摘要
常规位移有限元的结构振动方程是n个二阶常微分方程组。采用一般变分原理推导,将结构振动问题引入Hamilton体系,将得到2n个一阶常微分方程组。精细积分法宜于处理一阶方程,应用于线性定常结构动力问题求解,可以得到在数值上逼近精确解的结果。对于非齐次动力方程,当结构具有刚体位移时,系统矩阵将出现奇异。本文借鉴全元选大元高斯-约当法求解线性方程组的经验,提出全元选大元法求奇异矩阵零本征解的方法,该方法可以简便快速地寻求奇异矩阵零本征值对应的子空间。利用Hamilton体系已有研究成果及Hamilton系统的共轭辛正交归一关系,迅速将零本征值对应的子空间分离出来,通过投影排除奇异部分,然后用精细积分法求得问题的解。数值算例表明,该方法对Hamilton系统奇异问题,处理方便,计算量小,易于实现,同时保持了精细算法的优点。
There are n second-order ordinary differential equations （ODE） for structural dynamics of finite element method. It＇s obtained 2n first-order ODEs when the dynamic system is introduced into Hamilton system through the principle of general variation. The precise integration method is good for solving ODEs. It can give precise numerical results approaching to the exact solution at the integration points when it is applied to linear time-invariant dynamic system. The system matrix will be singular when the structure has rigid body displacement for non-homogeneous dynamic systems. A new method named complete pivot Gauss-Jordan elimination is proposed. It is used to derive to zero eigen-solutions for singular matrix. Based on this method, it is easy to separate the subspace corresponding to zero eigen-solutions from singular Hamilton matrix by using conjugate sympletic orthogonal normalization between Hamilton eigen-vectors. Then the singular portion can be excluded through projection. The singular solution is derived from analysis. The nonsingular solution is derived from the precise integration method. The numerical result demonstrates the validity and efficiency of the method.
出处
《计算力学学报》
EI
CAS
CSCD
北大核心
2009年第1期46-51,共6页
Chinese Journal of Computational Mechanics
基金
国家自然科学重点基金(10632032)
国家自然科学基金(10672100)资助项目
关键词
精细积分
Hamilton奇异矩阵
共轭辛正交
非齐次方程
precise integration method
singular Hamilton matrix
conjugate sympletic orthogonal normalization~ non-homogeneous dynamic systems
complete pivot gauss-jordan elimination
