15,451 research outputs found
Homogenization of the one-dimensional wave equation
We present a method for two-scale model derivation of the periodic
homogenization of the one-dimensional wave equation in a bounded domain. It
allows for analyzing the oscillations occurring on both microscopic and
macroscopic scales. The novelty reported here is on the asymptotic behavior of
high frequency waves and especially on the boundary conditions of the
homogenized equation. Numerical simulations are reported
The time horizon for stochastic homogenization of the one-dimensional wave equation
The wave equation with stochastic coefficients can
be classically homogenized on bounded time intervals; solutions
converge in the homogenization limit to solutions of a wave
equation with constant coefficients. This is no longer true on large
time scales: Even in the periodic case with periodicity ε, classical
homogenization fails for times of the order ε−2. We consider the
one-dimensional wave equation and are interested in the critical
time scale ε−β from where on classical homogenization fails. In
the general setting, we derive upper and lower bounds for β in
terms of the growth rate of correctors. In the specific setting
of i.i.d. coefficients with matched impedance, we show that the
critical time scale is ε−
Homogenization of the Prager model in one-dimensional plasticity
We propose a new method for the homogenization of hysteresis models of plasticity. For the one-dimensional wave equation with an elasto-plastic stress-strain relation we derive averaged equations and perform the homogenization limit for stochastic material parameters. This generalizes results of the seminal paper by Francu and KrejcÃ. Our approach rests on energy methods for partial differential equations and provides short proofs without recurrence to hysteresis operator theory. It has the potential to be extended to the higher dimensional case
An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form
The present study concerns the numerical homogenization of second order
hyperbolic equations in non-divergence form, where the model problem includes a
rapidly oscillating coefficient function. These small scales influence the
large scale behavior, hence their effects should be accurately modelled in a
numerical simulation. A direct numerical simulation is prohibitively expensive
since a minimum of two points per wavelength are needed to resolve the small
scales. A multiscale method, under the equation free methodology, is proposed
to approximate the coarse scale behaviour of the exact solution at a cost
independent of the small scales in the problem. We prove convergence rates for
the upscaled quantities in one as well as in multi-dimensional periodic
settings. Moreover, numerical results in one and two dimensions are provided to
support the theory
High frequency homogenization for travelling waves in periodic media
We consider high frequency homogenization in periodic media for travelling
waves of several different equations: the wave equation for scalar-valued waves
such as acoustics; the wave equation for vector-valued waves such as
electromagnetism and elasticity; and a system that encompasses the
Schr{\"o}dinger equation. This homogenization applies when the wavelength is of
the order of the size of the medium periodicity cell. The travelling wave is
assumed to be the sum of two waves: a modulated Bloch carrier wave having
crystal wave vector \Bk and frequency plus a modulated Bloch
carrier wave having crystal wave vector \Bm and frequency . We
derive effective equations for the modulating functions, and then prove that
there is no coupling in the effective equations between the two different waves
both in the scalar and the system cases. To be precise, we prove that there is
no coupling unless and (\Bk-\Bm)\odot\Lambda \in 2\pi
\mathbb Z^d, where is the
periodicity cell of the medium and for any two vectors the product is defined to be
the vector This last condition forces the
carrier waves to be equivalent Bloch waves meaning that the coupling constants
in the system of effective equations vanish.
We use two-scale analysis and some new weak-convergence type lemmas. The
analysis is not at the same level of rigor as that of Allaire and coworkers who
use two-scale convergence theory to treat the problem, but has the advantage of
simplicity which will allow it to be easily extended to the case where there is
degeneracy of the Bloch eigenvalue.Comment: 30 pages, Proceedings of the Royal Society A, 201
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