Patch Transfer Function (PTF): a substructuring approach for linear acoustics Morvan Ouisse 1 , Christian Cacciolati 2 , Jean-Louis Guyader 3 Laboratoire Vibrations Acoustique, INSA Lyon, F69621 Villeurbanne, France, 1 Email: morvan.ouisse@insa-lyon.fr 2 Email: christian.cacciolati@insa-lyon.fr 3 Email: jean-louis.guyader@insa-lyon.fr Introduction For industrial structures, large acoustics problems like noise radiated by cars are very difficult to solve using classical methods like full BEM, because of the complexity of the structure. The coupling between FEM and BEM procedures and the ways to reduce calculation time have been developed from the 70’s [1] and this is still a research area of first interest [2]. The PTF approach has been first presented in reference [3]: it considers several acoustic subsystems, which are analysed separately, considering each coupling area as a rigid surface, which is divided in elements called patches. For these calculations, any available method (FEM, BEM…) can be used in order to build a database of transfer functions between sources and patches, which are acoustic impedance transfer functions. Then, continuity relations can be written on the interface in order to couple the sub domains: this approach is close to the mobility and impedance concepts which are widely used in mechanics, it is based on linearity properties of acoustic phenomenon. Patch Transfer Functions The PTF method has been described in details in ref. [3], to which readers are invited to reefer for more details. The coupling area between the two considered subsystems has to be divided in surface elements called “patches”. For a sake a simplicity, one will suppose here that there is an internal medium, with rigid faces opened by holes allowing the sound to be radiated to an external medium. Analyses of uncoupled subsystems are performed using classical tools (FEM, BEM…), in order to built a database of Patch Transfer Functions Z , which are defined as the ratio EHWZHHQPHDQSUHVVXUHRQSDWFK DQGPHDQQRUPDOYHORFLW\ RQ SDWFK  OLNH LQGLFDWHG LQ HTXDWLRQ  7KLV FDOFXODWLRQ LV GRQHFRQVLGHULQJDQRUPDOXQLWYHORFLW\LPSRVHGRQSDWFK  while other patches are considered as rigid surfaces. β α αβ = n V p Z (1) Some PTFs are also defined between excitation areas (which are vibrating surfaces divided in patches) and coupling patches, and between patches and listening points (microphones locations). Writing continuity relationships on the coupling area leads to a N p by N p linear system if the coupling area is divided in N p patches: ( ) ∑ ∑ = γ γ γβ = α α αβ αβ = + s p N 1 n int N 1 ext n int ext V Z ~ V Z Z (2) In this equation ext and int indicates that PTFs calculations are obtained from internal and external analyses. The unknowns are the mean coupling velocities α ext n V on coupling patches. The right term is related to the N s source patches. Inversion of the linear system leads to the value of pressure at listening points. One should notice that this linear system is full, but its reduced size does not result in numerical difficulties. This analysis is performed at each considered frequency step. Convergence: number of modes When one of the sub domains is closed with only some holes, the internal analysis can be efficiently done using modal analysis: analytic modes for simple geometry, FEM for complex one. Then, the modal composition can be used to calculate the PTFs. When such a method is used, one has to take care of the number of modes used for this calculation, since they are used to built a linear system which should not be singular. An example is given on Figure 1: a parallelepiped box is considered, divided in two boxes with the coupling area between: this is a strong coupling case. In that situation the mode order in the longitudinal direction of the higher considered mode has to be larger than the number of patches, otherwise the PTFs will be linearly dependant. The figure shows the limit case between the two configurations, while the reference calculation comes from the analytical modal analysis of the box. Figure 1: Influence of the number of modes Convergence: patch size Another parameter of importance in the PTF approach is the patch size. On Figure 2, the same structure as the previous one is considered, using a large number of modes, while the patch size varies. One can observe that the meshing size of the coupling area which is necessary for convergence is the CFA/DAGA'04, Strasbourg, 22-25/03/2004 293