Transient vibrational power flows in slender random structures: Theoretical modeling and numerical simulations

Abstract

This paper deals with some recent developments for the modeling and numerical simulation of high-frequency (HF) vibrations of randomly heterogeneous slender structures. A time-domain approach is proposed to take account of the multiple reflection and scattering of vibrational waves in this frequency range. The mathematical model is derived from the semiclassical analysis of strongly oscillating (HF) solutions of quantum or classical wave systems, including acoustic, electromagnetic, or in the present case, elastic waves. This theory shows that the phase space energy density associated to these waves satisfies a radiative transfer equation in a random medium at length scales comparable to the small wavelength. The proposed model also considers energetic boundary and interface conditions consistent with the boundary and interface conditions imposed on the solutions of the underlying wave system. They are given in the form of power flow reflection/transmission operators for the energy rays impinging on a boundary or an interface. Nodal/spectral discontinuous “Galerkin” finite element methods and Monte-Carlo methods are implemented to integrate the radiative transfer equations supplemented with boundary and interface conditions. Some numerical simulations are presented to illustrate the theory: the first one deals with an assembly of random thick beams, and the second one with an assembly of random thick shells. This research applies to the prediction of the linear transient responses of engineering structures to impact loads or shocks, as encountered in the aerospace industry for example.