基于Bathe隐式算法的结构动力学显式算法
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摘要
提出了一种新的计算时具有结构动力特性相关性的无条件稳定的结构动力学时间积分算法。该算法不仅位移与速度均具有显式表达的特点,而且具有了Bathe复合时间积分方案的优点。该新算法的数值特征与Bathe隐式复合积分算法的数值特性相同,但新算法不需要任何时间步长内的子步细分,相对于新提出的算法而言,时间步长内的子步细分成了Bathe隐式复合积分算法的缺点。为了进一步了解所提出的新方法的谱特性,对新算法的稳定性和精度进行了全面的分析,包括数值耗散和色散。此外,当采用提出的算法计算多自由度系统时,给出了两个积分参数的推导过程和表达式。通过计算线性和非线性问题并与采用现有算法的结果比较,验证了新算法的正确性和有效性。
A new structure-dependent unconditionally stable time-integration method was presented for structural dynamic analysis. The proposed method not only benefits from an explicit formulation, but also inherits the advantage of the Bathe composite scheme. The numerical characteristics of the proposed algorithm are the same as those in the Bathe composite scheme, except that the suggested method does not require any time-subdividing, which is one of the drawbacks of the composite scheme. A comprehensive stability and accuracy analysis, including dissipation and dispersion, was carried out in order to gain an insight into the spectral properties of the proposed method. Also, when the proposed algorithm was used to analyse the multi degrees of freedom system, the derivation and expression of the two integral parameters were given. Finally, the correctness and effectiveness of the proposed algorithm was verified by comparing the results of linear and nonlinear problems calculated by the suggested method with those calculated by other existing algorithms.
A new structure-dependent unconditionally stable time-integration method was presented for structural dynamic analysis. The proposed method not only benefits from an explicit formulation, but also inherits the advantage of the Bathe composite scheme. The numerical characteristics of the proposed algorithm are the same as those in the Bathe composite scheme, except that the suggested method does not require any time-subdividing, which is one of the drawbacks of the composite scheme. A comprehensive stability and accuracy analysis, including dissipation and dispersion, was carried out in order to gain an insight into the spectral properties of the proposed method. Also, when the proposed algorithm was used to analyse the multi degrees of freedom system, the derivation and expression of the two integral parameters were given. Finally, the correctness and effectiveness of the proposed algorithm was verified by comparing the results of linear and nonlinear problems calculated by the suggested method with those calculated by other existing algorithms.
引文
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[2]WILSON E L.A computer program for the dynamic stress analysis of underground structures[D].Oakland:University of Califormia,1968.
[3]BATHE K J,BAIG M M I.On a composite implicit time integration procedure for nonlinear dynamics[J].Computers&Structures,2005,83(31/32):2513-2524.
[4]ZHAI W M.Two simple fast integration methods for largescale dynamic problems in engineering[J].International Journal for Numerical Methods in Engineering,1996,39(24):4199-4214.
[5]邢誉峰,郭静.与结构动特性协同的自适应Newmark方法[J].力学学报,2012,44(5):904-911.XING Yufeng,GUO Jing.A self-adaptive Newmark method with parameters dependent upon structural dynamic characteristics[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(5):904-911.
[6]CHANG S Y.Explicit pseudodynamic algorithm with unconditional stability[J].Journal of Engineering Mechanics,2002,128(9):935-947.
[7]CHANG S Y.An explicit method with improved stability property[J].International Journal for Numerical Methods in Engineering,2009,77(8):1100-1120.
[8]CHANG S Y,YANG Y S,HSU C W.A family of explicit algorithms for general pseudodynamic testing[J].Earthquake Engineering and Engineering Vibration,2011,10(1):51-64.
[9]CHANG S Y.Comparisons of structure-dependent explicit methods for time integration[J].International Journal of Structural Stability and Dynamics,2015,15(3):1450055.
[10]CHEN C,RICLES J M.Development of direct integration algorithms for structural dynamics using discrete control theory[J].Journal of Engineering Mechanics,2008,134(8):676-683.
[11]CHEN C,RICLES J M.Stability analysis of direct integration algorithms applied to MDOF nonlinear structural dynamics[J].Journal of Engineering Mechanics,2009,136(4):485-495.
[12]DORF R C,BISHOP R H.Modern control systems[M].9th ed.New Jersey:Prentice-Hall,2001.
[13]GUI Y,WANG J T,JIN F,et al.Development of a family of explicit algorithms for structural dynamics with unconditional stability[J].Nonlinear Dynamics,2014,77(4):1157-1170.
[14]杜晓琼,杨迪雄,赵永亮.一种无条件稳定的结构动力学显式算法[J].力学学报,2015,47(2):310-319.DU Xiaoqiong,YANG Dixiong,ZHAO Yongliang.An unconditionally stable explicit algorithm for structural dynamics[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):310-319.
[15]REZAIEE-PAJAND M,HASHEMIAN M.Time integration method based on discrete transfer function[J].International Journal of Structural Stability and Dynamics,2016,16(5):1550009.
[16]CHANG S Y.Elimination of overshoot in forced vibration responses for chang explicit family methods[J].Journal of Engineering Mechanics,2017,144(2):04017177.
[17]NAMADCHI A H,FATTAHI F,ALAMATIAN J.Semiexplicit unconditionally stable time integration for dynamic analysis based on composite scheme[J].Journal of Engineering Mechanics,2017,143(10):04017119.
[18]CHEN C,RICLES J M.Large-scale real-time hybrid simulation involving multiple experimental substructures and adaptive actuator delay compensation[J].Earthquake Engineering&Structural Dynamics,2012,41(3):549-569.
[19]CHEN C,RICLES J M.Improving the inverse compensation method for real-time hybrid simulation through a dual compensation scheme[J].Earthquake Engineering&Structural Dynamics,2009,38(10):1237-1255.
[20]孟凡涛,赵建锋,于广明.实时子结构混合试验中的数值积分方法对比分析[J].地震工程与工程振动,2011,31(5):60-67.MENG Fantao,ZHAO Jianfeng,YU Guangming.Study on numerical integration methods in realtime hybrid testing experiment[J].Earthquake Engineering and Engineering Dynamics,2011,31(5):60-67.
[21]赵建锋,孟凡涛,于广明.实时子结构混合试验中的Chang方法研究[J].地震工程与工程振动,2014,34(1):37-43.ZHAO Jianfeng,MENG Fantao,YU Guangming.Study on Chang method in real-time substructure hybrid testing[J].Earthquake Engineering and Engineering Dynamics,2014,34(1):37-43.
[22]YU K P.A new family of generalized-αtime integration algorithms without overshoot for structural dynamics[J].Earthquake Engineering&Structural Dynamics,2008,37(12):1389-1409.
[23]KOLAY C,RICLES J M.Improved explicit integration algorithms for structural dynamic analysis with unconditional stability and controllable numerical dissipation[J].Journal of Earthquake Engineering,2017(3):1-22.
[24]CHANG S Y.Unusual overshooting in steady-state response for structure-dependent integration methods[J].Journal of Earthquake Engineering,2017,21(8):1220-1233.
[2]WILSON E L.A computer program for the dynamic stress analysis of underground structures[D].Oakland:University of Califormia,1968.
[3]BATHE K J,BAIG M M I.On a composite implicit time integration procedure for nonlinear dynamics[J].Computers&Structures,2005,83(31/32):2513-2524.
[4]ZHAI W M.Two simple fast integration methods for largescale dynamic problems in engineering[J].International Journal for Numerical Methods in Engineering,1996,39(24):4199-4214.
[5]邢誉峰,郭静.与结构动特性协同的自适应Newmark方法[J].力学学报,2012,44(5):904-911.XING Yufeng,GUO Jing.A self-adaptive Newmark method with parameters dependent upon structural dynamic characteristics[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(5):904-911.
[6]CHANG S Y.Explicit pseudodynamic algorithm with unconditional stability[J].Journal of Engineering Mechanics,2002,128(9):935-947.
[7]CHANG S Y.An explicit method with improved stability property[J].International Journal for Numerical Methods in Engineering,2009,77(8):1100-1120.
[8]CHANG S Y,YANG Y S,HSU C W.A family of explicit algorithms for general pseudodynamic testing[J].Earthquake Engineering and Engineering Vibration,2011,10(1):51-64.
[9]CHANG S Y.Comparisons of structure-dependent explicit methods for time integration[J].International Journal of Structural Stability and Dynamics,2015,15(3):1450055.
[10]CHEN C,RICLES J M.Development of direct integration algorithms for structural dynamics using discrete control theory[J].Journal of Engineering Mechanics,2008,134(8):676-683.
[11]CHEN C,RICLES J M.Stability analysis of direct integration algorithms applied to MDOF nonlinear structural dynamics[J].Journal of Engineering Mechanics,2009,136(4):485-495.
[12]DORF R C,BISHOP R H.Modern control systems[M].9th ed.New Jersey:Prentice-Hall,2001.
[13]GUI Y,WANG J T,JIN F,et al.Development of a family of explicit algorithms for structural dynamics with unconditional stability[J].Nonlinear Dynamics,2014,77(4):1157-1170.
[14]杜晓琼,杨迪雄,赵永亮.一种无条件稳定的结构动力学显式算法[J].力学学报,2015,47(2):310-319.DU Xiaoqiong,YANG Dixiong,ZHAO Yongliang.An unconditionally stable explicit algorithm for structural dynamics[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):310-319.
[15]REZAIEE-PAJAND M,HASHEMIAN M.Time integration method based on discrete transfer function[J].International Journal of Structural Stability and Dynamics,2016,16(5):1550009.
[16]CHANG S Y.Elimination of overshoot in forced vibration responses for chang explicit family methods[J].Journal of Engineering Mechanics,2017,144(2):04017177.
[17]NAMADCHI A H,FATTAHI F,ALAMATIAN J.Semiexplicit unconditionally stable time integration for dynamic analysis based on composite scheme[J].Journal of Engineering Mechanics,2017,143(10):04017119.
[18]CHEN C,RICLES J M.Large-scale real-time hybrid simulation involving multiple experimental substructures and adaptive actuator delay compensation[J].Earthquake Engineering&Structural Dynamics,2012,41(3):549-569.
[19]CHEN C,RICLES J M.Improving the inverse compensation method for real-time hybrid simulation through a dual compensation scheme[J].Earthquake Engineering&Structural Dynamics,2009,38(10):1237-1255.
[20]孟凡涛,赵建锋,于广明.实时子结构混合试验中的数值积分方法对比分析[J].地震工程与工程振动,2011,31(5):60-67.MENG Fantao,ZHAO Jianfeng,YU Guangming.Study on numerical integration methods in realtime hybrid testing experiment[J].Earthquake Engineering and Engineering Dynamics,2011,31(5):60-67.
[21]赵建锋,孟凡涛,于广明.实时子结构混合试验中的Chang方法研究[J].地震工程与工程振动,2014,34(1):37-43.ZHAO Jianfeng,MENG Fantao,YU Guangming.Study on Chang method in real-time substructure hybrid testing[J].Earthquake Engineering and Engineering Dynamics,2014,34(1):37-43.
[22]YU K P.A new family of generalized-αtime integration algorithms without overshoot for structural dynamics[J].Earthquake Engineering&Structural Dynamics,2008,37(12):1389-1409.
[23]KOLAY C,RICLES J M.Improved explicit integration algorithms for structural dynamic analysis with unconditional stability and controllable numerical dissipation[J].Journal of Earthquake Engineering,2017(3):1-22.
[24]CHANG S Y.Unusual overshooting in steady-state response for structure-dependent integration methods[J].Journal of Earthquake Engineering,2017,21(8):1220-1233.
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