Finite Element Approximations of Bending Actuated Devices

从相对较小的能量产生复杂变形的能力在微工程和生物医学科学中有着巨大的应用。本研究的重点是开发和实现能够预测和优化弹性薄膜变形的数学算法，选择弹性薄膜是因为它们在生产鲁棒和轻质微尺度设备方面具有潜力。所考虑的变形是通过暴露于具有不同膨胀特性的聚合物或具有残余应力的制造凝胶的外部刺激而触发的。例如，基于这些技术的设备被用作药物递送囊泡、细胞封装装置、传感器、生物肌肉和组织生长的替代品。除了这些在生物医学科学中的应用之外，自主可折叠结构的开发，如航天器或可展开飞机上的自展开太阳帆、光伏装置、执行器、微电机、微抓手、微阀、微游泳器等，都是工程界非常受欢迎的兴趣，这项研究可能会产生影响。提出的研究重点是薄器件，其中弯曲是主要机制，可能大，变形。推导了三维超弹性薄固体的二维极限数学模型。它们的特点是能量密度以弯曲为主，以几何形式表示为薄膜的第二种基本形式。除了这种四阶系统的非发散形式所固有的困难之外，在大变形的情况下必须考虑到完全非线性的几何约束。本研究的目的是在不可用于目标应用的情况下推导数学模型，并设计，分析和实现基于有限元的近似算法。涵盖了从数学分析到实际高度并行执行有限元算法的整个过程。因此，从微分几何和变分学中借鉴的分析工具与数值分析和精细的计算努力相结合，以实现高效实用的算法。从相对较小的能量产生复杂变形的能力在微工程和生物医学科学中有着巨大的应用。由提议的研究产生的算法——特别是它们相对轻松的实现——很可能在这些领域产生影响。仅举几个应用，基于聚合物双层或预紧膜的设备被用作药物递送囊泡、细胞封装设备、传感器、生物肌肉和组织生长的代理。除了这些在生物医学科学中的应用之外，自主折叠结构的开发，如航天器或可展开飞机上的自展开太阳帆、光伏设备、工程支架、执行器、微电机、微钳、微阀、微游泳器等，都是工程界非常受欢迎的研究兴趣。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 deformations from relatively small energies has tremendous applications in micro-engineering and biomedical science. This research focuses on developing and implementing mathematical algorithms able to predict and optimize the deformation of elastic films, chosen for their potential in the production of robust and light-weight micro-scale devices. The deformations considered are triggered by exposing to external stimuli either polymers with different expansion characteristics or manufactured gels with residual stresses. Devices based on these technologies are, for instance, employed as drug delivery vesicles, cell encapsulation devices, sensors, bio-muscles and as proxies for tissue growth. In addition to these applications in biomedical science, the development of autonomous foldable structures such as self-deployable sun sails in spacecraft or deployable aircrafts, photovoltaic devices, actuators, micromotors, microgrippers, microvalves, microswimmers are very popular interests in the engineering community this research is likely to have impact on.The proposed study focuses on thin devices where bending is the principal mechanism for, possibly large, deformations. The mathematical models are derived as the two dimensional limit of thin three dimensional hyper-elastic solids. They are characterized by energy densities dominated by bending, expressed geometrically as the film's second fundamental form. In addition to the difficulties inherent in the non-divergence form of this fourth order system, fully non-linear geometrical constraints must be taken into account in the context of large deformations. The aim of this research is to derive mathematical models when not available for the targeted application, and to design, analyze and implement finite element based algorithms for their approximations. The entire process from the mathematical analysis to the actual highly parallel implementation of finite element algorithms is covered. Hence, analytical tools borrowed from differential geometry and calculus of variation are blended with numerical analysis and delicate computational efforts to achieve efficient and practical algorithms. The ability to generate complex deformations from relatively small energies has tremendous application in micro-engineering and biomedical science. The algorithms resulting from the proposed research - and in particular their relatively effortless implementations - are likely to have impact in these areas. To mention a few applications, devices based on bilayers of polymers or prestrained films are employed as drug delivery vesicles, cell encapsulation devices, sensors, bio-muscles and as proxies for tissue growth. In addition to these applications in biomedical science, the development of autonomous foldable structures such as self-deployable sun sails in spacecraft or deployable aircrafts, photovoltaic devices, engineered scaffolds, actuators, micromotors, microgrippers, microvalves, microswimmers are very popular research interests in the engineering community.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.