稀疏混合不确定变量优化方法及应用
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摘要
针对复杂产品设计中稀疏混合不确定变量导致的设计边界识别困难、计算结果失真等问题,提出一种基于Chebyshev逼近的稀疏混合不确定变量优化方法.首先采用极大似然估计方法构造稀疏混合不确定变量在给定分布下的概率密度函数,初步确定其在给定分布下对应的分布参数;其次基于贝叶斯信息准则计算对应分布的信息损失,进一步确定稀疏混合不确定变量的最合适的分布及分布参数.再次,为解决传统区间分析方法中区间扩张导致的计算失真问题,采用Chebyshev逼近优化目标函数并利用改进的HL-RF算法求解,获取可靠性指标及失效概率,在满足设计需求的同时,有效实现产品轻量化、刚度保持的设计目标.最后,以数值算例及高速压力机滑块的优化设计验证了所提方法的有效性.
Sparse hybrid uncertainties in the distribution of random variables brings the problem of hard border detection and calculation results distortion in complex product design. A sparse hybrid uncertain variable optimization method based on Chebyshev approximation was proposed. Firstly, the maximum likelihood estimation was utilized to construct the probability density function of a sparse hybrid uncertain variable under a given distribution, and its distribution parameter corresponding to the given distribution was preliminarily determined.Secondly, based on the Bayesian information criterion, the information loss of the corresponding distribution was calculated, and the most suitable distribution and parameters of the sparse mixed uncertain variables were further determined. Thirdly, in order to solve the computational distortion caused by interval expansion of the traditional interval analysis method, Chebyshev approximation was used to optimize the objective function and the improved HL-RF algorithm was used to obtain the reliability index and the failure probability value; while meeting the design requirements, the design goal of light weight and rigidity retention was effectively realized. Finally, the effectiveness of the proposed method is verified by numerical examples and the optimization design of a high-speed press slider.
Sparse hybrid uncertainties in the distribution of random variables brings the problem of hard border detection and calculation results distortion in complex product design. A sparse hybrid uncertain variable optimization method based on Chebyshev approximation was proposed. Firstly, the maximum likelihood estimation was utilized to construct the probability density function of a sparse hybrid uncertain variable under a given distribution, and its distribution parameter corresponding to the given distribution was preliminarily determined.Secondly, based on the Bayesian information criterion, the information loss of the corresponding distribution was calculated, and the most suitable distribution and parameters of the sparse mixed uncertain variables were further determined. Thirdly, in order to solve the computational distortion caused by interval expansion of the traditional interval analysis method, Chebyshev approximation was used to optimize the objective function and the improved HL-RF algorithm was used to obtain the reliability index and the failure probability value; while meeting the design requirements, the design goal of light weight and rigidity retention was effectively realized. Finally, the effectiveness of the proposed method is verified by numerical examples and the optimization design of a high-speed press slider.
引文
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[22]DAVID F F.Counterexamples to parsimony and BIC[J].Annals of the Institute of Statistical Mathematics,1991,43(3):505-514.
[23]WU J L,ZHANG Y Q,CHEN L P.A Chebyshev interval method for nonlinear dynamic systems under uncertainty[J].Applied Mathematical Modelling,2013,37(6):4578-4591.
[24]BERNARDO J M,SMITH A F.Bayesian theory[M].1st ed.New York:John Wiley&Sons,1994.
[25]MADSEN H O,KRENK S,LIND N C.Methods of structural safety[M].1st ed.Englewood Cliffs:Prentice-Hall,1986.
[26]CHENG J,DUAN G F,LIU Z Y,et al.Interval multi-objective optimization of structures based on radial basis function,interval analysis,and NSGA-Ⅱ[J].Journal of Zhejiang University-SCIENCEA:Applied Physics and Engineering,2014,15(10):774-788.
[2]SESHADRI P,CONTINE P,IACARINO G,et al.Adensity-matching approach for optimization under uncertainty[J].Computer Methods in Applied Mechanics and Engineering,2016,305(15):562-578.
[3]LIN T P,GEA H C,JALURIA Y.A modified reliability index approach for reliability-based design optimization[J].Journal of Mechanical Design,2011,133(4):044501.
[4]DEB K,GUPTA S,DAUM D,et al.Reliability-based optimization using evolutionary algorithms[J].IEEE Transactions On Evolutionary Computation,2009,13(5):1054-1074.
[5]SINHA K.Reliability-based multi-objective optimization for automotive crashworthiness and occupant safety[J].Structural and Multidisciplinary Optimization,2007,33(3):255-268.
[6]姜潮.基于区间的不确定性优化理论与算法[D].长沙:湖南大学,2008.JIANG Chao.Theories and algorithms of uncertain optimization based on interval[D].Changsha:Hunan University,2008.
[7]CHENG J,TANG M Y,LIU Z Y,et al.Direct reliability-based design optimization of uncertain structures with interval parameters[J].Journal of Zhejiang University-Science A:Applied Physics and Engineering,2016,17(11):841-854.
[8]KUNDU A,ADHIKARI S,FRISWELL M I.Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty[J].International Journal for Numerical Methods in Engineering,2014,100(3):183-221.
[9]CHENG J,LIU Z Y,TANG M Y,et al.Robust optimization of uncertain structures based on normalized violation degree of interval constraint[J].Computers and Structures,2017,182(1):41-54.
[10]JIANG C,LI W X,HAN X.Structural reliability analysis based on random distributions with interval parameters[J].Computers and Structures,2011,89(23/24):2292-2302.
[11]JIANG C,LI W X,HAN X.A hybrid reliability approach based on probability and interval for uncertain structures[J].Journal of Mechanical Design,2012,134(3):031001.
[12]姜潮,韩旭,谢慧超.区间不确定性优化设计理论有方法:第一版[M].北京:科学出版社,2017:22-29.
[13]JIANG C,HAN X,LIU G P.A sequential nonlinear interval number programming method for uncertain structures[J].Computer Methods in Applied Mechanics and Engineering,2008,197(49-50):4250-4265.
[14]张德权,韩旭,姜潮,等.时变可靠性的区间PHI2分析方法[J].中国科学:物理学力学天文学,2015,45(5):54-61.ZHANG De-quan,HAN Xu,JIANG Chao,et al.The interval PHI2 analysis method for time dependent reliability[J].SCIENTIA SINICA:Physica,Mechanica and Astronomica,2015,45(5):54-61.
[15]JIANG C,LU G Y,HAN X,et al.A new reliability analysis method for uncertain structures with random and interval variables[J].International Journal of Mechanics and Materials in Design,2012,8(2):169-182.
[16]HAN X,JIANG C,LIU L X,et al.Response-surfacebased structural reliability analysis with random and interval mixed uncertainties[J].Science China Technological Sciences,2014,57(7):1322-1334.
[17]LIM W,JANG J,KIM S,et al.Reliability-based design optimization of an automotive structure using a variable uncertainty[J].Proceedings of the Institution of Mechanical Engineers,Part D:Journal of Automobile Engineering,2016,230(10):1314-1323.
[18]PENG X,LI J Q,JIANG S F.Unified uncertainty representation and quantification based on insufficient input data[J].Structural and Multidisciplinary Optimization,2017,56(6):1305-1317.
[19]PENG X,WU T J,LI J Q,et al.Hybrid reliability analysis with uncertain statistical variables,sparse variables and interval variables[J].Engineering Optimization,2018,50(4):1-17.
[20]SANKARARAMAN S,MAHADEVAN S.Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data[J].Reliability Engineering and System Safety,2011,96(7):814-824.
[21]SANKARARAMAN S,MAHADEVAN S.Distribution type uncertainty due to sparse and imprecise data[J].Mechanical Systems and Signal Processing,2013,37(1-2):182-198.
[22]DAVID F F.Counterexamples to parsimony and BIC[J].Annals of the Institute of Statistical Mathematics,1991,43(3):505-514.
[23]WU J L,ZHANG Y Q,CHEN L P.A Chebyshev interval method for nonlinear dynamic systems under uncertainty[J].Applied Mathematical Modelling,2013,37(6):4578-4591.
[24]BERNARDO J M,SMITH A F.Bayesian theory[M].1st ed.New York:John Wiley&Sons,1994.
[25]MADSEN H O,KRENK S,LIND N C.Methods of structural safety[M].1st ed.Englewood Cliffs:Prentice-Hall,1986.
[26]CHENG J,DUAN G F,LIU Z Y,et al.Interval multi-objective optimization of structures based on radial basis function,interval analysis,and NSGA-Ⅱ[J].Journal of Zhejiang University-SCIENCEA:Applied Physics and Engineering,2014,15(10):774-788.