# Homogenization Techniques for Lower Dimensional Structures

@inproceedings{Dobberschtz2012HomogenizationTF, title={Homogenization Techniques for Lower Dimensional Structures}, author={S{\"o}ren Dobbersch{\"u}tz}, year={2012} }

This thesis is concerned with extensions and applications of the theory of periodic unfolding in the field of (mathematical) homogenization. The first part extends the applicability of homogenization in domains with evolving microstructure to the case of evolving hypersurfaces: We consider a diffusion-reaction equation inside a perforated domain, where also surface diffusion and reaction takes place. Upon a transformation to a referential geometry, we (formally) obtain a transformed set of…

## Figures and Tables from this paper

## 10 Citations

Homogenization of Thermoelasticity Systems Describing Phase Transformations

- Mathematics
- 2018

This thesis is concerned with the mathematical homogenization of thermoelasticity models with moving boundary describing solid-solid phase transformations occurring in highly heterogeneous, two-phase…

Applications of periodic unfolding on manifolds

- Mathematics
- 2018

Abstract We show how the newly developed method of periodic unfolding on Riemannian manifolds can be applied to PDE problems: we consider the homogenization of an elliptic model problem. In the…

Homogenization of a diffusion‐reaction system with surface exchange and evolving hypersurface

- Mathematics
- 2015

This paper is concerned with the homogenization of a diffusion‐reaction process in a domain undergoing an evolution of the microstructure. The main novelty is the consideration of a chemical process…

The construction of periodic unfolding operators on some compact Riemannian manifolds

- Mathematics
- 2014

Abstract. The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the “flat” Euclidean…

Homogenization of a locally periodic time-dependent domain

- MathematicsCommunications on Pure & Applied Analysis
- 2020

We consider the homogenization of a Robin boundary value problem in a locally periodic perforated domain which is also time-dependent. We aim at justifying the homogenization limit, that we derive…

Corrector Estimates for the Homogenization of a Two-Scale Thermoelasticity Problem With a Priori Known Phase Transformations

- Mathematics
- 2017

We investigate corrector estimates for the solutions of a thermoelasticity problem posed in a highly heterogeneous two-phase medium and its corresponding two-scale thermoelasticity model which was…

Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an H^{1,p} Setting

- Mathematics
- 2013

The processes of chemical transport in porous media are extensively studied in the fields of applied mathematics, material science, chemical engineering etc. A porous medium (e.g. concrete, soil,…

Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with a priori known phase transformations

- Mathematics
- 2016

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two‐phase medium. The medium in question consists of a connected matrix with disconnected, initially…

Homogenization of evolutionary Stokes-Cahn-Hilliard equations for two-phase porous media flow

- MathematicsAsymptot. Anal.
- 2017

This work considers homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects and obtains upscaled equations for the considered model by a rigorous two-scale convergence approach.

## References

SHOWING 1-10 OF 104 REFERENCES

Stochastic two-scale convergence in the mean and applications.

- Mathematics
- 1994

It is well known that the modelling of physical processes in strongly inhomogeneous media leads to the study of differential equations with rapidly varying coefficients. More precisely, if the scale…

Homogenization and two-scale convergence

- Mathematics
- 1992

Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2…

Derivation of boundary conditions at a curved contact interface between a free fluid and a porous medium via homogenisation theory

- Mathematics
- 2009

In soil chemistry or marine microbiology (for example when dealing with marine aggregates), one often encounters situations where porous bodies are suspended in a fluid. In this context, the question…

A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media

- Mathematics
- 2010

Homogenisation of a locally periodic medium with areas of low and high diffusivity

- MathematicsEuropean Journal of Applied Mathematics
- 2011

We aim at understanding transport in porous materials consisting of regions with both high and low diffusivities. We apply a formal homogenisation procedure to the case where the heterogeneities are…

Two-Scale Convergence On Periodic Surfaces And Applications

- Mathematics
- 1995

This paper is concerned with the homogenization of model problems in periodic porous media when important phenomena occur on the boundaries of the pores. To this end, we generalize the notion of…

Homogenization of a reaction-diffusion system modeling sulfate corrosion in locally-periodic perforated domains

- Mathematics
- 2009

We discuss a reaction–diffusion system modeling concrete corrosion in sewer pipes. The system is coupled, semi-linear, and partially dissipative. It is defined on a locally-periodic perforated domain…

Two-scale models for reactive transport and evolving microstructure

- Mathematics
- 2008

Reactive transport in materials with a complex microstructure is a phenomenon that frequently occurs in nature and in technical applications. An example are chemical reactions taking place in the…

Effective Transmission Conditions for Reaction-Diffusion Processes in Domains Separated by an Interface

- MathematicsSIAM J. Math. Anal.
- 2007

An effective model is derived which consists of the reaction-diffusion equations on two domains separated by an interface together with appropriate transmission conditions across this interface, based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers.