基于配点法的谱随机有限元理论与应用研究
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摘要
基于配点法的谱随机有限元法(CSFEM)方法实现了随机分析和有限元分析的解耦,使确定性有限元分析可用作“黑箱”,一般需要的样本计算比蒙特卡罗法少。但CSFEM在处理高维问题时,其样本(配点)是基于张量积法的,配点数会随着维数增长呈几何级数增加,有时甚至比蒙特卡罗法所需的配点数还要多。
稀疏网格法提供了在多维空间中采用最少的点来构造插值函数的方法,而且这种方法的插值误差非常接近张量积法。本文研究了随机配点法的各种配点选取方法,详细介绍了稀疏网格法的基本原理和配点选取方法。并将稀疏网格法与CSFEM方法相结合,可以大量地减少高维问题所需的配点数,克服了配点数随维数呈几何级数增长的问题,使得CSFEM方法能够更好的处理高维问题。本文以混凝土框架结构以及单桩沉降随机响应为例,研究了CSFEM方法在结构工程和岩土工程中的应用。其中单桩沉降分析中用稀疏网格法来选取配点,并与采用张量积法选取配点得到的结果进行对比。结果表明,利用稀疏网格法来选取配点,只需要较之普通CSFEM方法少很多的配点就可以得到与其相接近的计算精度。
Collocation based stochastic finite element method (CSFEM) can uncouple the finite element analysis with stochastic analysis, which means the finite element code can be treated as a black box. It needs much fewer samples than the Monte Carlo method does. But when CSFEM is dealing with problems with high dimension, the number of collocation points will grow very fast, sometimes even more than Monte Carlo’s.
Sparse grid method uses fewer points to construct multidimensional interpretation functions with very small error. Several methods for selecting collocation points are introduced. The sparse grid method is illustrated in length. Combining CSFEM with the sparse grid method can reduce the number of collocation points in high dimension. It extends the applicability of CSFEM to problems with high dimension. Numerical examples involving a concrete frame and a single pile are studied. The sparse grid method is applied to the analysis of the settlement of single pile. Compared to the tensor product method for the collocation selection, the sparse grid method is more efficient. Results show that the sparse grid method save a lot of computational effort.
稀疏网格法提供了在多维空间中采用最少的点来构造插值函数的方法,而且这种方法的插值误差非常接近张量积法。本文研究了随机配点法的各种配点选取方法,详细介绍了稀疏网格法的基本原理和配点选取方法。并将稀疏网格法与CSFEM方法相结合,可以大量地减少高维问题所需的配点数,克服了配点数随维数呈几何级数增长的问题,使得CSFEM方法能够更好的处理高维问题。本文以混凝土框架结构以及单桩沉降随机响应为例,研究了CSFEM方法在结构工程和岩土工程中的应用。其中单桩沉降分析中用稀疏网格法来选取配点,并与采用张量积法选取配点得到的结果进行对比。结果表明,利用稀疏网格法来选取配点,只需要较之普通CSFEM方法少很多的配点就可以得到与其相接近的计算精度。
Collocation based stochastic finite element method (CSFEM) can uncouple the finite element analysis with stochastic analysis, which means the finite element code can be treated as a black box. It needs much fewer samples than the Monte Carlo method does. But when CSFEM is dealing with problems with high dimension, the number of collocation points will grow very fast, sometimes even more than Monte Carlo’s.
Sparse grid method uses fewer points to construct multidimensional interpretation functions with very small error. Several methods for selecting collocation points are introduced. The sparse grid method is illustrated in length. Combining CSFEM with the sparse grid method can reduce the number of collocation points in high dimension. It extends the applicability of CSFEM to problems with high dimension. Numerical examples involving a concrete frame and a single pile are studied. The sparse grid method is applied to the analysis of the settlement of single pile. Compared to the tensor product method for the collocation selection, the sparse grid method is more efficient. Results show that the sparse grid method save a lot of computational effort.
引文
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[3] Shinozuka.M.Monte Carlo solution of structural dynamics.Computers and Structures[J]. 1972,Volume 2.lssue 5-6,855-874.
[4] Cambou, Bernard. Applications of first-order uncertainty analysis in the finite elements method in linear elasticity[C].Proc.Second Int.Conf.on Applications of Statistics and Probability in Soil and Struct.Engng.London England.1971,117-122.
[5] Houstis.E.N.The complexity of numerical methods for elliptic partial differential equations[J].Journal of Computational and applied mathematic.1978,Vol 4,191-197.
[6] Ingra.T.S,Baecher.G.B.Uncertainty in bearing capacity of sands[J].Journal of Geotechnical Engineering,ASCE.1983,Vol 109,899-914.
[7] Handa K.Anderson K.Application of finite element methods in statixtical analysis of structures[C].Proc.3rd Int.Conf.on Struct.Safcty and Reliability Trondheim, Norway.1981, 409-417.
[8] Hisada T,Nakagiri S.Stochastic finite element method developed for struct.safety and reliability[C]. Proc.3rd Int.Conf.on Struct.Safcty and Reliability Trondheim, Norway.1981, 395-408.
[9] Nakagiri S,Hisada T.Stochastic finite element method applied to struct.analysis with uncertain parameters[C].Proc.Int.Conf.on FEM Australia.1982,206-211.
[10] Hisada T,Nakagiri S.Role of the stochastic finite element method in struct.safety and reliability[C].Proc.4th Int.Conf.on struct.Safaty and Reliability Kobe,Japan.1985,213-219.
[11] Hisada T,Nakagiri S.Stochastic finite element analysis of uncertain struct.system[C]. Proc.4th Int.Conf.on FEM.Australia.1982, 133-138.
[12] Shinozuka M,Deodatis G.Response variability of stochastic finite element systems[J].Engng.Mech.,ASCE.1988,114,3,499-519.
[13] Yamazaki F,Shinozuka M,Dasgupta G.Neumann expansion for stochastic finite element analysis[J]. Engng.Mech. ASCE.1988, 114, 8, 1335-1354.
[14] Kiureghian.Der.Stochastic finite element method in structural reliability[J].Probabilistic Engng.Mech.1988,3,2,83-91.
[15]吴世伟,李同春.重力坝最大可能失效模式初探[J].水利学报.1990(4):36-44.
[16]李同春,吴世伟.基于三维SFEM的拱坝可靠度分析[C].工程结构可靠性全国第二届学术交流会议论文集.重庆,1989:114-125.
[17]吴世伟等.拱坝的失效模式与可靠度[J].河海大学学报,1992,20,2:88-96.
[18] Takada T,Shinozuka M.Local integration method in stochastic finite element analysis[C].Proc.Int.Conf.Struct.Safety and Reliability.1989:1072-1080.
[19] Ghanem R,Spanos PD.Stochastic finite elements:A spectral approach[J].Springer-Verlag, 1991.
[20]黄淑萍.基于配点法的谱随机有限元分析-随机响应面法[J],计算力学学报.2005(2):173-180.
[21] Huang shuping.Simulation of Random Processes Using Karhunen-Loeve Expansion [Dissertation].Singapore: Nat Univ of Singapore, 2001.
[22] Huang SP,Liang,B,Phoon.K.K.Geotechnical probabilistic analysis by collocation-based stochastic response surface method-An EXCEL add-in implementation[J].Georisk,2009,3(2):75-86.
[23] Isukapalli,SS.Uncertainty analysis of transport-transformation models[Dissertation].New Jersey:The state university of New Jersey,1998.
[24] Berveiller,M.Elements finis stochastiques:approaches intrusive et nonintrusives pour des analyses de fiabilite[Dissertation].France:Universite blaise pascal-clermont ferrand,2005.
[25] Li,D.Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables[J].Comp.&Geotech,doi.2010,10(1016):10006.
[26] Liu P L,Kiureghian A D.Reliability of geometrically nonlinear structrures Probabilistic Methods in Civil Engng[J].ASCE.1988:164-168.
[27] Phoon.K.K.et al.Reliability analysis of pile settlement [J].J.Geotech.Engrg, ASCE, 1990, Vol.116.No.11:1717-1735.
[28]邹进彰.岩土工程中稳定随机分析及可靠度计算[M].武汉:武汉水利电力学院,1988.
[29]刘宁.基于三维弹塑性随机有限元的结构可靠度的计算[M].南京:河海大学,1994.
[30] Bazant Z P,Wang T S.Spectral finite element analysis of random shrinkage in concrete[J].Struct Engng ASCE.1984,(9):2196-2211.
[31] Liu W K,Besterfield G,Mani A.Probabilistic finite element methods in nonlinear dynamics[J].Computer Methods in applied Mech and Engng.1986,(57):61-81.
[32]吕泰仁,何慧莉,吴世伟.结构自振频率的统计特性[C].工程结构可靠性全国第三届学术交流会议.南京,1992:368-372.
[33]陈厚群,梁爱虎.重力坝动力可靠度的随机有限元分析方法[C].工程结构可靠性全国第三届学术交流会议.1992:324-331.
[34] Zhu W Q,Ren Y J,Wu W Q.Stochastic FEM based on local averages of random vector fields[J].Engng Mech ASCE.1992,(3):496-511.
[35]蒋鸣晓,任永坚,丁皓江.概率断裂中随机有限元法的应用[J].浙江大学学报.1995,29(1):53-59.
[36]于国安.盘式制动器制动力矩可靠性及制动力矩在线测量装置的研究[D].沈阳:东北大学,1996.
[37] Hong C S,Park J S and Kim C G.Stochastic finite element method and system reliabilityanalysis for laminated composite structrres Recent Advances in Solids and Structures[J].ASME.1995,(18):165-172.
[38] Zhang J,Ellingwood B.SFEM for reliability of structures with material nonlinearities[J].Struct Engng.1996,122(6):701-704.
[39] Gao L W,Haldar A.Nonlinear SFEM-based reliability for space structures[J].Structural Safety & Reliability Balkema,Rotterdam.1994:325-331.
[40]黄广龙.岩土工程中的不确定性及柔性桩沉降的可靠性分析[D].杭州:浙江大学,1998.
[41]马克生.柔性桩复合地基沉降可靠度分析[D].杭州:浙江大学,2000.
[42]傅旭东.单桩沉降可靠度的随机有限元分析[J].铁道学报.2002(1):70-75.
[43]王卿,黄淑萍.随机响应面法在单桩沉降可靠性分析中的研究[J].低温建筑技术.2008.2(40).
[44]黄淑萍,景鹏.单桩沉降可靠度的配点谱随机有限元分析[J].计算力学学报.(2010年9月录用).
[45] Baskar Ganapathysubramanian,Nicholas Zabaras.Sparse grid collocation schemes for stochastic natural convection problems[J].Journal of Computational Physics,2007,225,652-685.
[46] Smolyak.S.Quadrature and interpolation formulas for tensor products of certain classes of functions [J].Dokl.Akad.Nauk SSSR.Vol.4,1963,pp.240-243.
[47] Grzegorz W.Wasilkowski.Explicit cost bounds of algorithms for multivariate tensor productproblems [J].Journal of complexity.1995, 11, 1-56.
[48]刘次华.随机过程[M].武汉:华中科技大学,2000.
[49] Phoon.K.K,Huang SP.Applying wavelets to extension of Karhunen-Loeve expansion for simulation[J].Probabilistic Engineering mechanics.2002,17(3),293-303.
[50] Phoon.K.K,Huang SP.Simulation of non-Gaussian processes using Karhunen-Loeve Expansion[J].Computers & Structures.2002,80(16),1049-1060.
[51] Huang SP,S.T.Quek.Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes[J].International Journal for Numerical Mechods in Engineering.2001(52,1029-1043).
[52] Huang SP.Simulation of Random Processes using Karhunen-Loeve Expansion[Dissertation].Singapore,Nat Univ of Singapore,2001.
[53] Villadsen.J,Michelsen.Solution of differential equation models by polynomial approximation[M].Prentice-Hall,New Jersy,1978.
[54] Isukapalli SS,Roy A.Stochastic response surface methods(CSFEMS) for uncertainty propagation:application to environmental and biological systems[J].Risk analysis.1998,18(3),51-63.
[55] Huang SP,Mahadevan,S.Collocation-based stochasticfinite element analysis for random fieldproblems[J].Probabilistic Engineering Mechanics.2007,Vol.22,194-205.
[56] Xiu Dongbing,Jan S.Hesthaven.High-order collocation methods for differential equations with random inputs[J].SIAM J.SCI.COMPUT.2005,Vol.27.No.3,pp.1118-1139.
[57] Babuska.Ivo.A stochastic collocation method for elliptic partial differential equations with random input data[J].Numer.2007,Vol.45,No.3,PP.1005-1034.
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