Direct and inverse problems for composite materials

Ou, M. YvonneComposite materials or composites, in general, can be considered as a combination of more than one elements. Studies of composites have been lasting for hundreds of years. On the one hand, composites may exhibit superior physical properties than any one of its component material and people can take advantages of such optimal designed composites to meet different needs in scientific studies or real-life applications. On the other hand, the determination of each component from a composite based on its "overall" properties has far-reaching impact as well, one example would be in the application of medical imaging, where different tissue of the human body possesses its unique physical property, and such uniqueness can be used to characterize this material by non-destructive experiments. In this thesis, direct and inverse problems in composite materials will be studied. For the direct problem, we investigated how does the microstructural information of a porous medium influences the permeability constant of the porous material; and for the inverse problem, we are interested in the numerical approaches that recover material properties based on indirect measurements of the composites. ☐ The thesis is structured as follows: • In part I, we studied a direct problem in composite material. Specifically, we investigated the problem of characterizing the dependence of permeability of a porous medium on its micro-structural information, by considering fluid flow in porous media as a limit case of a two-phase Stokes flow, or in other words, a composite material consisting two fluids with different viscosities. • In part II, we studied some inverse problems in composite materials. First of all, we studied the inverse problem of reconstruction the one-dimensional T2 time distribution in NMR relaxometry, that characterizes material properties in human tissues. In this study, we proposed a new idea in Tikhonov regularization, which we call it the Multi-Regularization, or Multi-Reg, and developed two different numerical schemes that achieve the goal of Multi-Reg. Secondly, we studied inverse problems in the Bayesian framework, the first problem is the one-dimensional T2 values recovery in the Bayesian framework; and the second problems is the application of electrical impedance tomography (EIT) technique in composite materials where EIT is used to locate a crack or defect in a conductive composite. ☐ Discussions and future work are also included in each part of the thesis.University of Delaware, Department of Mathematical SciencesPh.D