Finite Element Approximations of Developable Surfaces with Curved Folds

利用相对较小的能量产生复杂而强大的变形的能力在许多领域都有大量的应用，包括航空航天、纳米技术和生物技术等战略领域。这个研究项目探索折叠装置的潜在好处，其中折叠不一定发生在直线上。与更传统的折纸型变形相比，弯曲折痕极大地扩展了可实现构型的范围和刚性。太阳能电池板、太阳帆、太空望远镜、安全气囊、襟翼装置和可消化机器人等可展开表面的设计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，但允许从非必要的直折痕弯曲。近年来，这些弯曲折纸受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，它们能够产生刚性构型和扑动机制，它们能够使用少量能量进行大变形，并且它们在小尺度和大尺度上都同样适用。该研究项目的成果包括推导出抗弯曲和允许沿弯曲位置折叠的薄材料的简化板模型，设计和分析近似相应板变形的动力学和平衡的有限元算法，并并行实现所提出的算法，说明它们在基准测试以及与从业者相关的配置上的效率。在本研究的中心，伽马收敛的概念被用来证明简化的模型，但也发展为分析相关的数值方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.