Novel Computational Methods for Design Under Uncertainty with Arbitrary Dependent Probability Distributions

Design of complex systems and engineered artifacts is often confronted with uncertainties in manufacturing processes, material properties, and operating environments. Traditional design approaches often rely on heuristically derived safety factors and do not quantitatively address the statistical variation of a system response. This project will promote scientific progress through foundational research on the design optimization of complex engineering systems and structures in the presence of statistically dependent uncertainty. Novel methods will be created to determine the best design alternative, considering uncertain system behavior, influenced by dependent input variables. Potential engineering applications include ground vehicle design for improved durability and crashworthiness, fatigue- and fracture-resistant design for aerospace applications, and design of microelectronic packaging under harsh environments. The results from this research will contribute to national prosperity through the development of complex systems and products that are more durable, robust, and reliable. Beyond engineering, the results from this research will benefit the U.S. economy and society through potential application in areas such as energy, finance and management, and transportation and logistics, where optimization under uncertainty plays a vital role. This research is multi-disciplinary, spanning several fields, including engineering design, applied mathematics, and probability and statistics. It will foster broad participation of underrepresented groups in research and positively impact engineering education. The chief goal of this project is to conduct research in the creation of efficient computational algorithms and practical computational tools for robust and reliability-based design optimization (RDO and RBDO) of high-dimensional complex systems subject to random input resulting from an arbitrary dependent probability distribution. The research plan comprises three scientific objectives: (1) novel mathematical developments of a generalized analysis-of-variance expansion, leading to a generalized spline dimensional decomposition (GSDD) for tackling dependent random variables directly; (2) new scalable algorithms of the GSDD method for calculating relevant probabilistic response characteristics and design sensitivities of a high-dimensional, complex mechanical system; and (3) innovative GSDD-driven optimization algorithms for efficiently solving high-dimensional RDO and RBDO problems, including stochastic shape and topology designs. This research is innovative for several reasons. First, the GSDD method will account for truly arbitrary, dependent probability distributions of random input, heretofore unavailable to the scientific community. Second, it will address discontinuous or non-smooth performance functions, if they exist, using hundreds of random/design variables, thereby diminishing the curse of dimensionality to a great extent. Third, the synchronous formulation of the statistical moment, reliability, and design sensitivity analyses, which requires a single or at most a few stochastic simulation(s) for all possible designs, will markedly accelerate the design optimization process, potentially producing breakthrough solutions to RDO/RBDO problems. The implementation of the probabilistic methods will lead to next-generation computational tools, bridging stress analysis, stochastic simulation, and optimization to form a seamless design pipeline of the future. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.