  Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in
the plane with same winding number. A curvature flow with a nonlocal term is
constructed to evolve $X_0$ into $\widetilde{X}$. It is proved that this flow
exits globally, preserves both the local convexity and the elastic energy of
the evolving curve. If the two curves have same elastic energy then the
curvature flow deforms the evolving curve into the target curve $\widetilde{X}$
as time tends to infinity.
