  For fourth-order geometric evolution equations for planar curves with the
dissipation of the bending energy, including the Willmore and the Helfrich
flows, we consider a numerical approach. In this study, we construct a
structure-preserving method based on a discrete variational derivative method.
Furthermore, to prevent the vertex concentration that may lead to numerical
instability, we discretely introduce Deckelnick's tangential velocity. Here, a
modification term is introduced in the process of adding tangential velocity.
This modified term enables the method to reproduce the equations' properties
while preventing vertex concentration. Numerical experiments demonstrate that
the proposed approach captures the equations' properties with high accuracy and
avoids the concentration of vertices.
