Robust and Efficient Numerical Methods for Electromagnetic Wave Propagation in Complex Media

该项目将开发新的数学模型和稳健的计算方法来模拟波在复杂介质中的传播，如超材料和石墨烯。这种跨学科的研究通过发现和理解超材料和石墨烯带来的纳米光学和隐身技术中的新现象，在纳米技术和材料方面具有直接的应用。石墨烯具有吸收范围宽、衰减快、稳定性高等特点，可用于产生皮秒激光脉冲。石墨烯可用于传感器以同时检测质量、气体、张力、疾病和爆炸物；石墨烯可用于移动设备的低成本显示屏、具有快速充电能力的锂离子电池、燃料电池动力汽车的储氢以及低成本燃料电池和水淡化等。所有这些应用都得益于通过降低物理实验成本来求解相关数学模型的准确和高效的数值算法。该项目每年将为一名博士生提供支持。本项目的重点是开发和分析稳健和高效的有限元方法来解决复杂介质中的电磁波传播问题。理论分析和实用算法的发展目标如下：(1)进一步探索最近发展起来的一些用于超材料的完美匹配层(PML)模型，发展和分析使用边缘单元和间断Galerkin方法求解它们的时域有限元方法；(2)探索石墨烯模型和高效的有限元算法来模拟石墨烯中的波传播；(3)发展稳健和高效的超材料和石墨烯中含时Maxwell方程的后验误差估计器，探索高阶三角形和四面体边缘单元可能的超收敛点；(4)发展和分析了具有随机输入的Maxwell方程的有效数值方法，并将其应用于随机超材料。该项目中开发的算法和代码将使人们更好地了解超材料和石墨烯及其物理效应，以便研究人员可以在应用程序中设计和使用它们。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。

This project will develop novel mathematical modeling and robust computational methods for simulating wave propagation in complex media such as metamaterials and graphene. This interdisciplinary research has direct applications in nanotechnology and materials through the advancement of discovery and understanding of new phenomena in nanooptics and stealth technology brought by metamaterials and graphene. Graphene can be used to generate picosecond laser pulses because of its wide absorption range, fast decay, and high stability properties. Graphene can be used in sensors to concurrently sense mass, gas, tension, diseases, and explosives; graphene can be used in low-cost display screens of mobile devices, lithium-ion batteries with fast recharge capacity, hydrogen storage for fuel cell-powered cars, and low-cost fuel cells and water desalination, etc. All of these applications benefit from accurate and efficient numerical algorithms for solving the associated mathematical models by reducing the cost of physical experiments. This project will provide support for one PhD student per year. The focus of this project is to develop and analyze robust and efficient finite element methods for solving electromagnetic wave propagation problems in complex media. Theoretical analysis and practical algorithms will be developed with the following objectives: (1) Further explore some perfectly matched layer (PML) models recently developed for metamaterials, develop and analyze time-domain finite element methods using both edge elements and discontinuous Galerkin methods for solving them; (2) Explore graphene models and efficient FEM algorithms to simulate wave propagation in graphene; (3) Develop robust and efficient a posteriori error estimators for time-dependent Maxwell’s equations in metamaterial and graphene, explore possible superconvergence points for high-order triangular and tetrahedral edge elements; (4) Develop, and analyze efficient numerical methods for Maxwell's equations with random inputs with applications for random metamaterials. The developed algorithms and codes in the project will lead to a better understanding of metamaterials and graphene, and their physical effects, so that researchers can design and use them in applications.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.