Research On Fast Multipole Method Of Fundamental Solutions For Large-scale Acoustic Problems And Acoustic Sensitivity Analysis

The method of fundamental solutions (MFS), as a boundary meshless method, is an effective alternative to the boundary element method (BEM), it shares the same advantages of the BEM over finite element method (FEM), such as dimension reduction, high accuracy and suitable for infinite domain problems. It also has certain advantages as the following:singular integrals are avoided by placing the source at some auxiliary points off the problem domain, and implement of the MFS is significantly simplified compared with the BEM. Therefore, the MFS is very suitable for solving acoustic problem. However, like conventional BEM, the system matrix in the MFS is also dense and Non-symmetric matrix, and it requires O(N2) operations in computation and memory storage, where N is the number of unknowns. Using the conventional method for solution of the linear system, it requires O(N)-O(N) operations, and has low computational efficiency. Because of the high computational cost of the conventional MFS, MFS can only solve small-scale acoustic problems, but can not effectively deal with large-scale acoustic problems. Therefore, it is essential to develop a new method for large-scale acoustic problems. Based on the depth study of fast multipole method and conventional MFS, this dissertation develops a new method called fast multipole method of fundamental solutions (FMMFS) with O(N) computational efficiency for low-frequency acoustic problems in practical engineering, and this method solves two-dimensional, three-dimensional full space and half space large-scale acoustic problems quickly and efficiently, and it is also applied to realize large-scale acoustic sensitivity analyses.In chapter one, the history and research situation of acoustic radiation, acoustic sensitivity analysis have been reviewed. The problems existed in them have also been analyzed. Finally, the research topics in the dissertation are determined based on the literature review.In chapter two, the formulas of the conventional MFS are given in this chapter. With the help of iterative solver called GMRES based on Krylov subspace, the computational efficiency of MFS is to some extent improved. Based on FORTRAN, the MFS program for both2D and3D acoustic problems are obtained. The simulations are used to demonstrate the program for both radiation and scattering problems, and the simulations also reveal the inherent shortcomings of high consumption of computation and memory, which indicate that the conventional MFS is not suitable for large-scale acoustic problem.In chapter three, a new fast multipole method of fundamental solutions is presented for two-dimensional (2D) large-scale acoustic problems. According to the theories of multipole expansions for2D acoustical Green function, the formulations and algorithms of the FMMFS are developed. Numerical results demonstrate the effectiveness, accuracy and efficiency of the FMMFS for solving2D acoustic problems, and the O(N) complexity of the FMMFS is verified by theoretical analysis and numerical simulation. A multiple scattering model is solved effectively on a personal computer. The results demonstrate that the FMMFS has the advantage for large-scale acoustic problems, and it also shows the great potential for large-scale engineering applications.In chapter four, the FMMFS is extended from2D to3D acoustic problems. According to the theories of multipole expansions for3D full-space acoustical Green function, the formulations and algorithms of the FMMFS are developed. Compared with conventional methods, numerical simulations demonstrate the effectiveness, accuracy and efficiency of the FMMFS for solving3D full-space acoustic problems. Large-scale acoustical models are solved effectively on a personal computer. The results demonstrate that the FMMFS has great potential for3D full-space large-scale engineering applications.In chapter five, firstly, the conventional MFS for3D half-space acoustic problems is presented, then, similar to the3D full-space FMMFS, a fast multipole method of fundamental solutions for3D half-space acoustic problems is proposed according to the theories of multipole expansions for half-space Green function. Numerical simulations also demonstrate the effectiveness, accuracy and efficiency of the FMMFS for solving3D half-space acoustic problems. Finally, the analysis of the half-space acoustic model of the building and noise barriers further illustrates the potential of the presented method for solving large-scale practical problems.In chapter six, the FMMFS developed in this dissertation are applied to realize acoustic sensitivity analyses based on direct differentiation method for3D large-scale noise barriers model. The developed fast algorithms are employed in the design and optimization analyses of noise barriers, through calculating the sound pressure sensitivity value with respect to the design height of noise barriers. More engineering applications are also expected to be solved using the developed algorithms in the future.In chapter seven, researches in this dissertation have been summarized, and the topics need further study have been proposed.