[en] Nematic liquid crystal elastomers (LCEs) contract in the director direction but expand in other directions, perpendicular to the director, when heated. If the expansion of an LCE is constrained, compressive stress builds up in the LCE, and it wrinkles or buckles to release the stored elastic energy. Although the instability of soft materials is ubiquitous, the mechanism and programmable modulation of LCE instability has not yet been fully explored. We describe a finite element method (FEM) scheme to model the inhomogeneous deformation and instability of LCEs. A constrained LCE beam working as a valve for microfluidic flow, and a piece of LCE laminated with a nanoscale poly(styrene) (PS) film are analyzed in detail. The former uses the buckling of the LCE beam to occlude the microfluidic channel, while the latter utilizes wrinkling or buckling to measure the mechanical properties of hard film or to realize self-folding. Through rigorous instability analysis, we predict the critical conditions for the onset of instability, the wavelength and amplitude evolution of instability, and the instability patterns. The FEM results are found to correlate well with analytical results and reported experiments. These efforts shed light on the understanding and exploitation of the instabilities of LCEs. (paper)