Recent research developments and collaboration with the Colorado School of Mines Electrical Engineering department
One exciting and ongoing development since my arrival at Mines has been a collaboration between our own department (Applied Mathematics and Statistics) and the Electrical Engineering department - particularly with Atef Elsherbeni and Mohammed Hadi. One of their current research interests is the development of finite difference time domain (FDTD) methods for simulating electromagnetic (EM) wave propagation around conducting and dielectric materials.
Greg Fasshauer and I have enjoyed working with these two professors and their groups in an effort to adapt current radial basis function-generated finite difference (RBF-FD) methods from our own research to their FDTD push. The ability of RBF-FD to handle curved boundaries and interfaces between different materials well (as explored in my dissertation work) has proven quite helpful in designing efficient EM wave simulation methods in scenarios of interest. Snapshots from an example of one such scenario are included below in Figure 1 (a planar wave encounters a cylindrically-shaped perfect electric conductor).
We're currently preparing our preliminary results for presentation at the 2018 International Applied Computational Electromagnetics Society (ACES) Symposium to be held March 24-29 in Denver, Colorado. I'll provide more updates on our investigation during our drafting process.
Figure 1a: EM model snapshot; t ~ 1 nanosecond. The color plot above shows the relative strength of the magnetic field component (perpendicular to the cross-section of the perfectly-conducting cylinder shown by the dashed white circle) of a planar EM wave as it approaches a conducting feature. This and the following snapshots happen to be from a relatively low-frequency-band Fourier reference solution to one of our basic test problems (resulting in the faint, artificial "rays" you may perceive at the periphery of the model domain).
Figure 1b: EM model snapshot (magnetic field component perpendicular to page); t ~ 1.66 nanoseconds
Figure 1c: EM model snapshot (magnetic field component perpendicular to page); t ~ 2.33 nanoseconds