Parallel Algorithm Research On The Fast Multipole Boundary Element Method And Its Engineering Applications

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.