  Recent applications (e.g. active gels and self-assembly of elastic sheets)
motivate the need to efficiently simulate the dynamics of thin elastic sheets.
We present semi-implicit time stepping algorithms to improve the time step
constraints that arise in explicit methods while avoiding much of the
complexity of fully-implicit approaches. For a triangular lattice
discretization with stretching and bending springs, our semi-implicit approach
involves discrete Laplacian and biharmonic operators, and is stable for all
time steps in the case of overdamped dynamics. For a more general
finite-difference formulation that can allow for general elastic constants, we
use the analogous approach on a square grid, and find that the largest stable
time step is two to three orders of magnitude greater than for an explicit
scheme. For a model problem with a radial traveling wave form of the reference
metric, we find transitions from quasi-periodic to chaotic dynamics as the
sheet thickness is reduced, wave amplitude is increased, and damping constant
is reduced.
