262 M. K. Ramancha et al. Note that, in this study, the input earthquake ground motion record is appropriately scaled to drive the dam into adequate levels of constitutive nonlinearity. A simple FE response sensitivity analysis using the direct differentiation method (DDM), a local sensitivity analysis, is then performed to ensure that the resulting measured responses are sufficiently sensitive to all the eleven material parameters governing both linear and non-linear behavior of the dam. Low sensitivity of a measured response to a certain parameter implies that the parameter cannot be estimated in the context of a parameter estimation problem. 29.4.2 Estimation The noisy input and output data set, generated in the simulation phase, are used in the estimation phase to recursively estimate the unknown parameter vector θ using the UKF. The same FE model of the dam, as the one used to simulate the response in the simulation phase, is employed in the parameter estimation phase. The probability distribution of the initial value (θ0|0) of the unknown parameter vector θ is assumed as 0|0 ∼N θ0|0, Pθθ 0|0 → ⎧ ⎪⎪⎨ ⎪⎪⎩ θ0|0 11×1 = 1.40 Gtrue, 0.55 Ktrue, 0.85Xtrue 0 , 1.20Dtrue, 0.80Wtrue, 0.80Rtrue, 1.15λtrue, 1.05θtrue, 0.95βtrue, 0.80αtrue, 0.60 Ttrue T Pθθ 0|0 11×11 = 0.25×diag θ0|0 2 (29.3) The mean parameter values θ0|0 in Eq. (29.3) are defined in terms of the true parameter values θtrue. However, in a realworld problem, the data are obtained from the sensors mounted on the real system (not simulated numerically); therefore, the true parameter values are unknown and do not even exist (since the selected FE model class may not contain the real structure). The diagonal elements of the covariance matrix of the artificial process noise Qare assumed to be equal to 10−3 ×θ (i) 0|0 2 , i =1, 2, . . . ,11. The choice of Qgoverns the convergence and tracking performance of the filter [9]. In real-world problems, the characteristics of the noise in the measurement data are not known exactly. Therefore, the covariance matrix of the measurement noise Ris set different than that of the (known) added noise to account for the unknown noise level. The time histories of the posterior mean estimates of all eleven parameters obtained using the UKF and normalized with respect to their corresponding true values are shown in Fig. 29.4. The blue line in each plot represents the normalized mean estimate θk|k, k =1, 2, 3, . . . and the grey shaded region represents the estimation uncertainty Pθθ k|k , k =1, 2, 3, . . . in the corresponding mean estimate, namely the mean ±two standard deviations. The estimates of the two linear-elastic concrete material parameters, Gand K, and of a nonlinear-material parameter, the tensile strength of concrete (T), converge smoothly to their corresponding true values (see blue lines in Fig. 29.4). In addition, their estimation uncertainty decreases asymptotically and very fast (see grey shaded areas in Fig. 29.4) as more information about these parameters is assimilated step by step from the measured input and output response. Even though the sensitivity of the measured responses to all parameters is high, it is observed in Fig. 29.3 that the estimates of other parameters controlling the nonlinear concrete behavior, X, R, α, λ, D, W, β and θ, do not converge to their corresponding true values. However, it is important to note that the measurement responses predicted by the FE model characterized by the parameter estimates obtained at the last step of the filter (tk =20s) are in excellent agreement with the true measurement responses. In this study, the FE predicted seismic response of the dam obtained using the posterior mean estimates of the parameters is compared to the true response, obtained from the FE model usingθ=θ true, and the relative-root-mean-square error (RRMS) [3] is used as metric to measure the discrepancy between two time series. The evolution of the RRMS error for each sensor (aA, aB, . . . aF, dA, dB, . . . dF) during filtering is shown in Fig. 29.5. In this figure, the RRMS error at each time step tk is computed by comparing the entire time history of FE predicted response obtained using the posterior mean estimates of the material parameters θ at tk θk|k and the entire time history of the true response (obtained usingθ =θtrue). As expected, at the start of filtering (i.e., at k =0), the error between the predicted response (obtained using θ0|0) and the true response (obtained using θtrue) is very high (with a RRMS error of over 100%). During filtering, the UKF adjusts the estimates of the material parameters to decrease the error between the predicted response and the true response. It is important to observe that although the estimates of some material parameters, X, R, α, λ, D, W, β andθ, do not converge to their corresponding true values, the RRMS errors between the predicted responses and the corresponding true responses decrease progressively to very small values. This implies that the filter finds different sets of parameter values (non-true) that each yields a very good match of the time histories of the measured response quantities. In addition, a very good agreement between predicted and
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