基于动力系统特性和群理论的一维周期结构瞬态响应的高效算法

An Efficient Algorithm Based on Dynamic System Properties and Group Theory for Transient Responses of 1D Periodic Structures

基于凝聚技术、周期结构的动力特性和群理论,提出了一种求解一维周期结构瞬态响应的高效数值算法.高效求解线性方程是动力响应求解过程中的关键问题.基于结构的周期特性和凝聚技术,减小结构对应线性方程的规模.利用周期结构动力系统中线性方程的特性,证明了在给定时间步长内,作用在某个单胞的外力只会对临近的有限个单胞产生影响.基于这个性质,一维周期结构动力响应的求解可转换为一系列小规模子结构的响应分析.进一步地,将小规模子结构的动力响应转化为循环周期结构的响应分析,而循环周期结构对应的线性方程可基于群理论高效求解.数值算例表明,该算法计算效率高且节省存储要求.

Based on the condensation technology,the dynamic periodic structure properties and the group theory,an efficient numerical method for computing the transient responses of 1 D periodic structures was proposed. Efficiently solving linear equations is an issue for computing the dynamic responses. Based on the periodic properties of the structure and with the condensation technology,the scale of the linear equation corresponding to the structure was reduced. By means of the properties of linear equations for dynamic periodic systems,it was proved that the force on any chosen unit cell can only influence a finite number of adjacent unit cells within a time step. Then,the dynamic response computation of 1 D periodic structures was converted into the computation of a series of small-scale substructures. Subsequently,the dynamic response computation of the substructures can be converted into the computation of the cyclic-periodic structures. Then,the cyclic-periodic structures were solved efficiently in light of the group theory. Numerical examples illustrate the high efficiency and memory saving of the proposed method.