Finite Element Approximations of Developable Surfaces with Curved Folds

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The ability to generate complex and robust deformations from relatively small energies has a tremendous number of applications in many areas including the strategic areas of aerospace, nanotechnology and biotechnology. This research project explores the potential benefits of folding devices where folding does not necessarily occur on straight lines. Compared to more traditional origami-type deformations, curved creases greatly expand the range and rigidity of achievable configurations. The design of deployable surfaces such as solar panels, solar sails, space telescopes, airbags, flapping devices, and ingestible robots are few examples benefiting from this technology. This project encompasses the mathematical modeling of folding devices, the design of numerical algorithms predicting their deformations, and a mathematical analysis guaranteeing the efficiency of the predictions.The PI will consider thin materials resisting shear and stretch but allowing for bending away from non-necessarily straight creases. In recent years, these curved origamis received significant attention from the scientific community in view of the fascinating variety of shapes they can exhibit, their ability to produce rigid configurations and flapping mechanisms, their capacity to undergo large deformations using a small amount of energy, and their applicability at small and large scales alike. The outcomes of this research program include the derivation of a reduced plate model for thin materials resisting bending and allowing for folding along curved locations, the design and analysis of finite element algorithms approximating the dynamics and equilibriums of the corresponding plate deformations, and a parallel implementation of the proposed algorithms illustrating their efficiency on benchmarks as well as on configurations relevant to practitioners. Central in this study, the concept of gamma convergence is used to justify the reduced model but also developed for the analysis of the associated numerical methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.