The Computational Modeling of Crystalline Materials Using a Stochastic Variational Principle ⋆ Dennis Cox 1 , Petr Klouˇ cek 2 , and Daniel R. Reynolds 2 1 Department of Statistics, Rice University, 6100 Main Street, Houston, TX 77005, USA 2 Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005, USA Abstract. We introduce a variational principle suitable for the computational modeling of crystalline materials. We consider a class of materials that are de- scribed by non-quasiconvex variational integrals. We are further focused on equlib- ria of such materials that have non-attainment structure, i.e., Dirichlet bound- ary conditions prohibit these variational integrals from attaining their infima. Consequently, the equilibrium is described by probablity distributions. The new variational principle provides the possibility to use standard optimization tools to achieve stochastic equilibrium states starting from given initial deterministic states. 1 I NTRODUCTION Recently, there has been a great deal of research into the modeling and use of so-called Smart Materials. These materials have the ability to undergo internal physical transfor- mations, which may be used to do work in ways and places that traditional engineer- ing materials cannot. Such new materials include composites, ceramics, liquid crystals, biomaterials, ferromagnetics and shape memory alloys. Advances in micro-machines, damping mechanisms, high-resolution displays and superconductors are a few of the applications for which they are being designed. These materials are singled out by their unique ability to undergo a temperature dependent crystalline phase transformation, known as the Martensitic Transformation. At higher temperatures Shape Memory Alloys exhibit a stiff, cubic crystalline lattice structure known as the Austenitic phase. In lower temperatures, these alloys change to exhibit a more easily deformable tetragonal lattice structure, having many crystallo- graphically equivalent states. The reason for the name comes from the process where, if one begins by establishing a reference configuration in the Austenitic phase, then cools ⋆ The first author was supported in part by the grant NSF DMS–9971797. The other two authors were supported in part by the grant NSF DMS–0107539, by the Los Alamos National Labo- ratory Computer Science Institute (LACSI) through LANL contract number 03891–99–23, as part of the prime contract W–7405–ENG-36 between the Department of Energy and the Re- gents of the University of California, by the grant NASA SECTP NAG5–8136, by the grant from Schlumberger Foundation, and by the grant from TRW Foundation. The computations in this paper were performed on a 16 processor SGI Origin 2000, which was partly funded by the NSF SCREMS grant DMS–9872009. P.M.A. Sloot et al. (Eds.): ICCS 2002, LNCS 2330, pp. 461-469, 2002.  Springer-Verlag Berlin Heidelberg 2002