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

复杂系统和工程制品的设计经常面临制造过程、材料特性和操作环境的不确定性。传统的设计方法通常依赖于启发式得出的安全系数，并且不能定量地处理系统响应的统计变化。该项目将通过在存在统计相关不确定性的情况下对复杂工程系统和结构的设计进行优化的基础研究来推动科学进步。考虑到不确定的系统行为，受相关输入变量的影响，将创造新的方法来确定最佳设计方案。潜在的工程应用包括提高耐用性和耐撞性的地面车辆设计，航空航天应用的抗疲劳和抗断裂设计，以及在恶劣环境下的微电子封装设计。这项研究的结果将通过开发更耐用、更强大和更可靠的复杂系统和产品来促进国家繁荣。除了工程，这项研究的结果将通过在能源、金融和管理以及运输和物流等领域的潜在应用而造福美国经济和社会，在这些领域，不确定性下的优化发挥着至关重要的作用。这项研究是多学科的，跨越了几个领域，包括工程设计、应用数学和概率统计。这个项目的主要目标是研究如何创建有效的计算算法和实用的计算工具，用于高维复杂系统的稳健和基于可靠性的设计优化(RDO和RBDO)，该系统受任意相关概率分布的随机输入影响。该研究计划包括三个科学目标：(1)广义方差分析展开的新的数学发展，导致直接处理相依随机变量的广义样条维分解(GSDD)；(2)用于计算高维、复杂机械系统的相关概率响应特征和设计灵敏度的GSDD方法的新的可扩展算法；以及(3)创新的GSDD驱动的优化算法，用于高效地解决高维RDO和RBDO问题，包括随机形状和拓扑设计。这项研究之所以具有创新性，有几个原因。首先，GSDD方法将考虑到随机输入的真正任意的、相关的概率分布，这是科学界迄今无法获得的。其次，它将使用数百个随机/设计变量来处理不连续或不光滑的性能函数，从而在很大程度上减少维度的诅咒。第三，对所有可能的设计进行一次或至多几次随机模拟(S)的统计矩、可靠性和设计灵敏度分析的同步公式，将显著加快设计优化过程，潜在地产生RDO/RBDO问题的突破性解。概率方法的实施将导致下一代计算工具、桥梁应力分析、随机模拟和优化，以形成未来的无缝设计管道。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.