American Institute of Aeronautics and Astronautics 1 Probabilistic Sensitivity Analysis Methods for Design under Uncertainty Huibin Liu * and Wei Chen. † Integrated Design Automation Laboratory, Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Tech B224, Evanston, IL 60208 Agus Sudjianto ‡ V-Engine Engineering Analytical Powertrain, Ford Motor Company Sensitivity analysis (SA) is an important procedure in engineering design to obtain valuable information about the model behavior to guide a design process. For design under uncertainty, probabilistic sensitivity analysis (PSA) methods have been developed to provide insight into the probabilistic behavior of a model. In this paper, the goals of PSA at different design stages are investigated. In the prior-design stage, PSA can be utilized to identify those probabilistically non-significant variables and reduce the dimension of a random design space. It can reduce the computational cost associated with uncertainty assessment without much sacrifice on the optimum solution. For post-design analysis, probabilistic sensitivity analysis can be used to identify where to spend design resources for the largest potential improvement of a performance. Based on the interested distribution range of a random response, the PSA methods can be categorized into two types: the global response probabilistic sensitivity analysis (GRPSA) and the regional response probabilistic sensitivity analysis (RRPSA). Four widely-used PSA methods: Sobol’ indices, Wu’s sensitivity coefficients, the MPP based sensitivity coefficients, and the Kullback-Leibler entropy based method are selected for comparison. The merits behind each method are reviewed in details. Their advantages, limitations, and applicability are investigated. Their effectiveness and applicability under different design scenarios are compared in two numerical examples and two engineering design problems. Key words: probabilistic sensitivity analysis, robust design, reliability-based design, sensitivity coefficient, main effect, total effect, variance-based methods, Kullback-Leibler entropy I. Introduction ENSITIVITY analysis (SA) has been widely applied in engineering design to explore the model response behavior, to evaluate the accuracy of a model, to test the validity of the assumptions made, etc. In deterministic design, sensitivity analysis is used to find the rate of change in a model output due to changes in the model inputs. That is usually performed by varying input variables one at a time near a given central point, which involves partial derivatives and often called local sensitivity analysis. It has been widely acknowledged that uncertainty is inevitable in a product development process. Robust design 1,2 and reliability-based design 3,4 are two widely used probabilistic design methods that have gained wide attentions to ensure the quality of a product under uncertainty. Robust design is used to minimize the effect of variations in controllable and/or uncontrollable factors without eliminating the sources of variations, while the reliability-based design has been widely applied to ensure that a system performance meets the pre-specified target with a required probability level. Though it is important to seek the optimal solution in design under uncertainty, sensitivity analysis is also important for designers to gain insights about the complex model behavior and make informed decisions regarding where to spend the engineering effort to reduce the variability of a system. When uncertainty is considered, sensitivity analysis has different meanings. We assume that the uncertainty in a design performance is described probabilistically by its mean (µ), variance (σ 2 ), the probability density function (PDF), or the cumulative distribution function (CDF), etc. Correspondingly, the sensitivity analysis under uncertainty needs to be performed on the probabilistic characteristics of a model response with respect to the * Graduate Research Assistant. † Corresponding author, Associate Professor, Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Tech B224, Evanston, IL 60208-3111, Phone: (847)491-7019, weichen@northwestern.edu , Associate Fellow of AIAA. ‡ Manager S