35 Applications of Reduced Order and Surrogate Modeling in Structural Dynamics 299 0 0.2 0.4 0.6 0.8 1 35 40 45 50 Log evidence 10 2 4 6 8 11 9 7 5 3 1 10 2 4 6 8 11 9 7 5 3 1 Reference iteration of algorithm (k) iteration of algorithm (k) Discrepancy of approximated distridutions in succesive iterations (b) Log evidence approximation Discrepancy between approximated and target distribution Hellinger distance (a) Discrepancy (Hellinger distance) of different distributions Fig. 35.2 Results for implementation of AK-SSD algorithm for a modal synthesis problem; (a) discrepancy between different distributions across iterations and (b) approximation of log evidence of intermediate densities to approximate gradually the target density. At each AK-SSD iteration, the current metamodel is used to approximate the intermediate and target densities and if convergence is not achieved knowledge gained from current iteration is leveraged, primarily through an adaptive design of experiments, to update the metamodel formulation and proceed to the next iteration. The Hellinger distance between the approximated densities in consecutive iterations is used to quantify convergence, whereas new experiments are obtained using a hybrid DoE strategy that balances goals of improvement of global metamodel accuracy and addition of experiments in critical regions for the posterior sampling. AK-SSD was initially formulated for rare event estimation but can be seamlessly extended to Bayesian posterior sampling as shown in Fig. 35.2 for a modal synthesis problem of an eight story structure, updating stiffness characteristics through eigenfrequency and modal information for the first three modes of vibration. Part (a) of Fig. 35.2 shows the Hellinger distance discrepancy between the approximated target densities in consecutive iterations or between the approximated and actual target density, while part (b) shows the iteration-wise metamodel-based evidence estimate versus the reference one. As the iteration number k increases discrepancy reaches a plateau, facilitating an efficient convergence to the target density, something further validated by the agreement of the estimated evidence. References 1. Gidaris, I., Taflanidis, A.A.: Parsimonious modeling of hysteretic structural response in earthquake engineering: calibration/validation and implementation in probabilistic risk assessment. Eng. Struct. 49, 1017–1033 (2013) 2. Jensen, H.A., Muñoz, A., Papadimitriou, C., Millas, E.: Model-reduction techniques for reliability-based design problems of complex structural systems. Reliab. Eng. Syst. Safe. 149, 204–217 (2016) 3. Tehrani, M., Harvey Jr., P., Gavin, H., Mirza, A.: Inelastic condensed dynamic models for estimating seismic demands for buildings. Eng. Struct. 177, 616–629 (2018) 4. Gidaris, I., Taflanidis, A.A., Mavroeidis, G.P.: Kriging metamodeling in seismic risk assessment based on stochastic ground motion models. Earthq. Eng. Struct. Dynam. 44(14), 2377–2399 (2015) 5. Mai, C.V., Spiridonakos, M.D., Chatzi, E.N., Sudret, B.: Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogenous input models and polynomial chaos expansions. Int. J. Uncertain. Quantif. 6, 4 (2016) 6. Bakalis, K., Vamvatsikos, D., Fragiadakis, M.: Seismic risk assessment of liquid storage tanks via a nonlinear surrogate model. Earthq. Eng. Struct. Dynam. 46(15), 2851–2868 (2017) 7. Jensen, H., Millas, E., Kusanovic, D., Papadimitriou, C.: Model-reduction techniques for Bayesian finite element model updating using dynamic response data. Comput. Methods Appl. Mech. Eng. 279, 301–324 (2014) 8. Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989) 9. Angelikopoulos, P., Papadimitriou, C., Koumoutsakos, P.: X-TMCMC: adaptive kriging for Bayesian inverse modeling. Comput. Methods Appl. Mech. Eng. 289, 409–428 (2015) 10. Zhang, J., Taflanidis, A.A.: Adaptive Kriging stochastic sampling and density approximation and its application to rare-event estimation. ASCE-ASME J. Risk Uncertain. Eng. Syst. A Civil Eng. 4(3), 04018021 (2018)
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