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

本项目旨在发展人造周期复合材料（超材料）中麦克斯韦方程组的非渐近均匀化理论。目前，有一个共识是，足够大的晶格单元尺寸对于在这种结构中发生一些重要的物理效应是必要的。（这些效应中最有趣的是高频磁性。）经典的均质化理论在零细胞尺寸限制下工作良好，但难以应用于大型超材料细胞。相反，所提出的理论是非渐近的，并且不涉及关于细胞大小的任何级数展开。材料中的电磁场由一组有限的函数（模态）近似表示，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形插值和分形插值定义。建立了这些插值之间的线性映射，并定义了扩展的材料张量。在一定正则基下，该扩展张量具有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的离散Hodge算子（对应于向量场的离散1-形式和对应于通量的2-形式之间的线性映射）的实现。在过去的十年中，由于各种潜在的应用，包括超透镜、电磁隐身、电磁诱导透明、高效天线等，超材料吸引了前所未有的关注。这些效应的实验证明仅限于原理证明，在光学频率上的证明迄今为止还不完整。亚波长光学成像已实现仅在准静态（近场）制度，标准的更传统的近场光学；所谓的“地毯斗篷”隐藏的是表面凸起，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依赖于有效介质描述的应用更容易实现。后者主要包括超透镜效应和隐形效应。这表明，要取得进一步的进展，超材料科学核心的理论和数学问题必须得到明确的解决。主要问题可以陈述如下。给定超材料电池的组成和工作频率，确定该超材料是否可以像任何天然光学材料一样，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，为这样的描述制定一个严格的方法。所提出的研究旨在解决这一问题，在最困难的情况下，当复合材料的细胞尺寸是光波长的微不足道的一部分。这种方法一旦开发出来，将使科学界能够在超材料领域描绘出可能与不可能。所提出的研究的智力价值在于发展了一种新的非渐近均匀化范式，以及与之相关的新计算方法，并将所提出的方法应用于电磁超材料，使人们能够更深入地了解它们的性质和局限性。作为一个新的研究领域，非渐近均质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、传热和可能的弹性。

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.