Improved Integration Algorithm for Application to Hybrid Simulation of Numerical and Experimental Models

Hybrid simulation which combines experimental and numerical modeling is a powerful and relatively new test method for evaluating the seismic performance of structural systems. In this method only critical components of structure are tested experimentally while the rest of the structure is numerically modeled in the computer. In this method the response of the structure is achieved by numerically integrating the equation of motion of the whole system. Among numerical integration methods, operator splitting (OS) method is of great interest for hybrid simulation, since not only its results are more accurate and stable in comparison with explicit methods but also its application for hybrid simulation is much more easier than implicit methods; the reason is that in OS method it is not required to conduct iteration on experimental element or estimate its tangent stiffness matrix during the simulation, the tasks which limit the application of implicit methods for hybrid simulation. But OS method suffers from the shortcoming that the use of initial stiffness matrix in its corrector step decreases the accuracy of results in nonlinear range. This paper presents a modified form of OS method which is termed modified operator splitting (MOS) integration method in which by proposing a new procedure in the predictor step, the accuracy of this step is increased. When the accuracy of the predictor step increases, the difference between predictor and corrector displacements decreases and as a result the effect of initial stiffness approximation becomes less important. This would finally result in the improved accuracy of the whole simulation, as is shown in the paper. The performance, accuracy and stability characteristics of the proposed integration method were studied through numerical simulations, where it was assumed that the restoring force of the system is achieved experimentally and no information about the experimental stiffness is available. The results showed that for the wide range of considered systems including various natural periods, various ductility ratios and various degrees of freedom, MOS results are more accurate than OS method. This shows that the employed method of the predictor step of MOS method has successfully decreased the length of the corrector step with initial stiffness assumption. All the employed error indices also verified that not only the results of MOS are in great harmony with the reference solution but also its accuracy is improved over regular OS method, especially in simulations involving severe nonlinearity. Furthermore results of multi degree of freedom systems with high frequency modes show that MOS results are quite stable as long as the accuracy of important modes of the system is maintained, which is usually the case. As in a real hybrid simulation, experimental errors also affect the accuracy and stability of integration methods, in this paper a hybrid simulation algorithm is numerically modeled and the effect of actuator time delay on the performance of MOS method is investigated. It was observed that in the presence of actuator delay, which is known to be one of the most important sources of experimental errors in hybrid simulation, MOS integration method has solved the equation of motion in an accurate and stable manner with very small level of errors in comparison with the reference solution.