Abstract
The purpose of this paper is to expound some recent developments for the modeling and numerical simulation of high-frequency (HF) vibrations of randomly heterogeneous structures, and outline some perspectives of future research. The mathematical-mechanical model is based on a microlocal analysis of quantum or classical linear wave systems. The theory shows that the energy density associated to their mean-zero solutions—including the strongly oscillating (HF) ones—satisfies a Liouville-type transport equation, or a radiative transfer equation in a random medium at length scales comparable to the small wavelength. Its main limitation to date lies in the consideration of energetic boundary and interface conditions consistent with the boundary and interface conditions imposed to the solutions of the wave system. The corresponding power flow reflection/transmission operators are derived formally and rigorously for elastic media, including slender structures, near the doubly-hyperbolic and hyperbolic-elliptic sets, ignoring however the glancing set. Yet a radiative transfer model in bounded media with general transverse or diffuse boundary conditions for the power flows is detailed in the paper. Some direct numerical simulations are presented to illustrate the theory.
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Savin, É. (2011). High-frequency vibrational power flows in randomly heterogeneous coupled structures. In: Belyaev, A., Langley, R. (eds) IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties. IUTAM Bookseries, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0289-9_17
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DOI: https://doi.org/10.1007/978-94-007-0289-9_17
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