基于多项式混沌展开的结构动力特性高阶统计矩计算
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摘要
结构参数的不确定性会导致其动力特性的不确定性,量化动力特性的不确定性能为结构动力设计分析提供准确的动力信息.统计矩是表征结构动力特性不确定性非常重要的统计量,比如均值和方差.传统的Monte-Carlo(蒙特-卡洛)模拟方法需要大量次数的模型运算来保证结果的收敛,其用于复杂结构的动力特性统计矩计算因耗时太高而使用受限.该文采用多项式混沌展开替代模型来取代计算花费高的有限元模型,然后在替代模型框架下快速有效地计算结构动力特性的统计矩.该方法在建立替代模型之初只需要少量次数有限元分析,后续的统计矩计算无需有限元模型,因此从根本上解决了动力特性统计矩计算花费高的难题.该文的多项式混沌展开方法适用于参数服从任意概率分布,能够有效地计算高阶统计矩,为量化结构动力特性不确定性提供更多统计矩信息.最后通过平铝板算例验证了此方法的有效性.
Uncertainty of structural parameters leads to uncertainty of structural dynamic characteristics.Quantification of uncertainty of dynamic characteristics provides accurate dynamic information for structural dynamic analysis. Statistical moments( e. g.,mean and variance) mainly represent the uncertainty of structural dynamic properties. The Monte-Carlo simulation( MCS) requires a large number of model evaluations to ensure the convergence of the results,which hinders its application to the large-scale,complex engineering structures. The polynomial chaos expansion( PCE) surrogate model was used to replace the computationally expensive finite element model( FEM),and then the statistical moments of structural dynamic characteristics were efficiently calculated. The presented PCE-based method only needs a small set of model runs before the model formulation and subsequently does not require the FEM for calculations of the statistical moments. Therefore,the issue of the high computational cost associated with the computations of dynamic characteristic statistical moments was solved. The method is suitable for parameters with arbitrary probability distribution and has high computational efficiency in calculating the high-order statistical moments. Finally,the effectiveness of the developed method was verified through an example of an aluminum plate.
Uncertainty of structural parameters leads to uncertainty of structural dynamic characteristics.Quantification of uncertainty of dynamic characteristics provides accurate dynamic information for structural dynamic analysis. Statistical moments( e. g.,mean and variance) mainly represent the uncertainty of structural dynamic properties. The Monte-Carlo simulation( MCS) requires a large number of model evaluations to ensure the convergence of the results,which hinders its application to the large-scale,complex engineering structures. The polynomial chaos expansion( PCE) surrogate model was used to replace the computationally expensive finite element model( FEM),and then the statistical moments of structural dynamic characteristics were efficiently calculated. The presented PCE-based method only needs a small set of model runs before the model formulation and subsequently does not require the FEM for calculations of the statistical moments. Therefore,the issue of the high computational cost associated with the computations of dynamic characteristic statistical moments was solved. The method is suitable for parameters with arbitrary probability distribution and has high computational efficiency in calculating the high-order statistical moments. Finally,the effectiveness of the developed method was verified through an example of an aluminum plate.
引文
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[11] MOENS D,VANDEPITTE D. Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures[J]. Computer Methods in Applied Mechanics and Engineering,2007,196(21/24):2486-2496.
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[14] WAN H P,REN W X,TODD M D. An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions[J]. International Journal for Numerical Methods in Engineering,2017,109(5):739-760.
[15] WAN H P,REN W X. Parameter selection in finite-element-model updating by global sensitivity analysis using Gaussian process metamodel[J]. Journal of Structural Engineering,2015,141(6):04014164.
[2]万华平,任伟新,钟剑.桥梁结构固有频率不确定性量化的高斯过程模型方法[J].中国科学:技术科学,2016,46(9):919-925.(WAN Huaping,REN Weixin,ZHONG Jian. Gaussian process model-based approach for uncertainty quantification of natural frequencies of bridge[J]. Scientia Sinica:Technologica,2016,46(9):919-925.(in Chinese))
[3] SHINOZUKA M,ASTILL C J. Random eigenvalue problems in structural analysis[J]. AIAA Journal,1972,10(4):456-462.
[4] SZEKELY G S,SCHUELLER G I. Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties[J]. Computer Methods in Applied Mechanics and Engineering,2001,191(8/10):799-816.
[5] WAN H P,MAO Z,TODD M D,et al. Analytical uncertainty quantification for modal frequencies with structural parameter uncertainty using a Gaussian process metamodel[J]. Engineering Structures,2014,75:577-589.
[6]陈塑寰,张宗芬.结构固有频率的统计特性[J].振动工程学报,1991,4(1):90-95.(CHEN Shuhuan,ZHANG Zongfen. Statistics of structural natural frequeneies[J]. Journal of Vibration Engineering,1991,4(1):90-95.(in Chinese))
[7]刘春华,秦权.桥梁结构固有频率的统计特征[J].中国公路学报,1997,10(4):50-55.(LIU Chunhua,QIN Quan. Statistics of natural frequencies for bridge structures[J]. China Journal of Highway and Transport,1997,10(4):50-55.(in Chinese))
[8] ADHIKARI S,FRISWELL M I. Random matrix eigenvalue problems in structural dynamics[J]. International Journal for Numerical Methods in Engineering,2007,69(3):562-591.
[9] QIU Z P,WANG X J,FRISWELL M I. Eigenvalue bounds of structures with uncertain-but-bounded parameters[J]. Journal of Sound and Vibration,2005,282(1/2):297-312.
[10] MODARES M,MULLEN R,MUHANNA R. Natural frequencies of a structure with bounded uncertainty[J]. Journal of Engineering Mechanics,2006,132(12):1363-1371.
[11] MOENS D,VANDEPITTE D. Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures[J]. Computer Methods in Applied Mechanics and Engineering,2007,196(21/24):2486-2496.
[12] WANG C,GAO W,SONG C M,et al. Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties[J]. Journal of Sound and Vibration,2014,333(9):2483-2503.
[13] GAUTSCHI W. Orthogonal Polynomials:Computation and Approximation[M]. Oxford:Oxford University Press,2004.
[14] WAN H P,REN W X,TODD M D. An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions[J]. International Journal for Numerical Methods in Engineering,2017,109(5):739-760.
[15] WAN H P,REN W X. Parameter selection in finite-element-model updating by global sensitivity analysis using Gaussian process metamodel[J]. Journal of Structural Engineering,2015,141(6):04014164.