Finite Element Approximations of Developable Surfaces with Curved Folds

从相对较小的能量产生复杂和稳健变形的能力在许多领域中具有大量的应用，包括航空航天、纳米技术和生物技术的战略领域。该研究项目探讨了折叠设备的潜在好处，其中折叠不一定发生在直线上。与更传统的折纸式变形相比，弯曲折痕大大扩展了可实现的配置的范围和刚度。可展开表面的设计，如太阳能电池板，太阳帆，太空望远镜，安全气囊，扑翼装置和可摄取机器人，都是受益于这项技术的少数例子。该项目包括折叠装置的数学建模，预测其变形的数值算法设计，以及保证预测效率的数学分析。PI将考虑薄材料抵抗剪切和拉伸，但允许弯曲远离不必要的直折痕。近年来，这些弯曲的折纸得到了科学界的极大关注，因为它们可以展示出迷人的各种形状，它们能够产生刚性配置和拍打机制，它们能够使用少量能量进行大变形，以及它们在小尺度和大尺度上的适用性。这项研究计划的成果包括推导出一个减少板模型薄材料抵抗弯曲，并允许折叠沿着弯曲的位置，有限元算法的设计和分析近似的动态和平衡的相应板变形，并并行实施所提出的算法，说明其效率的基准以及配置相关的从业者。在这项研究的中心，伽玛收敛的概念是用来证明简化模型，但也开发了相关的数值方法的分析。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.