We consider the numerical approximation of theL2–gradient flow of general curvatureenergies∫G(|~κ|) for a curve inRd,d≥2. Here the curve can be either closed, or it can be open andclamped at the end points. These general curvature energies, and the considered boundary conditions,appear in the modelling of the power loss within an optical fibre. We present two alternative finiteelement approximations, both of which admit a discrete gradient flow structure. Apart from beingstable, in addition, one of the methods satisfies an equidistribution property. Numerical resultsdemonstrate the robustness and the accuracy of the proposedmethods.