Structural Spectra and Applications to Heterogeneous Media

技术的进步得益于对异质材料的理解及其控制电磁、弹性和声学现象的能力的提高。在这个项目中，研究人员和他的合作者开发了三个研究领域，通过引入和应用与材料异质性几何结构相关的新光谱问题来支持这些改进。在第一组项目的周期性和准周期性的静电频谱的理论开发的结构化的几何形状，使控制波在光子，声子和超材料晶体的传播。该理论将应用于通信和能量传输设备的通带和带隙的系统设计。在第二组项目中，开发了一种新的谱理论，用于约束算子作用于异质材料内部的电磁场，并用于提供新的有效方法来快速数值求解电介质和磁介质的多尺度问题。最后一个研究项目开发了一个频谱公式的能量集中在非均匀介质中，并将其应用于确定最坏情况下的负载由复合材料和多晶介质的结构。这将使我们更好地了解为什么金属和复合材料接头在基础设施和航空中失败。这些研究项目符合赖特帕特森空军基地材料和制造局的研究兴趣，也是PI和先前NSF奖项中启动的实验室之间互动的延续。预计该奖项支持的研究生将受益于与赖特帕特森空军基地材料和制造局科学家的互动。带隙理论的早期工作解决了无限对比度下带隙的渐近理论。在这个项目中，研究人员将利用新的结构光谱与夹杂物的几何形状和晶格参数，以开发新的技术，复杂的运营商价值的功能，提供明确的公式，在有限的对比度带隙。这里提出的方法是不同的，从早期的，并将提供新的显式公式的带隙和通带的大小给出的对比度，形状和配置的散射，和晶格参数。在该项目的第二个推力中，研究人员和他的合作者将开发与矢量波方程的限制算子相关的新的谱方法，以提供新的和更快的方法，用于非均匀介质中电磁波传播的多尺度问题的数值解。将开发一种新的能量集中的谱公式，并应用于确定由复合材料和多晶介质制成的结构的最坏情况载荷。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.