The Periodic Unfolding Method in Domains with Holes

Cioranescu, Doina; Damlamian, Alain; Donato, Patrizia; Griso, Georges; Zaki, Rachad

doi:10.1137/100817942

Cited by 162 publications

(251 citation statements)

References 15 publications

Supporting

Mentioning

243

Contrasting

Unclassified

Order By: Relevance

“…Applying estimates in Lemmata 2.3, 2.5 and Convergence Theorem [7,8], see Theorem 5.3 in Appendix, implies the convergences for u ε , v ε , w ε in (12). The strong convergence of r ε is achieved by showing that T…”

Section: Unfolded Limit Equations We Defineωmentioning

confidence: 92%

See 1 more Smart Citation

Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

Fatima

Muntean

Ptashnyk

2012

Applicable Analysis

View full text Add to dashboard Cite

Section: Unfolded Limit Equations We Defineωmentioning

confidence: 92%

“…(i) To show the positivity of a weak solution we consider u ε− as test function in (5), (7) , and r ε− in (8), where φ − = min{0, φ} with φ…”

Section: Microscopic Equationsmentioning

confidence: 99%

Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

Fatima

Muntean

Ptashnyk

2012

Applicable Analysis

View full text Add to dashboard Cite

“…Applying standard extension arguments (see, e.g., [1,16] or Lemma 3.1) and using the same notation for the original and extended sequences, from estimates (15) …”

Section: Convergence Resultsmentioning

confidence: 99%

“…The two-scale convergence, e.g., [3,31], and the periodic unfolding method, e.g., [15], are applied to obtain the macroscopic equations. The main mathematical difficulty in the derivation of the macroscopic problem arises from the strong coupling between the equations of poroelasticity and the system of reaction-diffusion-convection equations.…”

mentioning

confidence: 99%

Homogenization of Biomechanical Models for Plant Tissues

Piatnitski¹,

Ptashnyk

2017

Multiscale Model. Simul.

View full text Add to dashboard Cite

Abstract. In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusionconvection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques, we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field.

show abstract

“…It removes problems with the choice of space for test functions, therefore it is more natural. A comprehensive survey of the application of this method to the problems in domains with holes is described by Ciornescu, Damlamian, Donato, Griso and Zaki [Cio+12].…”

mentioning

confidence: 99%