Mathematical and Numerical Study of Electromagnetic Waves Interacting with Metamaterials

这个项目关注的是数学分析和设计鲁棒和有效的计算算法来模拟与负折射率超材料（NIMs）的波相互作用。这些负折射率超材料具有一些在天然电磁材料中从未见过的奇异特性（如近场再聚焦）。数值NIM分析对于建立NIM的复杂几何纳米结构的设计具有重要的作用。由于其色散性，以及引入的电和磁极化电流，这些NIM模型远比自由空间中得到充分研究的麦克斯韦方程组复杂得多。准确有效地解决这些问题是相当具有挑战性的，很少有关于可靠的数学分析和建模的工作。该项目的最终目标是开发一套有效的时域有限元方法，该方法在数学上合理，准确且快速收敛，用于模拟波与超材料的相互作用。自2000年以来，超材料的研究是许多学科中最热门的话题之一，在天线、波导和雷达的设计、纳米光刻和亚波长分辨率成像（用于更好地理解从非侵入性地球物理探测和肿瘤检测中获得的图像）、近场控制和操纵（用于检测低水平的化学和生物制剂，操纵分子）、隐形斗篷（用于隐形技术）。为负折射率超材料开发强大而高效的算法将有利于电气工程、材料、光学、物理、纳米技术和生物医学技术等更广泛的领域。此外，拟议的项目将帮助PI招募和培训研究生（该项目将支持目前由PI监督的女博士生）从事计算数学方面的职业。

This project is concerned with the mathematical analysis and design of robust and efficient computational algorithms for modeling wave interactions with negative index metamaterials (NIMs). These negative index metamaterials have some exotic properties (such as near fieldrefocusing) never seen before in those natural electromagnetic materials.The numerical NIM analysis plays an important role for the design of the nano-structures with complicated geometries that establish a NIM.these NIM models are far more complicated than the well-studied Maxwell's equations in free space due to its dispersiveness, and the introduced electric and magnetic polarization currents.Solving them accurately and efficiently is quite challenging and very few work has been done in regards of solid mathematical analysis and modeling. The ultimate goal of this project is to develop an efficient set of time domain finite element methods that are mathematically sound, accurate and fast convergent for simulating wave interactions with metamaterials.Study of metamaterials is one of the hottest topics in many disciplinaries since 2000, with potential revolution in design of antenna, waveguides and radar, nanolithography and imaging at subwavelength resolution (used for better understanding of the images obtained from noninvasive geophysical probing and tumor detection), near field control and manipulation (used for detecting low levels of chemical and biological agents, manipulation of molecules), and invisibility cloak (used for stealth technology). Developing robust and efficient algorithms for negative index metamaterials will benefit broader areas such as electrical engineering, materials, optics, physics, nano-technology, and biomedical technology. Furthermore, the proposed project will help the PI recruit and train graduate students (this project will support a female Ph.D student currently supervised by the PI) to pursue careers in computational mathematics.