Collaborative Research: Optimal Design of Responsive Materials and Structures

这个项目的动机是两种技术进步的融合。首先是3D打印和其他新颖的制造技术。第二个是开发活性材料，其性质可以通过电场或磁场和热来改变。现在，3D打印形状记忆合金和液晶弹性体等活性材料已经成为可能。这为响应结构铺平了道路，这种结构的形状可以由外部刺激控制。此外，将它们与结构材料结合可以赋予它们许多应用的功能，包括软机器人、可穿戴和假肢设备、微流体、危险化学品的清理、靶向药物输送和组织工程。然而，目前还没有已知的方法来系统地设计这种装置。该项目将为响应结构和元材料的系统设计开发一种方法，这些材料是不同材料和空隙的复杂组合，特别是以最低成本寻求最佳功能的最佳设计。这些优化设计问题导致了大量的数学问题。相反，对这些问题更好的数学理解可以带来新的设计方法。通过为响应结构和超材料的设计和合成提供可靠的方法，这项研究将产生重大的技术影响。它还将提供培训两名研究生和几名本科生研究人员的经费。它将为K-12学生参与STEM创造新的机会，并在代表性不足的群体中推广STEM教育。研究人员将研究一些数学问题，这些问题的动机是将结构和响应材料（其响应功能取决于外部刺激的材料）整合到集成功能材料和结构中，这些材料和结构可以改变形状，并可以与结构材料结合以赋予它们功能。这类材料包括形状记忆合金、光敏弹性体或定向可控的液晶弹性体。这种结构的设计是具有挑战性的。在结构材料中，拓扑优化与增材制造相结合已被证明是一种非常强大的工具，数学分析在实现这一目标方面发挥了非常重要的作用。事实上，最直接的表述是微积分变化中的不适定问题，这已经通过松弛（例如，均匀化方法）和正则化（例如，周长惩罚）来解决。使用响应材料的最优设计问题的原始公式仍然是病态的，它们的松弛和正则化是开放的。例如，虽然结构材料的优化设计通常会导致最小-最大问题，但扩展到响应材料需要最大化最小的线性组合。轨迹优化、单边约束（由于响应的限制）以及围绕可制造性的问题也引起了人们的兴趣。该研究将为集成功能材料和结构的设计和合成提供坚实的数学基础。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。

This project is motivated by the confluence of two technological advances. The first is 3D printing and other novel manufacturing technologies. The second is the development of active materials whose properties can be altered by electrical or magnetic fields and heat. It is now becoming possible to 3D print active materials like shape-memory alloys and liquid crystal elastomers. This paves the way for responsive structures whose shape can be controlled by external stimuli. Further, combining them with structural materials can endow them with functions that are of use for many applications including soft robotics, wearable and prosthetic devices, microfluidics, cleanup of hazardous chemicals, targeted drug delivery, and tissue engineering. However, there is no known way to systematically design such devices. This project will develop a methodology for the systematic design of responsive structures and meta-materials which are complex assemblies of distinct materials and voids, especially optimal design where one seeks the best function at the least cost. These optimal design problems lead to substantial mathematical problems. Conversely, a better mathematical understanding of these problems can lead to new design approaches. By providing robust methodologies for the design and synthesis of responsive structures and meta-materials, this research will have a significant technological impact. It will also provide for the training of two graduate students and several undergraduate researchers. It will generate new opportunities for engaging K-12 students in STEM, and for promoting STEM education amongst underrepresented groups.The investigators will study mathematical questions motivated by the vision of incorporating structural and responsive materials (materials whose response function depends on external stimuli) into integrated functional materials and structures which can change shape and can be combined with structural materials to endow them with function. Such materials include shape-memory alloys, photo-sensitive elastomers, or liquid crystal elastomers with controlled orientation. The design of such structures is challenging. In structural materials, topology optimization combined with additive manufacturing has proven to be an extremely powerful tool, and mathematical analysis played a very important role in making it so. Indeed, the most straightforward formulation is an ill-posed problem in the calculus variations, and this has been addressed using relaxation (for example, the homogenization method) and regularization (for example, perimeter penalization). Naive formulations of optimal design problems using responsive materials are still ill-posed and their relaxation and regularization are open. For example, while optimal design with structural materials typically leads to min-max problems, extension to responsive materials requires maximizing a linear combination of minima. Trajectory optimization, unilateral constraints (due to limits in response), and issues surrounding manufacturability are also of interest. The research will provide a robust mathematical foundation that can form the basis for methodologies for the design and synthesis of integrated functional materials and structures.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.