Stochastic Optimization for Design under Uncertainty with Dependent Probability Measures

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Many complex systems and engineering structures are plagued by uncertainties in manufacturing processes and operating environments. Conventional design approaches rely on heuristically derived safety factors and do not account quantitatively for the statistical variation of a system response. In this project, the principal investigator will conduct fundamental research on design optimization of complex systems in the presence of statistically dependent uncertainty. Novel methods will be developed to determine the best design alternative considering that the system behavior is uncertain and driven by dependent input variables. Potential engineering applications include ground vehicle design for improved durability and crashworthiness, fatigue- and fracture-resistant design for civil and aerospace applications, and reliable design of microelectronic packaging under harsh environments. Beyond engineering, the results from this research will benefit the U.S. economy and society through potential application in areas such as energy, finance, management, scheduling, and transportation and logistics, where optimization under uncertainty plays a vital role. This research is multi-disciplinary, encompassing several disciplines, including engineering, computer science, mathematics, and statistics. It will help broaden participation of underrepresented groups in research and positively impact engineering education.The objectives of this project are to build a solid mathematical foundation, devise efficient numerical algorithms, and develop practical tools for design optimization subject to uncertainty characterized by dependent probability distributions. The effort will involve (1) a new theoretical development of the generalized polynomial dimensional decomposition method for a high-dimensional stochastic response; (2) new formulae and scalable algorithms for calculating the statistical moments and reliability, followed by design sensitivity analysis; and (3) new reliability-based and robust optimization algorithms for shape and topology designs. Due to innovative calculation of the expansion coefficients, the generalized decomposition method will be efficiently implemented regardless of the size of the stochastic design problem. The innovative formulation of the statistical moment and reliability analyses and design sensitivities, which requires a single or at most a few stochastic simulations for all possible designs, will markedly accelerate the optimization process, potentially producing breakthrough solutions to stochastic design problems.