We consider applications, illustration and concrete numerical treatments of some homogenization results on Stokes flow in porous media. In particular, we compute the global permeability tensor corresponding to an unidirectional array of circular fibers for several volume-fractions. A 3-dimensional problem is also considered.
The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The...
In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are -periodic and of size . We show that, as , the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...
In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...
Integral representation of relaxed energies and of Γ-limits of functionals are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.
We study the initial boundary value problem for the system of thermoelasticity in a sequence of perforated cylindrical domains , s = 1,2,... We prove that as s → ∞, the solution of the problem converges in appropriate topologies to the solution of a limit initial boundary value problem of the same type but containing some additional terms which are expressed in terms of quantities related to the geometry of . We give an explicit construction of that limit problem.
For a fixed bounded open set , a sequence of open sets and a sequence of sets , we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on , satisfying Neumann boundary conditions on and Dirichlet boundary conditions on . We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on and locally.