  In this paper, we consider numerical approximation of constrained gradient
flows of planar closed curves, including the Willmore and the Helfrich flows.
These equations have energy dissipation and the latter has conservation
properties due to the constraints. We will develop structure-preserving methods
for these equations that preserve both the dissipation and the constraints. To
preserve the energy structures, we introduce the discrete version of gradients
according to the discrete gradient method and determine the Lagrange
multipliers appropriately. We directly address higher order derivatives by
using the Galerkin method with B-spline curves to discretize curves. Moreover,
we will consider stabilization of the schemes by adding tangential velocities.
We introduce a new Lagrange multiplier to obtain both the energy structures and
the stability. Several numerical examples are presented to verify that the
proposed schemes preserve the energy structures with good distribution of
control points.
