This page uses data in peerreviewed 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.
FrequencyDomain IntegralEquation Solvers
Notes
 In the last two decades, parallel versions of fast iterative
algorithms were able to “ride the supercomputer wave” to solve 10^{4} times
larger problems, while traditional methodofmoments (MOM) simulators as
well as sequential fast iterative algorithms stagnated.
 Supercomputer performance increased almost 10^{6} times while fast
iterative algorithm performance increased only about 10^{4}
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 integralequation solution algorithms, even the fast iterative
algorithms, do not generally scale as O(N), where N
is the number of spatial unknowns.
 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 wellconditioned integral
equations) using 10^{9} unknowns is a lot easier/more
unrealistic/less useful than simulating scattering
from a jetengine inlet (an open waveguide with
complex boundaries inside) using 10^{9} unknowns.
(return to top)
Serial MOM
[1]
S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409418, May 1982.
[2]
A. Taflove and K. Umashankar, “Radar cross section of general threedimensional scatterers,” IEEE Trans. Electromagn. Compat., vol. 25, no. 4, pp. 433440, Nov. 1983.
[3]
D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. 32, no. 1, pp. 7785, Jan. 1984.
[4]
S. M. Rao, et al., “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antennas Propag., vol. 39, no. 5, pp. 627637, May 1991.
[5]
K. R. Aberegg, A. Taguchi, and A. F. Peterson, “Application of higherorder vector basis functions to surface integral equation formulations,” Radio Sci., vol. 31, no. 5, pp. 12071213, Sep.Oct. 1996.
[6]
P. YlaOijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 178185, Jan. 2007.
[7]
J. Markkanen, P. YlaOijala, and A. Sihvola, “Discretization of volume integral equation formulations for extremely anisotropic materials,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 51955202, Nov. 2012.
Parallel MOM
[8]
T. Cwik, J. Partee, and J. Patterson, “Method of moment solutions to scattering problems in a parallel processing environment,” IEEE Trans. Magn., vol. 27, no. 5, pp. 38373840, Sep. 1991.
[9]
T. Cwik, J. Patterson, and D. Scott, “Electromagnetic scattering calculations on the Intel Touchstone Delta,” in Proc. Supercomputing, pp. 538542, Nov. 1992.
[10]
T. Cwik, R. van de Geijn, and J. Patterson, “Application of massively parallel computation to integral equation models of electromagnetic scattering,” J. Opt. Soc. America A, vol. 11, no. 4, pp. 15381545, Apr. 1994.
[11]
J. M. Putnam, D. D. Car, and J. D. Kotulski, “Parallel CARLOS3D—an electromagnetic boundary integral method for parallel platforms,” Eng. Anal. Boundary Elements, vol. 19, no. 1, pp. 4955, Jan. 1997.
[12]
A. Rubinstein, et al., “A parallel implementation of NEC for the analysis of large structures,” IEEE Trans. Electromagn. Compat., vol. 45, no. 2, pp. 177188, May 2003.
[13]
F. Wei and A. E. Yılmaz, “A hybrid message passing/shared memory parallelization of the adaptive integral method for multicore clusters,” Parallel Computing, vol. 37, no. 6, pp. 279301, JuneJuly 2011.
Serial Fast Iterative Algorithm
[14]
M. F. Catedra, J. G. Cuevas, and L. Nuno, “A scheme to analyze conducting plates of resonant size using the conjugategradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 17441752, Dec. 1988.
[15]
M. F. Catedra, E. Gago, and L. Nuno, “A numerical scheme to obtain the RCS of threedimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 528537, May 1989.
[16]
C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett., vol. 7, no. 10, pp. 466470, July 1994.
[17]
H. Gan and W. C. Chew, “A discrete BCGFFT algorithm for solving 3D inhomogeneous scatterer problems,” J. Eletromagn. Waves Applicat., vol. 9, no. 10, pp. 13391357, 1995.
[18]
J. M. Song and W. C. Chew, “Multilevel fastmultipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett., vol. 10, no. 1, pp. 1419, Sep. 1995.
[19]
J. Song, C.C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 14881493, Oct. 1997.
[20]
C. F. Wang and J. M. Jin, “Simple and efficient computation of electromagnetic fields in arbitrarily shaped inhomogeneous dielectric bodies using transposefree QMR and FFT,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 553558, May 1998.
[21]
T. F. Eibert, “A diagonalized multilevel fast multipole method with spherical harmonics expansion of the kspace Integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 814817, Feb. 2005.
[22]
S. M. Seo and J.F. Lee, “A fast IEFFT algorithm for solving PEC scattering problems,” IEEE Trans. Magn., vol. 41, no. 5, pp. 14761479, May 2005.
[23]
K. Zhao, M. N. Vouvakis, and J.F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 763773, Nov. 2005.
[24]
M. Vikram, et al., “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 20942104, July 2009.
[25]
A. Heldring, et al., “Sparsified adaptive cross approximation algorithm for accelerated method of moments computations,” IEEE Trans. Antennas Propag., vol. 61, no. 1, pp. 240246, Jan. 2013.
Parallel Fast Iterative Algorithm
[26]
E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving largescale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 12251251, Sep.Oct. 1996.
[27]
J. M. Song, et al., “Fast Illinois solver code (FISC),” IEEE Antennas Propagat. Mag., vol. 40, no. 3, pp. 2734, June 1998.
[28]
S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big?,” IEEE Antennas Propagat. Mag., vol. 45, no. 2, pp. 4358, Apr. 2003.
[29]
M. L. Hastriter, “A study of MLFMA for largescale scattering problems,” Ph.D. dissertation, Univ. Illinois at UrbanaChampaign, June 2003.
[30]
Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of largescale scattering problems,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 23352345, Aug. 2008.
[31]
J. M. Taboada, et al., “High scalability FMMFFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propagat. Mag., vol. 51, no. 6, pp. 2028, Dec. 2009.
[32]
Ö. Ergül and L. Gürel, “Accurate solutions of extremely large integralequation problems in computational electromagnetics,” Proc. of the IEEE, vol. 101, no. 2, pp. 342349, Feb. 2013.
[33]
B. Michiels, et al., “Weak scalability analysis of the distributedmemory parallel MLFMA,” IEEE Trans. Antennas Propag., vol. 61, no. 11, pp. 55675574, Nov. 2013.
[34]
F. Wei and A. E. Yılmaz, “A more scalable and efficient parallelization of the adaptive integral method—part II: BIOEM application,” IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 727738, Feb. 2014.
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TimeDomain IntegralEquation Solvers
Fig. 2: Number of unknowns solved in select papers
is plotted as a function of the date of publication.
Notes

Timedomain integralequation based simulators have lagged behind
their frequencydomain 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
frequencydomain methods, which are relatively easy to implement in
software, do not suffer from instabilities (though they suffer from
iterative solver nonconvergence), and have lower memory requirement (as
they can simulate each frequency sample independently).

Envelopetracking methods can be considered a specialized version
of timedomain integralequation methods that are more efficient for
bandpass problems [32].

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
wellconditioned integral equations) using 10^{9} unknowns is a lot easier/more
unrealistic/less useful than simulating scattering
from a jetengine inlet (an open waveguide with
complex boundaries inside) using 10^{9} unknowns.
(return to top)
FrequencyDomain IntegralEquation Solvers
[1]
S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409418, May 1982.
[2]
A. Taflove and K. Umashankar, “Radar cross section of general threedimensional scatterers,” IEEE Trans. Electromagn. Compat., vol. 25, no. 4, pp. 433440, Nov. 1983.
[3]
D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. 32, no. 1, pp. 7785, Jan. 1984.
[4]
M. F. Catedra, J. G. Cuevas, and L. Nuno, “A scheme to analyze conducting plates of resonant size using the conjugategradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 17441752, Dec. 1988.
[5]
M. F. Catedra, E. Gago, and L. Nuno, “A numerical scheme to obtain the RCS of threedimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 528537, May 1989.
[6]
T. Cwik, J. Patterson, and D. Scott, “Electromagnetic scattering calculations on the Intel Touchstone Delta,” in Proc. Supercomputing, pp. 538542, Nov. 1992.
[7]
H. Gan and W. C. Chew, “A discrete BCGFFT algorithm for solving 3D inhomogeneous scatterer problems,” J. Eletromagn. Waves Applicat., vol. 9, no. 10, pp. 13391357, 1995.
[8]
J. M. Song and W. C. Chew, “Multilevel fastmultipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett., vol. 10, no. 1, pp. 1419, Sep. 1995.
[9]
E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving largescale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 12251251, Sep.Oct. 1996.
[10]
C. F. Wang and J. M. Jin, “Simple and efficient computation of electromagnetic fields in arbitrarily shaped inhomogeneous dielectric bodies using transposefree QMR and FFT,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 553558, May 1998.
[11]
J. M. Song, et al., “Fast Illinois solver code (FISC),” IEEE Antennas Propagat. Mag., vol. 40, no. 3, pp. 2734, June 1998.
[12]
S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big?,” IEEE Antennas Propagat. Mag., vol. 45, no. 2, pp. 4358, Apr. 2003.
[13]
M. L. Hastriter, “A study of MLFMA for largescale scattering problems,” Ph.D. dissertation, Univ. Illinois at UrbanaChampaign, June 2003.
[14]
Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of largescale scattering problems,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 23352345, Aug. 2008.
[15]
J. M. Taboada, et al., “High scalability FMMFFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propagat. Mag., vol. 51, no. 6, pp. 2028, Dec. 2009.
[16]
Ö. Ergül and L. Gürel, “Accurate solutions of extremely large integralequation problems in computational electromagnetics,” Proc. of the IEEE, vol. 101, no. 2, pp. 342349, Feb. 2013.
[17]
F. Wei and A. E. Yılmaz, “A more scalable and efficient parallelization of the adaptive integral method—part II: BIOEM application,” IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 727738, Feb. 2014.
TimeDomain IntegralEquation Solvers
[18]
C. L. Bennett and H. Mieras, “Time domain scattering from open thin conducting surfaces,”
Radio Sci., vol. 16, no. 6, pp. 12311239, Nov.Dec. 1981.
[19]
S. M. Rao and D. R. Wilton, “Transient scattering by conducting surfaces of arbitrary shape,”
IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 5661, Jan. 1991.
[20]
D. A. Vechinski and S. M. Rao, “A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape,”
IEEE Trans. Antennas Propag., vol. 40, no. 6, pp. 661665, June 1992.
[21]
M. J. Bluck and S. P. Walker, “Timedomain BIE analysis of large threedimensional electromagnetic scattering problems,”
IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 894901, May 1997.
[22]
S. J. Dodson, S. P. Walker, and M. J. Bluck, “Costs and cost scaling in timedomain integralequation analysis of electromagnetic scattering,”
IEEE Antennas Propagat. Mag., vol. 40, no. 4, pp. 1221, Aug. 1998.
[23]
B. Shanker, et al., “Analysis of transient electromagnetic scattering phenomena using a twolevel plane wave timedomain algorithm,”
IEEE Trans. Antennas Propag., vol. 48, no. 4, pp. 510523, Apr. 2000.
[24]
J. L. Hu, C. H. Chan, and Y. Xu, “A fast solution of the timedomain integral equation using fast Fourier transformation,”
Microw. Opt. Technol. Lett., vol. 25, no. 3, pp. 172175, May 2000.
[25]
A. E. Yılmaz, et al., “A hierarchical FFT algorithm (HILFFT) for the fast analysis of transient electromagnetic scattering phenomena,”
IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 971982, July 2002.
[26]
B. Shanker, et al., “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,”
IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 628641, Mar. 2003.
[27]
A. E. Yılmaz, J.M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,”
IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 26922708, Oct. 2004.
[28]
H. Bagci, et al., “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cableloaded structures,”
IEEE Trans. Electromagn. Compat., vol. 49, no. 2, pp. 361381, May 2007.
EnvelopeTracking IntegralEquation Solvers
[29]
A. Mohan and D. S. Weile, “A hybrid method of momentsmarching on in time for the solution of electromagnetic scattering problems,”
in Proc. Antennas Propag. Soc. Int. Symp., June 2004.
[30]
A. Mohan and D. S. Weile, “A hybrid method of momentsmarching on in time for the solution of electromagnetic scattering problems,”
IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 12371242, Mar. 2005.
[31]
G. Kaur and A. E. Yılmaz, “On the performance of envelopetracking surfaceintegral equation solvers,” in
Proc. IEEE Antennas Propagat. Soc. Int. Symp., pp. 27162719, July 2011.
[32]
G. Kaur and A. E. Yılmaz, “Envelopetracking integral method for the electric field integral equation,” in
Proc. ACES, Mar. 2012.
[33]
G. Kaur and A. E. Yilmaz, “Envelopetracking adaptive integral method for bandpass transient scattering analysis,” submitted for publication, 2014.
(return to top)
LayeredMedium FrequencyDomain IntegralEquation Solvers
Fig. 3: Number of unknowns solved in select papers
is plotted as a
function of the date of publication.
Notes
 The historical increase in number of unknowns solved for
layeredmedium 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.
 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
wellconditioned integral equations) using 10^{9} unknowns is a lot easier/more
unrealistic/less useful than simulating scattering
from a jetengine inlet (an open waveguide with
complex boundaries inside) using 10^{9} unknowns.
(return to top)
Free Space
[1]
S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409418, May 1982.
[2]
A. Taflove and K. Umashankar, “Radar cross section of general threedimensional scatterers,” IEEE Trans. Electromagn. Compat., vol. 25, no. 4, pp. 433440, Nov. 1983.
[3]
D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. 32, no. 1, pp. 7785, Jan. 1984.
[4]
M. F. Catedra, J. G. Cuevas, and L. Nuno, “A scheme to analyze conducting plates of resonant size using the conjugategradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 17441752, Dec. 1988.
[5]
M. F. Catedra, E. Gago, and L. Nuno, “A numerical scheme to obtain the RCS of threedimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 528537, May 1989.
[6]
T. Cwik, J. Patterson, and D. Scott, “Electromagnetic scattering calculations on the Intel Touchstone Delta,” in Proc. Supercomputing, pp. 538542, Nov. 1992.
[7]
H. Gan and W. C. Chew, “A discrete BCGFFT algorithm for solving 3D inhomogeneous scatterer problems,” J. Eletromagn. Waves Applicat., vol. 9, no. 10, pp. 13391357, 1995.
[8]
J. M. Song and W. C. Chew, “Multilevel fastmultipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett., vol. 10, no. 1, pp. 1419, Sep. 1995.
[9]
E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving largescale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 12251251, Sep.Oct. 1996.
[10]
C. F. Wang and J. M. Jin, “Simple and efficient computation of electromagnetic fields in arbitrarily shaped inhomogeneous dielectric bodies using transposefree QMR and FFT,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 553558, May 1998.
[11]
J. M. Song, et al., “Fast Illinois solver code (FISC),” IEEE Antennas Propagat. Mag., vol. 40, no. 3, pp. 2734, June 1998.
[12]
S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big?,” IEEE Antennas Propagat. Mag., vol. 45, no. 2, pp. 4358, Apr. 2003.
[13]
M. L. Hastriter, “A study of MLFMA for largescale scattering problems,” Ph.D. dissertation, Univ. Illinois at UrbanaChampaign, June 2003.
[14]
Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of largescale scattering problems,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 23352345, Aug. 2008.
[15]
J. M. Taboada, et al., “High scalability FMMFFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propagat. Mag., vol. 51, no. 6, pp. 2028, Dec. 2009.
[16]
Ö. Ergül and L. Gürel, “Accurate solutions of extremely large integralequation problems in computational electromagnetics,” Proc. of the IEEE, vol. 101, no. 2, pp. 342349, Feb. 2013.
[17]
F. Wei and A. E. Yılmaz, “A more scalable and efficient parallelization of the adaptive integral method—part II: BIOEM application,” IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 727738, Feb. 2014.
Layered Medium  Structure in Single Layer
[18]
J. R. Mosig, “Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation,”
IEEE Trans. Microw. Theory Tech., vol. 36, no. 2, pp. 314323, Feb. 1988.
[19]
G. A. E. Vandenbosch and A. R. Van de Capelle, “Use of subsectional edge expansion functions (SEEFs) to analyse rectangular microstrip antennas with the method of moments,”
IEE Proc.H, vol. 39, no. 2, pp. 159164, Apr. 1992.
[20]
Z. H. Xiong, “Electromagnetic modeling of 3D structures by the method of system iteration using integral equations,”
Geophysics, vol. 57, no. 12, pp. 15561561, Dec. 1992.
[21]
M.J. Tsai, C.L. Chen, N. G. Alexopoulos, and T.S. Horng, “Multiple arbitrary shape viahole and airbridge transitions in multilayered structures,”
IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2504–2511, Dec. 1996.
[22] N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from 3D PEC targets situated in a halfspace environment,”
Microw. Opt. Technol. Lett., vol. 21, no. 6, pp. 399405, Jun. 1999.
[23] T. J. Cui and W. C. Chew, “Fast algorithm for electromagnetic scattering by buried 3D dielectric objects of large size,”
IEEE Trans. Geosci. Remote Sens., vol. 37, no. 5, pp. 25972608, Sep. 1999.
[24] N. Geng, A. Sullivan, and L. Carin, “Multilevel fastmultipole algorithm for scattering from conducting targets above or embedded in a lossy half space,”
IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 15612608, Jul. 2000.
[25] B. Hu and W. C. Chew, “Fast inhomogeneous plane wave algorithm for scattering from objects above the multilayered medium,”
IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 10281038, May 2001.
[26] X. M. Millard and Q. H. Liu, “A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media,”
IEEE Trans. Antennas Propagat., vol. 51, no. 9, pp. 23932401, Sep. 2003.
[27] T. Moselhy, X. Hu, and L. Daniel, “pFFT in FastMaxwell: a fast impedance extraction solver for 3D conductor structures over substrate,” in
Proc. Conf. on Design, Automation and Test in Europe, 2007, pp. 11941199.
[28] L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGSFFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,”
IEEE Trans. Geosci. Remote Sens., vol. 48, no. 3, pp. 11801185, Mar. 2010.
[29] Y. P. Chen, J. L. Xiong, and W. C. Chew, “A mixedform thinstratified medium fastmultipole algorithm for both low and midfrequency problems,”
IEEE Trans. Antennas Propagat., vol. 59, no. 6, pp. 23412349, Jun. 2011.
[30] K. Yang and A. E. Yılmaz, “A three dimensional adaptive integral method for layered media,”
IEEE Trans. Geosci. Remote Sens., vol. 50, no. 4, pp. 11301139 Apr. 2012.
[31] K. Yang and A. E. Yılmaz, “FFT accelerated analysis of scattering from complex dielectrics embedded in uniaxial layered media,”
IEEE Geosci. Remote Sensing Lett., vol. 10, no. 4, pp. 662666, Jul. 2013.
Layered Medium  Structure in Multiple Layers
[32]
K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. II. Implementation and results for contiguous halfspaces,”
IEEE Trans. Antennas Propagat., vol. 38, no. 3, pp. 345352, Mar. 1990.
[33]
M.J. Tsai, F. De Flaviis, O. Fordham, and N. G. Alexopoulos, “Modeling planar arbitrarily shaped microstrip elements in multilayered media,”
IEEE Trans. Microw. Theory Tech., vol. 45, no. 3, pp. 330337, Mar. 1997.
[34]
E. Jorgensen, O. S. Kim, P. Meincke, and O. Breinbjerg, “Higher order hierarchical discretization scheme for surface integral equations for layered media,”
IEEE Trans. Geosci. Remote Sens., vol. 42, no. 4, pp. 764772, Apr. 2004.
[35]
F. Ling, V. I. Okhmatovski, W. Harris, S. McCracken, and A. Dengi, “Largescale broadband parasitic extraction for fast layout verification of 3D RF and mixedsignal onchip structures,”
IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 264273, Jan. 2005.
[36]
V. Okhmatovski, M. T, Yuan, I. Jeffrey, and R. Phelps, “A threedimensional precorrected FFT algorithm for fast method of moments solutions of the mixedpotential integral equation in layered media,”
IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 35053517, Dec. 2009.
[37]
K. Yang and A. E. Yılmaz, “FFTaccelerated analysis of scattering from threedimensional structures residing in multiple layers,” in
Proc. Computational Electromagnetics Int. Workshop, 2013, pp. 48.
(return to top)
Criteria for Inclusion
To be included in these plots, a paper had to meet four
criteria:
 Did the paper use a 3D surface or volumeintegralequation
formulation? Papers that used 2D formulations, thin or thickwire
formulations, hybrid integraldifferentialequation formulations, etc.
were not included.
 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.
 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).
 Were the results published in a peerreviewed 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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