Sensitivity Developments for Sensitivity-Based RBDO With Correlated Input Variable and Varying Input Standard Deviation

In conventional reliability-based design optimization (RBDO), the input variable is statistically independent, and its standard deviation (STD) is a fixed constant. However, the independent input variable and the constant STD may not correctly describe reliability problems because, for certain problems, the input variable is correlated, and its STD varies as the mean changes. For this kind of design problem, the input correlation should be considered, and the input STD should be changed as the design iteration proceeds and the corresponding design variable — the input mean — changes. Specifically, the varying input STD causes the input variability to fluctuate so that the probability of failure (PoF) of the design becomes a moving target. Hence, RBDO with varying input STD suffers difficulty in convergence unless accurate design sensitivity is available. In sensitivity-based RBDO, the sensitivity for the most probable point (MPP) search and the design sensitivity of probabilistic constraints are required. In this paper, it is found that the input correlation significantly affects the sensitivity for the MPP search and that the varying STD has large impact on the design sensitivity for RBDO. Therefore, sensitivities considering both the input correlation and the varying input STD have been developed for an efficient and effective optimization process. For the reliability analysis method, the enriched performance measure approach (PMA+) is used, as it is efficient and more stable during the RBDO process. To represent the input correlation, a copula is used in the sensitivity derivations. Using a mathematical example, the accuracy and efficiency of the developed sensitivities are verified. The RBDO result for the mathematical example indicates that the developed methods provide accurate sensitivities in the optimization process.