We consider the effects of chordwise flexibility on the aerodynamic performance of flapping wings using numerical simulation. The two-dimensional Navier-Stokes equations are solved using a Fourier pseudo-spectral method with no-slip boundary conditions imposed by the volume penalization method. The flexible wing is modelled with a non-linear beam equation. Our numerical simulations of heaving plates show that the maximum thrust is achieved at a stroke frequency lower than resonant, which is in agreement with experiments. The oscillatory part of the force only increases in amplitude when the frequency increases. We also consider aerodynamic interactions between two heaving foils.