Keywords: dynamics, frequency domain, medium frequencies, VTCR, multiscale, heterogeneities.

The study of the dynamic behaviour of industrial structures is essential for their design either for human security, or for the safety of structures as satellites in launchers, or for acoustic comfort in vehicles for instance. The problem is that when predicting the transient state of the response due to a shock characterised by a large frequency content, the current tools, based on finite element techniques and explicit numerical schemes, fail to describe the medium frequency range. Indeed taking these frequencies into account would mean a very expensive calculation has to be carried out with a very refined spatial mesh and, consequently, a refined time discretisation and this would lead to numerical difficulties [1].

Nevertheless, taking the medium-frequency content into account can prove necessary since, although the displacements are small over this frequency range, the velocity and therefore the kinetic energy can be significant. The present work thus develops a frequency domain approach for the computation of transient wave propagation in complex engineering structures. In this frequency domain approach, one needs to solve a forced vibration problem over a frequency range which includes the low- and the medium-frequency ranges. In order to use the tools with the best performance, we propose a partition of the frequency range being studied: the low and the medium frequency ranges.

The low-frequency range does not present any major difficulties, at least with regard to modeling and calculation, even for complex structures. For high frequencies, some computational tools do exist, such as the statistical energy analysis (SEA) method in which the spatial aspects disappear almost entirely. By contrast, the modeling and calculation of medium frequency vibrations continue to raise some problems. The difficulty lies in the fact that the wavelengths of the phenomena studied are very small compared to the characteristic dimensions of the structure. Consequently, if one were to extend the low frequency methods, disregarding the serious numerical difficulties which would occur, the finite element calculation to be performed would still require an unreasonable amount of degrees of freedom. The alternative approach we use here to handle the medium frequency range, is the "variational theory of complex rays" (VTCR) that was first introduced in [2] for the calculation of medium-frequency vibrations. The capabilities of the method are demonstrated in [3,4].

The present frequency domain approach for transient dynamics has been applied on assemblies of beams and homogeneous plates in [5,6]. These examples show the importance of taking the medium frequencies into account in the response and the reduced numerical effort it involves. The aim of this paper is to extend this method to more complex structures as heterogeneous plates that present holes or stiffeners. To study such structures, it is necessary to use the extension of the VTCR for heterogeneities introduced in [7].

1

F. Ilhenburg and I. Babuska, "Dispersion analysis and error estimation of Galerkin finite element methods for Helmholtz equation", Int. J. Numer. Methods Engrg., 38:3745-3774, 1995. doi:10.1002/nme.1620382203

2

P. Ladevèze, "A new computational approach for structure vibrations in the medium frequency range", C. R. Acad. Sci. Paris Sér. IIb, 322(12):849-856, 1996.

3

P. Ladevèze, L. Arnaud, P. Rouch and C. Blanzé, "The variational theory of complex rays for the calculation of medium-frequency vibrations", Engrg. Comput., 18(1/2):193-214, 2001. doi:10.1108/02644400110365879

4

P. Rouch and P. Ladevèze, "The variational theory of complex rays: a predictive tool for medium-frequency vibrations", Comput. Methods Appl. Mech. Engrg., 192:3301-3315, 2003. doi:10.1016/S0045-7825(03)00352-9

5

P. Ladevèze and M. Chevreuil, "A new computational method for transient dynamics including the low- and the medium-frequency ranges", Int. J. Numer. Methods Engrg., 64:503-527, 2005. doi:10.1002/nme.1379

6

P. Ladevèze and M. Chevreuil, "A new computational method for transient analyses including the low- and the medium-frequency ranges of engineering structures", Computers and Structures, submitted for publication.

7

P. Ladevèze, L. Blanc, P. Rouch and C. Blanzé, "A multiscale computational method for medium-frequency vibrations of assemblies of heterogeneous plates", Computers and Structures, 81(12):1267-1276, 2003. doi:10.1016/S0045-7949(03)00041-5

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