Sensitivity Developments for RBDO With Dependent Input Variable and Varying Input Standard Deviation

In reliability-based design optimization (RBDO), dependent input random variables and varying standard deviation (STD) should be considered to correctly describe input distribution model. The input dependency and varying STD significantly affect sensitivity for the most probable target point (MPTP) search and design sensitivity of probabilistic constraint in sensitivity-based RBDO. Hence, accurate sensitivities are necessary for efficient and effective process of MPTP search and RBDO. In this paper, it is assumed that dependency of input random variable is limited to the bivariate statistical correlation, and the correlation is considered using bivariate copulas. In addition, the varying STD is considered as a function of input mean value. The transformation between physical X-space and independent standard normal U-space for correlated input variable is presented using bivariate copula and marginal probability distribution. Using the transformation and the varying STD function, the sensitivity for the MPTP search and design sensitivity of probabilistic constraint are derived analytically. 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. In addition, a 14D engineering example is tested to verify the practicality and scalability of the developed sensitivity methods.