School of Mathematics,

University of Minnesota

Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetry-related states on a fine scale to minimize energy.

We have developed an approximation theory for crystal microstructure which gives an analysis of the stability of macroscopic variables with respect to small energy perturbations. This theory has been applied to develop and analyze several numerical methods for the approximation of martensitic microstructure and ferromagnetic microstructure.

Recently, we have developed mathematical models and computational methods for microvalves and micropumps based on active thin films of the crystals described above. A promising application for such microvalves and micropumps is to implanted drug delivery systems. One advantage of implanted micromachines is that stronger medicines could be used if they could be applied locally, in contrast to most current drug delivery systems in which the theraputic agent is applied uniformly over the bloodstream.