Keywords: state of the art, review, medium frequency, vibro-acoustics, dispersion, pollution error.

Over the last decade, the computational simulation of structural vibrations and acoustic radiation or scattering has become a major field of interest both at the research and industrial levels. According to major scientists, simulating accurately waves in the medium frequency range is still an unsolved problem. For the low frequency range, the finite element method (FEM) and the boundary element method (BEM) are still most widely used to solve wave propagation problems, even for complex structures. For a high frequency range, other techniques were developed where the spatial aspect disappears almost entirely such as the Statistical Energy Analysis (SEA) methods for instance.

Despite considerable recent efforts to extend the possible frequency range of both the FE and the SEA methods, fully satisfactory numerical procedures to cover the medium frequency range still do not exist. The key issue to improve the accuracy of the approximate numerical solution, or, alternatively, to simulate properly the medium frequency range, for wave propagation in bounded domains, or for near-field sub-domains in unbounded domains, is to control the dispersion error. It is particularly crucial for industrial purpose which request robust methods for the medium frequency range within a reasonable computational cost, having in mind a sufficiently accurate numerical modelling for updating with experimental data. It is emphasised that the expression `medium frequency range' actually depends on the context, i.e. on the natural frequency spectrum of the range of interest.

Many numerical methods have been proposed to eliminate the dispersion, none of them being totally `dispersion-free'. The first approaches were based on the idea of stabilising the finite element method itself but did not improve significantly the accuracy of the solution. High order approximations were then proposed, based on the finite element method (hp formulations) or on meshless methods. By now, everybody seems however to agree that it is even more advantageous to incorporate information on the solution itself into the numerical discrete subspace, as achieved for instance in the wave envelope approach, the Discontinuous Galerkin FEM, the Trefftz based approach, the Variational Theory of Complex Rays. Note that the generalised FEM and the meshless methods are considered again in this respect.

This paper aims at reviewing the main numerical methods which are under development to analyse the wave propagation, with a special emphasis on the medium frequency range. The paper shows that the issue of computing short wave seems to be still unsolved since none of the methods is really `dispersion-free' or outperforms the other ones. Some criteria allow an objective comparison. A first criterion consists in the answer to the following questions: (i) is the method general for any geometry?, (ii) does the method allow a coupling for an infinite medium?, (iii) does the method allow a vibro-acoustic coupling ? Secondly, other criteria are of importance such as the accuracy, the maximum frequency range or the computational time. A special attention also has to be paid to the fact that some materials exhibit frequency dependent properties.

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