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Multiscale Modeling of Materials — the Role of Analysis

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Trends in Nonlinear Analysis

Abstract

We present two case studies how analysis can be used to derive a hierarchy of models to capture multiscale behavior of materials. The determination, via Γ-convergence, of the thin film limit of micromagnetism delivers a reduced two-dimensional model for soft ferromagnetic films which justifies previously known theories for small fields and extends them to the regime of field penetration. The analytic evaluation of the quasiconvex envelope of the microscopic energy density of nematic elastomers allows efficient numerical computations with finite elements and shows the existence of a new “smectic” phase. In both cases, the numerical solution of the coarse-grained model is complemented by a reconstruction of the microscopic pattern associated with the reduced field.

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Author information

Authors and Affiliations

  1. Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, 04103, Leipzig, Germany

    Sergio Conti, Antonio DeSimone & Stefan Müller

  2. Mathematics Department, University of Maryland, College Park, MD, 20742-4015, USA

    Georg Dolzmann

  3. Institut für Angewandte Mathematik, Wegelerstraße 10, 53115, Bonn, Germany

    Felix Otto

Authors
  1. Sergio Conti
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  2. Antonio DeSimone
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  3. Georg Dolzmann
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  4. Stefan Müller
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  5. Felix Otto
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Editor information

Editors and Affiliations

  1. Mathematics Department and Center for Scientific Computing, University of Warwick, CV4 7AL, Coventry, UK

    Markus Kirkilionis

  2. Institute of Scientific Computing, University of Heidelberg, Im Neuheimer Feld 368, 69120, Heidelberg, Germany

    Susanne Krömker

  3. Institute of Applied Mathematics, University of Heidelberg, Im Neuheimer Feld 293, 69120, Heidelberg, Germany

    Rolf Rannacher

  4. Department of Mathematics, University of Heidelberg, Im Neuheimer Feld 288, 69120, Heidelberg, Germany

    Friedrich Tomi

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Conti, S., DeSimone, A., Dolzmann, G., Müller, S., Otto, F. (2003). Multiscale Modeling of Materials — the Role of Analysis. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_11

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  • DOI: https://doi.org/10.1007/978-3-662-05281-5_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07916-0

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