Stochastic Emulation using Generalized Stratified Sampling for Performance-Based Risk Optimization of Structures
Authors: Isabela D. Rodrigues, Seymour M. J. Spence, Henrique M. Kroetz, André T. Beck
Organizations: Department of Structural Engineering, São Carlos School of Engineering, University of São Paulo, São Carlos, 13566-590, São Paulo, Brazil · Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, 48109, Michigan, USA · Center for Marine Studies, Federal University of Paraná, Pontal do Paraná, 13566-590, Paraná, Brazil
Metamodels are instrumental in reducing the computational burden associated with nested reliability analyses and optimization loops in Performance-Based Risk Optimization (PBRO) of structures under stochastic loads. In this context, stochastic emulators are particularly useful because they approximate response distributions while accounting for the intrinsic stochasticity of the simulator. Among these methods, Stochastic Polynomial Chaos Expansion (SPCE) is especially attractive because it does not require replications of nonlinear analyses at fixed input conditions. However, SPCE may present limitations in accurately representing extreme responses in the tails of structural response distributions. To address this limitation, this study proposes a framework that combines Generalized Stratified Sampling (GSS) with SPCE. The GSS scheme partitions the input space into strata according to the intensity of the hazard, improving the representation of extreme responses, while independent SPCE emulators are trained within each stratum. The conditional exceedance probabilities estimated in each stratum are then recombined using the total probability theorem to evaluate the probabilistic constraints. The proposed GSS-SPCE framework is applied to the optimal design of buckling-restrained brace cross-sectional areas in a two-story steel building. The objective is to minimize the initial construction cost while satisfying prescribed probabilistic performance constraints. Results show that the proposed framework accurately estimates structural response distributions, including their tail regions, while substantially reducing the number of nonlinear model evaluations required for PBRO.
Categorical structural optimization under aleatoric uncertainty is challenging because each design variable must be selected from a finite catalog of admissible instances, while each candidate design may require expensive stochastic finite-element evaluations. Existing latent-space optimization strategies can reduce the dimensionality of catalog attributes, but they often treat the reduced space as a continuous search domain. The resulting continuous optimum must then be rounded off to a nearby catalog instance, which may alter the objective value, constraint status, or physical interpretation of the design. To address this issue, this paper proposes the \textbf{C}ategorical \textbf{O}ptimization with \textbf{B}ayesian \textbf{A}nchored \textbf{L}atent \textbf{T}rust Regions (\textbf{COBALT}) framework for high-dimensional categorical Optimization Under Uncertainty. COBALT first embeds the physical catalog into a low-dimensional latent representation and locks the mapped instances as a discrete anchored graph. A data-independent random tree decomposition is then used to provide bounded-complexity additive modeling over high-dimensional categorical variables. On this anchored domain, an additive SAAS-GP surrogate is fitted to heteroscedastic MC-FEA observations, and a trust-region discrete graph acquisition search selects the next admissible catalog configuration without continuous relaxation or rounding-off. The proposed strategy is applied to robust design optimization of complex bar structures, considering structural weight, strain energy, and local buckling performance. By evaluating only valid catalog designs through the MC-FEA oracle, COBALT preserves physical admissibility throughout the active learning loop and improves the efficiency of robust categorical structural optimization.
Modern stochastic optimization pipelines increasingly rely on learned generative models to represent uncertainty, while downstream decisions are evaluated almost entirely through Monte Carlo scenarios. This shifts the operational object of uncertainty from an explicit probability law to the sampler induced by the learned generator. Reliability therefore depends on two errors: sampler misspecification and finite-simulation error. We propose Sampler-Robust Optimization (SRO), which optimizes decisions against the worst-case sampler induced by perturbing the learned generator. This sampler-first formulation aligns with simulation-based decision pipelines and admits a sharpness-aware interpretation: it favors decisions whose performance is stable under generator perturbations, rather than merely under the nominal sampler. Under a coverage assumption, we show that the empirical worst-case objective provides a high-probability upper certificate for the true population objective, with finite-simulation error partially absorbed by the robustification used to guard against sampler misspecification. The framework accommodates generative models with or without explicit densities and admits efficient minimax procedures. Portfolio-optimization experiments show that SRO produces more stable decisions and improves out-of-sample performance under distribution shift.
Structural optimization problems often involve a large number of decision variables and highly non-convex feasible regions, making convergence to the true Pareto front extremely challenging. Even when convergence is achievable, it typically requires thousands of function evaluations, resulting in significant computational cost. This highlights the need for efficient and robust optimization algorithms for real-world engineering applications. In this study, we introduce a novel constrained multi-objective evolutionary algorithm, termed DPCME. The algorithm employs two interacting populations that exchange information, enabling effective global exploration and reducing the risk of convergence to local optima. To further enhance performance, a recent repair-based constraint-handling technique is incorporated, and alternative repair approaches are proposed and systematically evaluated. The proposed algorithm is tested on three engineering problems: the 72-bar truss, the 120-bar truss, and a chemical tanker structure, each involving hundreds of nonlinear failure constraints. Its performance is evaluated against state-of-the-art constrained multi-objective optimization algorithms from the latest PlatEMO package. A total of 43 algorithms are initially tested, from which the 12 best-performing methods are selected for detailed comparison. The results demonstrate that DPCME achieves superior or competitive convergence and diversity across all test cases, and that the inclusion of repair-based constraint handling further improves its performance.