Inverse Homogenization for Microstructured Media

University Of Utah, Salt Lake City UT

Investigators

Abstract

Abstract: DMS-0508901, Elena Cherkaev, University of Utah Title: 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 medium on 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.