Probabilistic analytical target cascading using kernel density estimation for accurate uncertainty propagation

Probabilistic analytical target cascading (PATC) has been developed to incorporate uncertainty of random variables in a hierarchical multilevel system using the framework of ATC. In the decomposed ATC structure, consistency between linked subsystems has to be guaranteed through individual subsystem optimizations employing special coordination strategies such as augmented Lagrangian coordination (ALC). However, the consistency in PATC has to be attained exploiting uncertainty quantification and propagation of interrelated linking variables that are the major concern of PATC and uncertainty-based multidisciplinary design optimization (UMDO). In previous studies, the consistency of linking variables is assured by matching statistical moments under the normality assumption. However, it can induce significant error when the linking variable to be quantified is highly nonlinear and non-normal. In addition, reliability estimated from statistical moments may be inaccurate in each optimization of the subsystem. To tackle the challenges, we propose the sampling-based PATC using multivariate kernel density estimation (KDE). The framework of reliability-based design optimization (RBDO) using sampling methods is adopted in individual optimizations of subsystems in the presence of uncertainty. The uncertainty quantification of linking variables equivalent to intermediate random responses can be achieved by multivariate KDE to account for correlation between linking variables. The constructed KDE based on finite samples of the linking variables can provide accurate statistical representations to linked subsystems and thus be utilized as probability density function (PDF) of linking variables in individual sampling-based RBDOs. Stochastic sensitivity analysis with respect to multivariate KDE is further developed to provide an accurate sensitivity of reliability during the RBDO. The proposed sampling-based PATC using KDE facilitates efficient and accurate procedures to obtain a system optimum in PATC, and the mathematical examples and roof assembly optimization using finite element analysis (FEA) are used to demonstrate the effectiveness of the proposed approach.