Fast Directional Boundary Element Method: Theory And Its Applications

Boundary element method(BEM) is believed as a very powerful and promising algorithm. In comparison with the domain discretizing algorithms including the currently widely used finite element method(FEM), finite difference method(FDM),its advantages lies in its dimension reduction, high accuracy and suitability for treating infinite and semi-infinite domain problems. However, the computational cost of traditional BEM grows quadratically with the degree-of-freedom N because of its fully populated system matrix, making it computational inhibitable for large scale problems.In the past three decades, many fast algorithms were proposed to alleviate this limitation by bringing down the computational cost from O(N2) to O(N log N).Among them, the kernel independent wideband fast algorithms, such as the fast directional algorithm(FDA), directional fast multipole method(d FMM), etc., are very competitive due to their salient advantages including high efficiency, simplicity,ease to implement and ease to extend to various kinds of kernel functions. Thus they have great potentials in dealing with large-scale real-world problems.In this thesis, the fast directional algorithm is used to accelerate the BEM, resulting in an accurate and efficient fast BEM solver suitable for wideband problems, which is named as the fast directional boundary element method(FDBEM). Then it is applied to a series of engineering problems. The main theoretical contributions of this thesis lies in the following aspects:1. An SVD-based accelerating technique is proposed to further accelerate the kernel independent fast multipole method(KIFMM), which is the degenerated fast directional algorithm(FDA) for low frequency problems.2. A further accelerating technique for FDA is proposed, which could improve the efficiency by about 40% for wideband problems. Then the improved FDA is applied to accelerate the locally-corrected Nystr?m BEM, resulting in a efficient and accurate fast directional BEM(FDBEM) for wideband problems.3. The FDA for oscillatory tensor kernels is developed and applied to elastody-namic problems.The FDBEM is applied to electrostatic, acoustic and elastodynamic problems.Then, combining with topological sensitivity analysis, it is used in structural health monitoring. Coupling with FEM, it is applied to vibro-acoustic coupling analyses. Numerical results show that, the FDBEM developed in this thesis can be used as an accurate and efficient solver for large-scale wideband problems in multiple engineering fields.