Inverse Homogenization for Microstructured Media

摘要：DMS-0508901，Elena Cherkaev，University of Utah题目：微结构化介质的逆均匀化该项目开发了一种从有效的声学或电磁测量中提取有关微结构或纳米结构材料的精细尺度几何结构信息的方法。使用声波或电磁波来感测微结构或纳米结构介质的技术难点在于微几何形状的复杂性和精细尺度。所施加的声波的长度远大于微结构的变化，使得在测量数据中仅存在结构的有效或均匀化响应。所提出的“逆均匀化”方法从有效的测量中获得微观结构参数。该方法是基于Stieltjes分析表示的随机微结构介质的有效特性，并有可能重建的频谱函数在此表示从均匀化材料的有效响应。该方法将介质在不同尺度上的性质联系在一起：粗尺度上的复介电常数为微尺度反演提供数据。在计算上，这个问题是不适定的，需要正则化来开发稳定的数值算法。恢复的微观结构信息可以用于表征其他运输和介质的物理性质，如渗透性，扩散，导热性和水力传导性等。该项目开发了一种新的方法来重建的微观和纳米结构的信息，从测量复合材料的有效性能。它汇集了最近在两个不同的领域获得的结果：均匀化，逆和成像理论，这使得有可能制定新的应用这两个研究领域。问题是从复合材料对所施加的声场或电磁场的已知响应中找到复合材料的结构。当所施加的场的波长远大于微观结构尺度时，则不能分辨结构的精细尺度特征，并且微观几何形状被均匀化。该项目开发了一种“逆均匀化”方法，该方法利用在一定频率范围内的声学或电磁测量来恢复关于介质的微观结构参数的信息。拟议工作的结果在设计人工微和纳米结构复合材料方面是有用的。它们还可通过超声评估骨结构和密度来应用于骨质疏松症，并可用于无创监测血凝块并评估其年龄和结构。

Abstract: DMS-0508901, Elena Cherkaev, University of UtahTitle: Inverse Homogenization for Microstructured Media The project develops a method of extracting information about the fine scale geometric structure of micro- or nano-structured material from effective acoustic or electromagnetic measurements. The technical difficulty of using sound or electromagnetic waves in sensing micro- or nano-structured media is the complexity and fine scale of the microgeometry. The length of the applied sound waves is much larger than the variations of the microstructure, so that only an effective or homogenized response of the structure is present in measured data. The proposed "inverse homogenization" method derives microstructural parameters from effective measurements. The method is based on the Stieltjes analytic representation of the effective properties of a randomly microstructured medium, and a possibility to reconstruct the spectral function in this representation from the effective response of the homogenized material. The approach ties together properties of the mediumon different scales: Complex permittivity on the coarse scale provides data for microscale inversion. Computationally, the problem is ill-posed and requires regularization to develop stable numerical algorithms. The recovered microstructural information can be used for characterizing other transport and physical properties of the medium, such as permeability, diffusion, thermal and hydraulic conductivity, etc.The project develops a novel approach to reconstruction of micro- and nano-structural information from measurements of effective properties of composite materials. It brings together results recently obtained in two different fields: homogenization, and inverse and imaging theory, which makes possible to formulate new applications for both research areas. The problem is to find a composite material's structure from its known response to the applied acoustic or electromagnetic field. When the wavelength of the applied field is much larger than the microstructural scale, then fine scale features of the structure cannot be resolved, and the microgeometry is homogenized. The project develops an "inverse homogenization" method that utilizes acoustic or electromagnetic measurements over a range of frequency to recover information about the microstructural parameters of the medium.The results of the proposed work are useful in designing artificial micro- and nano-structured composites. They also have applications to osteoporosis through ultrasound evaluation of bone structure and density and to noninvasive monitoring of blood clots and evaluation of their age and structure.