Keywords: high-frequency, vibration, transport, discontinuous finite elements, Monte-Carlo method.

The vibrational response of complex engineering structures to low-frequency (LF) loads is physically well represented by global standing waves, i.e. their eigenmodes. However such a description is no longer relevant for high-frequency (HF) loads, for which these systems exhibit a typical transport and diffusive behavior. The purpose of this paper is to expound some recent developments for the modeling and numerical simulation of high-frequency vibrations of heterogeneous structures, and outline some perspectives for future research. The mathematical-mechanical model is based on a microlocal analysis of quantum or classical wave systems: acoustics, elastodynamics, electromagnetics, phonons, etc.. 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 in the HF range. 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 have been derived elsewhere formally and rigorously for elastic media, including slender structures, near the doubly-hyperbolic and hyperbolic-elliptic sets, ignoring however the glancing set. Yet a transport model in bounded media with general transverse or diffusive boundary conditions for the power flows is detailed in this paper. Specular-like transverse reflections may be treated as a particular case of the proposed model. Some direct numerical simulations are presented to illustrate the theory: the first one deals with an assembly of thick beams, and the second one with an assembly of thick shells. Nodal-spectral discontinuous Galerkin (DG) finite element methods [1] and Monte-Carlo (MC) methods [2] are implemented to integrate the transport equations for arbitrary boundary conditions, including multiphysics coupling. The numerical fluxes in the DG methods are constructed in order to account for the power flow reflection-transmission processes at junctions and boundaries. The MC methods consider the multiple scattering of power flows on these interfaces as well. This research applies to the prediction of transient responses of structures to impact loads or shocks or the analysis of non-destructive evaluation techniques.

1

J.S. Hesthaven, T. Warburton, "Nodal Discontinuous Galerkin Methods", Springer, New York, 2008.

2

B. Lapeyre, E. Pardoux, R. Sentis, "Introduction to Monte-Carlo Methods for Transport and Diffusion Equations", Oxford University Press, Oxford, 2003.

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