网架结构动力分析的三次样条辛算法
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摘要
根据对偶互补的思想,建立了网架结构动力学的相空间非传统Hamilton型变分原理。这种变分原理不仅能反映这种动力学初值-边值问题的全部特征,而且它的欧拉方程具有辛结构。基于该变分原理,空间域采用有限元法与时间子域采用三次样条函数插值的时间子域法相结合,构造了求解网架结构动力响应的一种辛算法,给出了逐步递推计算格式。数值算例结果表明,这种新方法的稳定性、计算精度和效率都明显高于Wilson-θ法和Newmark-β法。
According to the basic idea of dual-complementarity,the unconventional Hamilton-type variational principle in phase space for dynamic analysis of space truss structure was introduced,which can fully characterize this kind of dynamic initial-boundary-value problems.In addition,its Euler equation is of symplectic structure character.Based on this vairiational principle,a symplectic algorithm was presented,combining the finite element method in space domain with the time subdomain method,in which the cubic spline interpolation was applied as approximation.The results of numerical examples show that the method is a highly efficient method with better computational performance and superior ability of stability compared with Wilson-θ and Newmark-β methods.
According to the basic idea of dual-complementarity,the unconventional Hamilton-type variational principle in phase space for dynamic analysis of space truss structure was introduced,which can fully characterize this kind of dynamic initial-boundary-value problems.In addition,its Euler equation is of symplectic structure character.Based on this vairiational principle,a symplectic algorithm was presented,combining the finite element method in space domain with the time subdomain method,in which the cubic spline interpolation was applied as approximation.The results of numerical examples show that the method is a highly efficient method with better computational performance and superior ability of stability compared with Wilson-θ and Newmark-β methods.
引文
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[10]富明慧,廖子菊,刘祚秋.结构动力方程的样条精细积分法[J].计算力学学报,2009,26(3):379-384.FU Ming-hui,LIAO Zi-ju,LIU Zuo-qiu.Spline precise time-integration of structural dyanmic analysis[J].Chinese Journal of Computational Mechanics,2009,26(3):379-384.
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[13]张毅刚,薛素铎,杨庆山.大跨度空间结构[M].北京:机械工业出版社,2005.
[2]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 Wan-xie.On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology,1994,34(2):131-136.
[3]钟万勰.子域精细积分及偏微分方程数值解[J].计算结构力学及其应用,1995,12(3):253-260.ZHONG Wan-xie.Subdomain precise integration method and numerical solution of partial differential equations[J].Computational Structural Mechanics and Applications,1995,12(3):253-260.
[4]顾元宪,陈飙松,张洪武.结构动力方程的增维精细积分法[J].力学学报,2000,32(4):447-456.GU Yuan-xian,CHEN Biao-song,ZHANG Hong-wu.Precise time-interration with dimension expanding method[J].Chinese Journal of Theoretical and Applied Mechanics,2000,32(4):447-456.
[5]张森文,曹开彬.计算结构动力响应的状态方程直接积分法[J].计算力学学报,2000,17(1):94-97.ZHANG Sen-wen,CAO Kai-bin.Direct integration of state equation method for dynamic response of structure[J].Chinese Journal of Computational Mechanics,2000,17(1):94-97.
[6]Wang M F,Zhou X Y.Modified precise time step integration method of structural dyanmic analysis[J].Earthquaske Engineering and Engineering Vibration,2005,4(2):287-293.
[7]王元丰,储德文.精细时程积分法在地震响应分析中的应用[J].土木工程学报,2004,37(1):20-23.WANG Yuan-feng,CHU De-wen.Application of precise time integration method for structural earthquake response[J].China Civil Engineering Journal,2004,37(1):20-23.
[8]郭泽英,李青宁.结构地震反应分析的Newmark精细耦合级数显式方法[J].工程力学,2009,26(1):204-208.GUO Ze-ying,LI Qing-ning.Explicit series solutions of coupled newmark and precise time step integration method for structural seismic response analysis[J].Engineering Mechanics,2009,26(1):204-208.
[9]高小科,邓子辰,黄永安.基于三次样条插值的精细积分法[J].振动与冲击,2007,26(9):75-77.GAO Xiao-ke,DENG Zi-chen,HUANG Yong-an.A high precise direct integration base on cubic spline interpolation[J].Journal of Vibration and Shock,2007,26(9):75-77.
[10]富明慧,廖子菊,刘祚秋.结构动力方程的样条精细积分法[J].计算力学学报,2009,26(3):379-384.FU Ming-hui,LIAO Zi-ju,LIU Zuo-qiu.Spline precise time-integration of structural dyanmic analysis[J].Chinese Journal of Computational Mechanics,2009,26(3):379-384.
[11]Luo E,Huang W J,Zhang H X.Unconventional Hamilton-type variational principles in phase space and symplectic algorithm[J].Science in China,Ser.G,2003,46(3):248-258.
[12]Finlayson B A,Scriven L E.On the search for variational principles[J].International Jouranal Heat and Mass Transfer,1967,10(6):799-821.
[13]张毅刚,薛素铎,杨庆山.大跨度空间结构[M].北京:机械工业出版社,2005.
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