基于椭球模型的机构非精确概率可靠性分析方法
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摘要
航天机构可靠性分析中经常遇到不确定参数样本数较少的问题。在此情形下样本数据的边界往往小于参数的边界,以此建立区间或椭球凸集模型,并采用均匀分布简单量化凸集变量进行非概率可靠性分析的传统方法与结果值得怀疑。针对非概率可靠性方法的不足,提出一种基于椭球模型的非精确概率可靠性分析方法。建立一种基于样本特征的椭球模型高维构建方法。将椭球模型标准化为圆球模型,采用无差别减小原则进行不确定性量化,推导出标准空间中变量在圆球域内的联合概率密度函数。采用基于椭球模型的重要抽样法进行非精确概率可靠度求解。通过算例和工程实例验证了方法的准确性和可行性。新方法较好地实现了可靠度分析中稳健性与精度的兼顾,可作为概率可靠性方法的一种有益补充。
One common problem is a small sample of uncertain parameters in the reliability analysis of the space mechanism. In thiscase, the boundary of the sample data is normally smaller than the boundary of the parameter. It is questionable to conductnon-probabilistic reliability analysis by establishing interval or ellipsoid convex set models with sample data and adopting uniformdistribution to simply quantify convex set variables. To address the weakness of non-probabilistic reliability methods, an impreciseprobabilistic reliability analysis method based on ellipsoid model is put forward. A high dimension construction method of ellipsoidmodel based on sample features is established. The ellipsoid model is standardized into a sphere model. The uncertainty is quantifiedby the principle of indifference reduction, and the joint probability density function of variables is derived. An important samplingmethod based on ellipsoid model is used to solve the imprecise probability reliability. The accuracy and feasibility of the method hasbeen verified by a numerical example and a project example. Achieved good balance between robustness and accuracy in reliabilityanalysis, it can be used as a useful supplement to probabilistic reliability method.
One common problem is a small sample of uncertain parameters in the reliability analysis of the space mechanism. In thiscase, the boundary of the sample data is normally smaller than the boundary of the parameter. It is questionable to conductnon-probabilistic reliability analysis by establishing interval or ellipsoid convex set models with sample data and adopting uniformdistribution to simply quantify convex set variables. To address the weakness of non-probabilistic reliability methods, an impreciseprobabilistic reliability analysis method based on ellipsoid model is put forward. A high dimension construction method of ellipsoidmodel based on sample features is established. The ellipsoid model is standardized into a sphere model. The uncertainty is quantifiedby the principle of indifference reduction, and the joint probability density function of variables is derived. An important samplingmethod based on ellipsoid model is used to solve the imprecise probability reliability. The accuracy and feasibility of the method hasbeen verified by a numerical example and a project example. Achieved good balance between robustness and accuracy in reliabilityanalysis, it can be used as a useful supplement to probabilistic reliability method.
引文
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[21]檀中强.基于椭球模型的不确定性量化与可靠性分析方法研究[D].杭州:浙江理工大学,2018.TAN Zhongqiang.A study of uncertainty quantification and reliability analysis method based on ellipsoid model[D].Hangzhou:Zhejiang Sci-Tech University,2018.
[22]JIANG C,HAN X,LU G Y,et al.Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique[J].Computer Methods in Applied Mechanics&Engineering,2011,200(33):2528-2546.
[23]JIANG C,NI B Y,HAN X,et al.Non-probabilistic convex model process:A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems[J].Computer Methods in Applied Mechanics&Engineering,2014,268(1):656-676.
[24]KHACHIYAN L G.Rounding of polytopes in the real number model of computation[J].Mathematics of Operations Research,1996,21(2):307-320.
[2]彭文胜,张建国,王丕东,等.混合不确定信息航天机构可靠性综合分析方法[J].宇航学报,2015,36(5):596-604.PENG Wensheng,ZHANG Jianguo,WANG Pidong,et al.Comprehensive reliability analysis of the aerospace mechanism with hybrid uncertainty information[J].Journal of Astronautics,2015,36(5):596-604.
[3]ELISHAKOFF I,ELISSEEFF P,GLEGG S A L.Nonprobabilistic,convex-theoretic modeling of scatter in material properties[J].Aiaa Journal,2012,32(4):843-849.
[4]JIANG C,ZHANG Q F,HAN X,et al.Multidimensional parallelepiped model:A new type of non-probabilistic convex model for structural uncertainty analysis[J].International Journal for Numerical Methods in Engineering,2015,103(1):31-59.
[5]王丕东,张建国,阚琳洁,等.基于时变区间和穿阈模型的机械时变可靠性分析方法[J].机械工程学报,2017,53(11):1-9.WANG Pidong,ZHANG Jianguo,KAN Linjie,et al.Time variant reliability approach with time-variant interval and upcrossing model[J].Journal of Mechanical Engineering,2017,53(11):1-9.
[6]ZHENG J,LUO Z,JIANG C,et al.Non-probabilistic reliability-based topology optimization with multidimensional parallelepiped convex model[J].Structural&Multidisciplinary Optimization,2017(1):1-17.
[7]KARUNA K,MANOHAR C S.Inverse problems in structural safety analysis with combined probabilistic and non-probabilistic uncertainty models[J].Engineering Structures,2017,150:166-175.
[8]YANG Zhengmao,ZHANG Yanjuan,MENG Wenjun,et al.A convex model approach for structure non-probabilistic reliability analysis[J].Journal of Risk&Reliability,2017,231(10):1-8.
[9]BEN-HAIM Y.A non-probabilistic concept of reliability[J].Structural Safety,1994,14(4):227-245.
[10]BEN-HAIM Y,ELISHAKOFF I.Discussion on:Anon-probabilistic concept of reliability[J].Structural Safety,1995,17(3):195-199.
[11]郭书祥,吕震宙.结构体系的非概率可靠性分析方法[J].计算力学学报,2002,19(3):332-335.GUO Shuxiang,LüZhenzhou.A procedure of the analysis of non-probabilistic reliability of structural systems[J].Chinese Journal of Computational Mechanics,2002,19(3):332-335.
[12]WANG X,WANG L,ELISHAKOFF I,et al.Probability and convexity concepts are not antagonistic[J].Acta Mechanica,2011,219(1):45-64.
[13]李昆锋,杨自春,孙文彩.结构凸集-概率混合可靠性分析的新方法[J].机械工程学报,2012,48(14):192-198.LI Kunfeng,YANG Zichun,SUN Wencai.New hybrid convex model and probability reliability method for structures[J].Journal of Mechanical Engineering,2012,48(14):192-198.
[14]JIANG C,BI R G,LU G Y,et al.Structural reliability analysis using non-probabilistic convex model[J].Computer Methods in Applied Mechanics&Engineering,2013,254(2):83-98.
[15]JIANG C,HAN S,JI M,et al.A new method to solve the structural reliability index based on homotopy analysis[J].Acta Mechanica,2015,226(4):1067-1083.
[16]毕仁贵.考虑相关性的不确定凸集模型与非概率可靠性分析方法[D].长沙:湖南大学,2015.BI Rengui.Non-probabilistic uncertainty convex model and reliability analysis method incorporating correlation[D].Changsha:Hunan University,2015.
[17]WANG P,ZHANG J,ZHAI H,et al.A new structural reliability index based on uncertainty theory[J].Chinese Journal of Aeronautics,2017,30(4):1451-1458.
[18]李永华,黄洪钟,刘忠贺.结构稳健可靠性分析的凸集模型[J].应用基础与工程科学学报,2004,12(4):383-391.LI Yonghua,HUANG Hongzhong,LIU Zhonghe.Convex model in robust reliability analysis of structure[J].Journal of Basic Science and Engineering,2004,12(4):383-391.
[19]BEN-HEIM Y.Maximum structural response using convex models[J].Journal of Engineering Mechanics,1996,122(4):325-333.
[20]王晓军,邱志平,武哲.结构非概率集合可靠性模型[J].力学学报,2007,39(5):641-646.WANG Xiaojun,QIU Zhiping,WU Zhe.Nonprobabilistic set-based model for structural reliability[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(5):641-646.
[21]檀中强.基于椭球模型的不确定性量化与可靠性分析方法研究[D].杭州:浙江理工大学,2018.TAN Zhongqiang.A study of uncertainty quantification and reliability analysis method based on ellipsoid model[D].Hangzhou:Zhejiang Sci-Tech University,2018.
[22]JIANG C,HAN X,LU G Y,et al.Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique[J].Computer Methods in Applied Mechanics&Engineering,2011,200(33):2528-2546.
[23]JIANG C,NI B Y,HAN X,et al.Non-probabilistic convex model process:A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems[J].Computer Methods in Applied Mechanics&Engineering,2014,268(1):656-676.
[24]KHACHIYAN L G.Rounding of polytopes in the real number model of computation[J].Mathematics of Operations Research,1996,21(2):307-320.
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