摘要
多项式混沌拓展(polynomial chaos expansion,PCE)模型现已发展为全局灵敏度分析的强大工具,却很少作为替代模型用于可靠性分析。针对该模型缺乏误差项从而很难构造主动学习函数来逐步更新的事实,在结构可靠性分析的框架下提出了基于PCE模型和bootstrap重抽样的仿真方法来计算失效概率。首先,对试验设计(experimental design)使用bootstrap重抽样步骤以刻画PCE模型的预测误差;其次,基于这个局部误差构造主动学习函数,通过不断填充试验设计以自适应地更新模型,直到能够精确地逼近真实的功能函数;最后,当PCE模型具有足够精确的拟合、预测能力,再使用蒙特卡洛仿真方法来计算失效概率。提出的平行加点策略既能在模型更新过程中找到改进模型拟合能力的"最好"的点,又考虑了模型拟合的计算量;而且,当失效概率的数量级较低时,PCE-bootstrap步骤与子集仿真(subset simulation)的结合能进一步加速失效概率估计量的收敛。本文方法将PCE模型在概率可靠性领域的应用从灵敏度分析延伸到了可靠性分析,同时,算例分析结果显示了该方法的精确性和高效性。
Polynomial chaos expansion(PCE) model has been widely used as a surrogate in global sensitivity analysis;on the contrary,it has rarely been applied in the area of reliability analysis,which is another important research topic in uncertainty quantification.Due to the lack of error term in PCE model,which makes it much harder for adaptive model-updating,we proposed a method comprised of PCE model and bootstrap procedure to calculate the failure probability.Firstly,by bootstrapping the experimental design,it enables us to characterize the prediction error of PCE model;then,a practical and intuitive learning function is suggested and it ensures that the potential points to be added into the experimental design are efficient and effective in improving the accuracy of the PCE surrogate;lastly,once the convergence criterion is achieved,Monte Carlo-based reliability analysis is implemented on the PCE model instead of the authentic one.The proposed multiple-points strategy not only identifies the best points,but considers the computational budget of model updating.Further,when the magnitude of failure probability is extremely small,the combination of PCE-bootstrap and subset simulation could accelerate the convergence of this estimate.In summary,the proposed method extends the application of PCE model from sensitivity analysis to reliability analysis;meanwhile,numerical examples demonstrate its accuracy and efficiency.
作者
王娟
马义中
WANG Juan;MA Yi-zhong(School of Economics and Management,Nanchang Hangkong University,Nanchang 330063,China;School of Economics and Management,Nanjing University of Science and Technology,Nanjing 210094,China)
出处
《数理统计与管理》
CSSCI
北大核心
2020年第1期104-115,共12页
Journal of Applied Statistics and Management
关键词
多项式混沌拓展
可靠性分析
失效概率
自助法
自适应的模型更新
polynomial chaos expansion(PCE)
reliability analysis
failure probability
bootstrap
adaptive model updating