a) Target shape (b) Without inversion, sagging (c) Simulation after inversion, under wind motion Fig. 1. Inverse design of an arched book page modeled by an artist.The target shape (a) is a half-cylinder which lies in contact with other pages on its two long edges. When simulated without prior inversion (i.e., simply initializing the natural shape with the target), the target shape (a) slides over the rest of the book (b). Thanks to our inverse process which simultaneously accounts for elasticity, gravity, and frictional contact (here with µ = 0.4), we manage to recover a natural rest shape such that the target shape (a) is at stable equilibrium under surrounding forces, even for soft material parameters. The page can then be further simulated in a consistent manner, for instance under wind motion (c).We propose an inverse strategy for modeling thin elastic shells physically, just from the observation of their geometry. Our algorithm takes as input an arbitrary target mesh, and interprets this configuration automatically as a stable equilibrium of a shell simulator under gravity and frictional contact constraints with a given external object. Unknowns are the natural shape of the shell (i.e., its shape without external forces) and the frictional contact forces at play, while the material properties (mass density, stiffness, friction coefficients) can be freely chosen by the user. Such an inverse problem formulates as an ill-posed nonlinear system subject to conical constraints. To select and compute a plausible solution, our inverse solver proceeds in two steps. In a first step, contacts are reduced to frictionless bilateral constraints and a natural shape is retrieved using the adjoint method. The second step uses this result as an initial guess and adjusts each bilateral force so that it projects onto the admissible Coulomb friction cone, while preserving global equilibrium. To better guide minimization towards the target, these two steps are applied iteratively using a degressive regularization of the shell energy. We validate our approach on simulated examples with reference material parameters, and show that our method still converges well for material parameters lying within a reasonable range around the reference, and even in the case of arbitrary meshes that are not issued from a simulation. We finally demonstrate practical inversion results on complex shell geometries freely modeled by an artist or automatically captured from real objects, such as posed garments or soft accessories.