自适应时间步长法及三维堤坝地震液化数值模拟
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摘要
三维大模型数值计算因巨大的单元和结点数目而非常耗时,在地震响应分析中受计算时间步长的限值则更加耗时。在饱和砂土动力液化计算平台上开发时域离散误差评估方法和时间步长自适应调整的计算程序,并成功应用于三维堤坝地震液化响应分析。时域离散误差包括土骨架的位移误差和单元孔压误差,通过定义孔压误差影响系数计算出混合误差,根据混合误差和设定的误差允许值进行计算步长的自适应调整。在三维堤坝地震液化数值模拟中,采用自适应时间步长法有效避免小步长精确但耗时、大步长省时而不精确的缺点。在大模型和超大模型计算中,最优调整每一步的计算时间步长,完美实现既节省时间又不失精度的时域离散策略。
Three-dimensional(3D)numerical simulation of a large model is a time-consuming process due to the huge number of elements and nodes.This is particularly true for seismic behavior analysis in which the accuracy is limited by the size of the time increment.In this study,the temporal discretization error estimator and time increment adaptive adjustment method is proposed on the basis of the numerical platform for seismic liquefaction analysis of saturated soil.The proposed method is successfully applied in 3Dembankment seismic response analysis.The temporal discretization error estimation includes errors in soil skeleton displacement and pore water pressure.These two types of errors are combined to form a mixed error as an effect of pore water pressure error.The time increment is adaptively adjusted by the relationship of mixed error and the given error tolerance.Through a numerical example of 3Dembankment,it is determined that the adaptive time-stepping method can be used to improve the accuracy of small time increments.The results of this method are comparable to those obtained using highly efficient but time-consuming large time increments.
Three-dimensional(3D)numerical simulation of a large model is a time-consuming process due to the huge number of elements and nodes.This is particularly true for seismic behavior analysis in which the accuracy is limited by the size of the time increment.In this study,the temporal discretization error estimator and time increment adaptive adjustment method is proposed on the basis of the numerical platform for seismic liquefaction analysis of saturated soil.The proposed method is successfully applied in 3Dembankment seismic response analysis.The temporal discretization error estimation includes errors in soil skeleton displacement and pore water pressure.These two types of errors are combined to form a mixed error as an effect of pore water pressure error.The time increment is adaptively adjusted by the relationship of mixed error and the given error tolerance.Through a numerical example of 3Dembankment,it is determined that the adaptive time-stepping method can be used to improve the accuracy of small time increments.The results of this method are comparable to those obtained using highly efficient but time-consuming large time increments.
引文
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[22]Oka F,Yashima A,Tateishi A,et al.A Cyclic Elastic-plastic Constitutive Model for Sand Considering a Plastic-strain Dependence of the Shear Modulus[J].Geotechnique,1999,49(5):661-680.(in Chinese)
[2]Chen L W,Yuan X M,Cao Z Z,et al.Liquefaction Macrophenomena in the Great Wenchuan Earthquake[J].Earthquake Engineering and Engineering Vibration,2009,8(2):219-229.
[3]Ishihara K,Yasuda S,Yoshida Y.Liquefaction Induced Flow Failure of Embankment and Residual Strength of Silty Sand[J].Soils and Foundations,1990,30(3):69-80.
[4]Manzari M T.Finite Deformation Analysis of Liquefaction Induced Flow Failure in Soil Embankments[J].Geotechnical and Geological Engineering,1996,14:83-110.
[5]Elgamal A,Parra E,Yang Z,et al.Numerical Analysis of Embankment Foundation Liquefaction Countermeasures[J].Journal of Earthquake Engineering,2002,6(4):447-471.
[6]汪明武,Iai S,Tobita T.液化场地堤坝地震响应动态土工离心试验及模拟[J].水利学报,2008,39:1346-1352.WANG Ming-wu,Iai S,Tobita T.Centrifuge Test and Numerical Analysis of Seismic Responses of Dyke on Liquefiable Soils Foundation[J].Journal of Hydraulic Engineering,2008,39:1346-1352.(in Chinese)
[7]HuangY,Yashima A,Sawada K,et al.A Case Study of Seismic Response of Earth Embankment Foundation on Liquefiable Soils[J].Journal of Central South University of Technology,2009,16(6):994-1000.
[8]Babuska I,Rheinboloat W C.Error Estimates for Adaptive Finite Element Computations[J].Siam Journal on Numerical Analysis,1978,15(4):736-754.
[9]Zienkiewicz O C,Zhu J Z.A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis[J].International Journal for Numerical Methods in Engineering,1987,24:337-357.
[10]Bergan P G,Mollestad E.An Automatic Time Stepping Algorithm for Dynamic Problems[J].Computer Methods in Applied Mechanics and Engineering,1985,49(3):299-318.
[11]Zienkiewicz O C,Xie Y M.A Simple Error Estimator and Adaptive Time Stepping Procedure for Dynamic Analysis[J].Earthquake Engineering and Structure Dynamics,1991,20:871-887.
[12]Zeng L F,Wiberg N E,Li X D.A Posteriori Local Error Estimation and Adaptive Time-Stepping for Newmark Integration in Dynamic Analysis[J].Earthquake Engineering and Structure Dynamics,1992,21:555-571.
[13]Zhang Z H,Yang Z J,Liu G H.An Adaptive Time-Stepping Procedure Based on the Scaled Boundary Finite Element Method for Elastodynamics[J].International Journal of Computational Methods,2012,9(1):13.
[14]王怀忠,孙钧.能量的局部误差估计和直接积分法的时步自适应算法[J].岩土工程学报,1995,17(5):32-41.WANG Huai-zhong,SUN Jun.Local Error Estimator of Energy and Adaptive Time Stepping Procedure for Direct Integration Method[J].Chinese Journal of Geotechnical Engineering,1995,17(5):32-41.(in Chinese)
[15]冯领香,魏建国,王森林,等.一种可自调步长的改进newmark算法[J].河北农业大学学报,2004,27(3):111-114.FENG Ling-xiang,WEI Jian-guo,WANG Seng-lin.The Improved Newmark Algorithm With Adaptive Step Size[J].Journal of Agricultural University of Hebei,2004,27(3):111-114.(in Chinese)
[16]Hulbert G M,Jang I.Automatic Time Step Control Algorithms for Structural Dynamics[J].Computer Methods in Applied Mechanics and Engineering,1995,126:155-178.
[17]Tang G P,Alshawabkeh A N,Mayes M A.Automatic Time Stepping with Global Error Control for Groundwater Flow Models[J].Journal of Hydrologic Engineering,2008,13(9):803-810.
[18]Hirthe E M,Graf T.Non-Iterative Adaptive Time Stepping Scheme with Temporal Truncation Error Control for Simulating Variable-density Flow[J].Advances in Water Resources,2012,49:46-55.
[19]Sloan S W,Abbo A J.Biot Consolidation Analysis with Automatic Time Stepping and Error Control,Part 1:Theory and Implementation[J].International Journal for Numerical and Analytical Methods in Geomechanics,1999,23(6):467-492.
[20]Sloan S W,Abbo A J.Biot Consolidation Analysis with Automatic Time Stepping and Error Control,Part 2:Applications[J].International Journal for Numerical and Analytical Methods in Geomechanics,1999,23(6):493-529.
[21]阮光雄,傅少君,陈胜宏.固结问题有限元法时步自适应研究[J].岩土力学,2005,26(4):591-599.Ruan Guang-xiong,FU Shao-jun,CHEN Sheng-hong.Study on Adaptive Time Step of Consolidation Problems by Finite Element Method[J].Rock and Soil Mechanics,2005,26(4):591-599.(in Chinese)
[22]Oka F,Yashima A,Tateishi A,et al.A Cyclic Elastic-plastic Constitutive Model for Sand Considering a Plastic-strain Dependence of the Shear Modulus[J].Geotechnique,1999,49(5):661-680.(in Chinese)