  We aimed to use finite element method to simulate the unique behaviors of
liquid crystal elastomer, such as semi-soft elasticity, stripe domain
instabilities etc. We started from an energy functional with the 2D
Bladon-Warner-Terentjev stored energy of elastomer, the Oseen-Frank energy of
liquid crystals, plus the penalty terms for the incompressibility constraint on
the displacement, and the unity constraint on the director. Then we applied
variational principles to get the differential equations. Next we used mixed
finite element method to do the numerical simulation. The existence,
uniqueness, well-posedness and convergence of the numerical methods were
investigated. The semi-soft elasticity was observed, and can be related to the
rotation of the directors. The stripe domain phenomenon, however, wasn't
observed. This might due to the relative coarse mesh we have used.
