98-31   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Andreas Prohl:

Multiscale resolution in the computation of crystalline microstructure

The minimization of non-convex problems arises in material sciences, where deformations of certain alloys exhibit laminated microstructures. For example, minimizing sequences of the non-convex Ericksen-James energy can be associated with deformations in martensitic materials that are observed in experiments. The application of conforming and classical nonconforming (Crouxeiz-Raviart) finite element methods in minimization problems that involve non-convex energy densities is well-known to give minimizers with their quality being highly dependent on the underlying triangulation. --- For conforming finite elements, bounds can be given that ensure that minimizers create an energy that is not bigger than ${\cal O}(h^{1/2})$ (with ${\rm inf}_{v \in {\cal A}} {\cal E}(v) = 0$ for the continuous model). The proof is accomplished via explicit construction of a discrete deformation that exhibits laminated microstructure and satisfies this upper bound. In the present work, this bound is sharpened to ${\cal O}\bigl(h^{2/3}\, (1+{\rm log} \frac{1}{h})\bigr)$ by constructing discrete deformations that have needle-like wedge microstructure in the vicinity of the boundary --- in addition to laminated microstructure in the interior of the domain ---, and thus enjoy the improved upper energy bound.

Keywords: finite element method, non-convex minimization, multi-well problem, microstructure, multiscale, nonlinear elasticity, shape-memory alloy, materials science.

Mail an Jens Burmeister
[Thu Feb 19 18:56:34 2009]
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