Keywords: vibrations, high frequency, transport, radiative transfer, diffusion.

Realistic, complex industrial structures exhibit typical transport and diffusive behaviour in the high-frequency range of vibration, as demonstrated by the experiments described in [1] among others. Whereas their dynamics are physically well represented by global standing waves (the eigenmodes) in the low-frequency range, this approach fails dramatically as the frequency increases, because of the multiple interactions of waves with material heterogeneities, and reflections from the boundaries and at the interfaces between structural components with different stiffnesses (stiffeners on plates and shells, bulkheads, stringers ...). Predictive methods for high-frequency vibrations are required in the automotive and aerospace industries for applications to noise control, acoustic comfort and discretion, or pyrotechnic shock simulation for example. Statistical Energy Analysis (SEA [2]) is the most popular method among mechanical engineers to deal with such problems. It is based on some rigorous arguments developped in the late 1950s, but also on several phenomenological assumptions which can hardly be satisfied exactly. Derivation of a vibrational conductivity analogy (VCA) for high-frequency structural dynamics has attracted several researchers in the early 1990s [3], but this model is based on heuristic, unproven arguments as well. On the basis of (rigorous) mathematical arguments, it can be shown that the evolution of the high-frequency vibrational energy density of visco-elastic media is depicted by Liouville transport equations [4], or radiative transfer equations in the presence of random heterogeneities [5]. These results generalize the well-known method of geometrical optics. The transport regime is obtained under the assumptions that (i) wavelengths are short with respect to the macroscopic features of the medium (the high-frequency range), (ii) they are comparable to the correlation lengths of random heterogeneities, and (iii) fluctuations of the latter are small and rather isotropic.

Radiative transfer equations are costly to solve numerically. However they can be approximated by diffusion equations in a rather systematic way for long propagation distances with respect to the scattering mean free path, which is the mean distance travelled by an energy ray before its direction is significantly altered by scattering on the heterogeneities. This paper deals with diffusion approximations of the stochastic Liouville equations and the radiative transfer equations. Heat conduction-like diffusion equations are directly derived from the latter in the limit of a large ratio of the typical propagation distance to the scattering mean free paths. Fokker-Planck diffusion equations are derived from the former when the correlation length of the heterogeneities is small with respect to the medium size, but still large with respect to the wavelength. Some numerical simulations for a random Mindlin plate are presented to illustrate the approach. These results are used to discuss the relevance of the Vibrational Conductivity Analogy invoked in the structural acoustics literature.

1

É. Savin, "Midfrequency vibrations of a complex structure: Experiments and comparison with numerical simulations", AIAA Journal, 40(9), 1876-1884, 2002. doi:10.2514/2.1867

2

R.H. Lyon, R.G. DeJong, "Theory and Application of Statistical Energy Analysis", Butterworth-Heinemann, Boston MA, United States, 1995.

3

D.J. Nefske, S.H. Sung, "Power flow finite element analysis of dynamic systems: basic theory and application to beams", ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, 111(1), 94-100, 1989.

4

P. Gérard, P.A. Markowich, N.J. Mauser, F. Poupaud, "Homogenization limits and Wigner transforms", Communications on Pure and Applied Mathematics, L(4), 323-379, 1997. doi:10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.3.CO;2-Q

5

G.C. Papanicolaou, L.V. Ryzhik, "Waves and Transport", in "Hyperbolic Equations and Frequency Interactions", L. Caffarelli, W. E, (Editors), IAS/Park City Mathematics Series Vol. 5, American Mathematical Society, Providence RI, United States, 305-382, 1999.

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