Collaborative Research: Optimal Design of Responsive Materials and Structures

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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.