The nonlinear problem of a curved flame interaction with a weak shock is considered. Collisions of curved flames propagating in open tubes with weak shocks of controlled intensity are studied numerically by means of two-dimensional simulations of the complete set of equations of flame dynamics including chemical kinetics, thermal conduction and fuel diffusion. It is shown that the: result of a flame-shock collision depends on the shock intensity and on the expansion coefficient of the burning matter. The critical intensity of a shock needed to invert the shape of a curved flame is calculated. The obtained critical intensity is in a good agreement with the analytical scalings found for the linear problem of flame-shock interaction. It is obtained that for sufficiently large intensities of the shock the flame-shock interaction leads to considerable increase of the flame curvature and velocity. Special attention is paid to the flame-shock collisions for the case of shock intensities close to the intensities of shocks generated by a flame in a closed burning chamber. It is shown that most of the laboratory flames are destabilized by weak shocks in closed burning chambers.