Structural Spectra and Applications to Heterogeneous Media

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Advancements in technology benefit from improvements in understanding heterogeneous materials and their ability to control electromagnetic, elastic, and acoustic phenomena. In this project the investigator and his collaborators develop three research areas that support these improvements by introducing and applying novel spectral problems associated with the geometry of material heterogeneity. In the first set of projects the theory of periodic and quasi periodic electrostatic spectra is developed for structured geometries enabling control of wave propagation in photonic, phononic, and metamaterial crystals. This theory will be applied to the systematic design of pass bands and band gaps for devices used in communication and energy transmission. In the second set of projects a novel spectral theory is developed for restriction operators acting on electromagnetic fields inside heterogeneous materials and applied to provide new efficient approaches to the fast numerical solution of multiscale problems for dielectric and magnetic media. A final research project develops a spectral formulation for energy concentration in heterogeneous media and applies it to identify the worst case loads for structures made from composite and polycrystalline media. This will give us a better understanding of why metallic and composite joints fail in infrastructure and aviation. These research projects are in line with the research interests of the the Materials and Manufacturing Directorate at Wright Patterson Air Force Base and a continuation of interaction between the PI and the labs initiated in the previous NSF award. It is anticipated that the graduate student supported by this award will benefit from the interaction with scientists at the Materials and Manufacturing Directorate at Wright Patterson Air Force Base. Earlier work on the theory of band gaps addresses the asymptotic theory of band gaps at infinite contrast. In this project the investigator will exploit new structural spectra associated with inclusion geometry and lattice parameters to develop new techniques for complex operator valued functions which deliver explicit formulas for band gaps at finite contrast. The approach proposed here is distinct from earlier ones and will provide new explicit formulas for the size of band gaps and pass bands given in terms of contrast, shape and configuration of scatters, and lattice parameters. In a second thrust of the project, the investigator and his collaborators will develop new spectral methods related to the restriction operator for the vector wave equation to deliver new and faster methods for the numerical solution of multiscale problems of electromagnetic wave propagation in heterogeneous media. A novel spectral formulation for energy concentration will be developed and applied to identify worst case loads for structures made from composite and polycrystalline media.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.