不确定性参数识别的区间响应面模型修正方法
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摘要
工程问题中总存在着不确定性,因此通过概率方法识别得到的结构参数更加可靠。但实际应用中往往缺乏足够的结构统计信息,同时可能要求了解的仅仅是参数和响应的极限值,此时采用非概率的区间分析方法则更为合适。结合响应面模型和区间算法提出了区间响应面模型,即先将响应面表达式转化为完全平方项,再引入区间变量,以避免区间运算过程中的区间扩张问题。利用区间响应的上下界建立优化目标函数,并构建区间优化反问题,然后直接基于区间响应面模型实现修正过程。该方法避免了复杂的区间参数灵敏度计算,使得区间模型修正问题得到简化,并大幅提高修正效率。通过对一个数值质量-弹簧系统和一组试验钢板的几何及材料参数的区间值进行识别,验证了所提出方法的可行性和可靠性。
By considering the uncertainties existing in engineering structures,the parameter identification using probabilistic methods may provide more reliable predictions.However sufficient probabilistic information of a structure is often impractical in the real world.Moreover,sometimes engineers require only the limits of parameters and responses.Due to these facts,the application of interval analysis for parameter identification could be more rational.This study develops a new concept of interval response surface model,which incorporates the conventional response surface model with the interval arithmetic.The expression of the response surface is first transformed into a complete-square form and then interval variables are introduced into the transformed expressions.By this means,the usual interval overestimation due to interval arithmetic operations can be avoided providing correct interval predictions.Secondly,the infimum and supremum of interval parameters are used to construct the objective functions for interval inverse optimization.Then using the interval response surface models,the interval model updating procedure is implemented.The proposed method can avoid the complicated sensitivity computation of interval parameters and thus highly simplifies the interval model updating process in the interest of improving the updating efficiency.Lastly,the proposed method has been validated using a numerical mass-spring system and then a set of experimental steel plates.The intervals of their geometric and material parameters are identified and the updating results have proved the feasibility and reliability of the method.
By considering the uncertainties existing in engineering structures,the parameter identification using probabilistic methods may provide more reliable predictions.However sufficient probabilistic information of a structure is often impractical in the real world.Moreover,sometimes engineers require only the limits of parameters and responses.Due to these facts,the application of interval analysis for parameter identification could be more rational.This study develops a new concept of interval response surface model,which incorporates the conventional response surface model with the interval arithmetic.The expression of the response surface is first transformed into a complete-square form and then interval variables are introduced into the transformed expressions.By this means,the usual interval overestimation due to interval arithmetic operations can be avoided providing correct interval predictions.Secondly,the infimum and supremum of interval parameters are used to construct the objective functions for interval inverse optimization.Then using the interval response surface models,the interval model updating procedure is implemented.The proposed method can avoid the complicated sensitivity computation of interval parameters and thus highly simplifies the interval model updating process in the interest of improving the updating efficiency.Lastly,the proposed method has been validated using a numerical mass-spring system and then a set of experimental steel plates.The intervals of their geometric and material parameters are identified and the updating results have proved the feasibility and reliability of the method.
引文
[1]Friswell M I,Mottershead J E.Finite Element Model Updating in Structural Dynamics[M].Kluwer Academic Publishers,Dordrecht,Netherlands,1995.
[2]Fang S E,Perera R,Roeck G D.Damage identification of a reinforced concrete frame by finite element model updating using damage parameterization[J].Journal of Sound and Vibration,2008,313:544—559.
[3]Alvin K,Oberkampf W,Diegert K,et al.Uncertainty quantification in computational structural dynamics:a new paradigm for model validation[A].Proceedings of the 16th International Modal Analysis Conference[C],1998:1 191—1 197.
[4]冯新,李国强,周晶.土木工程结构健康诊断中的统计识别方法综述[J].地震工程与工程振动,2005,25(2):105—113.Feng Xin,Li Guoqiang,Zhou Jing.State-of-the-art of statistical identification for structural health diagnosis in civil engineering[J].Earthquake Engineering and Engineering Vibration,2005,25(2):105—113.
[5]Moens D.A non-probabilistic finite element approach for structural dynamic analysis with uncertain parameters[D].Catholic University of Leuven,Belgium,2002.
[6]Friswell M I.The adjustment of structural parameters using a minimum variance estimator[J].Mechanical Systems and Signal Processing,1989,3(2):143—155.
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[8]韩芳,钟冬望,汪君.基于贝叶斯法的复杂有限元模型修正研究[J].振动与冲击,2012,31(1):39—43.HAN Fang,ZHONG Dongwang,WANG Jun.Complicated finite element model updating based on Bayesian method[J].Journal of Vibration and Shock,2012,31(1):39—43.
[9]Khodaparast H H,Mottershead J E,Friswell M I.Perturbation methods for the estimation of parameter variability in stochastic model updating[J].Mechanical Systems and Signal Processing,2008,22(8):1 751—1 773.
[10]王登刚,秦仙蓉.结构计算模型修正的区间反演方法[J].振动工程学报,2004,17(2):205—209.Wang Denggang,Qin Xianrong.Interval method for computational model updating of dynamic structures[J].Journal of Vibration Engineering,2004,17(2):205—209.
[11]Moore R,Kearfott R,Cloud M.Introduction to Interval Analysis[M].Society for Industrial and Applied Mathematics,Philadelphia,2009.
[12]Rao S S,Berke L.Analysis of uncertain structural systems using interval analysis[J].AIAA Journal,1997 35(4):727—735.
[13]Li S L,Li H,Ou J P.Model updating for uncertain structures with interval parameters[A].Asia-Pacific Workshop on Structural Health Monitoring[C].Yokohama,Japan,2006.
[14]Jiang C,Han X,Liu G.R.Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J].Computer Methods in Applied Mechanics and Engineering,2007,196:4 791—4 800.
[15]Dong W,Shah H.Vertex method for computing functions of fuzzy variables[J].Fuzzy Sets and Systems,1987,24(1):65—78.
[16]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):297—312.
[17]Khodaparast H H,Mottershead J E,Badcock K J.Interval model updating with irreducible uncertainty using the Kriging predictor[J].Mechanical Systems and Signal Processing,2011,25(4):1 204—1 226.
[18]Myers R H,Montgomery D C,Anderson-Cook C M.Response Surface Methodology:Process and Product Optimization Using Designed Experiments[M].3rd Edition,New York:John Wiley&Sons,2009.
[19]Cundy A L.Use of response surface metamodels in damage identification of dynamic structures[D].Virginia Polytechnic Institute and State University,2002.
[20]Fang S E,Perera R.A response surface methodology based damage identification technique[J].Smart Materials and Structures,2009,18(6):065009.
[21]Fang S E,Perera R.Damage identification by response surface model updating using D-optimal design[J].Mechanical Systems and Signal Processing,2011,25(2):717—733.
[22]Fox R,Kapoor M.Rates of change of eigenvalues and eigenvectors[J].AIAA Journal,1968,6(12):2 426—2 429.
[2]Fang S E,Perera R,Roeck G D.Damage identification of a reinforced concrete frame by finite element model updating using damage parameterization[J].Journal of Sound and Vibration,2008,313:544—559.
[3]Alvin K,Oberkampf W,Diegert K,et al.Uncertainty quantification in computational structural dynamics:a new paradigm for model validation[A].Proceedings of the 16th International Modal Analysis Conference[C],1998:1 191—1 197.
[4]冯新,李国强,周晶.土木工程结构健康诊断中的统计识别方法综述[J].地震工程与工程振动,2005,25(2):105—113.Feng Xin,Li Guoqiang,Zhou Jing.State-of-the-art of statistical identification for structural health diagnosis in civil engineering[J].Earthquake Engineering and Engineering Vibration,2005,25(2):105—113.
[5]Moens D.A non-probabilistic finite element approach for structural dynamic analysis with uncertain parameters[D].Catholic University of Leuven,Belgium,2002.
[6]Friswell M I.The adjustment of structural parameters using a minimum variance estimator[J].Mechanical Systems and Signal Processing,1989,3(2):143—155.
[7]Beck J L,Katafygiotis L S.Updating models and their uncertainties.I:Bayesian statistical framework[J].Journal of Engineering Mechanics.1998,124(4):455—461.
[8]韩芳,钟冬望,汪君.基于贝叶斯法的复杂有限元模型修正研究[J].振动与冲击,2012,31(1):39—43.HAN Fang,ZHONG Dongwang,WANG Jun.Complicated finite element model updating based on Bayesian method[J].Journal of Vibration and Shock,2012,31(1):39—43.
[9]Khodaparast H H,Mottershead J E,Friswell M I.Perturbation methods for the estimation of parameter variability in stochastic model updating[J].Mechanical Systems and Signal Processing,2008,22(8):1 751—1 773.
[10]王登刚,秦仙蓉.结构计算模型修正的区间反演方法[J].振动工程学报,2004,17(2):205—209.Wang Denggang,Qin Xianrong.Interval method for computational model updating of dynamic structures[J].Journal of Vibration Engineering,2004,17(2):205—209.
[11]Moore R,Kearfott R,Cloud M.Introduction to Interval Analysis[M].Society for Industrial and Applied Mathematics,Philadelphia,2009.
[12]Rao S S,Berke L.Analysis of uncertain structural systems using interval analysis[J].AIAA Journal,1997 35(4):727—735.
[13]Li S L,Li H,Ou J P.Model updating for uncertain structures with interval parameters[A].Asia-Pacific Workshop on Structural Health Monitoring[C].Yokohama,Japan,2006.
[14]Jiang C,Han X,Liu G.R.Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval[J].Computer Methods in Applied Mechanics and Engineering,2007,196:4 791—4 800.
[15]Dong W,Shah H.Vertex method for computing functions of fuzzy variables[J].Fuzzy Sets and Systems,1987,24(1):65—78.
[16]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):297—312.
[17]Khodaparast H H,Mottershead J E,Badcock K J.Interval model updating with irreducible uncertainty using the Kriging predictor[J].Mechanical Systems and Signal Processing,2011,25(4):1 204—1 226.
[18]Myers R H,Montgomery D C,Anderson-Cook C M.Response Surface Methodology:Process and Product Optimization Using Designed Experiments[M].3rd Edition,New York:John Wiley&Sons,2009.
[19]Cundy A L.Use of response surface metamodels in damage identification of dynamic structures[D].Virginia Polytechnic Institute and State University,2002.
[20]Fang S E,Perera R.A response surface methodology based damage identification technique[J].Smart Materials and Structures,2009,18(6):065009.
[21]Fang S E,Perera R.Damage identification by response surface model updating using D-optimal design[J].Mechanical Systems and Signal Processing,2011,25(2):717—733.
[22]Fox R,Kapoor M.Rates of change of eigenvalues and eigenvectors[J].AIAA Journal,1968,6(12):2 426—2 429.
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