Modeling, Algorithms and Computation of Electromagnetic Wave Interacting with Dispersive Interface

该项目的目标是开发新的数学和模拟工具，用于研究电磁波与任意弯曲的色散界面的相互作用。在开发这种相互作用的有效和可靠的数值方法方面存在巨大的挑战。物理上，波解及其导数在色散界面上的跳跃与时间有关。在数值上，现有的算法由于不能处理这种时变跳变而遭受了严重的精度下降。在计算上，当与处理弯曲界面的阶梯近似相耦合时，该界面误差将被显著放大。由于这些挑战，在超材料模拟中通常采用每波长约100个网格点的极其昂贵的网格分辨率。在这个项目中，研究人员将严格分析色散界面上电磁场分量的时间相关性和交叉耦合。对于常用的色散材料和超材料模型，将推导出新的公式，将随时间变化的跳跃条件转换为与时间无关的条件，并将交叉耦合降至最低。在这些数学模型的基础上，只使用简单的笛卡尔网格，就可以开发一种处理任意曲线色散界面的二阶精确界面算法。这种更高的精度将保证更高的数值分辨率，因此可以显著减轻现有模拟的计算负担。弥散介质在自然界中普遍存在，例如在生物组织、岩石、土壤和等离子体中。色散介质的数值模拟对于诸如用于乳腺癌早期检测的微波成像、基于双负超材料的亚波长成像系统和隐身设备等广泛的电磁和光学应用是至关重要的。拟议的数学建模、算法开发和数值计算将解决位于计算数学、物理和电气工程之间的跨学科领域的关键科学挑战。计划中的研究活动将为计算数学带来新的进步，并为各种实际工程设备和系统的表征、分析和设计提供可靠的模拟工具。这些工具反过来可以为分析或校准一些基本的物理定律提供更好的手段，例如支配次衍射成像系统分辨率极限的定律。此外，这个项目将提供一个跨学科的研究培训环境，可以激励和促进更多的学生追求科学和工程的职业生涯。

The goal of the proposed project is to develop novel mathematical and simulation tools for studying electromagnetic wave interacting with arbitrarily curved dispersive interface. Great challenges exist in developing efficient and reliable numerical methods for such interactions. Physically, jumps in wave solution and its derivatives across the dispersive interface are time dependent. Numerically, the existing algorithms suffer a serious accuracy reduction due to their incapability to handle such time variant jumps. Computationally, this interface error will be significantly amplified when coupling with the staircasing approximation in treating curved interface. Due to these challenges, an extremely expensive mesh resolution of about 100 grid points per wavelength was commonly practiced in the metamaterial simulations. In this project, the investigator will rigorously analyze the time dependence and cross coupling of electromagnetic field components at the dispersive interface. Novel formulations will be derived for commonly used dispersive material and metamaterial models to convert time dependent jump conditions into time independent ones and to minimize the cross coupling. Building on these mathematical modeling, a second order accurate interface algorithm will be developed to deal with arbitrarily curved dispersive interface, by using only a simple Cartesian grid. This higher order of accuracy will promise a higher numerical resolution, so that the computational burden of the existing simulations can be significantly relieved. Dispersive media are ubiquitous in nature, such as in biological tissues, rocks, soils, and plasma. The numerical simulation of dispersive media is crucial to a wide range of electromagnetic and optical applications, such as microwave imaging for early detection of breast cancer, double negative metamaterial based subwavelength imaging system, and cloaking devices. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in an interdisciplinary filed lying at the interface of computational mathematics, physics, and electric engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the characterization, analysis, and design of various practical engineering devices and systems. These tools in turn may offer a better means for analyzing or calibrating some basic physical laws, such as the one governing the resolution limit of the sub-diffraction imaging system. In addition, this project will provide an interdisciplinary research training environment which could inspire and promote more students to purse careers in science and engineering.