实际工程设计优化、设计空间搜索、灵敏度分析、可靠性分析等问题，若单次模拟比较费时，直接用原模型进行数千、甚至数百万次模拟是不可能完成的任务。多项式混沌展开方法是解决这类问题的有效方法，其方法表达和程序实现是应用中关注的重点问题。该文介绍了多项式混沌展开方法的数学理论，并将之用于结构可靠性分析。首先，将结构可靠性分析的功能响应函数以多项式混沌展开表示，其中统一采用Hermite多项式。给出Hermite多项式的一种适合计算机程序生成的通项形式，实现多项式混沌展开的计算程序的通用化，以及多项式次数的自适应选择。其次，利用具有显式功能函数的结构可靠性分析算例，考察所构建的代理模型的正确性和适用性。结果表明，多项式混沌展开的次数越高，模型的精度就越高，具有良好的收敛性，同时表明仅就考察代理模型而言，利用显式功能函数是最简便的方式。
Abstract
Practical engineering issues such as design optimization, design space exploration, sensitivity analyses, and reliability analyses need many simulations. If a single simulation is very time-consuming, engineers cannot perform the thousands or even millions of simulations needed for such analyses. The polynomial chaos expansion (PCE) method is an effective method that allows analyses of complex problems. This paper introduces the mathematical theory of the PCE method and presents a structural reliability analysis example. The performance response function for the structural reliability analysis is expressed as a PCE using Hermite polynomials. A general form of the Hermite polynomial, which is suitable for use in a computer program, is used to generalize the PCE analysis program and the adaptive selection of the polynomial order. Then, the accuracy and applicability of the surrogate model are verified using structural reliability analysis examples with explicit performance functions. The results show that the model has an excellent convergence rate with higher order PCE giving higher accuracy. The examples also show that the direct use of explicit performance functions is the easiest way to investigate PCE surrogate models.
关键词
优化设计 /
结构可靠性 /
多项式混沌展开 /
Hermite多项式 /
功能函数
Key words
optimal design /
structural reliability /
polynomial chaos expansion (PCE) /
Hermite polynomial /
performance function
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基金
国家自然科学基金重大项目（52090081）；清华大学水沙科学与水利水电工程国家重点实验室开放基金资助项目（sklhse-2021-C-03）
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