Simple and efficient implementation of discrete plates and shells

Efficient computation of curvature-based energies is important for practical implementations of geometric modeling and physical simulation applications. Building on a simple geometric observation, we propose a hinge-based bending model that is simple to implement, efficient to compute, and offers a great number of effective material parameters. Our formulation builds on two mathematical observations: (a) the bending energy of a thin flexible plate (resp. shell) can be expressed as a quadratic (resp. cubic) polynomial of surface positions provided that the surface does not stretch; (b) a general class of anisotropic materials---those that are orthotropic---is captured by appropriate choice of a single stiffness per hinge. We provide two approaches for deriving our isometric bending model (IBM): a purely geometric view and a derivation based on finite elements. By offering a highly efficient treatment of force Jacobians, our model impacts the speed of a general range of surface animation applications, from isotropic cloth and thin plates, over orthotropic fracturing of thin shells, to Willmore-type surface fairing. The present notes are condensed from previous articles by the authors: [Bergou et al. 2006], [Garg et al. 2007], and [Wardetzky et al. 2007]---augmented by a finite element treatment that was not present in these earlier works.