Comparison of the Fast Multipole Method with Hierarchical Matrices for the Helmholtz-BEM
D. Brunner; M. Junge, P. Rapp, M. Bebendorf; and L. Gaul

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 58, No. 2, pp. 131-160, 2010
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Keywords Acoustics, Boundary Element Method, Fast Multipole Method, Hierarchical Matrices, Adaptive Cross Approximation
Abstract The simulation of the hydroacoustic sound radiation of ship-like structures has an ever-growing importance due to legal regulations. Using the boundary element method, the overall dimension of the problem is reduced and only integrals over surfaces have to be considered. Additionally, the Sommerfeld radiation condition is automatically satisfied by proper choice of the fundamental solution. However, the resulting matrices are fully populated and the set-up time and memory consumption scale quadratically with respect to the degrees of freedom. Different fast boundary element methods have been introduced for the Helmholtz equation, resulting in a quasilinear complexity. Two of these methods are considered in this paper, namely the fast multipole method and hierarchical matrices. The first one applies a series expansion of the fundamental solution, whereas the second one is of pure algebraic nature and represents partitions of the original system matrix by low-rank approximations in outer-product form. The two methods are compared for a structure, which is partly immersed in water. The memory consumption, the set-up time and the time required for a matrix-vector product are investigated. Different frequency regimes are considered. Since the diagonal multipole expansion is known to be unstable in the low-frequency regime, two types of expansions are necessary for a wideband analysis.
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