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

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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.