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在广泛的应用中出现的普遍问题。通常，波与随机或结构化介质相互作用，人们对波通过介质传播或被介质局部化时的有效行为感兴趣。此外，当某些参数变化时，诸如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液等各种系统的有效电磁特性都表现出临界行为。研究人员对电磁场与这种复合材料的相互作用进行了基本的数学研究，并对模型问题进行了最先进的数值实验。特别是，发展了一种基于谱理论和解析延拓的方法来获得准静态极限下的有效复介电常数的严格界。最近，这种方法发现，双组分介质的有效输运性质与统计力学中的序参数具有相同的分析性质，例如伊辛模型中的磁化。这些强大的关系只在静态情况下被利用，并提出了许多重要的问题，但统计力学的思想也形成了分析波动现象临界行为的自然框架。静态案例中显示的大部分解析结构都适用于复合材料中的亥姆霍兹方程。本文从解析和数值两方面对这一观测结果及其对统计力学应用的影响进行了研究。这些研究可能在电磁场及其与复合系统的相互作用的数学、计算和物理方面产生根本性的进步。研究生和本科生参与数学和计算分析相结合的重点项目。在许多学科的广泛问题中，光、雷达或微波等电磁场与半导体、充油岩石、骨或心脏组织、雷达吸收涂层或航运集装箱等非均匀材料相互作用。例子出现在物理、材料科学、电气和生物工程、化学、生物学、地球物理和天体物理，是通信和医疗技术的核心。研究人员对电磁场与复合介质的相互作用进行了基本的数学研究，并对模型问题进行了最先进的数值实验。在许多重要的例子中，有效的电磁性质与系统中的某些参数密切相关。例如，波是否可以通过某些类型的结构介质(称为光子晶体)传播，关键取决于波长与介质结构或其他性质的比例。最近发现，复合材料中电输运的数学基础与统计力学的数学基础几乎是相同的。统计力学处理相变，如水在零摄氏度的临界温度冻结，并为提出关于这些系统的关键问题提供了分析框架。研究人员通过数学分析和复杂的计算将统计力学的思想发展并应用到电磁系统中。他们寻求基本的新观点，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。该项目积极地让研究生和本科生参与对总体目标至关重要的分析和计算项目。结果，特别是关于关键性质的结果，可能会潜在地影响广泛的应用领域，包括纳米级结构的介质。

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.