Despite decades of advances in computational electromagnetics (CEM), some problems still elude efficient and accurate solutions. Problems occurring in the analysis of microelectromechanial systems, electromagnetic compatibility, and ultra wide band antenna design involve complex geometry, small structures, and large propagation distances across homogeneous regions. Solutions to problems with these characteristics are most efficiently computed (in principle) by time-domain integral equation (TDIE) techniques. Unfortunately, while TDIEs have been researched for more than thirty years, there is still no TDIE scheme for CEM that can efficiently model curvilinear geometry while ensuring stability. Thus, current techniques are either of low order of convergence or unreliable stability. Mathematical error analysis, work estimates and computer programs created under this proposal will demonstrate a new method, based on convolution quadrature, that promises to deliver both superlinear accuracy and unconditional stability.
Electromagnetic theory governs the behavior of light, radio waves, and microwaves. Understanding the behavior of electromagnetic waves is of the utmost concern to national security (in the design of radar and surveillance systems and protecting them from interference), energy systems (in the collection and distribution of electric power), and the economy (in cell phones and television broadcasts). Unfortunately, as electromagnetic systems become more complicated and powerful, they become harder to simulate and therefore design. For instance, modern problems in shielding devices from electromagnetic interference involve broad bandwidths, long propagation distances, and small features. While it has been known for decades that this combination of difficulties can be mitigated using time domain integral equation formulations, no such formulation has been produced that can accurately model arbitrary geometry without producing a solution with exponentially increasing error. This proposal produces such a scheme by introducing a new method for modeling the time dependence of the solution.
Computational electromagnetics is the branch of modeling research that seeks to understand how microwaves, radio waves, and light are created, travel, and scatter. Currently, several methods for such modeling exist, and are used by circuit design companies, antenna designers, computer manufacturers, and the military to help design and optimize devices ranging from waveguides to aircraft. For reasons relating to their formulation, some of the extant methods are better for some applications, and some are better for others. A method capable of analyzing electromagentic energy deposition in the brain during cell phone usage is probably not the best choice for analyzing scattering from the stealth bomber. Despite the existence of several methods for electromagnetic modeling, no method in widespread use is capable of the efficient simultaneous simulation of broadband problems involving motion and small apertures or objects. Such problems arise in the study of broadband antennas, electromagnetic interference, military applications, and nanotechnology. Ideal methods for such problems would be based on the analysis of time domain integral equations (TDIEs), but early attempts at the creation of TDIE solvers yielded unstable, physically meaningless solutions. At the start of this research, there existed stable TDIE solvers, but they were either not completely reliable, or limited (by the very nature of their formulation) in their ability to model geometry accurately. The intellectual impact of this work is found in the fact that it introduced a new method for modeling electromagnetic phenomena with TDIEs. The method is reliably stable, and its stability and accuracy were proven with both mathematical analysis and numerical experiment. Broader impacts spring primarily from applications: Applications to different types of scattering and radiation problems were demonstrated, and initial implementations of methods hybridized with earlier techniques were developed. Anticipated future applications include the modeling of moving scatterers, efficiency enhancements for consumer goods, and accelerated prototype development for electrical devices of all kinds.
