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Homogenization Method and Multiscale Modeling
Eindhoven University of Technology, The Netherlands. (Mathematics)ORCID iD: 0000-0002-1160-0007
Kyushu University, Japan.
2011 (English)Book (Other academic)
Abstract [en]

This mini-course addresses graduate students and young researchers in mathematics and engineering sciences interested in applying both formal and rigorous averaging methods to real-life problems described by means of partial differential equations (PDEs) posed in heterogeneous media. As a background application scenario we choose to look at the interplay between reaction, diffusion and flow in periodic porous materials, but broadly speaking, a similar procedure would apply for, e.g., acoustic and/or electromagnetic wave propagation phenomena in composite (periodic) media as well. We start off with the study of oscillatory elliptic PDEs formulated firstly in fixed and, afterwards, in periodically-perforated domains. We remove the oscillations by means of a (formal) asymptotic homogenization method. The output of this procedure consists of a “guessed” averaged model equations and explicit rules (based on cell problems) for computing the effective coefficients. As second step, we introduce the concept of two-scale convergence (and correspondingly, the two-scale compactness) in the sense of Allaire and Nguetseng and derive rigorously the averaged PDE models and coefficients obtained previously. This step uses the framework of Sobolev and Bochner spaces and relies on basic tools like weak convergence methods, compact embeddings as well as extension theorems in Sobolev spaces. We particularly emphasize the role the choice of microstructures (pores, perforations, subgrids, etc.) plays in performing the overall averaging procedure. Finally, we focus our attention on a two-scale partly dissipative reaction-diffusion system with periodically distributed microstructure modeling chemical attack on concrete structures. We present a two-scale finite difference scheme able to approximate the unique weak solution to the two-scale system and prove its convergence. We illustrate numerically the typical micro-macro behavior of the active concentrations involved in the corrosion process and give details on how a two-scale FD scheme can be implemented in C. The main objective of the course is to endow the audience with a rather flexible mathematical homogenization tool so that he/she can quickly start applying this averaging methodology to other PDEs scenarios describing physico-chemical processes in media with microstructures.

Place, publisher, year, edition, pages
Fukuoka, Japan: Kyushu University , 2011. , 76 p.
Series
COE lecture note series, ISSN 1881-4042 ; 34
Keyword [en]
Formal asymptotic homogenization, two-scale convergence, materials with microstructures, concrete corrosion, two-scale finite difference method, programming in C
National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-39828OAI: oai:DiVA.org:kau-39828DiVA: diva2:901195
Available from: 2016-02-06 Created: 2016-02-06 Last updated: 2017-11-22Bibliographically approved

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Muntean, Adrian

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CiteExportLink to record
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Citation style
  • apa
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