  We present an approach to solving problems in micromechanics that is amenable
to massively parallel calculations through the use of graphical processing
units and other accelerators. The problems lead to nonlinear differential
equations that are typically second order in space and first order in time.
This combination of nonlinearity and nonlocality makes such problems difficult
to solve in parallel. However, this combination is a result of collapsing
nonlocal, but linear and universal physical laws (kinematic compatibility,
balance laws), and nonlinear but local constitutive relations. We propose an
operator-splitting scheme inspired by this structure. The governing equations
are formulated as (incremental) variational problems, the differential
constraints like compatibility are introduced using an augmented Lagrangian,
and the resulting incremental variational principle is solved by the
alternating direction method of multipliers. The resulting algorithm has a
natural connection to physical principles, and also enables massively parallel
implementation on structured grids. We present this method and use it to study
two examples. The first concerns the long wavelength instability of finite
elasticity, and allows us to verify the approach against previous numerical
simulations. We also use this example to study convergence and parallel
performance. The second example concerns microstructure evolution in liquid
crystal elastomers and provides new insights into some counter-intuitive
properties of these materials. We use this example to validate the model and
the approach against experimental observations.
