298 A. A. Taflanidis et al. 0 20 40 60 time (sec) -0.04 -0.02 0 0.02 0.04 drift ratio for 1st floor High-fidelity FEM model Reduced order Model 10 -2 10-3 10-2 10-1 100 Probability of drift exceeding corres[ponding threshold drift ratio threhsold 1st floor 3rd floor (a) calibration (b) risk estimation High-fidelity FEM model Reduced order Model Fig. 35.1 (a) Calibration results (time-history comparisons for one of the six excitations used in the calibration) for a reduced order model matching a high-fidelity FEM model (developed in OpenSees modeling environment) for a three-story benchmark structure and (b) comparison of seismic risk estimates from the reduced order and high-fidelity models considering an ensemble of 1000 ground motions 35.2 Reduced Order Modeling in Seismic Loss Assessment Modern seismic loss estimation practices (and more broadly seismic risk assessment) require simulation of structural behavior for different levels of earthquake shaking though time-history analysis. Under moderate and strong excitations this behavior is strongly inelastic/hysteretic and evaluating it through high-fidelity Finite Element Models (FEMs) introduces a significant computational burden. Reduced order modeling has been suggested to alleviate this burden [1, 3]. The reduced order model in this setting is developed using data form the original high-fidelity model. Modal analysis (or static condensation) is first leveraged to condense the initial equations of motion to a specific set of degrees of freedom per story. For planar structural models, which is the focus here, this can be established by using one degree of freedom per story. The restoring forces prescribed by the linear stiffness matrix are then substituted with hysteretic ones, equivalently viewed as nonlinear springs connecting different degrees of freedom. Hysteretic models that can be considered for this task include piecewise-linear models, Massing models [1] or Bouc-Wen models [3]. The characteristics of these models that describe the initial (linear) response are selected directly based on the modal analysis results to match exactly the condensed stiffness matrix. Characteristics that describe the nonlinear response can be subsequently calibrated [3] by comparing the reduced order model time-history to the time-history of the original FEM for a range of different excitations. This is posed as a model parameter identification problem with data coming from the high-fidelity FEM simulations. The excitations used in this calibration should be carefully chosen to facilitate identification of all relevant model parameters, something accommodated when nonlinearities associated with all degrees of freedom can be observed in the available data. Earlier work in this field used simplified excitations for this purpose [3] while recent work by the first and third authors has demonstrated how this can be more efficiently accomplished using seismic excitations with different intensity levels and frequency content. As shown in Fig. 35.1, the calibrated model can be then used to efficiently provide loss assessment estimates. 35.3 Surrogate Modeling for Posterior Sampling The importance of Bayesian inference in the analysis of engineering systems has dramatically increased over the past few decades. For applications with complex posterior Probability Density functions (PDFs) this inference needs to rely on stochastic sampling techniques for approximating this PDF. To alleviate the computational burden for applications entailing complex numerical models, the implementation of surrogate modeling techniques for approximating the likelihood function, involved in the posterior PDF definition, has been investigated [9]. The strategy by Angelikopoulos et al. [9] relied on use of the metamodel for local approximation and only for samples for which its predicted metamodel accuracy was sufficiently high (exact numerical model was used for samples that did not satisfy this requirement). The first two authors recently developed [10] the adaptive Kriging stochastic sampling and density approximation algorithm (AK-SSD) for approximating target densities using metamodeling techniques. AK-SSD utilizes solely metamodel predictions for all calculations within the stochastic sampling task and establishes high computational efficiency by leveraging a global metamodel that is iteratively developed to provide higher accuracy in domains of interest. AK-SSD assumes a sequential sampling setting, using a series
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