【作者】 罗博文； 戴鸿哲；

【Author】 Luo Bowen;Dai Hongzhe;Harbin Institute of Technology;

【机构】 哈尔滨工业大学；

【摘要】 近年来,谱随机有限元法(SSFFM)在结构随机分析中得到越来越多的应用。但是随着结构模型的愈加精细、对随即响应的精度要求越来越高,采用SSFEM求解结构的随即响应所对应的线性方程组的规模越来越大,对当前计算机造成了巨大的压力。区域分解法(DDM)通过将整个结构转化为多个子结构和一个界面方程,压缩了核心计算方程的规模,大幅节约了计算成本。但即使如此,界面方程在大多数场合下较大,计算时间较长,因此本文提出了一种应用分层迭代技术的随机有限元区域分解求解器,通过重新排列界面方程,方程的结构为类似于SSFEM方程的二乘二递归层次,利用解的分层关系将原方程组求解转化成分层的方案,得到相对接近的近似解,整个求解过程作为预处理共轭梯度法(PCGM)求解界面方程的预处理函数。求解过程的所有矩阵由子域的矩阵和对应的布尔矩阵表示,避免了直接显式表达矩阵带来的存储压力。通过两个数值算例检验了该方法的精度和计算效率,结果表明,本文提出的分层前提条件可以在不影响计算精度的情况下,大大提高计算效率。更多还原

【Abstract】 Spectral stochastic finite element method(SSFFM) has been more and more used in stochastic analysis recently.However,with the scale of the systems of linear equations become larger and larger,it is more and more difficult for nowadays computers to solve them.The domain decomposition method(DDM) can greatly reduce the computation cost by transforming the whole structure into several substructures and an interface equation.The interface equation is large in most occasions and it takes a lot of time to calculate it,so in this paper,a hierarchical precondition is proposed to solve the interface equation,by rearranging the order of the equation,the equation can have a structure of two-by-two recursive hierarchy which is similar to the SSFEM equation although the interface equation has lost the block sparse structure,the lower-right submatrix in every level of structure can be simplified as block diagonal,every diagonal block in the submatrix has the same structure as the deterministic mean-value problem.The whole matrix of the equation is expressed by the subdomains’ matrices and the corresponding mapping matrix which is called ’Boolean matrix’.the submatrices of the equation are expressed by the same way which are mapped from the corresponding part of the subdomains’ matrices.Two numerical examples are used for examining the accuracy and computational efficiency of the method,the results show that the hierarchical precondition proposed in this paper can greatly improve the calculation efficiency without affecting the calculation accuracy.更多还原