A flexible membrane, pinned at both ends,  was studied using CFD and a moving grid technique. The objective was to understand the wing's flexibility effects on the dragonfly hovering flight. In the following videos we can see the vorticity evolution in a hovering flight: for three rigid airfoils -- elliptic, rigid and dimond-shaped -- and for a rigid plate and a flexible menbrane, both in the first mode of vibration. The Reynolds number is 157 and the videos were taken after 10 cycles.

This work resulted in a publication: Calculation of a deformable membrane airfoil in hovering flight.