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  • 1.
    Muntean, Adrian
    et al.
    Technische Universiteit Eindhoven.
    van Noorden, T. L.
    University of Erlangen-Nürnberg, Martensstraße 3, Erlangen 91058, Germany .
    Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity2013In: European journal of applied mathematics (Print), ISSN 0956-7925, E-ISSN 1469-4425, Vol. 24, no 5, p. 657-677Article in journal (Refereed)
    Abstract [en]

    We prove an upper bound for the convergence rate of the homogenization limit epsilon -> 0 for a linear transmission problem for a advection-diffusion(-reaction) system posed in areas with low and high diffusivity, where epsilon is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493-516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.

  • 2.
    van Meurs, Patrick
    et al.
    Eindhoven University of Technology, Netherlands.
    Muntean, Adrian
    Eindhoven University of Technology, Netherlands .
    Peletier, M.A.
    Eindhoven University of Technology, Netherlands.
    Upscaling of dislocation walls in finite domains2014In: European journal of applied mathematics (Print), ISSN 0956-7925, E-ISSN 1469-4425, Vol. 25, no 6, p. 749-781Article in journal (Refereed)
    Abstract [en]

    We wish to understand the macroscopic plastic behaviour of metals by upscaling the micromechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of Gamma-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.

  • 3.
    van Noorden, T.L.
    et al.
    Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
    Muntean, Adrian
    Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
    Homogenisation of a locally periodic medium with areas of low and high diffusivity2011In: European journal of applied mathematics (Print), ISSN 0956-7925, E-ISSN 1469-4425, Vol. 22, no 5, p. 493-516Article in journal (Refereed)
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