【作者】 雷霆；

【导师】 姚振汉；

【作者基本信息】 清华大学 ， 力学， 2006， 博士

【摘要】 边界元法作为有限元法、有限差分法等区域解法的重要补充,具有降维、精度高的特点,在各种工程领域有许多成功的应用。但传统边界元法形成的线性方程组的系数矩阵是满阵,因此一般只能应用于数千到数万自由度的小至中等规模问题的求解。然而,近20年以来,由于一些快速算法的出现并不断发展,情况正在发生改变。特别是由Greengard和Rokhlin首先提出的快速多极算法,可以将矩阵向量相乘操作的计算量级和存储量级同时降为O ( N ),其中N是未知量的个数,被美国工业与应用数学学会(SIAM)评为上个世纪十大算法之一。应用快速多极算法的边界元法被称为快速多极边界元法。而在此基础上对其可扩展性并行计算算法的研究则可以进一步扩大解题规模、提高计算速度、并通过网格加密和增加级数展开阶数两方面来提高求解的精度、从而进一步扩展边界元法的应用范围和优势领域。以此为目标,本文对适用于二维、三维弹性力学问题等的初始版本和新版本快速多极边界元法的并行算法作了研究,结合边界元二次等参单元的特点,提出了基于自适应树结构的快速多极边界元法的一种分布式并行计算格式,其中包括计算量预测、加权任务分配方式、通信关系的建立和数据通信过程等几部分的实现,并对几何形状不规则的结构在最多为64个处理器的并行机群上进行了测试并达到了满意的加速比,最大计算规模达到了二百多万自由度。在此基础上,本文通过数值算例对基于树结构的快速多极并行计算与常规矩阵并行计算进行了比较,指出了它们各自在计算规模上的合理应用范围。并给出了一种结合算法,从而在预处理方案、多子域问题等方面扩展了快速多极边界元并行算法的应用范围。最后,作为上述算法的具体应用,本文对短纤维复合材料进行了数值模拟,利用高性能计算的优势,对复杂形状纤维以及随机取向弯曲纤维复合材料中的应力分布规律进行了研究,给出了一些有参考价值的数值结果,并表明快速多极边界元法并行计算在复杂界面问题上的大规模计算与常用的有限元等区域方法相比具有明显的优势。更多还原

【Abstract】 As an effective alternative to existing domain methods, including the finite element method (FEM) and finite difference method (FDM), boundary element method (BEM) has main advantages of dimension reduction and high accuracy, and has many successful applications in various engineering fields. Because the conventional BEM usually results in dense and asymmetric matrix of its linear systems, it is not capable to solve large-scale problems, only the small-scale problems of several thousands DOFs and the middle-scale problems up to several ten thousands are solvable. However in the recent twenty years this situation is changing gradually due to the introduction of the fast multipole method (FMM) first presented by Greengard and Rokhlin, which has been named as one of the top-10 algorithms of the 20th century by SIAM. The computational complexity and memory requirements of matrix-vector multiplication can then be reduced to O ( N ), where N is the number of unknowns.BEM accelerated by the FMM is called fast multipole boundary element method (FM-BEM). Further investigation on scalable parallel computation algorithm of FM-BEM can provide the possibility to solve extreme large-scale problems, to speed up the computation and to enhance the accuracy by means of using refined mesh in the numerical discretization and increased expansion order in the approximation. In this way the BEM can be extended to much more application fields with considerable advantages.For these reasons, the parallel algorithm of the original and new version of the adaptive FMM, suitable for both 2D and 3D elasticity problems and so on, is studied in this dissertation, and focusing mainly on the 3-D quadratic boundary elements, a scheme on the parallelization of FM-BEM targeting on the distributed architectures is presented. The formulation features the prediction of the computational loads, weighted task partitioning method, communication list establishment and the data communications. The performance is measured on a 64-processor cluster, and asatisfactory speed-up is achieved even for structures with irregular boundaries. The scale of the solved problem reaches more than 2 million DOFs.Next, the developed parallel FM-BEM solver is compared with conventional parallel BEM solver on the same cluster environment. A conclusion is drawn on the suitable ranges of the solving scale of the two methods. In addition, a combined algorithm of the two different parallel formulations is presented for incorporating other pre-conditioner scheme or the multi-domain methods to extend the applying fields of the FM-BEM.Finally, as application of the presented method, some short fiber-reinforced composites are simulated with the power of high performance computing. The influences of fiber shapes, orientation distribution and curvature on the stress distribution are summarized. Some valuable numerical results are given, which shows that the proposed method is both accurate and efficient, and can solve problems of large size that are challenging to existing state-of-the-art domain methods.更多还原