Mathematical fundamentals of Metamaterials for multiscale Physics and Mechanics

超材料是人造材料，具有天然材料不具备的特性，例如，进入超材料板的光可以以与人们通常期望的相反的方式弯曲。这不仅仅是一个科学上的好奇心，它可以产生深远的影响，导致亚波长成像，聚焦，隐形斗篷以及其他效果，这反过来又可以导致具有意想不到和新颖特性的材料。迄今为止，人们对超材料的兴趣主要集中在光学和电磁学方面，但很明显，其基本思想应该适用于其他领域，如弹性、扩散、结构材料、声学甚至水波。有很多重要的应用：设计热斗篷来保持敏感的电子冷却，创造水下隐身的声学超材料，用于建筑物或关键部件的结构保护的波旁通系统，所有这些都在超材料的光学范围之外，因为它们目前存在。创造超材料的一个关键问题是它的设计，通常是一个具有精确微观结构几何形状的周期性介质，以及它的工作频率。随着超材料在光学方面开始达到一定的成熟，将这些知识连贯地转移到其他领域的时机已经成熟，也适时地用超材料中创造的令人兴奋的概念问题来丰富数学，并用复杂的数学技术丰富超材料工具箱。本提案旨在利用数学的变革工具和统一思想，将超材料的物理学引入弹性、声学、结构力学和扩散等研究领域，这些领域几乎没有对超材料进行过研究，但毫无疑问会产生影响和应用。通过与物理学家密切合作，它将通过将复杂的数值和理论方法带到前台，丰富和增强现有的超材料社区。

Metamaterials are materials that are man-made and can have properties that no natural material could have, for instance light entering a metamaterial slab can be bent in the opposite manner to that which one would usually expect. This is not merely a scientific curiosity, it can have profound implications leading to sub-wavelength imaging, focusing, invisibility cloaks amongst other effects and this, in turn, can lead to materials with unexpected and novel properties. Much of the interest in metamaterials has thus far been in optics and electromagnetism, but it is clear that the underlying ideas should be applicable in other contexts such as elasticity, diffusion, structured materials, acoustics and even water waves. There is an abundance of important applications: designing thermal cloaks for keeping sensitive electronics cool, creating acoustic metamaterials for underwater stealth, wave by-pass systems for structural protection of buildings or key components, all of which are outside the optical context of metamaterials as they currently exist.A key issue in creating a metamaterial is its design, normally as a periodic medium with a precise micro-structured geometry, and the frequency at which it operates. As Metamaterials are beginning to achieve a certain maturity in optics the time is ripe to move this knowledge coherently into other fields, it is also timely to enrich Mathematics with the exciting conceptual problems created in Metamaterials and enrich the Metamaterials toolkit with sophisticated Mathematical techniques. This proposal aims to use the transformative tools and unifying ideas of Mathematics to move the physics of Metamaterials into research areas such as Elasticity, Acoustics, Structural Mechanics and Diffusion where Metamaterials have barely been investigated, but where there will undoubtedly be impact and applications. By working closely with Physicists it will enrich and empower the existing Metamaterials community by bringing sophisticated numerical and theoretical methods to the fore.