Periodic Unfolding Method Research Articles

The Periodic Unfolding Method in Domains with Holes

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with nonhomogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding.

The periodic unfolding method for a class of imperfect transmission problems

The periodic unfolding method was introduced by D. Cioranescu, A. Damlamian and G. Griso for studying the classical periodic homogenization in fixed domains and more recently extended to periodically perforated domains by D. Cioranescu, A. Damlamian, P. Donato, G. Griso, and R. Zaki. Here, the method is adapted to two-component domains which are separated by a periodic interface. The unfolding method is then applied to an elliptic problem with a jump of the solution on the interface, which is proportional to the flux and depends on a real parameter. We prove some homogenization and corrector results, which recover and complete those previously obtained by the first author and S. Monsurr`o. Bibliography: 32 titles. Illustrations: 2 figures.

Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of typeMu= −N∇p+Fwith suitable matricesMandNwithFdepending on the right-hand side in the bulk term and in the boundary condition.

Homogenization of a class of Neumann problems in perforated domains

We consider elliptic problems in periodically perforated domains in R N , N 3, with nonhomogeneous Neumann conditions on the boundary of the holes. The aim is to give the asymptotic behavior of the solutions as the period e goes to zero. Two geometries are considered. In the first one, all the holes are small, i.e., their size is of order of er(e) with r(e) → 0. The second geometry is more general, there are small holes as before but also holes of size of the order of e (the last ones corresponding to the classical homogenization situation). Our study is performed by the periodic unfolding method from C. R. Acad. Sci. Paris Ser. I 335 (2002), 99-104, adapted to the case of holes of size er(e )( seeJ. Math. Pures Appl. 89 (2008), 248- 277). The use of this method allows us to study second-order operators with highly oscillating coefficients and so, to generalize here the results of RAIRO Model. Math. Anal. Numer. 4(22) (1988), 561-608. In both cases, if r(e) = exp(N/N − 1), an additional term appears in the right-hand side of the limit equation.

Homogenization of oscillating boundaries

A variational problem on a sequence of 2-dimensional domains with oscillating boundaries is studied. Using the periodic unfolding method, the homogenized problem is obtained in the limit as the period length approaches zero. Several extensions are also given. In this framework, a result of strong convergence is obtained which is new.

Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes

We consider elliptic problems in periodically perforated domains in R N , with nonhomogeneous Neumann conditions on the boundary of the holes. We are interested in the asymptotic behavior of the solutions as the period ε goes to zero. In a first case all the holes are “small”, i.e., are of size r ( ε ) with r ( ε ) / ε → 0 . In the second case, there are again small holes but also holes of size ε. We use the periodic unfolding method introduced in Cioranescu et al. (2002), which allows us to study second order operators with highly oscillating coefficients and so, to generalize here the results of Conca and Donato (1988). In both cases, if r ( ε ) = exp ( N / N − 1 ) , an additional term appears in the right-hand side of the limit equation. To cite this article: A. Ould Hammouda, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The Periodic Unfolding Method in Homogenization

The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...

The periodic unfolding method for perforated domains and Neumann sieve models

The periodic unfolding method, introduced in [D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104], was developed to study the limit behavior of periodic problems depending on a small parameter ε. The same philosophy applies to a range of periodic problems with small parameters and with a specific period (as well as to almost any combinations thereof). One example is the so-called Neumann sieve. In this work, we present these extensions and show how they apply to known results and allow for generalizations (some in dimension N ⩾ 3 only). The case of the Neumann sieve is treated in details. This approach is significantly simpler than the original ones, both in spirit and in practice.

Homogenization of thin piezoelectric perforated shells

We rigorously establish the existence of the limit homogeneous constitutive law of a piezoelectric composite made of periodically perforated microstructures and whose reference configuration is a thin shell with fixed thickness. We deal with an extension of the Koiter shell model in which the three curvilinear coordinates of the elastic displacement field and the electric potential are coupled. By letting the size of the microstructure going to zero and by using the periodic unfolding method combined with the Korn's inequality in perforated domains, we obtain the limit model.

Parameter estimation versus homogenization techniques in time-domain characterization of composite dielectrics

We compare an inverse problem approach to parameter estimation with homogenization techniques for characterizing the electrical response of composite dielectric materials in the time domain. We first consider an homogenization method, based on the periodic unfolding method, to identify the dielectric response of a complex material with heterogeneous micro-structures which are described by spatially periodic parameters. We also consider electromagnetic interrogation problems for complex materials assuming multiple polarization mechanisms with distributions of parameters. An inverse problem formulation is devised to determine effective polarization parameters specific to the interrogation problem. We compare the results of these two approaches with the classical Maxwell-Garnett mixing model and a simplified model with a weighted average of parameters. Numerical results are presented for a specific example involving a mixture of ethanol and water (modeled with multiple Debye mechanisms). A comparison between each approach is made in the frequency domain (e.g., Cole-Cole diagrams), as well as in the time domain (e.g., plots of susceptibility kernels).

Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method

The pointwise gradient constrained homogenization process, for Neumann and Dirichlet type problems, is analyzed by means of the periodic unfolding method recently introduced in [21]. Classically, the proof of the homogenization formula in presence of pointwise gradient constraints relies on elaborated measure theoretic arguments. The one proposed here is elementary: it is based on weak convergence arguments in L p spaces, coupled with suitable regularization techniques. Keywords: Homogenization, Gradient constrained problems, Periodic unfolding method Mathematics Subject Classification (2000): 49J45, 35B27, 74Q05

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

In this paper, we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the case of a circular microstructure in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure.

Periodic unfolding and Robin problems in perforated domains

The periodic unfolding method was introduced in 2002 by D. Cioranescu et al. for the study of classical periodic homogenization. In this Note, we extend this method to perforated domains introducing also a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with Robin condition on the boundary of the holes. To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Homogenization of Quasiconvex Integrals via the Periodic Unfolding Method

The homogenization problem in the general case of quasiconvex integral energies with polynomial growth, defined on vector-valued configurations, was studied by the $\G$-convergence methods in [A. Braides, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 9 (1985), pp. 313-321] and [S. Müller, Arch. Rational Mech. Anal., 99 (1987), pp. 189-212]. This paper presents a new proof by means of the periodic unfolding method introduced in [D. Cioranescu, A. Damlamian, and G. Griso, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 99-104]. It is an elementary proof since it reduces the homogenization process to a weak convergence problem in an $L^p$-type space.

Modelling of periodic electromagnetic structures bianisotropic materials with memory effects

We study the behavior of the electromagnetic field of a medium presenting periodic microstructures made of bianisotropic material. We reconsider the classical multi-scale homogenization technique by giving a new approach based upon the periodic unfolding method. The limiting homogeneous constitutive law is thus rigorously justified both in the time domain and in the frequency domain. In particular we show that the limit law differs from the initial one regarding the convolution term accounting for the memory effects.

Homogénéisation de coques minces piézoélectriques perforées

In this Note we establish the limit constitutive law of a piezoelectric material with periodically perforated microstructures and whose reference configuration is a thin shell with fixed thickness. The justification of the new associated model (we show that the limit global and local problems are more complicated than the intial one) is obtained using the periodic unfolding method introduced by Cioranescu, Damlamian and Griso. To cite this article: M. Ghergu et al., C. R. Mecanique 333 (2005).

Reiterated homogenization for elliptic operators

In this Note, using the periodic unfolding method (see D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 335 (1) (2002) 99–104), we study reiterated homogenization for equations of the form − div ( a ε ( x , D u ε ) ) = f , where a ε is Carathéodory and satisfies some monotone and growth conditions. We show that if we assume that T δ ( ε ) ′ ( T ε ( a ε ) ) ( x , y , z , ξ ) converges, for almost all ( x , y , z ) ∈ Ω × Y × Z , to a Carathéodory operator, then the sequences u ε and D u ε converge in a certain sense to the solution ( u 0 , u ˆ , u ˜ ) of a limit variational problem, as ε → 0 . In particular this contains the case a ε ( x , ξ ) = a ( x , x ε , { x / ε } Y δ ( ε ) , ξ ) , where a is periodic in the second and third arguments, and continuous in each argument. To cite this article: N. Meunier, J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Interior error estimate for periodic homogenization

The aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the periodic unfolding method. The interior error estimates are obtained by transposition without any supplementary hypothesis of regularity on correctors. This error is of order ɛ. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Homogenization of nonlinear integrals via the periodic unfolding method

We consider the periodic homogenization of nonlinear integral energies with polynomial growth. The study is carried out by the periodic unfolding method which reduces the homogenization process to a weak convergence problem in a Lebesgue space. To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Homogenization of nonlinear integrals via the periodic unfolding method

Homogenization of nonlinear integrals via the periodic unfolding method