砂土地震液化分析中Newmark时域离散的误差评估
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摘要
显式有限元存在计算精度低,对计算时间步长较敏感等缺点。基于后验误差评估的方法,给出了显式算法下Newmark时域离散误差的来源和评估方法。通过饱和砂土地震液化响应的数值算例评估了时间步长对计算结果的影响。数值分析结果表明:时间步长不同,结点位移和单元孔压的时程曲线明显不同,同时计算耗时也呈双曲线关系;相对误差主要分布在变形较大的区域,全域平均相对误差在动力响应剧烈的时间段内较大。通过对计算时间步长和离散误差的评估,可有效恰当的对计算时间步长进行取值,也为自动步长调整提供了依据。
The disadvantage of the explicit finite element method is low accuracy and sensitive to the time step size.In this paper,a posteriori error evaluation method was introduced for the explicit Newmark scheme,giving the source and estimation method of the Newmark temporal discretization error. Then,a numerical example of saturated sand liquefaction in earthquake was conducted to evaluate the influence of time step size on the calculation result.The results showed that the time step size has effects on the time histories of node displacements and element pore's water pressures. Moreover,the relation between the computation cost and the time step size is a hyperbolic curve.The relative error is mainly generated in the area with large deformation and the period with rapid dynamic response.The mean relative error in the entire area is larger in the time range with violent dynamic response. Through the estimated discrete error,a proper time step size can be determined to meet the desired accuracy. For automatic time step adjustment,the error estimation can provide a criterion to control the time step size.
The disadvantage of the explicit finite element method is low accuracy and sensitive to the time step size.In this paper,a posteriori error evaluation method was introduced for the explicit Newmark scheme,giving the source and estimation method of the Newmark temporal discretization error. Then,a numerical example of saturated sand liquefaction in earthquake was conducted to evaluate the influence of time step size on the calculation result.The results showed that the time step size has effects on the time histories of node displacements and element pore's water pressures. Moreover,the relation between the computation cost and the time step size is a hyperbolic curve.The relative error is mainly generated in the area with large deformation and the period with rapid dynamic response.The mean relative error in the entire area is larger in the time range with violent dynamic response. Through the estimated discrete error,a proper time step size can be determined to meet the desired accuracy. For automatic time step adjustment,the error estimation can provide a criterion to control the time step size.
引文
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[2]HOUBOLT J C.A recurrence matrix solution for the dynamic response of elastic aircraft[J].Journal of the Aeronautical Sciences,1950,17(9):540-550.
[3]WILSON E L.Nonlinear dynamic analysis of complex structures[J].Earthquake Engineering and Structural Dynamics,1973,1:241-252.
[4]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,34(2):131-136.ZHONG Wanxie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,34(2):131-136.
[5]郭泽英,李青宁,张守军.结构地震反应分析的一种新精细积分法[J].工程力学,2007,24(4):35-40.GUO Zeying,LI Qingning,ZHANG Shoujun.A new precise integration method for structural seismic response analysis[J].Engineering Mechanics,2007,24(4):35-40.
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[12]IAI S.Three dimensional formulation and objectivity of a strain place multiple mechanism model for sand[J].Soils and Foundations,1993,33(1):192-199.
[13]汪明武.可液化场地土工抗震性能动态离心机试验与数值模拟[M].北京:科学出版社,2012:110-130.WANG Mingwu.Dynamic centrifuge tests and numerical modelling for geotechnical earthquake engineering in liquefiable soils[M].Beijing:Science Press,2012:110-130.
[14]TANG Xiaowei,SHAO Qi.Numerical simulation on seismic liquefaction by adaptive mesh renement due to two recoveredelds in error estimation[J].Soil Dynamics and Earthquake Engineering,2013,49:109-121.
[15]WANG Mingwu,CHEN Guangyi,IAI S.Seismic performances of dyke on liquefiable soils[J].Journal of Rock Mechanics and Geotechnical Engineering,2013,5(4):294-305.
[16]ZHANG Xiwen,TANG Xiaowei,SHAO Qi,et al.The uplift behavior of large underground structures in liquefied field[J].Applied Mechanics and Materials,2011,90-93:2112-2118.
[17]兰景岩,刘红帅,吕悦军.渤海土类动力非线性参数及合理性[J].哈尔滨工程大学学报,2012,33(9):1079-1085.LAN Jingyan,LIU Hongshuai,LYU Yuejun.Dynamic nonlinear parameters of soil in the Bohai Sea and their rationality[J].Journal of Harbin Engineering University,2012,33(9):1079-1085.
[18]AKAI K,TAMURA T.Numerical analysis of multi-dimensional consolidation accompanied with elastic-plastic constitutive equation[C]//Proceedings of the Society of Civil Engineering,1978,269(3):95-104.
[19]OKA F,YASHIMA A,SHIBATA T,et al.FEM-FDM coupled liquefaction analysis of a porous soil using an elasticplastic model[J].Applied Scientific Reasearch 1994,52:209-245.
[20]OKA F,YASHIMA A,TATEISHI A,et al.A cyclic elasto-plastic constitutive model for sand considering a plasticstrain dependence of the shear modulus[J].Geotechnique,1999,49(5):661-680.
[21]UZUOKA R.Analytical study on the mechanical behavior and prediction of soil liquefaction andow[D].Kyoto:Kyoto University,2000:47-77.
[2]HOUBOLT J C.A recurrence matrix solution for the dynamic response of elastic aircraft[J].Journal of the Aeronautical Sciences,1950,17(9):540-550.
[3]WILSON E L.Nonlinear dynamic analysis of complex structures[J].Earthquake Engineering and Structural Dynamics,1973,1:241-252.
[4]钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,34(2):131-136.ZHONG Wanxie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,34(2):131-136.
[5]郭泽英,李青宁,张守军.结构地震反应分析的一种新精细积分法[J].工程力学,2007,24(4):35-40.GUO Zeying,LI Qingning,ZHANG Shoujun.A new precise integration method for structural seismic response analysis[J].Engineering Mechanics,2007,24(4):35-40.
[6]李爽,翟长海,刘洪波,等.Newmark积分方法在负刚度时的数值稳定性分析[J].哈尔滨工业大学学报,2011,43(8):1-5.LI Shuang,ZHAI Changhai,LIU Hongbo et al.On the stability of Newmark integration method for structure with negative stiffness[J].Journal of Harbin Institute of Technology,2011,43(8):1-5.
[7]ZIENKIEWICZ O C,XIE Y M.A simple error estimator and adaptive time stepping procedure for dynamic analysis[J].Earthquake Engineering&Structural Dynamics,1991,20(9):871-887.
[8]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&Structural Dynamics,1992,21(7):555-571.
[9]WIBERG N E,LI X D.A post-processing technique and an a posteriori error estimate for the newmark method in dynamic analysis[J].Earthquake Engineering&Structural Dynamics,1993,22(6):465-489.
[10]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.
[11]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.NGUYEN Quanghung,FU Shaojun,CHEN Shenghong.Study on adaptive time step of consolidation problems by finite element method[J].Rock and Soil Mechanics,2005,26(4):591-599.
[12]IAI S.Three dimensional formulation and objectivity of a strain place multiple mechanism model for sand[J].Soils and Foundations,1993,33(1):192-199.
[13]汪明武.可液化场地土工抗震性能动态离心机试验与数值模拟[M].北京:科学出版社,2012:110-130.WANG Mingwu.Dynamic centrifuge tests and numerical modelling for geotechnical earthquake engineering in liquefiable soils[M].Beijing:Science Press,2012:110-130.
[14]TANG Xiaowei,SHAO Qi.Numerical simulation on seismic liquefaction by adaptive mesh renement due to two recoveredelds in error estimation[J].Soil Dynamics and Earthquake Engineering,2013,49:109-121.
[15]WANG Mingwu,CHEN Guangyi,IAI S.Seismic performances of dyke on liquefiable soils[J].Journal of Rock Mechanics and Geotechnical Engineering,2013,5(4):294-305.
[16]ZHANG Xiwen,TANG Xiaowei,SHAO Qi,et al.The uplift behavior of large underground structures in liquefied field[J].Applied Mechanics and Materials,2011,90-93:2112-2118.
[17]兰景岩,刘红帅,吕悦军.渤海土类动力非线性参数及合理性[J].哈尔滨工程大学学报,2012,33(9):1079-1085.LAN Jingyan,LIU Hongshuai,LYU Yuejun.Dynamic nonlinear parameters of soil in the Bohai Sea and their rationality[J].Journal of Harbin Engineering University,2012,33(9):1079-1085.
[18]AKAI K,TAMURA T.Numerical analysis of multi-dimensional consolidation accompanied with elastic-plastic constitutive equation[C]//Proceedings of the Society of Civil Engineering,1978,269(3):95-104.
[19]OKA F,YASHIMA A,SHIBATA T,et al.FEM-FDM coupled liquefaction analysis of a porous soil using an elasticplastic model[J].Applied Scientific Reasearch 1994,52:209-245.
[20]OKA F,YASHIMA A,TATEISHI A,et al.A cyclic elasto-plastic constitutive model for sand considering a plasticstrain dependence of the shear modulus[J].Geotechnique,1999,49(5):661-680.
[21]UZUOKA R.Analytical study on the mechanical behavior and prediction of soil liquefaction andow[D].Kyoto:Kyoto University,2000:47-77.
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