. The method is called AK-MCS for Active learning reliability method combining Kriging and Monte Carlo Simulation. It is shown to be very efficient as the probability of failure obtained with AK-MCS is very accurate and this, for only a small number of calls to the performance function. Several examples from literature are performed to illustrate the methodology and to prove its efficiency particularly for problems dealing with high non-linearity, non-differentiability, non-convex and non-connex domains of failure and high dimensionality.
Many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior, making reliability analysis difficult. This paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space. The method begins with a Gaussian process model built from a very small number of samples, and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state. The resulting Gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding (or failing to exceed) the response level of interest. By locating multiple points on or near the limit state, more complex and nonlinear limit states can be modeled, leading to more accurate probability integration. By concentrating the samples in the area where accuracy is important (i.e., in the vicinity of the limit state), only a small number of true function evaluations are required to build a quality surrogate model. The resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions. This new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
We developed an on–line experimental system for conducting hybrid experiments in real time. It combines a computer, whic conducts vibration simulation and generates a control signal, and a hydraulic actuator, which conducts a vibration experimen driven by the control signal. This system compensates for actuator delay and thus enables experiments to be carried out i real time. We evaluated the stability of the experiments with respect to the mass of the structure under excitation, and w developed a new method for compensating actuator delay in order to increase the stability condition. In this method, the compensate control signal is generated from the simulation results by using not only displacement but also velocity and acceleration.
This method provides a stability criterion (allowable ratio of mass of the structure under excitation to that of a numerica model) about three times larger than that from the current method.
Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.
The theorem that an integrable function can be decomposed into summands of different dimensions is proved. The Monte Carlo algorithm is proposed for estimating the sensitivity of a function with respect to arbitrary groups of variables.
Real-time pseudodynamic (PSD) testing is an experimental technique for evaluating the dynamic behaviour of a complex structure. During the test, when the targeted command displacements are not achieved by the test structure, or a delay in the measured restoring forces from the test structure exists, the reliability of the testing method is impaired. The stability and accuracy of real-time PSD testing in the presence of amplitude error and a time delay in the restoring force is presented. Systems consisting of an elastic single degree of freedom (SDOF) structure with load-rate independent and dependent restoring forces are considered. Bode plots are used to assess the effects of amplitude error and a time delay on the steady-state accuracy of the system. A method called the pseudodelay technique is used to derive the exact solution to the delay differential equation for the critical time delay that causes instability of the system. The solution is expressed in terms of the test structure parameters (mass, damping, stiffness). An error in the restoring force amplitude is shown to degrade the accuracy of a real-time PSD test but not destabilize the system, while a time delay can lead to instability. Example calculations are performed for determining the critical time delay, and numerical simulations with both a constant delay and variable delay in the restoring force are shown to agree well with the stability limit for the system based on the critical time delay solution. The simulation models are also used to investigate the effects of a time delay in the PSD test of an inelastic SDOF system. The effect of energy dissipation in an inelastic structure increases the limit for the critical time delay, due to the energy removed from the system by the energy dissipation. Copyright © 2007 John Wiley & Sons, Ltd.
This paper presents a study for the development of a system capable of performing real-time pseudo dynamic testing. The system combines the basics of the pseudo dynamic test with a dynamic actuator, a digital displacement transducer and a digital servo-mechanism. The digital servo-mechanism has been introduced to ensure accurate displacement and velocity control, in which digital feedback control with a time interval of 2 msec has been performed continuously during actuator motion. Using the system, pseudo dynamic tests under sinusoidal and earthquake ground motion are carried out for a structure having a viscous damper, demonstrating that a perfectly real-time pseudo dynamic test can be achieved by incorporating the modified central difference method into an extra buffer operation of the digital servo-mechanism. The responses solved by the pseudo dynamic tests are compared with the responses of the test structure as well as those obtained from post-numerical analysis, and it is found that the real-time pseudo dynamic test conducted in this study is accurate.
This paper presents the implementation details of a real-time pseudodynamic test system that adopts an implicit time integration scheme. The basic configuration of the system is presented. Physical tests were conducted to evaluate the performance of the system and validate a theoretical system model that incorporates the dynamics and nonlinearity of a test structure and servo-hydraulic actuators, control algorithm, actuator delay compensation methods, and the flexibility of an actuator reaction system. The robustness and accuracy of the computational scheme under displacement control errors and severe structural softening are examined with numerical simulations using the model. Different delay compensation schemes have been implemented and compared. One of the schemes also compensates for the deformation of an actuator reaction system. It has been shown that the test method is able to attain a good performance in terms of numerical stability and accuracy. However, it has been shown that test results obtained with this method can underestimate the inelastic displacement drift when severe strain softening develops in a test structure. This can be attributed to the fact that the numerical damping effect introduced by convergence errors becomes more significant as a structure softens. In a real-time test, a significant portion of the convergence errors is caused by the time delay in actuator response. Hence, a softening structure demands higher precision in displacement control. Copyright © 2007 John Wiley & Sons, Ltd.