Research On Fast Multipole Boundary Element Method For Large-scale Acoustic Problems

The boundary element method (BEM) is a numerical method along with the development of the finite element method (FEM). The BEM is widely used to solve acoustic problems, since it has attractive advantages of boundary discretization, high accuracy and is especially suitable to handle infinite domain problems. However, the most serious problem is that the BEM leads to linear system of equations with general dense, non-symmetrical coefficient matrices. Solving the BEM system of equations needs expensive computational costs, when traditional solution techniques are used. As a result, the BEM has been limited to solve relatively small- and moderate-size problems, and is not available for large-scale problems. The computational ability of the BEM becomes a bottleneck problem. This restricts the large-scale engineering development and application of the BEM. Thus, it is crucial to develop a new fast BEM for solving large-scale acoustic problems. This dissertation focuses on the research of the fast multipole method (FMM) and BEM, and develops a new fast multipole BEM (FMBEM) for solving large-scale acoustic problems.The Burton-Miller formulation is employed to successfully remove the non-uniqueness problem associated with the conventional BEM for exterior Helmholtz equation. The major difficulty of the Burton-Miller formulation is that it includes a hypersingular integral. This dissertation proposes an improved form of the Burton-Miller formulation which it only contains weakly singular integrals, and avoids the difficulty of the hypersingular integral evaluation. Furthermore, the iterative efficiency of the presented method is significantly improved by adopting a simple and effective block diagonal preconditioner to improve the condition of the system matrix equations. The block diagonal preconditioner is very efficient and results in a large reduction of required iteration steps. Numerical results demonstrate the accuracy and efficiency of the improved BEM for acoustic problems, and show the conventional BEM needs O(N2) computational time and computer memory, where N is number of degrees of freedom (DOFs). Thus, the BEM is prohibitively expensive for solving large-scale acoustic problems.A new fast multipole BEM based on the improved Burton-Miller formulation is presented for solving large-scale two-dimensional (2D) acoustic problems. According to the theories of multipole expansions, the formulations and algorithms of the fast multipole BEM are developed. Furthermore, for overall improving the computational efficiency of the presented method, an effective sparse approximate inverse preconditioner is constructed based on the leaves of tree structure. Then the O(N) complexity of the fast multipole BEM is verified using the theoretical analysis. Numerical results demonstrate the accuracy and efficiency of the fast multipole BEM for solving 2D acoustic problems. Further numerical tests show that the presented method has O(N) computational efficiency and provides an order of magnitude increase in efficiency compared to the conventional BEM. A multiple scattering model with 240000 DOFs is solved effectively on a personal computer. The results demonstrate that the fast multipole BEM has the advantage for large-scale acoustic problems, and successfully solves the bottleneck problem of the BEM. This example shows the great potential of the presented method for large-scale engineering applications.The fast multipole BEM is extended from 2D to 3D acoustic problems. Based on the improved Burton-Miller formulation, a new wideband fast multipole BEM is presented for solving large-scale 3D full-space acoustic problems. According to the partial wave expansion method and plane wave expansion method, the formulations of the fast multipole BEM are developed for the low- and high-frequency problems, respectively. In order to further obtain overall computational efficiency in all frequencies, a seamless framework for adaptively combining the low- and high-frequency formulation is proposed. Furthermore, the practical formulation is presented to determine the number of truncation terms based on the empirical methods. The numerical examples including a model with 520000 DOFs clearly demonstrate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D acoustic problems in a wide frequency range, and show the potentially useful engineering applications.The fast multipole BEM is further extended from full-space to half-space acoustic problems. Based on the full-space algorithm, a new fast multipole BEM for solving large-scale 3D half-space acoustic problems is presented. Using the half-space Green's function, the formulations of the half-space fast multipole BEM are developed. In the new half-space algorithm, a tree structure of boundary elements can be constructed in the real domain only, instead of using a larger tree structure that contains both the real domain and its mirror image, which greatly simplifies the implementation of the half-space fast multipole BEM and reduces the computational time and memory storage. The numerical examples validate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D half-space acoustic problems. The analysis of the building and sound barrier noise further illustrates the potential of the presented method for solving large-scale practical problems.This dissertation mainly studies the fundamental theories and applications of the fast multipole BEM in acoustics, the research results demonstrate that the fast multipole BEM is efficient for solving large-scale acoustic problem, and show important academic value and extensive engineering prospect.