Scaling Laws and Optimal Design in Some Problems of Continuum Mechanics

在这个项目中研究的问题是能量驱动模式形成研究的前沿，更普遍的是变分法。该项目是跨学科的设计-问题跨越材料科学和流体动力学领域-并受到实际和未来工程应用的激励。例如，在材料科学中，有可能产生新的受生物启发和基于生长的设计机制，以构建高度灵活但可控的固体膜。关于流体动力学，除了设计新型高效热交换器的潜力之外，研究的动机是推导湍流中自然发生的输运的标度定律这一更广泛的科学挑战。经验表明，这些问题很难用传统的数值技术在计算机上解决。通常会有许多近乎最佳的答案，而且每个答案都可能相当复杂。严格的数学分析提供了了解某些模式总体上是真正最佳的可能性：使用其他非严格的方法无法收集此类信息。因此，该项目对更大的科学界来说是一个重要的贡献，他们对理解这种复杂模式为什么或如何实现全局最优感兴趣。该项目包括两个主要部分：(a)第一类问题涉及薄弹性片的力学，如自然产生的叶子和花朵，但也包括比标准纸张薄几个数量级的人造版本。弹性薄片越薄，它就越容易变形，特别是在弯曲力的作用下，但是从这些直观的物理原理中提取出实验中发现的大量起皱、折叠和皱缩模式背后的数学机制是一个明确而艰巨的挑战。研究者试图通过严格的数学分析推导出有效的粗粒度模型，在消失的厚度限制中控制最佳弹性模式。一个关键的任务是求出任意自然参数下最小能量的标度规律，然后问哪些模式可能达到这样的能量标度规律。(b)第二类问题涉及为实现最佳传热而设计不可压缩流体流动。当热物体浸没在流体介质中会自行冷却时，人们自然会问，是否可以通过对周围流体的智能搅拌来显著提高传热效率。通常情况下，任何搅拌协议实现一些增强传热，但确定哪种策略获得最大的传热总体是一个有趣的和开放的问题。在平流主导的极限下，可以通过增加可用的搅拌功率来达到，这种最佳策略的模式与数学材料科学中“能量驱动模式形成”的问题非常相似。研究者正在发展这些明显相关的问题之间的精确和普遍的联系，并正在探索其对自然发生的湍流输运的标度定律研究的分支。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 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.