Inverse Homogenization for Microstructured Media

摘要：DMS-0508901，犹他大学Elena Cherkaev标题：微结构介质的逆均质化该项目开发了一种方法，从有效的声学或电磁测量中提取关于微或纳米结构材料的精细几何结构的信息。利用声波或电磁波传感微结构或纳米结构介质的技术难点在于微观几何结构的复杂性和细微尺度。外加声波的长度远大于微结构的变化，因此在测量数据中只存在结构的有效响应或均匀响应。提出的“逆均质化”方法从有效的测量结果中得出微观结构参数。该方法基于随机微结构介质的有效性质的Stieltjes解析表示，以及根据均匀材料的有效响应重建该表示中的谱函数的可能性。该方法将介质在不同尺度上的性质联系在一起：粗略尺度上的复介电常数为微观尺度的反演提供了数据。在计算上，该问题是不适定的，需要正则化来开发稳定的数值算法。恢复的微结构信息可用于表征介质的其他传输和物理性质，如渗透性、扩散、热导率和水力传导性等。该项目开发了一种新的方法，通过测量复合材料的有效性能来重建微结构和纳米结构信息。它汇集了最近在两个不同领域取得的成果：均匀化，以及逆和成像理论，这使得有可能为这两个研究领域制定新的应用。问题是根据复合材料对外加声场或电磁场的已知响应来确定其结构。当外场波长远大于微结构尺度时，结构的细微尺度特征无法分辨，微几何形状趋于均匀。该项目开发了一种利用声波或电磁测量在一定频率范围内恢复介质微结构参数信息的反均质化方法。所提出的工作结果对人工微结构和纳米结构复合材料的设计是有用的。它们还通过超声评估骨骼结构和密度，以及无创监测血栓和评估其年龄和结构，应用于骨质疏松症。

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.