In the electromagnetic simulation of realistic structures, the spatial representation of the domain being analyzed depends not only on the frequency of interest but also on the need to capture possible fine geometric features. Such mixed scales cause havoc in standard integral equation based solvers on three fronts; (i) discretized integral equations become poorly conditioned as the size of the element becomes smaller, (ii) the function spaces used do not optimally represent the underlying physics, and (iii) the overall computational burden is exceedingly large. This largely limits the applicability of the existing methods. The proposed project seeks to develop a demonstrably unified, robust and accurate solution methodology that is well conditioned over a wide range of frequencies and, at the same time, has the flexibility to handle complicated (and possibly near singular) geometries. This is achieved by (i) developing a well conditioned integral equation scheme (that are Fredholm equations of the second kind) with provable bounds on convergence rates and accuracy to solve for electromagnetic quantities over a large range of spatial frequencies; (ii) enlarging the approximation space used for representing the unknown quantity so as to include the local physics; (iii) designing a scheme that permits seamless interplay between a variety of basis functions to model the unknown quantities to be used with the above integral equation scheme; (iv) deriving error bounds and convergence estimates on these schemes to demonstrate clear and easy user control over the error, and (iv) developing a domain decomposition framework so that these schemes can be integrated seamlessly with classical integral equation and finite element methods to solve electrically large problems. The educational objective is to develop a publicly available set of tutorials/teaching modules based on this research.
The rapid progress in simulation methods in concert with the Moore's law has made the analysis of electrically large problems possible on simple desktop machines in reasonable computational times. So much so that fullwave or rigorous simulation of realistic devices are within the realm of possibility. However, as one tends towards this goal, new and more challenging problems arise. In modeling mixed scale physics, it is necessary to correctly represent local physics, develop methods to overcome conditioning issues, and develop means to accelerate computation over multiple scales. This project addresses the resolution of these problems. The methods developed herein will have a wide footprint ranging from national security (design of conformal antennas) to sensor technology (surface enhanced raman and plasmonics) to metamaterials to nanotechnology (nano-structure crystal growth dynamics) to molecular dynamics. In addition to training graduate students in engineering and mathematics, existing channels are utilized to recruit women and minorities and undergraduate students are involved through senior design projects and potential REU supplements.
