Mathematical study and finite element simulation of wave propagation in metamaterials

自2000年以来，超材料的研究引起了许多科学家和工程师的关注。超材料中的波传播是许多学科感兴趣的领域，可能会彻底改变高效天线、波导和雷达的设计、纳米光刻和亚波长成像、近场控制和操作（用于检测低水平的化学和生物制剂，以及分子操作）、粒子检测和隐形斗篷（用于隐身技术）。开发强大而高效的超材料建模算法将有利于电气工程、材料科学、光学、物理、纳米技术和生物医学技术等多个领域。本计画的目标是透过数学模型与电脑模拟来研究波与超材料相互作用的问题。数学建模在超材料的设计和应用中起着重要的作用。2006年10月，杜克大学的一组研究人员利用计算机模拟发现了一种制造超材料的方法，可以制造出一种二维“隐形斗篷”，使物体在特定频率下不可见。然而，工程师提出的大多数超材料模型都没有得到很好的研究，缺乏可靠的数学分析和建模。超材料是有损耗和色散的，这导致控制方程比研究得很好的麦克斯韦要复杂得多。自由空间中的S方程。准确有效地逼近超材料方程的解是具有挑战性的（请注意，所有未知数都是三维空间中的物体，并且随时间变化），需要额外的研究。该项目将开发数学上稳健、准确、高效的有限元方法，可用于模拟波与超材料的相互作用。在提议的项目中，PI将培养有兴趣从事计算科学和工程职业的研究生。

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.