Mathematical study and finite element simulation of wave propagation in metamaterials

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Since 2000, the study of metamaterials has attracted the attention of many scientists and engineers. Wave propagation in metamaterials is of interest in various disciplines and could potentially revolutionize design of efficient antennas, waveguides and radars, nanolithography and subwavelength imaging, near field control and manipulation (useful for detecting low levels of chemical and biological agents, and manipulation of molecules), particle detection, and invisibility cloaking (useful for stealth technology). Developing robust and efficient algorithms for modeling metamaterials will benefit diverse areas such as electrical engineering, materials science, optics, physics, nano-technology, and biomedical technology. The goal of this project is to study problems of wave interaction with metamaterials through mathematical modeling and computer simulations. Mathematical modeling plays an important role in the design and application of metamaterials. In October 2006 using computer simulations, a group of researchers at Duke University found a way to fabricate the metamaterials to build a 2-D "invisibility cloak" that makes an object invisible to certain frequencies. However, most metamaterial models proposed by engineers are not well studied and there is a lack of solid mathematical analysis and modeling. Metamaterials are lossy and dispersive, which leads to governing equations which are much more complicated than the well-studied Maxwell?s equations in free space. Approximating solutions of the metamaterial equations accurately and efficiently is challenging (note that all unknowns are objects in three dimensional spaces and vary in time) and requires additional research. This project will develop mathematically robust, accurate, and efficient, finite element methods that can be be used for simulating wave interactions with metamaterials. During the proposed project the PI will train graduate students interested in pursuing careers in computational sciences and engineering.