Structural Spectra and Applications to Heterogeneous Media

技术的进步得益于对非均质材料及其控制电磁、弹性和声学现象的能力的理解的提高。在这个项目中，研究人员和他的合作者通过引入和应用与材料非均质性几何相关的新颖光谱问题，开发了三个支持这些改进的研究领域。在第一组项目中，周期和准周期静电谱理论被发展用于结构化几何结构，从而能够控制波在光子、声子和超材料晶体中的传播。这一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，发展了一种新的谱理论，用于非均匀材料中作用于电磁场的限制算子，并应用于快速数值求解介质和磁性介质的多尺度问题。最后一个研究项目开发了非均匀介质中能量集中的频谱公式，并将其应用于识别由复合和多晶介质制成的结构的最坏情况下的载荷。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领域失效。这些研究项目符合莱特·帕特森空军基地材料和制造局的研究兴趣，以及PI与前一次NSF奖项中启动的实验室之间的持续互动。预计该奖项支持的研究生将受益于与莱特帕特森空军基地材料和制造委员会的科学家的互动。早期关于带隙理论的工作解决了无限对比度下带隙的渐近理论。在这个项目中，研究人员将利用与夹杂几何和晶格参数相关的新结构谱来开发复算符值函数的新技术，这些技术可以给出有限对比度下带隙的显式公式。这种方法不同于以往的方法，它将根据对比度、散射体的形状和配置以及晶格参数提供新的带隙和通带大小的显式公式。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动方程的限制算子相关的新的谱方法，以提供新的、更快的方法来数值求解非均匀介质中的电磁波传播的多尺度问题。将开发一种新的能量集中频谱公式，并应用于识别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智力优势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.