NUMERICAL SIMULATION OF TRANSIENT VIBRATIONAL POWER FLOWS IN SLENDER RANDOM STRUCTURES ´ ERIC SAVIN 1 1 Aeroelasticity and Structural Dynamics Department, ONERA, France. E-mail: Eric.Savin@onera.fr This paperdeals with some recent developments for the modeling and numerical simulation of high-frequency (HF) vibrations of randomly heterogeneous slender structures. The mathematical model is derived from the semiclassical analysis of strongly oscillating (HF) solutions of quantum and classical wave systems, including acoustic, electromagnetic, or in the presentcaseelastic waves. This theory shows that the associated phase-space energy density satisfies a radiativetransfer equation in a random medium at lengthscales comparableto the small wavelength. The proposed model also considers energeticboundary and interface conditions consistent with theboundary and interface conditions imposed to the solutions of the wave system. They are given in the form of power flow reflection/transmission operators for theenergy rays impinging on a boundary or an interface. Specular-liketransverse boundary reflections, diffuse reflections, or fluid-structurecoupling may be treated as a particular case of the proposed model. Nodal/spectral discontinuous ’Galerkin’ finite element methods and Monte-Carlo methods are implemented to integrate the radiativetransfer equations with such boundary and interface conditions. Some numerical simulations are presented to illustrate thetheory: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 complexstructures to impact loads or shocks, or the analysis of non-destructive evaluation techniques. Keywords: High-frequency , vibration, radiativetransfer , discontinuous finite ele- ments, Monte-Carlo method. 1. Introduction Engineering structures exhibit typical transport and diffusive behaviors in the higher frequency (HF) range of vibra- tion (Savin, 2002). These regimes are described by linear transportequations for the energy density associated to the strongly oscillating (HF) solutions of the Navier-Cauchy equation for elas- tic wave propagation. More gener- ally this result holds for all symmet- ric hyperbolic systems, including quan- tum waves and classical acoustic or elec- tromagnetic waves (G´ erard et al., 1997, Guo & Wang, 1999, Papanicolaou & Ryzhik, 1999, Erd ¨ os & Yau, 2000, Bal, 2005, Powell & Vanneste, 2005, Lukkari- nen& Spohn, 2007).It has been special- ized to slender visco-elastic structures, typically beams, plates and shells, and fluid-saturated poro-visco-elastic media in Savin (2004, 2005a,b, 2007) for ap- plications toaerospace structures. The mathematical model is derived from the semiclassical analysis of HF solutions of such wave systems: it shows thatthe associated energy density in the phase- space position × wave vector satisfies a radiativetransfer equation (RTE) in a random medium at lengthscales com- parableto the small wavelength. The RTE allows to track theenergy paths Computational Stochastic Mechanics. Edited by G. Deodatis and P. D. Spanos Copyright c  2011 Published by Research Publishing :: www.rpsonline.com.sg ISBN: 978-981-08-7619-7 :: doi:10.3850/978-981-08-7619-7 P053 505