结构动力学方程的显式积分格式
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摘要
本文从空间解耦有限元常微分方程组出发,探讨了结构动力学方程的高精度显式积分格式。通过被积函数的拉格朗日多项式内插和分部积分导出了波动数值模拟的一组显式时步积分公式。这组公式是时间和空间解耦的,即波场内任一离散节点在任一时刻的波动数据可以用这组公式依据该节点及其邻近节点在该时刻之前的n+1个时刻的波动数据显式地算出(n为非负整数),阐明了这组公式的如下特点:第一,其截断误差的量级不超过O(Δtn+3),Δt为时间步距。第二,它不仅可用于线性波动的数值模拟,而且可用于本构方程具有强非线性情形。第三,这组公式也可推广应用于一系列数学物理暂态问题的数值求解。针对一个简单的时不变系统初步分析了此组积分格式的稳定性。但是,对其稳定性尚需作进一步研究。
Starting from the spacial decoupling finite element ordinary differential equations,this paper explores the explicit time integration method for structural dynamic equations.A group of explicit schemes for the numerical simulation of wave motion are derived via the Lagrange polynomial interpolation and integration by parts;the schemes are decoupling both in time and space,which mean that the motion of a discrete nodal point at a time can be computed explicitly by the schemes in term of the data of motion of the point and its neighboring nodal points at n+1 moments before the time.The schemes have the following features.Firsty,their truncation error is limited to O(Δtn+3),where Δt is time step and n is a non-negative integer.Seconly,they are not only suitable for the numerical simulation of linear wave motion,but also applicable for the numerical simulation of nonlinear constitutive equations.Finally,the schemes can be generalized to solve a series of transient mathematic-physical problems.The stability of the schemes are investigated preliminary by a simple time-invariant system,however,the stability should be further researched.
Starting from the spacial decoupling finite element ordinary differential equations,this paper explores the explicit time integration method for structural dynamic equations.A group of explicit schemes for the numerical simulation of wave motion are derived via the Lagrange polynomial interpolation and integration by parts;the schemes are decoupling both in time and space,which mean that the motion of a discrete nodal point at a time can be computed explicitly by the schemes in term of the data of motion of the point and its neighboring nodal points at n+1 moments before the time.The schemes have the following features.Firsty,their truncation error is limited to O(Δtn+3),where Δt is time step and n is a non-negative integer.Seconly,they are not only suitable for the numerical simulation of linear wave motion,but also applicable for the numerical simulation of nonlinear constitutive equations.Finally,the schemes can be generalized to solve a series of transient mathematic-physical problems.The stability of the schemes are investigated preliminary by a simple time-invariant system,however,the stability should be further researched.
引文
[1]廖振鹏.工程波动理论导论[M].2版.北京,科学出版社,2003:136-140
[2]Chew W C.Waves and fields in inhomogeneous media[M].2nd ed.Piscataway,NJ,IEEE Press,1995:211-270.
[3]Chen Houqun,Du Xiuli,Hou Shunzi,Appliciation of transmitting boundaries to nonlinear dynamic ananlysis of an arch dam-foundation-reserviorsystem[M]//(edited by Zhang Chuhan and J.P.Wolf)Dynamic Soil-Structure Interaction.Beijing:International Academic Publishers,1997:115-124.
[4]廖振鹏,黄孔亮,杨柏坡,等.暂态波透射边界条件[J].中国科学(A辑),1984,26(6):556-564.
[5]廖振鹏.法向透射边界条件[J].中国科学(E辑),1996,26(2):186-192.
[6]Liao ZP.Extrapolation non-reflecting boundary conditions[J].Wave Motion,1996,24:117-138.
[7]廖振鹏.透射边界和无穷远辐射条件[J].中国科学(E辑),2001,31(2),254-261
[8]廖振鹏,刘晶波.波动有限元模拟的基本问题[J].中国科学(B辑),1992,8:874-882.
[9]廖振鹏,周正华,张艳红.波动数值模拟中透射边界的稳定实现[J].地球物理学报,2002,45(4):533-545.
[10]Wagner R L,Chew W C.An analysis of Liao’s Absorbing bounary condition[J].J.Electromag.Waves Appl.,1995,9(7):993-1009.
[11]李小军,廖振鹏,杜修力.有阻尼体系动力问题的一种显式差分解法[J]地震工程与工程振动,1992,12(4):74-80..
[12]钟万勰.暂态历程的精细计算方法[J].计算结构力学及其应用,1995,12(1):1-6.
[13]Taflove A,Susan C H.Computational electrodynamics:the finite-difference time-domain method[M].Boston:Artech House,2000:1-30.
[14]Young J L,Gaitonde D,Shang J S.Toward the construction of a fouth-order difference scheme for transient EM wave simulation;Staggered gridapproach[J].Antennas Propagat,IEEE Trans,1997,45:1573-1580.
[2]Chew W C.Waves and fields in inhomogeneous media[M].2nd ed.Piscataway,NJ,IEEE Press,1995:211-270.
[3]Chen Houqun,Du Xiuli,Hou Shunzi,Appliciation of transmitting boundaries to nonlinear dynamic ananlysis of an arch dam-foundation-reserviorsystem[M]//(edited by Zhang Chuhan and J.P.Wolf)Dynamic Soil-Structure Interaction.Beijing:International Academic Publishers,1997:115-124.
[4]廖振鹏,黄孔亮,杨柏坡,等.暂态波透射边界条件[J].中国科学(A辑),1984,26(6):556-564.
[5]廖振鹏.法向透射边界条件[J].中国科学(E辑),1996,26(2):186-192.
[6]Liao ZP.Extrapolation non-reflecting boundary conditions[J].Wave Motion,1996,24:117-138.
[7]廖振鹏.透射边界和无穷远辐射条件[J].中国科学(E辑),2001,31(2),254-261
[8]廖振鹏,刘晶波.波动有限元模拟的基本问题[J].中国科学(B辑),1992,8:874-882.
[9]廖振鹏,周正华,张艳红.波动数值模拟中透射边界的稳定实现[J].地球物理学报,2002,45(4):533-545.
[10]Wagner R L,Chew W C.An analysis of Liao’s Absorbing bounary condition[J].J.Electromag.Waves Appl.,1995,9(7):993-1009.
[11]李小军,廖振鹏,杜修力.有阻尼体系动力问题的一种显式差分解法[J]地震工程与工程振动,1992,12(4):74-80..
[12]钟万勰.暂态历程的精细计算方法[J].计算结构力学及其应用,1995,12(1):1-6.
[13]Taflove A,Susan C H.Computational electrodynamics:the finite-difference time-domain method[M].Boston:Artech House,2000:1-30.
[14]Young J L,Gaitonde D,Shang J S.Toward the construction of a fouth-order difference scheme for transient EM wave simulation;Staggered gridapproach[J].Antennas Propagat,IEEE Trans,1997,45:1573-1580.
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