scholarly journals The Periodic Unfolding Method in Homogenization

2008 ◽  
Vol 40 (4) ◽  
pp. 1585-1620 ◽  
Author(s):  
D. Cioranescu ◽  
A. Damlamian ◽  
G. Griso
Keyword(s):  
Periodic Unfolding Method ◽  
Unfolding Method

The Periodic Unfolding Method

2018 ◽  
Author(s):  
Doina Cioranescu ◽  
Alain Damlamian ◽  
Georges Griso
Keyword(s):  
Periodic Unfolding Method ◽  
Unfolding Method

scholarly journals Homogenization of Perforated Elastic Structures

2020 ◽  
Vol 141 (2) ◽  
pp. 181-225
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Lipschitz Boundary ◽  
3 Dimensional ◽  
Priori Estimates ◽  
Homogenization Approach ◽  
Periodic Unfolding Method ◽  
Asymptotic Reduction ◽  
Periodic Repetition ◽  
Unfolding Method

Abstract The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

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

2006 ◽  
Vol 55 (1) ◽  
pp. 31-54 ◽  
Author(s):  
Doina Cioranescu ◽  
Alain Damlamian ◽  
Riccardo De Arcangelis
Keyword(s):  
Gradient Constraints ◽  
Periodic Unfolding Method ◽  
Unfolding Method

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

2011 ◽  
Vol 22 (4) ◽  
pp. 333-345 ◽  
Author(s):  
ANCA CAPATINA ◽  
HORIA ENE
Keyword(s):  
Boundary Condition ◽  
Porous Medium ◽  
Stokes System ◽  
Stokes Problem ◽  
Slip Boundary Condition ◽  
Perforated Domains ◽  
Slip Boundary ◽  
Periodic Unfolding Method ◽  
The Right ◽  
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.

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