Keywords: method of fundamental solutions, closed space, sound reduction.

Traditionally, sound propagation in two- and three-dimensional configurations has been modelled using numerical methods like the finite element method (FEM), the boundary element method (BEM) and the finite differences method (FDM). In recent years, a new class of numerical methods has been increasingly used, namely the meshless methods. The most well known among these are the RBF collocation method, the local Petrov-Galerkin method or the method of fundamental solutions (MFS). While the first two do not need the previous knowledge of fundamental solutions, the latter requires their previous definition. While it can be seen as a disadvantage in nonlinear problems or in cases where the fundamental solutions are very complex, it is clearly an advantage when these solutions can be defined with simplicity, making it possible to reach very high levels of accuracy.

In the present work, the authors apply the MFS to study the three-dimensional propagation of sound waves between two closed spaces interconnected by a localized heterogeneity. In this specific case, the system is composed of two closed rooms, limited by rigid walls, and separated by a thin rigid wall (null thickness) with the embedded heterogeneity. For such a configuration, it is well-known that a traditional single-domain approach using methods like the MFS or the BEM will lead to undetermined equation systems. To avoid this problem, the authors make use of the MFS together with decomposition of the domain in distinct regions (see, for example, Godinho et al. [1]). Boundary conditions of continuity of pressure and velocity in the opening between the rooms are then enforced to couple the two regions. When the opening is covered by a lightweight panel of known mass, it is possible to establish a relation between the sound pressure on both sides of the plate and the normal particle velocity at this surface. The numerical implementation of the MFS is performed in the frequency domain, making use of the fundamental solutions for sound propagation in an unbounded three-dimensional domain. Verification using known analytical solutions for simple geometric configurations revealed the excellent accuracy of the proposed method.

The MFS model was then used to simulate several test cases. The results obtained when the connection between the rooms is an opening, exhibited a behaviour which is consistent with the expected physical behaviour of the systems. These results revealed that the relative position of the source and of the opening are important to define the sound reduction provided by the separating wall. When the presence of a plate is simulated, the calculated results exhibit important differences from those predicted by a simplified model based on the mass law. Although the behaviour of the plate is also simulated similarly to the mass law, the complex wavefield generated within each of the rooms helps to understand those differences.

1

L. Godinho, A. Tadeu, P. Amado Mendes, "Wave propagation around thin structures using the MFS", Computers Materials & Continua (CMC), 5(2), 117-127, 2007.

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