Reliability-based design optimization using kriging surrogates and subset simulation

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.

💡 Research Summary

The paper presents a comprehensive methodology for solving reliability‑based design optimization (RBDO) problems when the underlying performance models are computationally expensive, such as finite‑element simulations in structural mechanics. Traditional double‑loop approaches that embed a reliability analysis (often FORM‑based) inside an outer optimization loop become prohibitive for nonlinear, high‑dimensional models, while surrogate‑based methods usually lack a rigorous way to quantify the impact of surrogate error on failure‑probability estimates.

To address these issues, the authors combine two powerful tools: Kriging (Gaussian‑process) surrogates and Subset Simulation. Kriging provides not only a prediction of the performance function but also a location‑dependent variance that reflects epistemic uncertainty due to limited training data. This variance can be propagated onto the limit‑state surface, yielding an explicit estimate of the error in the computed failure probability for any design point.

An adaptive refinement scheme is built on top of this error estimate. Starting from an initial Design of Experiments (DoE), the Kriging model is constructed and its error on the limit‑state is evaluated. If the error exceeds a prescribed tolerance, a population‑based refinement adds several new simulation points in regions where the surrogate is most uncertain (typically near the limit‑state). The new data are incorporated, the Kriging model is updated, and the error is re‑evaluated. This loop continues until the surrogate meets the accuracy requirement, guaranteeing that the subsequent reliability analysis is trustworthy.

Reliability analysis itself is performed with Subset Simulation, a variance‑reduction technique that decomposes a rare‑event probability into a product of larger conditional probabilities. By using the Kriging surrogate inside Subset Simulation, the authors avoid costly evaluations of the original model while still obtaining accurate estimates of low failure probabilities.

The optimization core is a conventional gradient‑based algorithm. Design variables are treated as hyper‑parameters of the random vector describing uncertainties, which enables analytical computation of the gradient of the failure probability with respect to the design variables (reliability sensitivity). This eliminates the need for finite‑difference approximations and accelerates convergence.

A key innovation is the introduction of an “augmented reliability space” where both design variables and random variables are combined into a single input vector for the surrogate. Consequently, a Kriging model built once can be reused throughout all iterations of the RBDO loop, dramatically reducing the number of surrogate reconstructions.

The methodology is validated on three benchmark structural problems (a simple beam, a 2‑D frame, and a composite structure). Comparisons are made with (i) FORM‑based double‑loop RBDO, (ii) sample‑average approximation methods, and (iii) existing Subset‑Simulation‑based RBDO. Results show that the proposed approach attains the same optimal designs and satisfies the prescribed failure‑probability constraints while reducing the number of expensive model evaluations by one to two orders of magnitude. Moreover, the adaptive refinement reliably drives the surrogate error below the target, leading to stable convergence of the optimization.

In summary, the paper delivers a fully integrated framework that (1) quantifies Kriging surrogate error and propagates it to failure‑probability estimates, (2) adaptively refines the surrogate only where needed, (3) employs Subset Simulation for efficient reliability analysis, and (4) reuses the surrogate in an augmented reliability space during gradient‑based optimization. This combination makes RBDO feasible for realistic engineering problems where each performance evaluation is costly. Future work is suggested on multi‑objective extensions, non‑Gaussian uncertainties, and scalability to very high‑dimensional design spaces.