Progress in Computational Electromagnetics

Last Update: 10/15/2016
Guneet Kaur, Jackson W. Massey, Kai Yang, and Ali E. Yılmaz
Department of Electrical and Computer Engineering
The University of Texas at Austin, Austin, TX 78712, USA

This page uses data in peer-reviewed publications to highlight the tremendous progress in computational electromagnetics. Disclaimer: The plots do not show all publications in the literature but rather a subset of them that meet these criteria. Despite our best efforts, we might have missed some publications, please contact the authors if you would like to suggest additional data points.

Frequency-Domain Integral-Equation Solvers


Fig. 1:  Number of unknowns solved (left) and the number of processes used in parallel (right) in select papers are plotted as a function of the date of publication. Also shown (middle) is the maximum number of Flop/s achieved by the number 1 and 500 supercomputer in the TOP500 list while solving a dense system of linear equations according to the LINPACK benchmark.

 Notes

  1. In the last two decades, parallel versions of fast iterative algorithms were able to “ride the supercomputer wave” to solve 104 times larger problems, while traditional method-of-moments (MOM) simulators as well as sequential fast iterative algorithms stagnated.
  2. Supercomputer performance increased almost 106 times while fast iterative algorithm performance increased only about 104 times in the last two decades. The main reasons for this difference are: (i) More processes are used to obtain the LINPACK benchmark data (typically the whole supercomputer is used for the TOP500 ranking vs. just a subset of it in the publications shown). (ii) Flop/s measured with the LINPACK benchmark generally does not represent the performance of a computer when executing the more complex tasks required by advanced algorithms. (iii) The computational costs of integral-equation solution algorithms, even the fast iterative algorithms, do not generally scale as O(N), where N is the number of spatial unknowns.
  3. Number of spatial unknowns is not equal to the complexity/realism/usefulness of the problem solved, especially for fast iterative algorithms; e.g., simulating scattering from a uniformly meshed sphere (a closed surface that can be simulated with well-conditioned integral equations) using 109 unknowns is a lot easier/more unrealistic/less useful than simulating scattering from a jet-engine inlet (an open waveguide with complex boundaries inside) using 109 unknowns.

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+- References

Time-Domain Integral-Equation Solvers


Fig. 2:  Number of unknowns solved in select papers is plotted as a function of the date of publication.

Notes

  1. Time-domain integral-equation based simulators have lagged behind their frequency-domain counterparts despite being more efficient for broadband analysis. Reasons for this historical lag include the relatively more complex nature of their software implementation, their tendency to be unstable at late times, and their higher memory requirement (to store data from previous time samples) compared to frequency-domain methods, which are relatively easy to implement in software, do not suffer from instabilities (though they suffer from iterative solver non-convergence), and have lower memory requirement (as they can simulate each frequency sample independently).

  2. Envelope-tracking methods can be considered a specialized version of time-domain integral-equation methods that are more efficient for band-pass problems [32].

  3. Number of spatial unknowns is not equal to the complexity/realism/usefulness of the problem solved, especially for fast iterative algorithms; e.g., simulating scattering from a uniformly meshed sphere (a closed surface that can be simulated with well-conditioned integral equations) using 109 unknowns is a lot easier/more unrealistic/less useful than simulating scattering from a jet-engine inlet (an open waveguide with complex boundaries inside) using 109 unknowns.

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+- References

Layered-Medium Frequency-Domain Integral-Equation Solvers


Fig. 3:  Number of unknowns solved in select papers is plotted as a function of the date of publication.

 Notes

  1. The historical increase in number of unknowns solved for layered-medium backgrounds, whether the structure of interest resides in multiple layers or a single layer, has been more limited. This is because the more complex Green functions for layered media do not permit the same type of algorithms that can be used with the free space Green function.
  2. Number of spatial unknowns is not equal to the complexity/realism/usefulness of the problem solved, especially for fast iterative algorithms; e.g., simulating scattering from a uniformly meshed sphere (a closed surface that can be simulated with well-conditioned integral equations) using 109 unknowns is a lot easier/more unrealistic/less useful than simulating scattering from a jet-engine inlet (an open waveguide with complex boundaries inside) using 109 unknowns.

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+- References

Criteria for Inclusion

To be included in these plots, a paper had to meet four criteria:

  1. Did the paper use a 3-D surface- or volume-integral-equation formulation? Papers that used 2-D formulations, thin- or thick-wire formulations, hybrid integral-differential-equation formulations, etc. were not included.
  2. Are the presented simulation results believable? Specifically:
    • Were the results of the simulations validated in the paper? Acceptable forms of validation included comparison to analytical solutions, numerical solutions, or to experimental/measured data. The reference data had to have been generated using an independent method.
    • Were computational costs, such as time or memory requirement, reported? Were they reasonable?
    • We think that publishing believable results in one paper does not imply all other results published in other papers using that method (or a close relative of it) are believable. Thus, papers that did not show any form of validation were omitted even though they were published by identical authors who may have validated the method they used (or a close relative of it) in an earlier publication. In other words, each paper was judged on its own merits.
  3. Is the number of unknowns solved close to the maximum being solved the year of publication? For example, in Fig. 1 we did not include papers that solved only 10 million unknowns in the last 5 years using parallel fast iterative algorithms in frequency domain (they were significantly behind the leading edge).
  4. Were the results published in a peer-reviewed journal or conference? Advertisement brochures, progress reports, arXiv documents, website announcements, or presentations at conferences without proceedings did not qualify. Dissertations were included in exceptional cases.

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