218 R. Astroza et al. the parameter-only approach with similar structures and levels of noise, when no model uncertainty is considered [17, 18]. In the dual approach, this covariance matrix is considered as the initial estimate for the simulation error covariance matrix, i.e., R0 =diag r0|0 =0.87×10−3 I 3 m/s 2 2 . In the SF, Prr 0|0 is assumed diagonal with entries computed according to Prr 0|0 = diag 0.2× r0|0 2 . The time-invariant covariance matrices of the process and measurement noises used in the SF are taken as T=U=1×10−20 I 3. The final estimates of the unknown model parameters for all the modeling uncertainty cases (Table 24.2) obtained with both approaches (parameter-only and dual) are reported in Table 24.3. Here, the normalized parameter estimates (with respect to the true parameter values) and the associated final coefficient of variation (CV) are shown, and Dand N denote the dual and parameter-only estimation approaches, respectively. Results obtained with the dual approach (D) shows significant improvement with respect to the parameter-only approach in terms of parameter estimates, an also allows keeping higher level of uncertainty in those parameters for which the information contained in the measured responses is limited. When using the parameter-only approach, divergence and convergence to unphysical values of the unknown model parameters are observed, undesired effects that are controlled by the dual approach. Parameters Ecol s and Ebeam s are estimated by the parameter-only estimation approach in the range (85.91–205.84%) of their true values, and the estimation using the dual adaptive filtering approach narrows that range to (95.05–153.95%). Results for parameters fcol y and f beam y are similar; using the parameter-only approach the estimates vary between 27.50% and 233.88% of the true values, and employing the dual adaptive filtering approach in the range (70.85–157.19%). Significant differences are also observed in the estimation of the post yield-related parameters, the less sensitive parameters for the acceleration response measurements considered [11]. For instance, the parameter-only approach estimates bcol in the range (10.01–400.51%), while with the dual adaptive filtering approach the estimates are in the range (30.56–155.09%) of the corresponding true parameter values, demonstrating a significant improvement. The final coefficient of variation estimates for bcol are between 0.51% and 5.74% for the parameter-only approach and between 1.54% and 15.72% for the dual adaptive filtering approach. Similar results are obtained for other post yield-related parameters (Rcol 0 , Rbeam 0 and b beam). Figure 24.5 shows the time histories of the normalized mean estimates of the unknown model parameters for case 11 (see Table 24.2). Results of the parameter-only and dual approaches are compared. At initial time steps, the amplitude of the response is low and the frame behaves in the linear-elastic range and therefore y only contains information about Ecol s and Ebeam s . As the base excitation and the response increase (at about 3 s), the frame behaves nonlinearly and yielding of some fibers occurs and fbeam y starts to be updated. At around 4 s, the measured response becomes sensitive to the other unknown model parameters (fcol y , Rbeam 0 , Rcol 0 , b beam, andbcol).WhenRis also estimated (i.e., dual approach), the convergence of the unknown model parameters tends to be considerably more stable and smoother, not exhibiting abrupt changes. In Fig. 24.5, dashed lines show plus/minus two standard deviations (±2σ). When the dual approach is used, unknown model parameter for which low information is contained in the measured response, the estimation uncertainty (±2σ) remains high. Errors in Observed Responses For each case of modeling uncertainty, the final estimates of the unknown model parameters are used with the FE model belonging to the corresponding model class Mj to predict the response of the structure when it is subjected to the seismic excitation. The responses of the calibrated models are compared to their true counterparts (defined by M0(θtrue, ϕ true)) using the relative root-mean-square error (RRMSE), which is defined by RRMSE (a, b) = 1 Ns Ns i=1(ai −bi) 2 / 1 Ns Ns i=1(ai) 2 ×100 (%)for signals a (reference) and b, where Ns denotes the total number of data samples. Figure 24.6 reports the RRMSEs between the true and FE-predicted measured (observed) acceleration responses for all cases studied (see Table 24.2) using the initial and final estimates of the unknown model parameters obtained from both approaches. The RRMSEs of the initial (non-updated) FE models range from 30.11% to 86.33%. The parameter-only approach reduces this misfit between the true and FE-predicted observed responses, with relative errors in the final FEpredicted responses ranging from 11.76% to 85.08%, but with none of the updated models achieving RRMSEs lower than 10% for the observed responses. On the contrary, when using the dual adaptive filtering approach, the RRMSEs decrease significantly, reaching values ranging between 0.68% and 25.02%, with 32% of the cases having relative errors below 10% for all measured responses, and 89% of the cases with relative errors below 20%, demonstrating an excellent performance in matching the observed responses.
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