Title: Application of Fast Multipole Method to Variational Boundary Element Method for large acoustic radiation problems
Authors: GERADIN MICHELSTÉPHANE Paquay
Citation: ISMA 2006 Proceedings p. 2289-2301
Publisher: K.U.Leuven Department of Mechanical Engineering
Publication Year: 2006
JRC N°: JRC32686
URI: http://publications.jrc.ec.europa.eu/repository/handle/JRC32686
Type: Articles in periodicals and books
Abstract: Several techniques can be used to simulate acoustic radiation problems. Among these, the finite element method coupled with infinite elements and the Boundary Element Method (BEM) are probably the two main methods actually used in industrial simulation codes. For both methods, the resolution of acoustic radiation problem with large wave number (with a lot of wave oscillations on the radiating surface) is quite difficult. For the particular case of BEM, in spite of its advantages, the main problem is that the resulting matrix system to be solved is full. So, when several tens of thousand unknowns have to be taken into account, it is not possible to store the matrix system in memory of classical computers. Even if the system could be stored in memory, the cost of a direct resolution of the system would be very high (N³ order). Another approach is then proposed. Since it only needs the construction of matrix-vector products during the iterative process, the GMRES algorithm can be used instead of a direct resolution technique. The storage of the full matrix system can be avoided if the matrix-vector products are built directly. However, the construction order of a matrix-vector product is N² and the presence of double surface integrals related to the variational BEM formulation does not help at all. To accelerate dramatically the computation procedure, the Fast Multipole Method (FMM) was implemented in the multiphysics simulation software OOFELIE and applied to the direct construction of matrix-vector products in GMRES. The FMM uses the multipole translation theory and, by a space division in base 2, it permits to reduce the order of matrix-vector construction to N log N. With this new algorithm implemented in OOFELIE, it is now possible to solve acoustic radiation or diffraction problems with several tens of thousand unknowns on classical computers. For example, acoustic source diffraction by a submarine is presented. In this case, even if excitation frequencies are quite small, a lot of elements on the radiating surface are necessary since the system is very large (about 40 meters long). For smaller structures also illustrated, it means that very high frequencies can be achieved for the simulation.
JRC Directorate:Space, Security and Migration

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