Analysis and Computation of Electromagnetic Transport in Composite Materials

电磁场与非均匀材料的相互作用是一个普遍存在的问题，在广泛的应用中出现。 波通常与随机或结构化介质相互作用，人们感兴趣的是波在介质中传播或被介质局部化时的有效行为。此外，光子晶体和带隙结构，随机电阻网络，海冰和其他多孔介质，以及电流变流体等各种系统都表现出临界行为，在其有效的电磁特性，随着一些参数的变化。 研究人员进行基本的，电磁场与这种复合材料的相互作用的理论研究，结合最先进的模型问题的数值实验。 特别地，基于谱理论和解析延拓的方法，在准静态极限下得到了有效复介电常数的严格界。 最近，这种方法导致了双组分介质的有效输运性质与统计力学中的序参量具有相同的分析性质，例如伊辛模型中的磁化强度。 这些强有力的关系只在静态情况下被利用，并提出了许多重要的问题，然而统计力学的思想也形成了一个分析波动现象临界行为的自然框架。 在静态情况下所显示的分析结构，在一个复合体中，大部分都可以转化为亥姆霍兹方程。 这一观察，以及它的影响，在这种情况下，使用matticalmechanics，研究分析和数值。 这些研究可能会在电磁场及其与复合系统的相互作用的数学、计算和物理学方面产生根本性的进步。 研究生和本科生参与结合数学和计算分析的重点项目。 在许多学科的广泛问题中，电磁场（如光、雷达或微波）与非均匀材料（如半导体、充油岩石、骨骼或心脏组织、雷达吸收涂层或集装箱）相互作用。 例子出现在物理学、材料科学、电子和生物工程、化学、生物学、电子物理学和天体物理学中，并且是通信和医疗技术的核心。 研究人员进行基本的数学研究电磁场与复合介质的相互作用，结合国家的最先进的numericalexperiment模型问题。 在许多重要的例子中，系统的有效电磁特性与系统中的某些参数密切相关。 例如，波是否能通过某些类型的结构化介质（称为光子晶体）传播，关键取决于波长与结构尺度或介质其他性质的比值。最近人们发现，复合材料中电输运的数学基础与统计力学中的物理基础几乎是一致的。 统计力学处理相变，如水在零摄氏度的临界温度下冻结，并提供了一个自然的框架，提出有关这些系统的关键问题。研究者们通过数学分析和复杂的计算，将电磁力学的思想发展并应用于电磁系统。 他们寻求基本的新视角，以及理解和预测电磁场如何与复合材料相互作用的新方法。 该项目积极涉及研究生和本科生从事分析和计算项目，是中央的整体目标。 结果，特别是关于关键属性，可能会影响广泛的应用领域，包括媒体结构的纳米尺度。

The interaction of an electromagnetic field with aninhomogeneous material is a pervasive problem arising in a broadarray of applications. Often a wave interacts with a random orstructured medium, and one is interested in the effective behavioras the wave propagates through or is localized by the medium. Moreover, systems as varied as photonic crystals and band gapstructures, random resistor netorks, sea ice and other porousmedia, and electrorheological fluids exhibit critical behavior intheir effective electromagnetic properties as some parameter isvaried. The investigators conduct fundamental, mathematicalstudies of the interaction of electromagnetic fields with suchcomposites, in conjunction with state of the art numericalexperiments on model problems. In particular, a method based onspectral theory and analytic continuation has been developed toobtain rigorous bounds on the effective complex permittivity inthe quasistatic limit. More recently, this approach has led tothe finding that the effective transport properties oftwo-component media share the same analytic properties as theorder parameters in statistical mechanics, such as themagnetization in an Ising model. These powerful relations haveonly been exploited in the static case and raise many importantquestions, yet the ideas of statistical mechanics form a naturalframework for analyzing critical behavior of wave phenomena aswell. Much of the analytic structure displayed in the static casecarries over to the Helmholtz equation in a composite. Thisobservation, as well as its implications for using statisticalmechanics in this context, is investigated analytically andnumerically. These investigations may yield fundamental advancesin the mathematics, computation, and physics of electromagneticfields and their interactions with composite systems. Graduateand undergraduate students are involved in key projects combiningmathematical and computational analysis. In a broad range of problems across many disciplines,electromagnetic fields such as light, radar, or microwavesinteract with inhomogeneous materials such as semiconductors,oil-filled rocks, bone or heart tissue, radar absorbing coatings,or shipping containers. Examples arise in physics, materialsscience, electrical and bio-engineering, chemistry, biology,geophysics, and astrophysics, and are central to communicationsand medical technologies. The investigators conduct fundamentalmathematical studies of electromagnetic fields interacting withcomposite media, in conjunction with state of the art numericalexperiments on model problems. In many important examples, theeffective electromagnetic properties depend critically on someparameter in the system. For example, whether or not a wave canpropagate through some types of structured media, called photoniccrystals, depends critically on the ratio of the wavelength to thescale of the structure or other properties of the medium. Recently it was found that the mathematics underlying electricaltransport in composites is almost identical to the mathematicsunderlying statistical mechanics. Statistical mechanics dealswith phase transitions such as the freezing of water at thecritical temperature of zero degrees Centigrade, and provides anatural framework for posing key questions about these systems. The investigators develop and apply the ideas of statisticalmechanics to electromagnetic systems using mathematical analysisand sophisticated computations. They seek fundamental newinsights, as well as novel ways of understanding and predictinghow electromagnetic fields interact with composites. The projectactively involves graduate and undergraduate students engaged inanalytical and computational projects that are central to theoverall goals. Results, particularly concerning criticalproperties, could potentially affect a broad range of applicationareas, including media structured on the nanometer scale.