A multilattice quasicontinuum for phase transforming materials: Cascading Cauchy Born kinematics 
M. Dobson, R. S. Elliott, M. Luskin and E. B. Tadmor
Journal of Computer-Aided Materials Design, 14 (supplement 1), 219-237 (2007).

ABSTRACT
The quasicontinuum (QC) method is applied to materials possessing a multilattice
crystal structure. Cauchy-Born (CB) kinematics, which accounts for the shifts of
the crystal basis, is used in continuum regions to relate atomic motions to
continuum deformation gradients. To avoid failures of the CB kinematics, QC is
augmented with a phonon stability analysis that detects lattice period
extensions and identifies the minimum required periodic cell size. This
augmented approach is referred to as Cascading Cauchy-Born kinematics. The
method is analyzed for both first- and second-order phase transformations, and
demonstrated numerically on a one-dimensional test problem.