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

The design industrial structures requires engineers to know their dynamic behaviour. The response, especially during the transient state, cannot be completely described using the current tools based on finite element techniques and explicit numerical schemes; indeed, the medium-frequency range is often ignored unless the calculation is carried out with a very refined spatial mesh and, consequently, a refined time discretisation. This would mean a prohibitive computation time. But 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. Transient dynamic analysis over the medium-frequency range for complex engineering structures is then an important challenge and appears useful especially for the study of pyrotechnical shocks. Such an analysis requires the use of additional computational tools. The present work, using new computational strategies in dynamics, answers this challenge for the transient part of the solution. The problem is solved in the frequency domain. One needs to solve a forced vibration problem over a frequency range which includes the low- and medium-frequency ranges. The low-frequency range is solved as usual while the medium-frequency range is handled using the Variational Theory of Complex Rays (VTCR). The final solution in the space-time domain is given using the inverse Fourier transform.

The main question is then to solve the forced vibration problem over a wide frequency range. The low-frequency range no longer presents any major difficulties, at least in what regards modeling and calculation, even for complex structures. As for high frequencies, some computational tools in which the spatial aspects disappear almost entirely, do exist, such as the SEA method. 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 studied phenomena 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, called the "Variational Theory of Complex Rays", was first introduced in [1] for the calculation of medium-frequency vibrations and the capabilities of the method are demonstrated in [2,3].

The present approach has been applied on beams in [4,5] and shows the importance of the medium frequencies. The aim of this paper is to extend this method to complex engineering structures as assemblies of plates, shells and beams submitted to an impact. To study such structures, it is necessary to use the VTCR over a relatively wide frequency range. In the approach, we propose to consider the time response as the superposition of two components that stem respectively from the low- and the mid-frequency contributions. These two components can then be computed separately: the low-frequency contribution can easily be obtained using any adapted technique and the medium-frequency contribution, obtained using the present approach, can be added afterwards. This enables us to use the fast Fourier transform on the medium-frequency range only and to take advantage of the quick fading-out of these medium frequencies in scattering media, and thus to dramatically reduce the computation cost. Several applications of assemblies of plates show the importance of the medium frequencies in the transient part of dynamic responses. They also feature the performance of the present approach by taking these medium frequencies into account with a reduced numerical effort.

1

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

2

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

3

P. Rouch, 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

4

P. Ladevèze, "A new computational method for medium-frequency vibrations and its extension to transient dynamics", In D. R. J. Owen, E. Oñate, and B. Suárez, editors, Proceedings of the seventh International Conference on Computational Plasticity - Complas 2003, CIMNE, Barcelona, Spain, on CD-ROM, 7-10 April 2003.

5

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

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