Newly restructured generalized polynomial chaos expansion (GPCE) methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions are reported. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multipoint single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus facilitating to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More importantly, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation. Numerical results affirm that the proposed methods yield accurate and computationally efficient optimal solutions of mathematical problems and design solutions for simple mechanical systems. Finally, the success in conducting stochastic shape optimization of a bogie side frame with 41 random variables demonstrates the power of the MPSS-DD-GPCE method in solving industrial-scale engineering design problems.

中文翻译：

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基于维数分解的广义多项式混沌展开的相关随机变量下的高维随机设计优化

报告了在存在具有任意依赖概率分布的输入随机变量的情况下用于高维设计优化的新重组广义多项式混沌展开 (GPCE) 方法。这些方法具有维度分解的 GPCE (DD-GPCE)，用于与高维随机响应相关的统计矩和可靠性分析；DD-GPCE 近似和评分函数之间的新综合，用于估计统计矩和失效概率的一阶设计灵敏度；和标准的基于梯度的优化算法，构建单步 DD-GPCE 和多点单步 DD-GPCE (MPSS-DD-GPCE) 方法。在这些新的设计方法中，多元正交基函数的组装与输入变量和多项式阶数之间选定的交互程度一致，从而有助于尽可能减少维数灾难。此外，当与得分函数结合使用时，DD-GPCE 近似会得出用于计算设计灵敏度的解析公式。更重要的是，统计矩、失效概率及其设计敏感性是通过单个随机分析或模拟同时确定的。数值结果证实，所提出的方法可以产生准确且计算高效的数学问题最优解和简单机械系统的设计解。最后，