An extended hierarchical statistical sensitivity analysis method for multilevel systems with shared variables

Statistical sensitivity analysis (SSA) is an effective methodology to examine the impact of variations in model inputs on the variations in model outputs at either a prior or posterior design stage. A hierarchical statistical sensitivity analysis (HSSA) method has been proposed in literature to incorporate SSA in designing complex engineering systems with a hierarchical structure. However, the original HSSA method only deals with hierarchical systems with independent subsystems. Due to the existence of shared variables at lower levels, responses from lower level submodels that act as inputs to a higher level subsystem are both functionally and statistically dependent. For designing engineering systems with dependent subsystem responses, an extended hierarchical statistical sensitivity analysis (EHSSA) method is developed in this work to provide a ranking order based on the impact of lower level model inputs on the top level system performance. A top-down strategy, same as in the original HSSA method, is employed to direct SSA from the top level to lower levels. To overcome the limitation of the original HSSA method, the concept of a subset SSA is utilized to group a set of dependent responses from lower level submodels in the upper level SSA. For variance decomposition at a lower level, the covariance of dependent responses is decomposed into the contributions from individual shared variables. To estimate the global impact of lower level inputs on the top level output, an extended aggregation formulation is developed to integrate local submodel SSA results. The importance sampling technique is also introduced to re-use the existing data from submodels SSA during the aggregation process. The effectiveness of the proposed EHSSA method is illustrated via a mathematical example and a multiscale design problem.