用于大规模断裂分析的快速多极边界元法  

Fast multipole BEM for analyses of large-scale fracture problems

作　　者：徐俊东[1] 王朋波[2] 姚振汉[1] 危银涛[2] 尹伟奇[3] 

机构地区：[1]清华大学工程力学系,北京100084 [2]清华大学汽车工程系,北京100084 [3]佛罗里达大学航天与机械工程系,美国佛罗里达32611

出　　处：《清华大学学报（自然科学版）》2008年第5期896-899,共4页Journal of Tsinghua University(Science and Technology)

基　　金：国家自然科学基金资助项目(10472051);中国博士后科学基金资助项目(20070420372)

摘　　要：将快速多极算法(FMM)应用到边界元法(BEM)中,对断裂力学问题进行大规模计算。基于对偶边界积分方程(DBIE)构造代数方程组,采用广义极小残值迭代法(GMRES)求解。利用自适应四叉树结构执行快速多极算法,系数矩阵不需要显式存储,与未知量向量的乘积通过树结构的递归操作获得,计算复杂度与存储需求均缩减为O(N)(N为问题的自由度数)。此外,该文提出了一种改进的预条件方案,使GMRES的求解时间与内存消耗进一步降低。数值算例表明:该方案在保证精度的前提下,使计算规模与计算效率有可观的提高;算例的最大规模达到了300万自由度。The fast multipole method （FMM） was used with the boundary element method （BEM） to predict fractures in large castings. The algebraic equation from the dual boundary integral equations （DBIE） was solved using the generalized minimum residual method （GMRES）. A quad-tree data structure was constructed to execute the FMM with the matrix-vector product obtained using recursive operations on the tree structure without explicitly forming the coefficient matrix. Both the computational complexity and the memory requirements are reduced to O（N）, where N is the number of degrees of freedom （DOFs）. An improved preconditioner for the GMRES was also developed to increase the run-time speed and memory storage efficiency. Numerical experiments show that the preconditioner significantly increases the computational scale and computing efficiency with up to 3 000 000 DOFs used in numerical examples.

关 键 词：快速多极算法 边界元 对偶边界积分方程 裂纹 

分 类 号：O346.11[理学—固体力学]