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Multiscale Modeling of Materials - the Role of Analysis
Sergio Conti, Antonio DeSimone, Georg Dolzmann, Stefan Müller and Felix Otto
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 Gamma-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.