Time-Domain Integral Equation Methods for Electromagnetic Scattering

Computational electromagnetics (CEM) is the science of simulating the behavior of electromagnetic fields on a computer. Several different methods exist for accomplishing this, and each has distinct advantages and disadvantages. CEM techniques may be categorized in a number of ways, but the primary categorizations are domain and equation type. The domain may be either time or frequency. Frequency domain CEM methods are more efficient for the simulation of narrowband phenomena and phenomena involving loss, but they can not handle nonlinear problems and are very inefficient for the simulation of broadband phenomena or moving objects. Time domain methods have the exact opposite properties; they are efficient over broad bands, can simulate nonlinear and moving phenomena, but the simulation of loss is difficult.

Equation type is either differential or integral. Differential equation methods solve for the actual fields, whereas integral equation methods solve for electric currents. Since the currents only exist on objects (instead of filling all of space) integral equations have no need for approximations at the end of the simulation domain. Moreover, if the object under study is homogeneous, the unknown current in the integral equation exists only on the surface, as opposed to filling the whole volume of the structure. Differential equation methods, on the other hand, lead to sparse systems of equations (i.e., most of the coefficients in the equations are zero) but they always require volumetric meshing and special approximate boundary conditions to terminate the simulation. (In principle, the fields are nowhere zero, and since computer memories are finite, something must be done at the edges of the simulation.) For this reason, integral equation methods are generally preferred for the study of homogeneous materials, and differential equation methods are best for inhomogeneous materials.

Most current CEM methods are variants of three primary approaches: the method of moments, the finite element method, and the finite difference time domain method. On the other hand, given the discussions above, there should be four methods, since there are two domains and two equation types. This research focuses on the missing fourth method: time domain integral equations (TDIEs). Of course, researchers in CEM have known for three decades that there should be a reliable TDIE scheme, but for three decades all attempts to create such a scheme have been unstable. In other words, every method proposed for using TDIEs to simulate electromagnetic fields have led to codes that exhibit such enormous numerical error that the simulation is eventually swamped by nonsense.

Prof. Weile’s work has demonstrated that the difficulties with previous approaches arose from ignoring the digital nature of computers, and has resulted in a new, stable numerical method that fixes the earlier problems using digital signal processing techniques. The work has far-reaching practical applications as well. While in principle any CEM technique can be applied to (almost) any problem, using the incorrect method for a given problem can lead to inefficiencies so profound that the given problem can not be solved in a reasonable amount of time, even on a supercomputer. Thus, for certain types of simulations, the new approach really means the difference between the ability to simulate something before building it, and an expensive, inefficient cut-and-try method of design.

One of the most important practical applications of the new method is in the study of electromagnetic interference (EMI). EMI is familiar to anyone who has operated a vacuum cleaner while watching TV: the brushes in the electric motor in the vacuum radiate broadband electromagnetic energy, which is re-radiated by the wiring in the house. This signal is then received by the TV antenna, resulting in the annoying ghosting and lines on the screen. EMI is even more familiar to companies that market electronic products: All electrical devices must, by FCC regulations, neither radiate too much energy, nor be to susceptible to EMI. The FCC periodically tests commercially available equipment for electromagnetic compatibility (EMC), and if a device fails the test, it must be taken off the market immediately. Obviously, such an event can bankrupt a company, so companies test their devices for EMC. This testing is time-consuming and expensive, and if a problem is found, fixing it may add prohibitively to the cost of the unit. Prof. Weile’s technique will allow for the simulation of EMI effects in devices before they are tested, so that difficulties can be ironed out even before the fabrication of the prototype. No currently available CEM method can do this efficiently for a number of technical reasons (differential equation methods have difficult simulating small holes and large propagation distances, and EMI problems are inherently broadband, so frequency domain methods are inefficient), so the economic impact of the new method could be profound.

EMI is not the only application of the new method, however. In particular, the new method would be the best way of simulating smart antennas, nonlinear antennas, and microwave circuits involving moving parts (i.e. MEMS). Any electromagnetic simulation problem involving small holes, large propagation distances, moving parts, nonlinear materials, or broadbands would best be simulated by the new method.

Stochastic Optimization of Electromagnetic Devices

Under construction.

Periodic Structure Modeling and Design

Under construction.