Collaborative Research: Optimal Design of Responsive Materials and Structures

这项计划是由两项技术进步共同推动的。首先是3D打印和其他新的制造技术。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成为可能。这为响应性结构铺平了道路，这些结构的形状可以通过外部刺激来控制。此外，将它们与结构材料相结合，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、危险化学品的清除、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设备。该项目将为响应性结构和超材料的系统设计开发一种方法，这些结构和超材料是由不同材料和空隙组成的复杂组件，特别是以最低成本寻求最佳功能的优化设计。这些优化设计问题导致了大量的数学问题。相反，对这些问题有更好的数学理解可以带来新的设计方法。通过为响应性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。它还将用于培训两名研究生和几名本科生研究人员。这将为K-12学生参与STEM创造新的机会，并在代表性不足的群体中促进STEM教育。调查人员将研究数学问题，其动机是将结构和响应材料(其响应功能依赖于外部刺激的材料)纳入可改变形状并可与结构材料结合以赋予其功能的综合功能材料和结构。这种材料包括形状记忆合金、光敏弹性体或取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，拓扑优化与加法制造相结合已被证明是一个非常强大的工具，数学分析在实现这一点上发挥了非常重要的作用。事实上，最直接的表述是微积分变分中的一个不适定问题，这个问题已经通过松弛(例如，齐次化方法)和正则化(例如，周长惩罚)来解决。使用响应材料的优化设计问题的朴素公式仍然是不适定的，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，但扩展到响应性材料需要最大化最小值的线性组合。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是令人感兴趣的。这项研究将提供坚实的数学基础，可以形成设计和合成集成功能材料和结构的方法学的基础。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.