Metamaterials and Inverse Problems

由该奖项支持的研究将探索由具有极端特性的成分材料组成的复合材料在单频和宽频率范围内可以表现出的电磁特性范围。在这里首次介绍的外来微观结构所表现出的任何有趣的行为，都应该激发人们对表现出类似有趣效果的实际微观结构的探索。该项目还解决了实际重要问题，即通过对物体表面的电磁场或弹性场进行少量测量，对物体中包含物的体积设置严格的上限和下限（界限），重点关注准静态状态下的电磁情况，其中时间谐波场的波长远大于微观结构，模量和场很复杂。此外，该项目引入了一种估算两相复合材料中相的体积分数的新方法，通过测量准静态状态下一组实频率下相的有效体积模量、体积模量和剪切模量。最后，该研究项目寻求在不太完美的情况下，由于相之间界面的不完美而对复合材料的有效模量进行修正。该公式应便于计算不完美界面下的有效模量，并给出了完美界面的数值解。超材料是一种人造复合材料，具有普通材料无法达到的性能。更好地理解这种超材料的宏观响应具有广泛的技术重要性。国防、汽车、航空航天、电子和其他制造业和电信业对新材料的需求不断增加。当这些新材料的特性与我们已知的任何材料都截然不同时，它们的影响可能是最大的。这个项目将有助于勾勒出材料电磁响应的可能性。在第二个方向上，对于生物医学、工程和反恐应用来说，使用非侵入性测试来确定人体内部的物质是至关重要的，如果人们能近乎确定地说出一些事情，那就更好了。该项目将给出包含体或两相复合材料中的一相所占体积的精确下限和上限。这可能在未来应用于确定癌症肿瘤的大小，或体内的空洞，或骨质疏松的骨质疏松。在第三个方向上，将发展新的数学近似，用于两种不同材料之间的界面缺陷的物理效应，这有望在新的复合材料的实际设计中使用。该奖项将帮助培养一名博士后和一名跨学科研究领域的研究生。

The research supported by this award will explore the range of electromagnetic properties that composite materials formed from constituent materials with extreme properties can exhibit, at a single frequency, and over a broad range of frequencies. Any interesting behavior that the exotic microstructures, introduced for the first time here, are found to exhibit should motivate the search for practical microstructures exhibiting similar interesting effects. The project also addresses the practically important problem of placing rigorous upper and lower limits (bounds) on the volume of an inclusion in a body using a small number of measurements of the electromagnetic or elastic fields at the surface of the body, focusing on the electromagnetic case in the quasistatic regime where the wavelength of the time harmonic fields is much larger than the microstructure and the moduli and fields are complex. Additionally, the project introduces a new approach for estimating the volume fractions of the phases in a two-phase composite, from measurements of the effective bulk modulus, and bulk and shear moduli of the phases at a set of real frequencies in the quasistatic regime. Finally the research project seeks to find corrections to the effective moduli of composites due to the imperfectness of the interfaces between phases, in the case where the imperfection is not too great. The formulae should facilitate the calculation of effective moduli with imperfect interfaces given numerical solutions for perfect interfaces.Metamaterials are man-made composite materials with properties that are unachievable in ordinary materials. A better understanding of the macroscopic response of such metamaterials has widespread technological importance. There is a constant need for new materials in the defense, automotive, aerospace, electronics and other manufacturing and telecommunication industries. The impact of such new materials is likely to be greatest when their properties are radically different from any material we know. This project will help shape a picture outlining what electromagnetic responses of materials are possible. In a second direction, for biomedical, engineering and counterterrorism applications it is vitally important to determine what is inside a body using non-invasive testing, and it is better if one can say things with near certainty. The project will give precise lower and upper limits on the volume occupied by an inclusion in a body, or by one phase in a two-phase composite. This may have future applications to determining the size of cancer tumors, or of voids in a body, or the porosity in an osteoporotic bone. In a third direction, new mathematical approximations for physical effects of imperfections at the interface between two different materials will be developed, which are expected to be of use in the practical design of new composite materials. The award will help train a postdoctoral associate and a graduate student in an interdisciplinary research area.