
Research > Solutions of Electromagnetics Problems
Solutions of Electromagnetics Problems Involving Hundreds of Millions of UnknownsAccurate simulations of reallife electromagnetics problems with integral equations require the solution of dense matrix equations involving millions of unknowns. Solutions of these extremely large problems cannot be achieved easily, even when using the most powerful computers with stateoftheart technology. Hence, many electromagnetics problems in the literature have been solved by resorting to various approximation techniques without controllable error. In this paper, we present fullwave solutions of scattering problems discretized with hundreds of millions of unknowns by employing a parallel implementation of the multilevel fast multipole algorithm. Various examples involving canonical and complicated objects are presented in order to demonstrate the feasibility of accurately solving largescale problems on relatively inexpensive computing platforms. Bistatic RCS (in dB) of a sphere of radius 210 wavelenghts discretized with 204,823,296 unknowns (a) from 0 degree to 180 degree and (b) from 174 degree to 180 degree , where 0 degree and 180 degree correspond to backscattering and forwardscattering directions, respectively. Normalized copolar bistatic RCS (RCS/wavelenght in dB) of the stealth airborne target Flamme at 44 GHz. Maximum dimension of the Flamme is 6 meters, corresponding to 880 wavelengths. The target is illuminated by plane waves propagating in the xy plane at (a) 30 degree and (b) 60 degree angles from the x axis, with the electric field polarized in the θ direction (horizontal polarization). Discretization of the problem with wavelength/10 mesh size leads to 204,664,320 unknowns. Solution of a scattering problem involving a metallic sphere of diameter 60 cm at 280 GHz. Normalized RCS (dB) is plotted as a function of bistatic angle from 0^{°} to 360^{°}, where 180^{°} corresponds to the forwardscattering direction. Computational values provided by the parallel MLFMA implementation with maximum 1% error agree well with an analytical Mieseries solution from 179^{°} to 180^{°}. Copolar bistatic RCS (dBms) of the rectangular box at 75 GHz on the xz plane. The box is illuminated by plane waves propagating on the xz plane at 30^{°} and 60^{°} angles from the z axis with the electric field polarized in the φ direction. Copolar bistatic RCS (dBms) of the wingshaped object at 150 GHz on the xy plane. The object is illuminated by plane waves propagating on the xy plane at 30^{°} and 60^{°} angles from the x axis with the electric field polarized in the θ direction. Solutions of scattering problems involving the NASA Almond at 850 GHz and the Flamme at 360 GHz. Both targets are discretized with more than 100 million unknowns. (a) Number of BiCGStab iterations and (b) total processing time are plotted with respect to the illumination angle measured from the x axis. Both φ and θ polarizations of the incident electric field are considered. Copolar bistatic RCS (dBms) of the NASA Almond at 850 GHz. RCS is plotted on the xy plane as a function of observation and illumination angles when the incident electric field is polarized in (a) φ and (b) θ directions. Copolar bistatic RCS (dBms) of the Flamme at 360 GHz. RCS is plotted on the xy plane as a function of observation and illumination angles when the incident electric field is polarized in (a) φ and (b) θ directions. Copolar (red) and crosspolar (blue) bistatic RCS (dBms) of the NASA Almond at 1.1 THz and 1.4 THz and the Flamme at 440 GHz and 620 GHz. RCS values lower than 70 dBms are omitted. Both targets are larger than 1000 wavelenghts and discretized with more than 300 million unknowns at the higher frequencies. The targets are illuminated by a plane wave propagating in the x direction with the electric field polarized in the φ direction.


