Collaborative Research: A Computational Framework for Non-asymptotic Homogenization with Applications to Metamaterials

协作研究：非渐近均质化的计算框架及其在超材料中的应用

• 批准号: 1216927
• 负责人: Igor Tsukerman
• 金额: $ 19.68万
• 依托单位: University of Akron
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216927&HistoricalAwards=false
• 关键词: Collaborative; Research; Computational; Framework; Non

The project is aimed at developing a non-asymptotic homogenizationtheory of Maxwell's equations in artificial periodic composites(metamaterials). Currently, there is a consensus that sufficientlylarge lattice cell sizes are necessary for some nontrivial physicaleffects to occur in such structures. (The most intriguing of theseeffects is high-frequency magnetism.) Classical homogenizationtheories work well in the zero-cell-size limit but are difficult toapply to large metamaterial cells. In contrast, the proposed theory isnon-asymptotic and does not involve any series expansions with respectto the cell size. The electromagnetic field in the material isapproximated by a finite set of functions (modes) usually but notnecessarily Trefftz functions such as Bloch waves. The coarse-grainedfields and flux densities are defined via curl-conforming anddiv-conforming interpolations, respectively. A linear map betweenthese interpolants is established and defines an extended materialtensor. In a certain canonical basis, this extended tensor has aclassical block of 36 local parameters and a novel block quantifyingnonlocal effects. From the differential-geometric perspective, thisconstitutive relationship can be viewed as a realization ofBossavit-Hiptmair's discrete Hodge operators (linear maps betweendiscretized 1-forms that correspond to vector fields and 2-forms thatcorrespond to fluxes).Over the last decade, metamaterials have attracted unprecedentedattention due to a variety of potential applications that includesuperlensing, electromagnetic cloaking, electromagnetically-inducedtransparency, efficient antennas, and more. Experimentaldemonstrations of these effects have been limited to proofs ofprinciple and at optical frequencies have so far been incomplete.Subwavelength optical imaging has been achieved only in thequasi-static (near-field) regime, standard for the more conventionalnear-field optics; the so-called "carpet cloak" conceals surface bumpsrather than 3D objects, and so on. Moreover, applications that do notdepend critically on the effective medium description of metamaterialsappear to be more easily achievable than the ones that do. The lattergroup includes, notably, superlensing and cloaking. This suggeststhat, to make further progress, theoretical and mathematical issues atthe heart of metamaterial science must be unambiguously resolved. Themain problem can be stated as follows. Given the composition of ametamaterial cell and the operating frequency, determine whether thismetamaterial can be reasonably described as a continuous medium withsome effective parameters, just like any natural optical material; ifthe answer is positive, develop a rigorous methodology for such adescription. The proposed research is aimed at solving this problem inthe most difficult case when the cell size of the composite is anappreciable fraction of the wavelength of light. The methodology, oncedeveloped, will allow the scientific community to delineate thepossible from the impossible in the field of metamaterials.The intellectual merit of the proposed research is in the developmentof a new paradigm of non-asymptotic homogenization, of newcomputational methods related to it, and in the application of theproposed methodology to electromagnetic metamaterials, allowing one togain a much deeper understanding of their properties and limitations.As a new area of research, non-asymptotic homogenization will alsohave a broader technical impact in other areas of applied physics andengineering, such as acoustics, heat transfer and possibly elasticity.

该项目旨在发展人工周期性复合材料（超材料）中麦克斯韦方程的非渐近均匀化理论。目前，有一个共识，即非常大的晶格单元的尺寸是必要的一些非平凡的物理效应发生在这样的结构。(The most最interesting有趣of these effects效果is high高frequency频率magnetic磁性.）经典的均匀化理论在零单元尺寸的限制下工作得很好，但很难应用于大的超材料单元。与此相反，所提出的理论是非渐近的，不涉及任何级数展开的尊重细胞的大小。材料中的电磁场近似于一组有限的函数（模式），通常但不一定是Trefftz函数，如Bloch波。粗粒场和通量密度分别通过旋度协调插值和凹度协调插值定义。在这些插值之间建立了一个线性映射，并定义了一个扩展的材料张量。在一定的正则基下，这个扩展张量具有36个局部参数的经典块和一个量化非局部效应的新块。从微分几何的角度来看，这种本构关系可以看作是Bossavit-Hiptmair离散Hodge算子的一种实现（对应于矢量场的离散1-形式和对应于通量的离散2-形式之间的线性映射）。在过去的十年中，由于各种潜在的应用，超材料吸引了前所未有的关注，包括双折射透镜，电磁隐身，电磁感应透明，高效率的天线，以及更多。这些效应的实验演示仅限于原理证明，而且到目前为止，在光学频率上的实验演示还不完整。（近场）制度，标准的更传统的近场光学;所谓的“地毯斗篷”隐藏的是表面凹凸而不是3D物体，等等。此外，不严格依赖于超材料的有效介质描述的应用似乎比那些更容易实现。后一组包括，值得注意的是，超透镜和隐形。这表明，为了取得进一步的进展，超材料科学的核心理论和数学问题必须明确解决。主要问题可以陈述如下。在已知超材料单元的组成和工作频率的情况下，确定这种超材料是否可以像任何自然光学材料一样，被合理地描述为具有某些有效参数的连续介质;如果答案是肯定的，则为这种描述开发一种严格的方法。提出的研究旨在解决这个问题在最困难的情况下，当细胞的复合材料的大小是一个可观的分数的光波长。该方法一旦开发出来，将使科学界能够在超材料领域中从不可能中描绘出可能。拟议研究的智力价值在于开发一种新的非渐近均匀化范式，与之相关的新计算方法，以及将拟议方法应用于电磁超材料，作为一个新的研究领域，非渐近均匀化也将在应用物理和工程的其他领域产生更广泛的技术影响，如声学，热传递和可能的弹性。