Uncertainty Quantification in Metamodel-Based Reliability Prediction

Optimization under uncertainty has been studied in two directions — (1) Reliability-based Design Optimization (RBDO), and (2) Robust Design Optimization (RDO). One of the crucial elements in an RBDO problem is reliability analysis. Reliability analysis is affected by different types of epistemic uncertainty, due to inadequate data and modeling errors, along with aleatory uncertainty in input random variables. When the original physics-based model is computationally expensive, a metamodel has often been used in reliability analysis, introducing additional uncertainty due to the metamodel. This work presents a framework to include statistical uncertainty and model uncertainty in metamodel-based reliability analysis. Inadequate data causes uncertainty regarding the statistics (distribution types and distribution parameters) of the input variables, and regarding the system model parameters. Model errors include model form errors, solution approximation errors, and metamodel uncertainty. Two types of metamodels have been considered in literature for reliability analysis: (1) metamodels that compute the system model output over the desired ranges of the input random variables; and (2) metamodels that concentrate only on modeling the limit state. This work focuses on the latter type, using Gaussian process (GP) metamodels for performing both component reliability (single limit state) and system reliability (multiple limit states) analyses. A systematic procedure for the inclusion of model discrepancy terms in the limit-state metamodel construction is developed using an auxiliary variable approach. An efficient single-loop sampling approach using the probability integral transform is used for sampling the input variables with statistical uncertainty. The variability in the GP model prediction (metamodel uncertainty) is also included in reliability analysis through correlated sampling of the model predictions at different inputs. Two mechanical systems — a cantilever beam with point-load at the free end and a two-bar supported panel with point load at its center, are used to demonstrate the proposed techniques.