Novel Decomposition Methods for Reliability-Based Topology Optimization

拓扑优化是一种使复杂工程系统材料分布理想化的计算设计框架。制造过程和操作环境中不可避免的不确定性常常困扰着这些工程系统，因此需要在设计过程中加以考虑。传统的确定性设计方法通常会导致效率低下和过度保守的设计，过度补偿不确定性，或者由于固有的不确定性而导致不知情的风险设计。该奖项支持存在不确定性的复杂工程结构拓扑优化的基础研究。具体来说，它将开发新的方法来确定具有低失效概率的复杂工程系统的理想材料分布，对应于一些关键的失效机制。方法和相关的数值工具将适用于广泛的，多学科的优化方法。这一发现将促进增材制造的发展，特别是3d打印，它能够制造具有复杂拓扑结构的产品，因此需要有效的拓扑设计方法来充分发挥其潜力。其他工程应用包括能源收集装置的耐用设计，民用和航空航天应用的抗疲劳设计，以及绿色能源工业的可靠设计。教育影响包括通过广泛的传播和推广项目吸引、吸引和培训K-12和本科生。在技术上，本课题旨在为复杂工程系统的大规模鲁棒拓扑优化（RTO）和基于可靠性的拓扑优化（RBTO）建立新的理论基础和数值算法。创新包括：(1)设计了一种新的自适应稀疏多项式维数分解方法，用于超高维随机系统的统计矩和可靠性分析；(2)提出了一种新的RTO和RBTO拓扑设计灵敏度分析方法，实现了不确定性及其设计灵敏度的并行评估；(3)结合水平集方法提出了一种新的拓扑优化算法，收敛速度快，几何形状清晰。此外，拟议的研究将纳入效用功能，以开发新的实用RBTO模型，用于工业应用。该项目将为解决不确定性下的大规模拓扑优化问题提供一种新颖、可行的范式转换。

Topology optimization is a computational design framework to idealize materials distribution of complex engineering systems. Uncertainties, unavoidable in manufacturing process and operating environments, often plague those engineering systems, thus need be taken into account during the design process. Conventional deterministic design approaches typically lead to inefficient and overly conservative designs that overcompensate for uncertainties, or unknowingly risky designs due to the inherent uncertainties. This award supports fundamental research on topology optimization of complex engineering structures in the presence of uncertainty. Specifically, it will develop novel methods to determine the ideal material distribution of complex engineering systems with low probabilities of failure corresponding to some critical failure mechanisms. The methods and associated numerical tools will be applicable to a broad, multidisciplinary optimization methodology. The findings will promote growth in additive manufacturing, especially 3D-printing, which is able to manufacture products with complex topology and thus demands efficient topology design methods to bring out its full potential. Other engineering applications include durable design for energy harvest devices, fatigue-resistant design for civil and aerospace applications, and reliable design for green energy industry. The education impact consists of attracting, engaging, and training K-12 and undergraduate students through extensive dissemination and outreach programs. Technically, this research project aims to create new theoretical foundations and numerical algorithms for large-scale, robust topology optimization (RTO) and reliability-based topology optimization (RBTO) of complex engineering systems. Innovations include: (1) a new adaptive-sparse polynomial dimensional decomposition method designed for statistical moments and reliability analyses of ultra-high-dimensional, stochastic systems; (2) a new topology design sensitivity analysis for RTO and RBTO to enable concurrent evaluation of uncertainties and their design sensitivities; and (3) a new topology optimization algorithm integrating the level-set method for both fast convergence and clear geometry. In addition, proposed research will incorporate utility functions for developing new practical RBTO model for industrial applications. The project will deliver a novel, feasible paradigm-shifting advance toward solving large-scale topology optimization problems under uncertainty.