Periodic unfolding and homogenization

Cioranescu, Doina; Damlamian, Alain; Griso, Georges

doi:10.1016/s1631-073x(02)02429-9

Cited by 382 publications

(399 citation statements)

References 8 publications

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“…The statement of the approximate model (3) derived with the two-scale transform was announced in (Lenczner, 1997). Then several proofs have been published, in (Lenczner and Senouci-Bereksi, 1999), (Casado-Díaz, 2000), (Cioranescu et al, 2002), (Lenczner and Smith, 2007), and (Cioranescu et al, 2008). Here we follow the proof in (Lenczner and Smith, 2007) where an effort has been made to formulate proofs in an algebraic way and to avoid any abstract reasoning in the sequences of formal transformations.…”

Section: Running Examplementioning

confidence: 99%

“…It combines mathematical and computer science tools. The mathematical tool is the two-scale transform originally introduced in (Lenczner, 1997;Casado-Díaz, 2000;Cioranescu et al, 2002) to model periodic and thin structures, and also referred as the unfolding method. We have extended its domain of application to cover in the same time homogenization of periodic media, see for instance Bensoussan et al (1978), and methods of asymptotic analysis for thin domains, see Ciarlet (1988).…”

Section: Introductionmentioning

confidence: 99%

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A symbolic transformation language and its application to a multiscale method

Belkhir¹,

Giorgetti²,

Lenczner³

2014

Journal of Symbolic Computation

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The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro-and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales.In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple TM package for rule-based programming, rewriting strategies and their combination with standard Maple TM code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.

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Section: Running Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A symbolic transformation language and its application to a multiscale method

Belkhir¹,

Giorgetti²,

Lenczner³

2014

Journal of Symbolic Computation

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“…For this purpose we need to study more carefully the behavior of u ε near Γ ε . This is carried out using an original adaptation of the unfolding method, [2], [9], [16], which is very related to the twoscale convergence method, [1], [21], [23]. The idea is to introduce suitable changes of variables which transform every periodic cell into a simpler reference set by using a supplementary variable (microscopic variable).…”

Section: Sketch Of the Proof Of Theorem 1 Sincementioning

confidence: 99%

On the Navier boundary condition for viscous fluids in rough domains

Casado–Díaz

Luna-Laynez

Suárez‐Grau

2012

SeMA

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In this paper we review some recent results concerning the study of the asymptotic behavior of viscous fluids in rough domains assuming Navier boundary conditions on the rough boundary. Our main interest is to study the relation between both the adherence and the Navier boundary conditions in the case of a boundary with weak rugosities. We show that the roughness acts on the fluid as a friction term. In particular, if the roughness is sufficiently strong, Navier condition implies adherence condition. This generalizes previous results of other authors.

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“…The primary objective of homogenization, i.e., of the micro-macro approach, is to replace a system with periodically varying coefficients by a limiting homogeneous system that facilitates computation. The approach that we take here is based on the periodic unfolding method presented in [8,9,7]. We first mention other efforts on homogenization of Maxwell's equations.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we use the periodic unfolding method introduced in [8] in the abstract framework of stationary elliptic equations. The study in this paper considers constitutive laws that take into account bianisotropy, chiral symmetry, thermal and memory effects.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

et al. 2006

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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 cases of circular and square microstructures 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. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTESThe original document contains color images. ABSTRACTIn this paper we employ the periodic unfolding method for simulating the electromagnetic eld in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the e ective parameters for a Debye dielectric medium in the cases of circular and square microstructures in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic eld is much larger than the relevant dimensions of the microstructure.

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Periodic unfolding and homogenization

Cited by 382 publications

References 8 publications

A symbolic transformation language and its application to a multiscale method

A symbolic transformation language and its application to a multiscale method

On the Navier boundary condition for viscous fluids in rough domains

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

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