Next-generation integral equation methods for wave scattering and propagation in periodic structures

研究人员和研究生将开发新的积分方程算法来求解复杂周期结构的时间谐波散射背景下的亥姆霍兹方程。这些算法将是高效的，高精度的，数学上严格的，并且与许多现有的积分方程解算器不同，对所有问题参数都是健壮的。在二维成功的基础上，该项目考虑了具有重要技术意义的多层介质和三维双周期散射体。这项工作包括开发新的表面求积方案，以及使用最新的快速直接求解器来处理多个入射角度。基础科学、工程进步和技术越来越依赖于波长尺度上的周期结构对波的引导和控制。例如，衍射光栅、滤光片、光子晶体和其他纳米材料、具有成本效益的太阳能电池、微波天线、雷达、吸音器、光刻和遥感。研究人员提出的计算机算法将使此类设备的建模和设计更加高效、准确和可靠。效率和稳健性是最重要的，因为预测真实世界的设备性能通常需要在不同参数下使用数千种解决方案。对这种复杂的设备进行更快速的建模和设计可以带来很多社会好处，包括更快的通信速度和更便宜的可再生能源。调查员将发布公开可用的软件，还将邀请当地农村高中的数学学生在音乐、声学、波浪和计算机信号分析方面进行动手探索。

The investigator and supported graduate student will develop new integral equation algorithms to solve the Helmholtz equation in the context of time-harmonic wave scattering from complex periodic structures. These algorithms will be efficient, high-order accurate, mathematically rigorous, and, unlike many existing integral equation solvers, robust for all problem parameters. Building upon success in two dimensions, the project considers technologically important multilayer media and three dimensional doubly-periodic scatterers. The work includes developing new surface quadrature schemes, and the use of recent fast direct solvers to handle multiple incident angles.Basic science, engineering progress, and technology is increasingly reliant upon the guidance and control of waves by periodic structures on the scale of the wavelength. Examples include diffraction gratings, filters, photonic crystals and other nano-scale materials, cost-effective solar cells, microwave antennae, radar, sound absorbers, lithography and remote sensing. The computer algorithms proposed by the investigator will make the modeling and design of such devices more efficient, accurate, and reliable. Efficiency and robustness are paramount because predicting real-world device performance often requires thousands of solutions at different parameters. The societal benefits of more rapid modeling and design of such complex devices are many, including faster communication, and cheaper renewable energy. The investigator will release publicly-available software, and also will engage local rural high-school mathematics students with hands-on explorations in music, acoustics, waves, and computer signal analysis.