Scaling Laws and Optimal Design in Some Problems of Continuum Mechanics

这个项目所研究的问题是能量驱动模式形成研究的前沿，更广泛地说，是变分演算研究的前沿。该项目在设计上是跨学科的--跨越材料科学和流体动力学领域的问题--并受到实际和未来工程应用的推动。例如，在材料科学中，有可能采用新的生物启发和基于生长的设计机制来构建高度灵活但可控的固体膜。在流体动力学方面，除了设计新型高效热交换器的可能性外，研究的动机是更广泛的科学挑战，即推导湍流流体中自然发生的传输的标度定律。经验表明，这些问题很难在计算机上用传统的数值方法来解决。通常有许多近乎最优的答案，而且每一个都可能相当复杂。严格的数学分析提供了知道某些模式是真正的整体最优的可能性：这种信息不能使用其他非严格的方法来收集。因此，该项目是对更大的科学界的一个重要贡献，该科学界对了解这种复杂的模式为什么或如何实现全球最优感兴趣。该项目包括两个主要部分：(A)第一类问题涉及弹性薄片的力学，例如自然产生的树叶和花朵，但也包括人造版本，其厚度可能比标准纸张薄几个数量级。弹性薄片越薄，就越容易变形，特别是通过弯曲力，但从这种直观的物理原理中提取实验中发现的大量褶皱、折叠和皱折图案背后的数学机制是一个明显而困难的挑战。研究人员试图通过严格的数学分析推导出有效的、粗粒度的模型，这些模型控制着在消失厚度限制下的最佳弹性模式。一个关键的任务是得到任何自然参数中最小能量的标度律，然后询问哪些模式可能达到这样的能量标度律。(B)第二类问题涉及如何设计不可压缩的流体流动，以达到最佳的换热效果。虽然浸没在流体介质中的热物体确实会自行冷却，但人们自然会问，通过智能搅拌周围的流体，热传递的效率是否可以显著提高。通常情况下，任何搅拌方案都能在一定程度上增强换热，但确定哪种方案总体上达到最大换热是一个有趣而又悬而未决的问题。在平流主导的极限中，可以通过增加可用来搅拌的能量来达到，这种最优策略表现出的模式与数学材料科学中的“能量驱动模式形成”问题非常相似。研究人员正在开发这些明显相关的问题类别之间的精确和一般联系，并正在探索其对自然发生的湍流传输的标度规律研究的影响。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。

The problems under investigation in this project are at the forefront of the study of energy-driven pattern formation and more generally the Calculus of Variations. The project is interdisciplinary by design - the problems cut across the fields of materials science and fluid dynamics - and is motivated by both practical and futuristic engineering applications. In materials science, for instance, there is the potential for new biologically-inspired and growth-based design mechanisms for building highly flexible but controlled solid membranes. Regarding fluid dynamics, besides the potential for the design of new and extremely efficient heat exchangers, the investigation is motivated by the broader scientific challenge of deriving the scaling laws of naturally occurring transport in turbulent fluids. As experience shows, these questions are very difficult to solve on a computer by applying conventional numerical techniques. There are oftentimes many nearly optimal answers, and each can be rather complex. Rigorous mathematical analysis presents the possibility of knowing that certain patterns are truly optimal overall: such information cannot be gleaned using other non-rigorous approaches. Thus, the project is an important contribution to the larger scientific community interested in understanding why or how such complex patterns achieve global optimality. This project consists of two main parts: (a) The first class of problems concerns the mechanics of thin elastic sheets, such as naturally occurring ones like leaves and flowers, but also man-made versions which can be orders of magnitude thinner than a standard sheet of paper. The thinner an elastic sheet becomes the more easily it can be deformed, in particular through bending forces, but extracting from such intuitive physical principles the mathematical mechanisms behind the vast array of wrinkling, folding, and crumpling patterns found in experiments is a clear and difficult challenge. The investigator seeks to derive through rigorous mathematical analysis the effective, coarse-grained models governing optimal elastic patterns in the vanishing thickness limit. A key task is to obtain the scaling law of the minimum energy in any natural parameters, and then to ask which patterns can possibly attain such energy scaling laws. (b) The second class of problems regards the design of incompressible fluid flows for achieving optimal heat transfer. While hot objects submerged in fluid media do cool on their own, it is natural to ask whether the efficiency of heat transfer can be significantly enhanced by intelligent stirring of the surrounding fluid. Oftentimes, any stirring protocol achieves some enhancement of heat transfer, but determining which strategies attain maximal heat transfer overall is an intriguing and open question. In the advection-dominated limit, which can be reached by increasing the amount of power available with which to stir, such optimal strategies exhibit patterns remarkably similar to those from problems of "energy-driven pattern formation" in mathematical materials science. The investigator is developing a precise and general link between these apparently related classes of problems, and is exploring its ramifications for the study of the scaling laws of naturally occurring turbulent transport.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.