A simple geometric model for elastic deformations

Abstract

We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust. Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.