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 field refocusing) 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.
Study of metamaterials is one of the hottest topics in many disciplinaries, with potential revolution in design of antenna, waveguides and radar, nanolithography and imaging at subwavelength resolution, near field control and manipulation, and invisibility cloak. Due to high cost of metamaterials, physical experiments using metamaterials are quite limited. Hence the computer simulation plays a very important role in the study of wave propagation involving metamaterials. For example, in October 2006, through computer simulations a group of researchers at Duke University found a way to fabricate the metamaterials to build an "invisibility cloak" that makes an object invisible to certain frequencies (details can be found in Science 314 (2006) 977-980). Though some commercial packages such as COMSOL can be used for metamaterial simulations, their roles are very limited, especially for three-dimensional time domain simulations due to the simple algorithms used. Our major goal in this project is to develop an efficient set of time domain finite element methods that are mathematically sound, accurate and efficient for simulating wave interactions with metamaterials. Through this NSF grant support, in the past several years, we have systematically studied the governing Maxwell's equations when metamaterials are involved. We studied several equivalent ways in writing the modeling equations and investigated some corresponding numerical algorithms to solve them. We have implemented most of the algorithms and developed the corresponding rigorous stability analysis and error estimates. Our research results have been published in more than 10 journal papers, a detailed list is shown below. In the long run, our developed robust and efficient algorithms for metamaterials will benefit broader areas such as electrical engineering, materials, optics, physics, nano-technology, and biomedical technology. We expect NSF's continuous support for our effort to make these algorithms more effective and productive through a few more years' hard work. On the other, the study of metamaterials presents a very good opportunity to attract students to pursue an advanced degree in computational and applied mathematics to enhance the competitiveness of our country in science and technology. Through this project, we have graduated one female PhD student, and one female M.Sc student. Currently, we have another one female working on her PhD thesis on metamaterials.
