Novel Design Framework for DualBand Frequency Selective Surfaces Using MultiVariant Differential Evolution
Abstract
:1. Introduction
2. Materials and Methods
2.1. Problem Definition
2.2. Optimization Process
2.3. MVDE Algorithm Description
2.4. Performance Evaluation
 Independent trials: 100;
 Iterations: 1000;
 Population: 100;
 Decision variables (solutions to the optimization problem): 30 and 50;
 Decision variables boundaries: [−10 10].
Algorithm 1 Pseudocode of the MultiVariant Differential Evolution Algorithm 

3. Optimization Results and Discussion
3.1. Optimization Setup
 Independent trials: 10;
 Iterations: 200;
 Population: 50;
 Decision variables (solutions to the optimization problem): 10 (UC1), 11 (UC2, UC3).
 $\mathbf{u}$ is the position vector for each member of the population of the utilized MVDE algorithm,
 ${S}_{11}^{i}$ (i = {2.45 GHz, 5.8 GHz}) is the system metric (magnitude of the reflection coefficient) of the designed EM structure at the specific frequencies of interest,
 ${T}_{dB}$ is the threshold criterion for an acceptable solution of the optimization problem provided by the members of the population (in our case ${T}_{dB}$=−10 dB),
 $\Psi $ is a positive number (multiplying factor in the objective function) that is triggered when the obtained solution is above the threshold criterion, and
 $OF$ is the objective function of the optimization problem.
3.2. Unit Cell Results
3.3. FSS Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
ACO  Ant Colony Optimization 
BBO  BiogeographyBased Optimization 
CMAES  Covariance Matrix AdaptationEvolution Strategy 
DE  Differential Evolution 
EA  Evolutionary Algorithm 
EH  Energy Harvesting 
EM  Electromagnetic 
FEM  Finite Element Method 
FDTD  Finite Difference Time Domain 
FSS  Frequency Selective Surface 
GA  Genetic Algorithm 
GWO  Gray Wolf Optimizer 
HHO  Harris Hawks Optimization 
HPBW  Half Power Bandwidth 
MFEM  Modular Finite Element Methods 
MIMO  MultipleInput MultipleOutput 
MOLACO  MultiObjective Lazy Ant Colony Optimization 
MVDE  Multi Variant Differential Evolution 
PCB  Printed Circuit Board 
PDE  Partial Differential Equation 
PSO  Particle Swarm Optimization 
RF  Radio Frequency 
SADE  SelfAdaptive Differential Evolution 
WDO  WindDriven Optimization 
Appendix A. Benchmark Functions
 Ackley Function:$$f\left(\mathbf{x}\right)=a\times exp\left(b\sqrt{\frac{1}{d}\sum _{i=1}^{d}{x}_{i}^{2}}\right)exp\left(b\sqrt{\frac{1}{d}\sum _{i=1}^{d}cosc{x}_{i}}\right)+a+exp\left(1\right)$$
 Griewank Function:$$f\left(\mathbf{x}\right)=\sum _{i=1}^{d}\frac{{x}_{i}^{2}}{4000}\prod _{i=1}^{d}cos\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$$Dimensions: d, Global Minimum: $f\left(\mathbf{x}\right)=0$ at $\mathbf{x}=[0,\dots ,0]$
 Rastrigin Function:$$f\left(\mathbf{x}\right)=10d+\sum _{i=1}^{d}\left[{x}_{i}^{2}10cos\left(2\pi {x}_{i}\right)\right]$$Dimensions: d, Global Minimum: $f\left(\mathbf{x}\right)=0$ at $\mathbf{x}=[0,\dots ,0]$
 Schaffer No. 4:$$f\left(\mathbf{x}\right)=0.5+\frac{co{s}^{2}\left(sin\left({x}_{1}^{2}{x}_{2}^{2}\right)\right)0.5}{\left[1+0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)\right]},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{Dimensions}:\phantom{\rule{4.pt}{0ex}}2$$
 Schwefel Function:$$f\left(\mathbf{x}\right)=418.9829d\sum _{i=1}^{d}{x}_{i}\left(\sqrt{{x}_{i}}\right)$$Dimensions: d, Global Minimum: $f\left(\mathbf{x}\right)=0$ at $\mathbf{x}=[418.9829,\dots ,418.9829]$
 Sphere Function:$$f\left(\mathbf{x}\right)=\sum _{i=1}^{d}{x}_{i}^{2},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{Dimensions}:\phantom{\rule{4.pt}{0ex}}d,\phantom{\rule{4.pt}{0ex}}\mathrm{Global}\phantom{\rule{4.pt}{0ex}}\mathrm{Minimum}:\phantom{\rule{4.pt}{0ex}}f\left(\mathbf{x}\right)=0\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}\mathbf{x}=[0,\dots ,0]$$
 Rozenbrock Function:$$f\left(\mathbf{x}\right)=\sum _{i=1}^{d1}\left[100{\left({x}_{i+1}{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}1\right)}^{2}\right]$$Dimensions: d, Global Minimum: $f\left(\mathbf{x}\right)=0$ at $\mathbf{x}=[1,\dots ,1]$
 De Jong Function No. 5:$$f\left(\mathbf{x}\right)=\left(0.002+\sum _{i=1}^{25}\frac{1}{i+{\left({x}_{1}{a}_{1i}\right)}^{6}+{\left({x}_{2}{a}_{2i}\right)}^{6}}\right)$$
 Hartmann 6D Function:$$f\left(\mathbf{x}\right)=\sum _{i=1}^{4}{\alpha}_{i}exp\left(\sum _{j=1}^{6}{A}_{ij}{({x}_{j}{P}_{ij})}^{2}\right)$$
 Powell Function:$$\left(\mathbf{x}\right)=\sum _{i=1}^{d4}\left[{({x}_{4i3}+10{x}_{4i2})}^{2}+5{({x}_{4i1}{x}_{4i})}^{2}+{({x}_{4i2}2{x}_{4i1})}^{4}+10{({x}_{4i3}{x}_{4i})}^{4}\right]$$Dimensions: d, Global Minimum: $f\left(\mathbf{x}\right)=0$ at $\mathbf{x}=[0,\dots ,0]$
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MVDE  GA  BBO  DE  CMAES  

D = 30  f1  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  7.994 $\times \phantom{\rule{4pt}{0ex}}{10}^{15}$  2.902 $\times \phantom{\rule{4pt}{0ex}}{10}^{02}$  1.262 $\times \phantom{\rule{4pt}{0ex}}{10}^{07}$  3.997 $\times \phantom{\rule{4pt}{0ex}}{10}^{14}$ 
f2  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  6.224 $\times \phantom{\rule{4pt}{0ex}}{10}^{05}$  3.886 $\times \phantom{\rule{4pt}{0ex}}{10}^{15}$  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  
f3  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  8.238 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  6.241 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  4.770 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.169 $\times \phantom{\rule{4pt}{0ex}}{10}^{+02}$  
f4  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  
f5  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.245 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  1.245 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  1.245 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  1.245 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  
f6  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.606 $\times \phantom{\rule{4pt}{0ex}}{10}^{35}$  1.061 $\times \phantom{\rule{4pt}{0ex}}{10}^{03}$  3.438 $\times \phantom{\rule{4pt}{0ex}}{10}^{14}$  1.242 $\times \phantom{\rule{4pt}{0ex}}{10}^{27}$  
f7  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.131 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.862 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  2.386 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  5.768 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  
f8  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  
f9  −3.042 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  −3.042 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  −3.042 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  −2.969 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  −1.364 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  
f10  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  6.950 $\times \phantom{\rule{4pt}{0ex}}{10}^{02}$  4.350 $\times \phantom{\rule{4pt}{0ex}}{10}^{02}$  1.429 $\times \phantom{\rule{4pt}{0ex}}{10}^{+02}$  9.343 $\times \phantom{\rule{4pt}{0ex}}{10}^{+03}$  
D = 50  f1  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.766 $\times \phantom{\rule{4pt}{0ex}}{10}^{09}$  1.486 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  3.109 $\times \phantom{\rule{4pt}{0ex}}{10}^{04}$  6.272 $\times \phantom{\rule{4pt}{0ex}}{10}^{10}$ 
f2  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  2.220 $\times \phantom{\rule{4pt}{0ex}}{10}^{16}$  1.436 $\times \phantom{\rule{4pt}{0ex}}{10}^{03}$  1.619 $\times \phantom{\rule{4pt}{0ex}}{10}^{08}$  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  
f3  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  3.980 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  4.341 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.541 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  6.965 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  
f4  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  2.926 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  
f5  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  2.075 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  2.075 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  2.075 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  2.075 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$  
f6  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.231 $\times \phantom{\rule{4pt}{0ex}}{10}^{14}$  6.051 $\times \phantom{\rule{4pt}{0ex}}{10}^{02}$  3.696 $\times \phantom{\rule{4pt}{0ex}}{10}^{07}$  5.605 $\times \phantom{\rule{4pt}{0ex}}{10}^{19}$  
f7  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  4.747 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  5.266 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  4.739 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  2.966 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  
f8  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  1.267 $\times \phantom{\rule{4pt}{0ex}}{10}^{+01}$  
f9  −2.981 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  −3.042 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  −3.042 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  −2.925 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  −1.465 $\times \phantom{\rule{4pt}{0ex}}{10}^{+00}$  
f10  0.000 $\times \phantom{\rule{4pt}{0ex}}{\mathbf{10}}^{+\mathbf{00}}$  2.251 $\times \phantom{\rule{4pt}{0ex}}{10}^{+02}$  9.063 $\times \phantom{\rule{4pt}{0ex}}{10}^{01}$  9.041 $\times \phantom{\rule{4pt}{0ex}}{10}^{+03}$  2.931 $\times \phantom{\rule{4pt}{0ex}}{10}^{+04}$ 
Algorithm  MVDE  GA  BBO  DE  CMAES  

D = 30  Friedman test  1.20  2.75  3.55  2.95  3.55 
Normalized Ranking  1  2  3.5  5  3.5  
D = 50  Friedman test  1.25  3.10  3.80  3.85  4.00 
Normalized Ranking  1  3  4  5  2 
UCs  Decision Variables  

UC1  Variable  ${L}_{p}$  ${W}_{p}$  $O{x}_{p}$  $O{y}_{p}$  ${L}_{s}$  ${L}_{sa}$  ${W}_{s}$  $O{x}_{s}$  $O{y}_{s}$  ${W}_{st}$  ${G}_{st}$ 
Value  29.78  25.30  5.55  4.56  9.14  8.49  1.70    3.80  1.01  1.47  
UC2  Variable  ${L}_{p}$  ${W}_{p}$  $O{x}_{p}$  $O{y}_{p}$  ${L}_{s}$  ${L}_{sa}$  ${W}_{s}$  $O{x}_{s}$  $O{y}_{s}$  ${W}_{st}$  ${G}_{st}$ 
Value  23.54  25.13  9.62  5.65  7.71  24.13  1.19  5.65  2.79  1.96  1.11  
UC3  Variable  ${L}_{p}$  ${W}_{p}$  $O{x}_{p}$  $O{y}_{p}$  ${L}_{s}$  ${L}_{sa}$  ${W}_{s}$  $O{x}_{s}$  $O{y}_{s}$  ${W}_{st}$  ${G}_{st}$ 
Value  25.22  39.85  9.88  3.35  8.73  11.23  1.70  3.40  9.44  1.72  1.54 
UCs  Frequency: 2.45 GHz  Frequency: 5.8 GHz 

UC1  $49.94+j\times 0.36$  $49.70j\times 0.35$ 
UC2  $48.35j\times 0.63$  $55.73+j\times 3.01$ 
UC3  $50.86+j\times 0.63$  $49.28j\times 2.13$ 
3 × 2 FSS Design  5 × 3 FSS Design  
MVDE  −33.71 dB @ 2.46 GHz −26.83 dB @ 5.82 GHz  −24.36 dB @ 2.45 GHz −31.15 dB @ 5.80 GHz 
DE  −29.57 dB @ 2.44 GHz −20.24 dB @ 5.81 GHz  −19.12 dB @ 2.47 GHz −20.88 dB @ 5.80 GHz 
Ref.  PCB Substrate  Unit Cell  Frequency Band  Layout  Occupied Area in ${\mathit{\lambda}}_{0}$  Reflection Coefficient  Reference Frequency 

[25]  FR4  Square Loop  WiFi 2.45 GHz  25 × 15  3.68 × 2.45  −28 dB (max. value)  3.05 GHz 
[40]  Rigid Polyurethane Foam  Cross and Fractal Square Patch  2–18 GHz  ∼ 13 × 13  7.64 × 7.64  <−10 dB  5.27–18 GHz 
[67]  0.12 mm ${\u03f5}_{r}$ = 3  Gridded Square Loop  WiFi 2.45 GHz  3 × 3 5 × 5  1.79 × 1.79 2.98 × 2.98  <−30 dB (for the unit cell)  2.2 GHz 
[69]  FR4  Single Patch  WiFi 2.45 GHz  4 × 3  2.94 × 2.21  −33.52 dB  2.45 GHz 
[70]  FR4  Cross with 4 apertures  WiFi 5.8 GHz  7 × 7  2.00 × 2.00  >−40 dB  5.8 GHz 
[71]  Rogers RO4003C  Pair of patches  WiFi 2.45 GHz  5 × 6  1.87 × 2.49  <−20 dB  2.45 GHz 
[72]  Rogers RO4003C  Pair of Bowtie dipoles  GSM1800, WiFi 2.45 GHz  5 × 4  1.11 × 1.80 (single unit cell)  <−30 dB (for the rectifier)  ∼ 2.4 GHz 
This work  FR4  Pair of Uslots  WiFi 2.45 GHz WiFi 5.8 GHz  3 × 2:  1.00 × 0.98  −33.71 dB −26.83 dB  2.46 GHz 5.82 GHz 
5 × 3:  1.49 × 1.50  −24.36 dB −31.15 dB  2.45 GHz 5.8 GHz 
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Boursianis, A.D.; Papadopoulou, M.S.; Nikolaidis, S.; Sarigiannidis, P.; Psannis, K.; Georgiadis, A.; Tentzeris, M.M.; Goudos, S.K. Novel Design Framework for DualBand Frequency Selective Surfaces Using MultiVariant Differential Evolution. Mathematics 2021, 9, 2381. https://doi.org/10.3390/math9192381
Boursianis AD, Papadopoulou MS, Nikolaidis S, Sarigiannidis P, Psannis K, Georgiadis A, Tentzeris MM, Goudos SK. Novel Design Framework for DualBand Frequency Selective Surfaces Using MultiVariant Differential Evolution. Mathematics. 2021; 9(19):2381. https://doi.org/10.3390/math9192381
Chicago/Turabian StyleBoursianis, Achilles D., Maria S. Papadopoulou, Spyridon Nikolaidis, Panagiotis Sarigiannidis, Konstantinos Psannis, Apostolos Georgiadis, Manos M. Tentzeris, and Sotirios K. Goudos. 2021. "Novel Design Framework for DualBand Frequency Selective Surfaces Using MultiVariant Differential Evolution" Mathematics 9, no. 19: 2381. https://doi.org/10.3390/math9192381