Scaling Laws and Optimal Design in Some Problems of Continuum Mechanics

在这个项目中正在调查的问题是在能源驱动的模式形成的研究，更普遍的变分法的前沿。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推动。例如，在材料科学中，有可能产生新的生物启发和基于生长的设计机制，用于构建高度灵活但可控的固体膜。关于流体动力学，除了设计新的和非常有效的热交换器的潜力外，这项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输运的标度律。经验表明，这些问题很难在计算机上通过应用传统的数值技术来解决。通常有许多接近最优的答案，每一个都可能相当复杂。严格的数学分析提供了了解某些模式是真正最优的可能性：这些信息无法使用其他非严格的方法收集。因此，该项目是一个重要的贡献，更大的科学界有兴趣了解为什么或如何这样的复杂模式实现全局最优。 该项目包括两个主要部分：（a）第一类问题涉及薄弹性片材的力学，例如自然发生的叶子和花朵，但也有人造版本，可以比标准纸张薄几个数量级。弹性片越薄，就越容易变形，特别是通过弯曲力，但是从这些直观的物理原理中提取实验中发现的大量扭曲，折叠和起皱模式背后的数学机制是一个明确而困难的挑战。调查人员试图通过严格的数学分析得出有效的，粗粒度的模型，在消失的厚度限制的最佳弹性模式。一个关键的任务是获得任何自然参数的最小能量标度律，然后问什么样的模式可能达到这样的能量标度律。(b)第二类问题涉及不可压缩流体流动的设计，以实现最佳传热。虽然浸没在流体介质中的热物体会自行冷却，但人们自然会问，通过智能搅拌周围的流体是否可以显着提高传热效率。通常情况下，任何搅拌方案都能实现一定程度的传热增强，但确定哪种策略总体上实现最大传热是一个有趣且开放的问题。在对流主导的极限，这可以通过增加可用的功率与搅拌量达到，这样的最佳策略表现出的模式非常类似的问题的“能量驱动的模式形成”在数学材料科学。该研究者正在开发这些明显相关的问题之间的精确和一般的联系，并正在探索其后果的自然发生的湍流transport.This奖项的标度律的研究反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.