Wideband fast multipole boundary element method: Applications to half-space/plane-symmetric acoustic wave problems

摘要

So far the boundary element method (BEM) has been applied extensively in solving acoustic wave problems, as it involves only surface discretization and solves infinite/semi-infinite problems naturally and exactly. Although the BEM reduces the problem dimensionality by one, the conventional method typically gives rise to fully populated and non-symmetric coefficient matrices which result in large storage requirements and prohibitive analysis time. The computational cost of the conventional method is O(N3) with direct solvers, or O(N2) with appropriate iterative solvers, and the storage requirements are O(N2), where N is the degree of freedom. This well-known drawback makes the BEM have difficulties with large models and thus be limited to numerical analyses of small bodies at low frequencies. In order to improve the efficiency, various fast approximate techniques, such as the fast multipole method (FMM), the fast wavelet transforms, the p-FFT method, the H-matrices and the adaptive cross approximation (ACA), have been proposed to accelerate the matrix-vector product in the BEM. Among these methods, the FMM approach seems to be one of the most widely accepted methods in the fast BEM community. Implemented with appropriate iterative solvers, the fast multipole BEM (FMBEM) reduces both the computational cost and storage requirement to O(N) for low-frequency acoustic wave problems, for example. In many applications, half-space/plane-symmetric acoustic wave problems are often analyzed, for instance the acoustic fields around noise barriers standing on the infinite ground. Although the FMBEM approach to full-space acoustic wave problems has been well studied in the literature, the applications of the FMBEM to half-space/plane-symmetric acoustic wave problems are still quite few and also need further studies. In order to accelerate the solution of half-space/plane-symmetric acoustic wave problems, Yasuda and Sakuma[1] proposed an efficient method by using the mirror image technique, where the half-space fundamental solution is not used, instead, half-space/plane-symmetric problems are mapped into full-space problems and the original full space fundamental solution is employed in the boundary integral equation. Using this method, the computational cost and storage requirement can be cut in half for problems with one symmetry plane. But this method leads to a bigger tree structure that is required to group all boundary elements and their mirror images. The situation becomes even worse when the distance between the structure and the infinite plane increases. Thus, some special techniques should be used to improve the efficiency for such cases. Bapat et al.[2] presented another half-space FMBEM approach using the half-space fundamental solution technique, where the tree structure only needs to group the boundary elements of the structure in the real domain. Although their method is simple to implement, the local expansion coefficients of the image domain should be calculated and kept in memory, which actually brings down the efficiency of the method. Moreover, the FMM approaches to acoustic wave problems in the frequency domain consists of the highand low-frequency methods, and the approaches proposed by Yasuda and Sakuma[1] and Bapat et al.[2] are based on these two methods, respectively. The computational cost is O(NlogN) in the high-frequency FMM, versus O(N) in the low-frequency FMM. However, either of them fails in some way outside its preferred frequency region. The high-frequency FMMis known to be not stable for low-frequency problems, while the low-frequency FMM is found to be not efficient for high-frequency problems [3]. Therefore, the wideband version which is accurate and efficient for any frequency seems necessary. In this study, a novel wideband FMBEM approach is presented for solving three dimensional half-space/ plane-symmetric acoustic wave problems. The half-space fundamental solution is employed in the boundary integral equation so that the tree structure required in the fast multipole algorithm should be built only for the boundary elements of the structure in the real domain. Moreover, the half-space fundamental solution is modified in order to apply the symmetric relations between the multipole expansion coefficients of the real and image domains to avoid calculating and saving the multipole/local expansion coefficients of the image domain. Also, the wideband adaptive multilevel fast multipole algorithm [4] is employed so that the present method is accurate, efficient and robust for both high- and low-frequency half-space/plane-symmetric acoustic wave problems. As for exterior acoustic wave problems, the Burton-Miller formulation [5] is applied to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method. Constant triangular elements are used to discretize the boundary surface so that the hypersingular boundary integrals can be evaluated directly and efficiently without using any regularization technique [6].