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Solutions of Electromagnetics Problems Involving Hundreds of Millions of Unknowns


Accurate simulations of real-life 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 state-of-the-art 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 full-wave 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 large-scale 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 back-scattering and forward-scattering directions, respectively.


  


Normalized co-polar 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 x-y 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 forward-scattering direction. Computational values provided by the parallel MLFMA implementation with maximum 1% error agree well with an analytical Mie-series solution from 179° to 180°.


  

Co-polar bistatic RCS (dBms) of the rectangular box at 75 GHz on the x-z plane. The box is illuminated by plane waves propagating on the x-z plane at 30° and 60° angles from the z axis with the electric field polarized in the φ direction.


  

Co-polar bistatic RCS (dBms) of the wing-shaped object at 150 GHz on the x-y plane. The object is illuminated by plane waves propagating on the x-y 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.


  


Co-polar bistatic RCS (dBms) of the NASA Almond at 850 GHz. RCS is plotted on the x-y plane as a function of observation and illumination angles when the incident electric field is polarized in (a) φ and (b) θ directions.


  


Co-polar bistatic RCS (dBms) of the Flamme at 360 GHz. RCS is plotted on the x-y plane as a function of observation and illumination angles when the incident electric field is polarized in (a) φ and (b) θ directions.


  

Co-polar (red) and cross-polar (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.


For more information:
  • Please see the full article published in the IEEE Transactions on Antennas and Propagation in 2008.
  • Please see the full article accepted for publication in the IEEE Antennas and Propagation Magazine, to appear in 2010.
How to cite these papers:
  • Ö. Ergül and L. Gürel, "Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems," IEEE Trans. Antennas Propagat., vol. 56, no. 8, pp. 2335-2345, Aug. 2008.
  •     
  • Ö. Ergül and L. Gürel, "Rigorous solutions of electromagnetics problems involving hundreds of millions of unknowns," IEEE Antennas Propagat. Mag., accepted for publication, June 2010
 
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