Improving the computational efficiency of seismic building-performance assessment through reduced order modeling and multi-fidelity Monte Carlo techniques

Abstract

Performance-based earthquake engineering offers a versatile framework for quantifying the seismic performance of structures. Its implementation requires a comprehensive description of the nonlinear structural behavior, facilitated typically via multiple nonlinear response history analyses (NLRHAs). This burden can be very high when high-fidelity finite element models (FEMs) are used to describe structural response. To alleviate it, approximations are commonly employed, using a moderate number of analyses, or even replacing altogether the NLRHA with a nonlinear static analysis. This contribution explores two alternative paths for accommodating the desired computational efficiency: (a) use of reduced order models that are calibrated to closely match the original FEM; (b) adoption of multi-fidelity Monte Carlo (MFMC) that combines the original FEM to guarantee unbiased predictions and the aforementioned reduced order models to accelerate the Monte Carlo implementation. Advancements are established for the MFMC implementation, in order to accommodate the efficient propagation of the different sources of uncertainty across the estimation of the different seismic performance statistics of interest. The accuracy and computational benefits are illustrated for two benchmark structures over two different output variables: repair cost (resiliency quantification) and embodied energy associated with repairs (sustainability quantification).

Appendices

Appendix 1: Computational details for estimation of output variable statistics conditional on EDP

This appendix discusses the estimation of conditional on EDP statistics for OVs, addressing computational details for the Consequence Module. Following the definitions in Sect. 2.3, assume that for the ith damageable assembly $n_{di}$, different damage states $d_{ik} , \, k = 0, \ldots ,n_{di}$ are designated, with damage state for k = 0 corresponding to undamaged conditions. The fragility of the ith damageable assembly is related to engineering demand parameter $EDP_{i}$. The probability that the damage measure related to the ith assembly (DM_i) exceeds damage state d_ik is quantified through its respective fragility function. When, as customary, a lognormal distribution is utilized for these functions, the resultant expression is:

$$P(DM_{i} > d_{ik} |EDP_{i} = edp_{i} ) = \Phi \left[ {\frac{{\ln (edp_{i} /\overline{b}_{ik} )}}{{\beta_{ik} }}} \right]$$

(18)

where Φ denotes the standard Gaussian cumulative distribution function, and $\overline{b}_{ik}$ and β_ik are the median and logarithmic standard deviation for the fragility function. This fragility quantification $P(DM_{i} > d_{ik} |EDP_{i} = edp_{i} )$ can be equivalently interpreted to correspond to $P(b_{ik} < edp_{i} )$, with $b_{ik}$ defined as the random variable for the threshold associated with damage state d_ik, which for the case of the fragility function of Eq. (18) follows a lognormal distribution with median $\overline{b}_{ik}$ and logarithmic standard deviation β_ik. This shows that the fragility quantification may be equivalently considered to define a probability model for the threshold $b_{ik}$ (treated as a random variable) defining damage state d_ik, with exceedance of the damage state corresponding to $edp_{i} > b_{ik}$ and undamaged conditions corresponding to $edp_{i} \le b_{i1}$. The probabilistic quantification of losses for each damageable assembly is accommodated through a probabilistic description for v_ik, the losses associated with the kth damage state for the ith assembly.

Let b and v denote vectors defined by augmenting all elements of $\{ b_{ik} {; }k = 1, \ldots ,n_{di} , \, i = 1,...,n_{as} \}$ and $\{ v_{ik} {; }k = 1, \ldots ,n_{di} , \, i = 1,...,n_{as} \}$, respectively. The fragility quantification for each $b_{ik}$ as well as the correlation across all elements of b defines ultimately the joint probability model p(b) for b, and similarly, the probabilistic description for v_ik and the correlation across damageable assemblies defines the joint probability model p(v) for v. According to Eq. (4), and using sample set $\{ {\mathbf{z}}^{im(s)} {; }s = 1, \ldots ,N_{g}^{im} \}$ or its parametric fit $LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )$ as discussed in Sect. 2.3, independent samples $OV|IM = im$ as well as samples for the losses associated with each damageable assembly $OV_{i} |IM = im$ can be generated according to the following process, termed herein as conditional OV sampling:

• Step 1: Generate an EDP|IM = im sample ${\mathbf{z}}^{im(r)}.$ This is done either by sampling (with replacement) one sample from set $\{ {\mathbf{z}}^{im(s)} {; }s = 1, \ldots ,N_{g}^{im} \}$ or, if parametric fit is employed, generating a sample from the fitted distribution $LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )$. These two variant implementations will be distinguished herein using terminology, EDP resampling and EDP approximation, respectively.

• Step 2: Generate a sample ${\mathbf{b}}^{(r)}$ from p(b) and a sample ${\mathbf{v}}^{(r)}$ from p(v).

• Step 3: For the ith damageable assembly, compare the corresponding edp_i value (for the ith component of EDP vector) from sample ${\mathbf{z}}^{im(r)}$ to the thresholds from sample ${\mathbf{b}}^{(r)}$ associated with the damaged states (of the assembly), and determine in which damage state the assembly belongs to (edp_i value exceeds the threshold for this damage state but does not exceed the threshold for the following one). The losses for the assembly are defined by the respective index of ${\mathbf{v}}^{(r)}$, which ultimately provides sample $ov_{i}^{im(r)}$ from $OV_{i} |IM = im$. Repeat this for all assemblies.

• Step 4: Combine $ov_{i}^{(r)}$ samples to obtain sample $ov_{{}}^{im(r)}$ from $OV|IM = im$.

Utilizing a sufficient number of samples $\{ ov_{{}}^{im(r)} ;r = 1,...,N_{r} \}$ the statistics for OV as well as the complementary cumulative distribution function can be estimated:

$$E[OV|IM = im] = \frac{1}{{N_{r} }}\sum\limits_{r = 1}^{{N_{r} }} {ov^{im(r)} }$$

(19)

$$Var[OV|IM = im] = \frac{1}{{N_{r} - 1}}\sum\limits_{r = 1}^{{N_{r} }} {\left( {ov^{im(r)} - \frac{1}{{N_{r} }}\sum\limits_{r = 1}^{{N_{r} }} {ov^{im(r)} } } \right)^{2} }$$

(20)

$$G_{OV} (ov|im) = \frac{1}{{N_{r} }}\sum\limits_{r = 1}^{{N_{r} }} {I[ov^{im(r)} > ov]}$$

(21)

where I[$\cdot$] represents the indicator function, corresponding to one if the relationship inside the brackets holds and to 0 else. Similar relationships hold for the statistics for OV_i.

If only the expected value of Eq. (19) is warranted, then an analytical approximation can be efficiently implemented. This is accommodated by first calculating

$$E[OV_{i} |EDP_{i} = edp_{i} ] = n_{i} \sum\limits_{k = 0}^{{n_{di} }} {\overline{v}_{ik} P[DM_{i} = d_{ik} |EDP_{i} = edp_{i} ]}$$

(22)

where n_i is the number of identical elements in the ith assembly,$\overline{v}_{ik}$ is the mean of v_ik, and, assuming perfect correlation (Porter et al. 2001; FEMA-P-58 2012) between the damage states, the conditional probability of being in each damage state is estimated as:

$$\begin{aligned} & P[DM_{i} = d_{i0} |EDP_{i} = edp_{i} ] = 1 - P(DM_{i} > d_{i1} |EDP_{i} = edp_{i} ) \\ & P[DM_{i} = d_{ik} |EDP_{i} = edp_{i} ] = P(DM_{i} > d_{ik} |EDP_{i} = edp_{i} ) - P(DM_{i} > d_{i(k + 1)} |EDP_{i} = edp_{i} );\quad \, k = 1,...,n_{di} - 1 \\ & P[DM_{i} = d_{in_{di}} |EDP_{i} = edp_{i} ] = P(DM_{i} > d_{{in_{di} }} |EDP_{i} = edp_{i} ) \\ \end{aligned}$$

(23)

This then leads to:

$$E[OV|EDP = edp] = \sum\limits_{i = 1}^{{n_{as} }} {E[OV_{i} |EDP_{i} = edp_{i} ]}$$

(24)

Utilizing this analytical approach, Steps 2–4 in the conditional OV sampling are replaced by using the average of the quantity presented in Eq. (24) over the EDP samples obtained in Step 1. If the lognormal fit is used for the EDP distribution, then even Step 1 can be integrated within the analytical estimation (Baker and Cornell 2008; Angeles et al. 2021). Similar extensions to estimate $Var[OV|EDP = edp]$ or higher order statistics through analytical or semi-analytical approximations involve significant computational burden for properly capturing correlations between damage states and damageable assemblies (Angeles et al. 2021). This approach is not recommended for applications with large number of damage states and assemblies. Note though that for the variance for individual OV_i’s, $Var[OV_{i} |EDP_{i} = edp_{i} ]$, a simple analytical expression similar to Eq. (22) can be obtained (Angeles et al. 2021), whereas estimation of $Var[OV|EDP = edp]$ can be efficiently analytically performed when correlations across assemblies are ignored (Angeles et al. 2021).

Appendix 2: Response approximation using nonlinear static analysis

This appendix reviews the approximation of the distribution $EDP|IM\sim LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )$ through static nonlinear analysis of FEMA P-58 (FEMA-P-58 2012) for low/medium-rise buildings (below 15 stories, with small higher mode contributions), with insignificant P-delta effects (drift ratios lower than 4%) and no irregularities in plan and elevation. Note that many advanced variants exist in the literature for improving loss estimation using nonlinear static analysis, for example (Nettis et al. 2021), and the simplified (in comparison) approach adopted here is chosen simply as a popular, and promoted in FEMA P-58 formulation. According to it, fundamental periods and mode shapes of the building are used to compute the vertical distribution of equivalent static forces with (total) base shear for IM = im given by $V = C_{1} C_{2} S_{a} (T_{1} )W_{1}$ (FEMA-P-58 2012) where C₁ is an adjustment factor for inelastic displacements (function of soil condition and yield base shear), C₂ is an adjustment factor for cyclic degradation (function of soil condition and yield base shear), S_a(T₁) is the 5% damped mean spectrum of the ground motion for the fundamental period of the building corresponding to the utilized IM, and W₁ is the effective modal weight for the first mode (not less than 80% of the total building weight). The yield base shear is calculated through nonlinear static pushover analysis based on the building FEM, adopting a bi-linearization based on the target displacement according to FEMA-440 standard (FEMA-440 2005). Shear V is distributed along the height of the building using equivalent static force distributions and elastic drift demands are then calculated through linear analysis. For each IM level, this information, along with the estimated peak ground acceleration, is used to compute the median estimates for the different engineering demand parameters, to accommodate the calculation of ${{\varvec{\upmu}}}^{im}$. More specifically, from peak ground acceleration, the peak floor accelerations are estimated, along with the peak ground velocity that is then used to estimate the peak floor velocities. Inelastic behavior is accounted for through an adjustment factor that depends on the relationship between the base shear per IM and the yield base shear calculated through the nonlinear pushover analysis (FEMA-P-58 2012). For the approximation of ${{\varvec{\Sigma}}}^{im}$, the dispersion for the median EDP predictions due to record to record variability is obtained from Table 5–6 of FEMA P-58-1 Methodology (FEMA-P-58 2012). Note that this dispersion differs for each IM since it depends on S_a(T₁). The complete covariance matrix ${{\varvec{\Sigma}}}^{im}$, needed to estimate higher-order OV statistics, is obtained by assuming, additionally, perfect correlation for the EDP vector.