A discrete-to-continuum approach to the curvatures of membrane networks and parametric surfaces

• The paper deals with a discrete-to-continuum approach to the curvatures of flexible membranes.
• The proposed approach makes use of Monge patches of polyhedral surfaces.
• The continuum limits of discrete definitions of the membrane curvatures are studied.
• A regularization technique is proposed for unstructured polyhedral membranes.

The present work deals with a scale bridging approach to the curvatures of discrete models of structural membranes, to be employed for an effective characterization of the bending energy of flexible membranes, and the optimal design of parametric surfaces and vaulted structures. We fit a smooth surface model to the data set associated with the vertices of a patch of an unstructured polyhedral surface. Next, we project the fitting function over a structured lattice, obtaining a ‘regularized’ polyhedral surface. The latter is employed to define suitable discrete notions of the mean and Gaussian curvatures. A numerical convergence study shows that such curvature measures exhibit strong convergence in the continuum limit, when the fitting model consists of polynomials of sufficiently high degree. Comparisons between the present method and alternative approaches available in the literature are given.