Metamaterials and Inverse Problems

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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.