321 research outputs found
Asymptotic analysis of boundary layer correctors in periodic homogenization
This paper is devoted to the asymptotic analysis of boundary layers in
periodic homogenization. We investigate the behaviour of the boundary layer
corrector, defined in the half-space , far away
from the boundary and prove the convergence towards a constant vector field,
the boundary layer tail. This problem happens to depend strongly on the way the
boundary intersects the underlying microstructure. Our
study complements the previous results obtained on the one hand for
, and on the other hand for satisfying a small divisors assumption. We tackle the case of arbitrary
using ergodicity of the boundary layer along
. Moreover, we get an asymptotic expansion of Poisson's
kernel , associated to the elliptic operator and , for .
Finally, we show that, in general, convergence towards the boundary layer tail
can be arbitrarily slow, which makes the general case very different from the
rational or the small divisors one.Comment: 39 page
Cell averaging two-scale convergence: Applications to periodic homogenization
The aim of the paper is to introduce an alternative notion of two-scale
convergence which gives a more natural modeling approach to the homogenization
of partial differential equations with periodically oscillating coefficients:
while removing the bother of the admissibility of test functions, it
nevertheless simplifies the proof of all the standard compactness results which
made classical two-scale convergence very worthy of interest: bounded sequences
in and are proven
to be relatively compact with respect to this new type of convergence. The
strengths of the notion are highlighted on the classical homogenization problem
of linear second-order elliptic equations for which first order boundary
corrector-type results are also established. Eventually, possible weaknesses of
the method are pointed out on a nonlinear problem: the weak two-scale
compactness result for -valued stationary harmonic maps.Comment: 20 pages, 2 Figure
Quantitative analysis of boundary layers in periodic homogenization
We prove quantitative estimates on the rate of convergence for the
oscillating Dirichlet problem in periodic homogenization of divergence-form
uniformly elliptic systems. The estimates are optimal in dimensions larger than
three and new in every dimension. We also prove a regularity estimate on the
homogenized boundary condition.Comment: 41 pages; updated to comment on results of arXiv:1610.0527
When a thin periodic layer meets corners: asymptotic analysis of a singular Poisson problem
The present work deals with the resolution of the Poisson equation in a
bounded domain made of a thin and periodic layer of finite length placed into a
homogeneous medium. We provide and justify a high order asymptotic expansion
which takes into account the boundary layer effect occurring in the vicinity of
the periodic layer as well as the corner singularities appearing in the
neighborhood of the extremities of the layer. Our approach combines the method
of matched asymptotic expansions and the method of periodic surface
homogenization, and a complete justification is included in the paper or its
appendix.Comment: 58 page
Blood-Flow Modelling Along and Trough a Braided Multi-Layer Metallic Stent
In this work we study the hemodynamics in a stented artery connected either
to a collateral artery or to an aneurysmal sac. The blood flow is driven by the
pressure drop. Our aim is to characterize the flow-rate and the pressure in the
contiguous zone to the main artery: using boundary layer theory we construct a
homogenized first order approximation with respect to epsilon, the size of the
stent's wires. This provides an explicit expression of the velocity profile
through and along the stent. The profile depends only on the input/output
pressure data of the problem and some homogenized constant quantities: it is
explicit. In the collateral artery this gives the flow-rate. In the case of the
aneurysm, it shows that : (i) the zeroth order term of the pressure in the sac
equals the averaged pressure along the stent in the main artery, (ii) the
presence of the stent inverses the rotation of the vortex. Extending the tools
set up in [Bonnetier et al, Adv. Math. Fluids, 2009, Milisic, Meth. Apl. Ann.,
2009] we prove rigorously that our asymptotic approximation is first order
accurate with respect to . We derive then new implicit interface conditions
that our approximation formally satisfies, generalizing our analysis to other
possible geometrical configurations. In the last part we provide numerical
results that illustrate and validate the theoretical approach
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