Consistent and convergent discretizations of Helfrich-type energies on general meshes

We show that integral curvature energies on surfaces of the type $E_0(M):={\int }_{M}f(x,n_M(x),D{n}_M(x))d{\mathcal{H}}^2(x)$ have discrete versions for triangular complexes, where the shape operator $D{n}_M$ is replaced by the piecewise gradient of a piecewise affine edge director field. We combine an ansatz-free asymptotic lower bound for any uniform approximation of a surface with triangular complexes and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director.