A new class of high-order integral solvers for wave propagation problems in composite media

This project is concerned with the design and the analysis of computational methodologies aimed at solving applied problems in materials science and engineering involving various physical observables (elastic and electromagnetic fields, acoustic fields in the frequency and the time domain) within and around complex structures (photonic or electronic devices, singular geometries with corners, edges or cracks, manmade structures built from metals or modern composite materials). The approaches also address frameworks containing complex materials—including composite elastic media, dielectrics, perfect and lossy conductors, as well as clouds of scatterers that can be described by media with dispersion and frequency-dependent absorption. Motivating applications for the solvers to be developed in this project include the radar clutter produced by chaff, photonic crystals and metamaterials, and communications. These are of fundamental significance in a wide class of areas concerning photonics (meta-materials, nanophotonics, meta-surfaces), antenna design (communications, remote sensing), electromagnetic interference and compatibility, and geophysical exploration. High-quality software implementation of the algorithms to be developed as part of this project will be released to the applied scientific community. This project will have a significant educational component, as both graduate and undergraduate students will be trained in scientific computing and mathematical modeling, and thus they will acquire the skills required to have a successful career in academia or industry.The computational methodologies underlying the proposed work are based on a class of density interpolations developed in recent years by the investigator and collaborators that is applicable to both Galerkin and Nystrom discretizations of all types of integral formulations. These methods combine the versatility of the Method of Fundamental Solutions with the robustness of integral formulations, are compatible with all kinds of meshes and quadratures, and can be seamlessly integrated with existing acceleration strategies such as Fast Multipole Methods. In practice, these types of solvers have demonstrated numerics that are fast and accurate for simulation of wave propagation problems in complex media, and thus they advance the state of the art in high-accuracy solution of partial differential equations. Owing to its ease of implementation and portability to solving a range of partial differential equations, the density interpolation method technology provides a vehicle to make integral equation solvers accessible to computational scientists from diverse backgrounds.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.