Renee E. George and Dr. R. Steven Turley, Physics and Astronomy

There are many applications of physics where it is necessary to solve Maxwell’s Equations for a wave interacting with or scattering from various objects. This problem is usually impossible to solve analytically, but several cases have been solved using numerical methods. Unfortunately some cases still cannot be solved within the desired accuracy. Designers of many important applications like high frequency circuits and high-gain antennas need to use a lot of guesswork in order to produce correctly functioning devices. Our research as been aimed at implementing a new method which will permit practical numerical solutions to these problems with the desired accuracy.

This case sits right in the middle of two regimes where solutions have been found. First, when the wavelength is much smaller than the characteristic length scale of the scatterer or circuit, approximations such as geometric and physical optics give sufficiently accurate answers. As the wavelength increases though, diffraction effects not adequately treated by these approximations begin to become significant. Second, when the wavelength is much larger than the object, one can use a technique called the Method of Moments. In this method, the current is represented as a sum of functions forming a set of basis functions of the following general form: I(x) = a1f1(x) + a2f2(x) + … + anfn(x) where fi is an appropriate function and I is the current. This splits a hard problem into many small easier problems, and increases the number of unknowns. It works well as long as the object is not much larger than the wavelength. For a long time physicists and engineers were unable to bridge the gap between these two regimes. There were just too many unknowns if designers tried to use the Method of Moments, and there were built-in inaccuracies if they tried to use tile approximate techniques. Since the required computer time in tile Method of Moments is proportional to the cube of the number of unknowns and the necessary memory is proportional to the square of this number, one quickly exhausts computer resources as size of the problem increases. Recently, researchers have developed basis functions which reduce the number of unknowns necessary to accurately model the current. So far, this research his been limited to problems where the characteristic length of the object is comparable to the wavelength in all dimensions.

We have extended what these researchers have done by finding the solution to problems involving wires whose diameters are small compared to the wavelengths. This involves the Method of Moments, but uses high order polynomials like Legendre polynomials as the basis functions along the wire and sinusoidal functions in the azimuthal direction. Previously physicists have only used constant or at most linear functions to model the current.

The easiest way to approach this for the first time was to simplify it to a problem of acoustic scattering off a hard surface and then make the surface straight and cylindrical (i.e. sound waves bouncing off a telephone pole). This seems very different from currents and wires, but the differences are essentially in the boundary conditions and the treatment of the vectors. It will involve a couple of extensions to my program to switch from Dirichlet to Neumann or electromagnetic boundary conditions and then treat the waves as vectors. After this it will take another extension to treat the wires as curved, but this will be straight forward since we used a general treatment of the basis functions in our mathematical derivation.

By looking at this simplified case, we developed basis functions that are appropriate for the problem in general. Then we developed a program to implement the solution, keeping in mind that we needed to solve it in the timeliest manner possible. We are presently removing the final bugs so that it works correctly. As soon is it is working, we will expect to see results like in figure I where the order refers to the order of the basis functions we are using.

Simpler solutions will still work better for small problems that don’t require much accuracy and only involve straight wires, but at some point in size and accuracy our much more general solution will surpass all the other programs. Preliminary models have indicated that we should see anywhere from a 10% to 100% increase in performance. With gains like this, previously intractable problems will be possible to solve on modest workstations. No longer will guesswork be necessary in problems where the wavelength of the radiation is comparable to the characteristic length of the object.