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Multiscale Modeling of Materials — the Role of Analysis

In: Trends in Nonlinear Analysis

Author

Listed:
  • Sergio Conti

    (Max-Planck-Institute for Mathematics in the Sciences)

  • Antonio DeSimone

    (Max-Planck-Institute for Mathematics in the Sciences)

  • Georg Dolzmann

    (University of Maryland, Mathematics Department)

  • Stefan Müller

    (Max-Planck-Institute for Mathematics in the Sciences)

  • Felix Otto

    (Institut für Angewandte Mathematik)

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.

Suggested Citation

  • Sergio Conti & Antonio DeSimone & Georg Dolzmann & Stefan Müller & Felix Otto, 2003. "Multiscale Modeling of Materials — the Role of Analysis," Springer Books, in: Markus Kirkilionis & Susanne Krömker & Rolf Rannacher & Friedrich Tomi (ed.), Trends in Nonlinear Analysis, chapter 11, pages 375-408, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-05281-5_11
    DOI: 10.1007/978-3-662-05281-5_11
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