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SCIENCE CHINA Information Sciences, Volume 63 , Issue 10 : 200301(2020) https://doi.org/10.1007/s11432-020-2927-2

A new optimization algorithm applied in electromagnetics — Maxwell's equations derived optimization (MEDO)

Donglin SU 1,2,4,5,*, Lilin LI 1,3,4, Shunchuan YANG 2,4,5, Bing LI 2,4,5, Guangzhi CHEN 2,4,5, Hui XU 1,4
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  • ReceivedJan 20, 2020
  • AcceptedMay 25, 2020
  • PublishedAug 26, 2020

Abstract

In this paper, a novel global optimization algorithm, named as Maxwell's equations derived optimization (MEDO), is proposed. Using the Maxwell's equations to analyse the behaviors of the time-varying current, the Ampere force is obtained from Fleming's left hand rule. MEDO introduces an `Ampere force' term, which is derived from Maxwell's equations and is rigorous in physics, to drive the variables to the global optimal solution in the search space. In addition, introducing `gravity' to MEDO can increase the stability of the optimizations. 11 classical benchmarks are tested, and results show that MEDO can always converge to numerical optimal solutions. To evaluate the proposed MEDO in solving the electromagnetic problems, four practical engineering applications are considered including the linear antenna array synthesis, frequency selected surface optimization, numerical dispersion reduction for finite-difference method, and parameters extraction of typical waveform. These examples are significant in electromagnetics, but tough to be solved because of their high dimensionality and strong nonlinearity. Numerical results show that MEDO can outperform several classic optimization methods, like wind driven optimization (WDO) and particle swarm optimization (PSO). Therefore, the electromagnetics-inspired MEDO is robust and of great potential in solving the electromagnetic optimization problems.


Acknowledgment

This work was supported by National Military Key Pre-research Project of the 13th Five-Year Plan (Grant No. 41409010101), National Natural Science Foundation of China (Grant Nos. 61427803, 61771032), and Civil Aircraft Projects of China (Grant No. MJ-2017-F-11).


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  • Figure 1

    Coaxial model to demonstrate the interesting division of the current. The coaxial is excited by a time-varying source and ended with a matched load. The load and the source are connected by a copper with small impedance.

  • Figure 2

    (Color online) Schematic of MEDO. The schematic is divided into several segments. $AKG$ is the main branch of the circuit, whose impedance is marked as $Z_1$. $ACDFGK$ is the paralleled branch in current loop $1$, and $AJIHGK$ is the paralleled branch in current loop $2$. Their impedances are recorded as $Z_2$ and $Z_3$. $GH$ is the individual in the optimization algorithm whose task is to find the optimal solution. Point $G$ is the position of the individual, which is known as variable of the optimization question. $KF$ is the domain of the variable, whose minimum and maximum values are denoted as $lb$ and $ub$. $HI$ is the area to be optimized. $F_G$ and $F_A$ refer to the external force acting on $GH$, which will be introduced in detail in the body.

  • Figure 3

    (Color online) Schematic diagram of calculating. (a) $R_3$; (b) $L_3$; and (c) $\frac{{\rm~d}S_{CDHJ}}{{\rm~d}t}$.

  • Figure 4

    (Color online) Change of the accuracy of different optimizations with the increase of the dimensions when calculating. (a) $F_1$; (b) $F_2$; (c) $F_3$; (d) $F_4$; (e) $F_5$; (f) $F_6$; (g) $F_7$; (h) $F_8$.

  • Figure 5

    (Color online) Geometry of the 16-elements linear antenna array positioned along the $x$-axis.

  • Figure 10

    (Color online) Structure of the FSS being designed. (a) Array and (b) crossed dipole element.

  • Figure 11

    (Color online) Insert loss comparison between MEDO and GA. MEDO can obtain a wider bandwidth than GA.

  • Figure 12

    (Color online) Resonant frequency comparison between the TE-based, AWDO-based, and MEDO-based FDM with the analytic results. MEDO-based FDM can obtain a more accurate resonant frequency.

  • Figure 15

    (Color online) Extraction results of $D_1$, $D_2$, and $D_3$ obtained by different optimization algorithms in 100 times.

  • Table 1  

    Table 1Description of 11 benchmark functions

    FunctionRangeDimension$F_{\rm~min}$
    $F_{1}=\sum_{i=1}^{n}~(x_{i}-2)^{2}$[$-$10,~10]400
    $F_{2}=\sum_{i=1}^{n}\left|x_{i}\right|+\prod_{i=1}^{n}\left|x_{i}\right|$[$-$10,~10]400
    $F_{3}=\sum_{i=1}^{n}\left(\sum_{j=1}^{i}\left(x_{j}\right)\right)^{2}$[$-$100,~100]400
    $F_{4}=\sum_{i=1}^{n}\left|x_{i}+0.5\right|^{2}$[$-$100,~100]400
    $F_{5}={\rm~r~a~n~d~o~m}[0,1)+\sum_{i=1}^{n}~i~x_{i}^{4}$[$-$1.28,~1.28]400
    $F_{6}=\sum_{i=1}^{n}\left[x_{i}^{2}-10~\cos~\left(2~\pi~x_{i}\right)+10\right]$[$-$5.12,~5.12]400
    $F_{7}=-20~\exp~(-0.2~\sqrt{\frac{1}{n}~\sum_{i=1}^{n}~x_{i}^{2}})-\exp~\left(\frac{1}{n}~\sum_{i=1}^{n}~\cos~\left(2~\pi~x_{i}\right)\right)+20+e$[$-$32,~32]400
    $F_{8}=\frac{1}{4000}~\sum_{i=1}^{n}~x_{i}^{2}-\prod_{i=1}^{n}~\cos~\left(\frac{x_{i}}{\sqrt{i}}\right)+1$[$-$600,~600]400
    $F_{9}=[1+\left(x_{1}+x_{2}+1\right)^{2}\left(19-14~x_{1}+3~x_{1}^{2}-14~x_{2}+6~x_{1}~x_{2}+3~x_{2}^{2}\right)]~$[$-$2,~2]23
    $\times~[30+\left(2~x_{1}-3~x_{2}\right)^{2}~\times\left(18-32~x_{1}+12~x_{1}^{2}+48~x_{2}-36~x_{1}~x_{2}+27~x_{2}^{2}\right)]$
    $F_{10}=0.5+\frac{\sin~^{2}(x_{1}^{2}-x_{2}^{2})-0.5}{(1+0.001(x_{1}^{2}+x_{2}^{2}))^{2}}$[$-$100,~100]20
    $F_{11}=-0.5+\frac{\sin~^{2}~\sqrt{x_{1}^{2}+x_{2}^{2}}}{(1+0.001(x_{1}^{2}+x_{2}^{2}))^{2}}$[$-$100,~100]2$-$1
  •   

    Algorithm 1 Time-varying effect optimization

    Require:${\rm~min}~f_{\rm~objective}(x)$;

    Preset: objective function, population size, maximum iterations;

    Preset: $\vert~\boldsymbol~g~\vert$, $B_0$, $\rho~S_{\rm~bar}$, $R_1$, $L_2$;

    Initialization: $x_0$, $v_0$;

    while ${\rm~iter}~<~\text~{maximum~number~of~iterations}$ do

    Calculate $f_{\rm~objective}(x)$;

    $u=-\nabla~f(x)$;

    Calculate $Z_{\text~{\rm~total}}$, $Z_2$, $Z_3$, $\frac{{\rm~d}~S}{{\rm~d}~t}$;

    $i_{3}=\frac{\frac{u}{Z_{\text~{\rm~total~}}}~Z_{2}+B_0~\frac{{\rm~d}~S}{{\rm~d}~t}}{Z_{2}+Z_{3}}$;

    $v_{\rm~new}=-g~\cdot~\text~{\rm~iter}~\cdot~v_{\rm~cur}+(\frac{B_0}{\rho~S}~i_{3}-x_{\text~{\rm~last}})~\cdot~\text~{iter}$;

    $x_{\rm~new}~=~x_{\rm~cur}~+~v_{\rm~new}$;

    if $\vert~x_{\rm~new}-x_{\rm~cur}\vert~<{\rm~threshold}$ then

    Exit.

    end if

    end while

  • Table 2  

    Table 2Values of the coefficients of MEDO when calculating different benchmark functions

    Function$B_0$$\rho~S$$\vert~\boldsymbol~g~\vert$$R_1$$L_2$
    $F_{1}$25E + 22E $-$ 31E + 63E + 9
    $F_{2}$05E + 22E $-$ 31E + 23E + 9
    $F_{3}$05E + 22E $-$ 31E + 23E + 9
    $F_{4}$5E $-$ 15E + 22E $-$ 33E + 23E + 9
    $F_{5}$01E + 38E $-$ 43E + 68E + 9
    $F_{6}$1.16E $-$ 95.63E + 22E $-$ 31E + 33E + 9
    $F_{7}$05E + 22E $-$ 35E + 23E + 9
    $F_{8}$05E + 22E $-$ 35E + 23E + 9
    $F_{9}$1.49E + 33E $-$ 55E + 23E + 9
    $F_{10}$2.8E $-$ 82.32E + 25E $-$ 35.1E + 23E + 5
    $F_{11}$1E $-$ 12E + 25E $-$ 35E + 23E + 5
  • Table 3  

    Table 3Comparison of the average of optimization results between MEDO, AWDO, PSO, DE, and GD

    Function MEDO AWDO PSO DE GD
    $F_1$ 2.06E $-$ 10 8.10E $-$ 09 1.159299 0.356657 0
    $F_2$ 0 0 14156.03 4.753654 1E + 38
    $F_3$ 0 0 5.95349 36.35761 8.16E $-$ 9
    $F_4$ 2.39E $-$ 16 5.88E $-$ 10 6.799546 36.06591 1.94E $-$ 27
    $F_5$ 6.08E $-$ 05 6.80E $-$ 05 65.75983 0.281079 22.54327
    $F_6$ 0 0 181.557 320.1764 350.34270
    $F_7$ 8.88E $-$ 16 8.88E $-$ 16 20.25262 2.878327 19.50377
    $F_8$ 0 0 753.0833 1.349412 1.05304
    $F_9$ 3.0001 3.0001 3.008 3 144.95598
    $F_{10}$ 0 0 1.78E $-$ 08 0 0.50283
    $F_{11}$ $-$0.9999997 $-$0.99922 $-$0.99998 $-$0.9966 $-$0.53942
  • Table 4  

    Table 4Comparison of the variance of optimization results between MEDO, AWDO, PSO, DE, and GD

    Function MEDOAWDO PSO DE GD
    $F_1$ 2.56E $-$ 19 1.73E $-$ 15 0.094204 0.051561 0
    $F_2$ 0 0 1.99E + 11 2.700029 1E + 78
    $F_3$ 0 0 54.079 421.2083 1.50E $-$ 18
    $F_4$ 1.16E $-$ 32 2.74E $-$ 18 42.17984 361.9966 3.61E $-$ 55
    $F_5$ 4.20E $-$ 09 4.84E $-$ 09 1891.318 0.008171 2079.07924
    $F_6$ 0 0 1477.551 356.0968 2886.41731
    $F_7$ 0 0 0.117238 0.172126 0.01412
    $F_8$ 0 0 1954.504 0.039167 5.29E $-$ 5
    $F_9$ 1.81E $-$ 8 1.34E $-$ 06 0.000105 8.05E $-$ 31 80171.13443
    $F_{10}$ 0 0 3.42E $-$ 16 0 0.00179
    $F_{11}$ 6.59E $-$ 14 7.02E $-$ 06 3.53E $-$ 10 2.17E $-$ 05 0.00708
  • Table 5  

    Table 5Comparison of the SLL obtained by MEDO and PSO

    OptimizationSLL (dB)
    MEDO$-$35.71
    PSO$-$31.29
  • Table 6  

    Table 6Comparison of the SLL obtained by MEDO and RGA

    OptimizationSLL (dB)Null depth ($75\degree$ and $105~\degree$) (dB)Null depth ($68\degree$ and $112~\degree$) (dB)
    MEDO$-$420.83 $-$61.65 $-$61.66
    RGA$-$15.18 $-$56.00 $-$86.07
  • Table 7  

    Table 7Excitation current amplitudes and element positions of 16-element linear array obtained by MEDO and PSO for Subsection sect. 4.1.1

    shortstackElement numbershortstackMEDO (for SLL suppression only)shortstackMEDO (for SLL suppression and null control)shortstackPSO (for SLL suppression only)
    Excitation Position Excitation Position Excitation Position
    amplitudespacingamplitudespacingamplitudespacing
    $1$st 0.151317 0.62926 0.257102 0.400000 0.210000 0.55053
    $2$nd 0.291622 0.80504 0.310612 0.724125 0.401480 0.66321
    $3$rd 0.468041 0.81434 0.333657 0.650454 0.320900 0.61984
    $4$th 0.584701 0.73644 0.510201 0.559664 0.487930 0.40000
    $5$th 0.564219 0.63420 0.382737 0.529326 0.865320 0.65443
    $6$th 0.69427 0.58506 0.551289 0.550030 1.000000 0.72970
    $7$th 0.719992 0.60010 0.661875 0.574347 0.975160 0.73800
    $8$th 0.595556 0.50668 0.685965 0.703852 0.819060 0.66374
    $9$th 0.777297 0.53528 0.713194 0.708551 0.647310 0.59657
    $10$th 0.762435 0.63004 0.713019 0.689588 0.637890 0.61989
    $11$th 0.710614 0.63960 0.527354 0.666631 0.503760 0.72864
    $12$th 0.629917 0.67286 0.618191 0.655562 0.270990 0.69000
    $13$th 0.435429 0.68788 0.321751 0.532389 0.178350 0.60816
    $14$th 0.330503 0.65466 0.267579 0.444508 0.067443 0.64391
    $15$th 0.213359 0.68206 0.450657 0.715799 0.049728 0.73366
    $16$th 0.136245 0.67226 0.423021 0.809753 0.024616 0.82829
  • Table 8  

    Table 8Excitation current amplitude distribution for original and corrected pattern of MEDO and QPSO

    Element numberOriginal patternCorrected pattern by MEDO Corrected pattern by QPSO
    $1$st 0.2769 0.01 0.1741
    $2$nd 0.3087 0.035661 0.1523
    $3$rd 0.4087 0 0
    $4$th 0.2993 0.151105 0.3372
    $5$th 0.5075 0 0
    $6$th 0.5807 0.2104 0.2692
    $7$th 0.775 0.25398 0.5821
    $8$th 0.7555 0.257913 0.5638
    $9$th 0.7742 0.5 0.6564
    $10$th 0.7219 0.520421 0.692
    $11$th 0.8832 0.60911 0.8583
    $12$th 0.7731 0.686959 0.7872
    $13$th 0.6126 0.765111 0.864
    $14$th 0.809 0.748587 0.8718
    $15$th 0.5706 0.788953 0.8739
    $16$th 0.5349 0.669952 0.6647
    $17$th 0.5196 0.551261 0.5375
    $18$th 0.3102 0.556071 0.4607
    $19$th 0.2837 0.50677 0.3564
    $20$th 0.1598 0.246381 0.1574
  • Table 9  

    Table 9Parameters of the two-layers FSS structure

    ClassParameterValue
    Substrate$\epsilon~_r$4.4
    Loss rangent0.02
    Structure 1$L_1$9.87 mm
    $w_1$1.35 mm
    $h_1$1.57 mm
    $T_{x1}$ 13 mm
    $T_{y1}$13 mm
    Structure 2$L_2$9.1 mm
    $w_2$1.77 mm
    $h_2$1.57 mm
    $T_{x2}$13 mm
    $T_{y2}$13 mm
  • Table 10  

    Table 10Optimization results comparison between MEDO and GA

    OptimalResonant Resonant Bandwidth
    gap (mm)frequency (1) (GHz)frequency (2) (GHz)($-$10 dB) (GHz)
    MEDO$8.2~$9.75 10.53 3.09
    GA$7.43$9.73 10.49 3.04
  • Table 11  

    Table 11Comparison of the parameters extraction results obtained by MEDO and AWDO

    ItemPreset value $(%)$AWDOMEDO
    BestVarianceBestVariance
    $D_1$4.003.941.03E $-$ 23.943.98E $-$ 3
    $D_2$36.036.11.09E $-$ 236.13.96E $-$ 3
    $D_3$5.004.951.09E $-$ 24.965.20E $-$ 3

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