AMS Colloquium: Making EM Problems More Efficient

Problems in the field of electromagnetics (EM) can get very complex and become very difficult to solve computationally. As a result, the cost of running simulations for these problems, especially regarding the time it takes to complete, can be rather high. But when these computations are the only means of solving these problems, as is often the case, the only way to minimize this cost is to develop a more efficient means of accomplishing the same computation. One of the more prominent methods to accomplish this is the finite-difference time-domain (FTDT) technique.

As explained by Dr. Atef Elsherbeni of the CSM Electrical Engineering and Computer Science (EECS) Department, the FTDT technique is a “formulation for efficient simulation for complex EM and antenna problems.” What the FTDT methodology provides is a way to solve Maxwell’s equations for electromagnetism in a time domain. It also partitions the geometry of a particular situation into a spatial grid, and presents the desired data as functions of time.

This is accomplished by using Maxwell’s time domain equations as the model’s starting point and uses the central difference method to return estimations with a fairly high order of precision. These are then coupled with the use of Yee cells, which are cubic representations of both electric and magnetic fields in a small area from a node (corner of the hypothetical cube). With the behavior of the fields in this small cells remaining basically constant from cell to cell, and with the ability to alter a cell’s properties based on what type of material it is representing, larger systems and surfaces can be modeled by grouping together many of these cells into the desired surface or object. The result is a surface or object whose EM properties can be easily computed due to the simplicity of the cells used and solutions that, while not exactly the same as the true nature of the object or surface, is similar enough to be an accurate estimation.

The advantages of FTDT are that it is has a relatively simple formulation procedure, and also responds to a wide variety of frequencies. It can also be used to model with different types and structures of materials, as well as maintaining rather basic geometries, allowing for virtually any situation to be modeled with this technique. However, these advantages of the FTDT method do come with a few setbacks, including a large execution time for a model to reach its point of convergence, plus the need for a good input interface to properly verify a model’s geometry and whatnot. Also, FTDT methods, while very useful if used correctly, are rather unstable and can lead to a divergent solution if the correct parameters for simulation are not set. Especially critical in this case, according to Elsherbeni, is the correct choice in timeframes. And lastly, numerical time domain results from the use of FTDT need to be examined and verified outside of the model before the model’s frequency domain results can be trusted, adding one more possibly time-consuming step to the process.

While it does have its slight disadvantages, the FTDT method still delivers very accurate estimations of EM systems with much less computational power needed to reach completion. It has already found its way into models for numerous EM related fields, including antennae, microwave activities, wave propagation, and radar technologies, and figures to find many more applications in the future.