The periodic unfolding method for a class of imperfect transmission problems

Donato, Patrizia; Nguyen, Kim Hang Le; Tardieu, R.

doi:10.1007/s10958-011-0443-2

Cited by 54 publications

(131 citation statements)

References 18 publications

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“…The results of this section are extracted from [4], [5], [6] and [9], with the slight difference that we consider here the time dependent case. As the time variable plays the role of a parameter, the extension is straightforward, which is why we do not recall any proof here.…”

Section: Some Results From the Unfolding Methods In Homogenizationmentioning

confidence: 99%

“…The connec- Figure 1: The periodic cell Y , containing a simplified biological cell without membrane tivity assumption is made here in order to simplify some proofs and notations. As explained in [9], this assumption can easily be lifted. However, once this connectivity assumption made, we need to impose that Γ m does not crosses ∂Y , if we want our model to remain realistic (the entire boundary of the biological cells must be recovered by a membrane).…”

Section: Model Problemmentioning

confidence: 99%

“…The next theorem, extracted from [6] and [9], describes the convergence properties of the unfolding operators acting on gradients :…”

Section: The Unfolding Operator and Gradientsmentioning

confidence: 99%

“…It is essentially a rewriting of a result of [9], in a form that is well suited for handling our particular problem :…”

Section: The Unfolding Operator On the Boundarymentioning

confidence: 99%

“…These limit models are similar to those that have already appeared in the literature when homogenizing elliptic problems with jump condition (see [14]- [15]) using two-scale convergence. Such problems have also been addressed through the periodic unfolding method in [9]. The paper will be organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

Coatléven

2015

ASY

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To cite this version:Julien Coatléven. Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method. Asymptotic Analysis, IOS Press, 2015, 93 (3) Abstract Diffusion Magnetic Resonance Imaging (dMRI) is a promising tool to obtain useful information on cellular structure when applied to biological tissues. A coupled macroscopic model has been introduced recently through formal homogenization to model dMRI's signal attenuation. This model was based on a particular scaling of the permeability condition modeling cellular membranes. In this article, we explore all the possible scalings and mathematically justify the corresponding limit models, using the periodic unfolding method. We also illustrate through numerical simulations the respective behavior of the limit models when compared to dMRI measurements.

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Section: Some Results From the Unfolding Methods In Homogenizationmentioning

confidence: 99%

Section: Model Problemmentioning

confidence: 99%

“…The next theorem, extracted from [6] and [9], describes the convergence properties of the unfolding operators acting on gradients :…”

Section: The Unfolding Operator and Gradientsmentioning

confidence: 99%

“…It is essentially a rewriting of a result of [9], in a form that is well suited for handling our particular problem :…”

Section: The Unfolding Operator On the Boundarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

Coatléven

2015

ASY

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Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains

Kovtunenko

Reichelt

Zubkova

2019

Math Methods in App Sciences

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This paper is devoted to the homogenization of a nonlinear transmission problem stated in a two-phase domain. We consider a system of linear diffusion equations defined in a periodic domain consisting of two disjoint phases that are both connected sets separated by a thin interface. Depending on the field variables, at the interface, nonlinear conditions are imposed to describe interface reactions. In the variational setting of the problem, we prove the homogenization theorem and a bidomain averaged model. The periodic unfolding technique is used to obtain the residual error estimate with a first-order corrector. KEYWORDS bidomain model, corrector estimates, diffusion problem, nonlinear transmission conditions, periodic unfolding technique MSC CLASSIFICATION 35B27; 35M10; 82C24This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Heat conduction in composite media involving imperfect contact and perfectly conductive inclusions

Amar

Andreucci

Timofte

2022

Math Methods in App Sciences

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We study the thermal properties of a composite material made up of a medium hosting an ε-periodic array of perfect thermal conductors. The thermal potentials of the two phases are coupled across the interface through a non-standard imperfect contact transmission condition, involving the external thermal flux and a proportionality coefficient D 0 ε α , where α ∈ R is a scaling parameter and D 0 > 0 accounts for the imperfect contact. We perform the homogenization for all the scalings α ∈ R and we compare the resulting models with the perfect contact transmission case addressed in [6,7], letting D 0 → 0, where this is meaningful.

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The periodic unfolding method for a class of imperfect transmission problems

Cited by 54 publications

References 18 publications

Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains

Heat conduction in composite media involving imperfect contact and perfectly conductive inclusions

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