American Institute of Aeronautics and Astronautics 1 Analytical Uncertainty Propagation via Metamodels in Simulation-Based Design under Uncertainty Wei Chen * Integrated DEsign Automation Laboratory (IDEAL), Northwestern University, Evanston, IL 60208-3111, USA Ruichen Jin † and Agus Sudjianto ‡ Ford Motor Company, Dearborn, MI 48124, USA In spite of the benefits, one of the most challenging issues for implementing optimization under uncertainty, such as the use of robust design approach, is associated with the intensive computational demand of uncertainty propagation, especially when the simulation programs are computationally expensive. In this paper, an efficient approach to uncertainty propagation via the use of metamodels is presented. Metamodels, created through computer simulations to replace expensive simulation programs, are widely used in simulation-based design. Different from existing techniques that apply sample-based methods to metamodels for uncertainty propagation, our method utilizes analytical derivations to eliminate the random errors as well as to reduce the computational expenses of sampling. In this paper, we provide analytical formulations for mean and variance evaluations via a variety of metamodels commonly used in engineering design applications. The benefits of our proposed techniques are demonstrated through the robust design for improving vehicle handling. In addition to the improved accuracy and efficiency, our proposed analytical approach can greatly improve the convergence behavior of optimization under uncertainty. Nomenclature R x : Random variables ( ) p R R x : Joint probability density function (PDF) of random variables ( ) i i p x : Individual (marginal) probability density function () y µ x , 2 () y σ x : Response mean and Variance () i B x : Multivariate tensor-product basis function ( ) il l h x : Univariate basis function b N : Number of multivariate basis functions 1,il C , 12 2,iil C : Univariate integrals involved in analytical uncertainty propagation 1. INTRODUCTION Development of efficient methods for uncertainty propagation has gained much attention in recent years due to the increasing awareness of the importance of nondeterministic optimization. By uncertainty propagation, we mean that the impact of input uncertainty on the variation of a model output (response) is studied. Despite the advancement of methods for uncertainty propagation, the rising fidelity of engineering analyses has significantly increased the computational expenses of simulation-based design and creates the barrier for applying nondeterministic optimization to real engineering design applications. One difficulty with the conventional statistical approach to uncertainty propagation is that it relies heavily on the use of data sampling to generate probabilistic distributions of a system output. Monte Carlo simulation, a random simulation based approach is very expensive. Even reduced sampling techniques, like Latin Hypercube Sampling (Box et al. 1978) and Taguchi’s orthogonal array (Phadke 1989), can still require a large amount of * Associate Professor, Department of Mechanical Engineering, 2145 Sheridan Rd., Tech B224, Evanston, IL 60208- 0311, weichen@northwestern.edu , Associate Fellow of AIAA. † Engineer ‡ Technical manager