基于概率空间剖分的结构非线性随机反应分析
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摘要
在概率密度演化理论的框架下,发展了基于概率空间剖分的多维空间选点方法。引入点集Voronoi域内的概率作为点集的赋得概率,对点集的F-偏差进行了以赋得概率替代均匀概率的修正。在此基础上,进行误差估计,提出了以点集的修正F-偏差、一阶偏差和二阶偏差均尽可能小为准则的点集选取方法——两步选点法。分析实例表明,基于概率空间剖分的选点方法具有较高的精度和效率。文中最后指出需要进一步研究的问题。
In the framework of probability density evolution theory,a point selection strategy via partition of probability-assigned space is developed.The assigned probability attached to a point set is defined as the probability over the Voronoi cell of the points.The F-discrepancy which is used to measure the goodness of point set is modified where the equi-probability is replaced by the assigned probability.On the above basis,the error estimate is studied and the criterion of minimizing the modified F-discrepancy and the first and second discrepancy is proposed,consequently,a two-step procedure is developed.Numerical examples indicate that the proposed approach is of acceptable accuracy and of high efficiency.The problems need to be further studied are pointed out.
In the framework of probability density evolution theory,a point selection strategy via partition of probability-assigned space is developed.The assigned probability attached to a point set is defined as the probability over the Voronoi cell of the points.The F-discrepancy which is used to measure the goodness of point set is modified where the equi-probability is replaced by the assigned probability.On the above basis,the error estimate is studied and the criterion of minimizing the modified F-discrepancy and the first and second discrepancy is proposed,consequently,a two-step procedure is developed.Numerical examples indicate that the proposed approach is of acceptable accuracy and of high efficiency.The problems need to be further studied are pointed out.
引文
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[10]Li J,Chen J B.The Principle of Preservation of Probability and the Generalized Density Evolution Equation[J].Structural Safety,2008,30(1):65-77.
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[16]Chen J B,Ghanem R,Li J.Partition of the Probability Space in Probability Density Evolution Analysis of Nonlinear Stochastic Structures[J].Probabilistic Engineering Mechanics,2007,doi:10.1016/j.probengmech.2007.12.017.
[17]Hua L K,Wang Y.Applications of Number Theory to Numerical Analysis[M].Berlin:Springer,1981.
[18]Wen Y K.Method for Random Vibration of Hysterestic Systems[J].Journal of the Engineering Mechanics Division,1976,102(2):249-263.
[19]Ma F,Zhang H,Bockstedte A,et al.Parameter Analysis of the Differential Model of Hysteresis[J].Journal of Applied Mechanics,2004,71(3):342-349.
[2]Schuёller G I.A State-of-the-art Report on Computational Stochastic Mechanics[J].Probabilistic Engineering Mechanics,1997,12(4):197-321.
[3]Crandall S.Random Vibration[M].Cambridge:MIT Press,1958.
[4]Lin Y K,Cai G Q.Probabilistic Structural Dynamics:Advanced Theory and Applications[M].New York:McGraw-Hill,1995.
[5]Lutes L D,Sarkani S.Random Vibrations:Analysis of Structural and Mechanical Systems[M].Amsterdam:Elsevier,2004.
[6]Kleiber M,Hien T D.The Stochastic Finite Element Method:Basic Perturbation Technique and Computer Implementation[M].Chichester:Wiley,1992.
[7]Ghanem R,Spanos P D.Stochastic Finite Elements:A Spectral Approach[M].New York:Springer-Verlag,1991.
[8]李杰,陈建兵.随机结构非线性动力响应的概率密度演化分析[J].力学学报,2003,35(6):716-722.
[9]李杰,陈建兵.随机动力系统分析中的广义密度演化方程[J].自然科学进展,2006,16(6):712-719.
[10]Li J,Chen J B.The Principle of Preservation of Probability and the Generalized Density Evolution Equation[J].Structural Safety,2008,30(1):65-77.
[11]李杰,艾晓秋.基于物理的随机地震动模型研究[J].地震工程与工程振动,2006,26(5):21-26.
[12]刘章军,李杰.脉动风速随机过程的正交展开[J].振动工程学报,2008,21(1):96-101.
[13]Bontempi F,Faravelli L.Lagrangian/Eulerian Description of Dynamic System[J].Journal of Engineering Mechanics,1998,124(8):901-911.
[14]Conway J H,Sloane N J A.Sphere Packings,Lattices and Groups(3rd Ed)[M].New York:Springer,1999.
[15]Fang K T,Wang Y.Number-theoretic Methods in Statistics[M].London:Chapman&Hall,1994.
[16]Chen J B,Ghanem R,Li J.Partition of the Probability Space in Probability Density Evolution Analysis of Nonlinear Stochastic Structures[J].Probabilistic Engineering Mechanics,2007,doi:10.1016/j.probengmech.2007.12.017.
[17]Hua L K,Wang Y.Applications of Number Theory to Numerical Analysis[M].Berlin:Springer,1981.
[18]Wen Y K.Method for Random Vibration of Hysterestic Systems[J].Journal of the Engineering Mechanics Division,1976,102(2):249-263.
[19]Ma F,Zhang H,Bockstedte A,et al.Parameter Analysis of the Differential Model of Hysteresis[J].Journal of Applied Mechanics,2004,71(3):342-349.
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