Diffusion regime for high-frequency vibrations of randomly heterogeneous structures

Abstract

The evolution of the high-frequency vibrational energy density of slender heterogeneous structures such as Timoshenko beams or thick shells is depicted by transport equations or radiative transfer equations (RTEs) in the presence of random heterogeneities. A diffusive regime arises when their correlation lengths are comparable to the wavelength, among other possible situations, and waves are multiply scattered. The purpose of this paper is to expound how diffusion approximations of the RTEs for elastic structures can be derived and to discuss the relevance of the vibrational conductivity analogy invoked in the structural acoustics literature. Its main contribution is the consideration of a heterogeneous background medium with varying parameters and the effects of polarization of elastic waves. The paper also outlines some of the remarkable features of the diffusive regime: depolarization of waves, energy equipartition, and asymptotic Fick's law.

A transport model and numerical simulation of the high-frequency dynamics of three-dimensional beam trusses

“…They are also called efficiencies in the dedicated literature. The above-presented relations constitute the boundary/interface conditions to be used to solve the transport equations (34). It should be noted that they remain valid everywhere within the beams if the convention q pp ab ¼ 0 and s pp ab ¼ d ab is adopted for all s 6 ¼ s 0 .…”

“…Discontinuities in the velocity fields at the junction do not allow one to use classical finite element method to solve numerically the Liouville equations (34) with the interface condition of Eq. (37) or Eq.…”

“…It can be used to determine the reflected and transmitted wavenumbers k p a from either side of the interface given an incident wavenumber |k|. Moreover, power flow (33) is conserved across the junction owing to the Rankine-Hugoniot condition written for transport equations (34). 49 For example, the energy density traveling in beam 1 and impinging the junction at s 0 with an incident wave vector k such that k Át 1 > 0 is partially reflected and partially transmitted, whereas the overall power flow remains constant.…”

“…The high-frequency energy density propagation in a beam is described by transport equations (34). In order to use this model for complex three-dimensional beam trusses, the reflection/transmission processes at the junctions have to be described.…”

“…Thus, they are well suited for late time simulations in order to exhibit the diffusive limit of the transport equations. 30,34 These methods have been applied to assemblies of twodimensional beams or plates, 35 as well as randomly heterogeneous shells. 36,37 The main objectives of this paper are twofold.…”