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Multi-objective crashworthiness optimization of perforated square tubes using modified NSGAII and MOPSO

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In this paper, multi-objective optimization of perforated square tubes is performed considering absorbed energy, peak crushing force and weight of the tube as three conflicting objective functions. In the multi-objective optimization problem (MOP), absorbed energy and peak crushing force are defined by polynomial models extracted using the software GEvoM based on the train and test data obtained from the numerical simulation of quasi-static crushing of the perforated square tubes using ABAQUS. To verify the numerical procedure, 16 different experimental tests are performed and then the experimental and numerical results are compared together. The comparison shows reasonable similarities between the numerical and experimental results. The MOP is solved using modified Non-dominated Sorting Genetic Algorithm II (NSGAII) and Multi-objective Particle Swarm Optimization (MOPSO) and then the solutions are combined for non-dominated sorting to obtain the non-dominated individuals of 3-objective optimization. 105 optimum points are extracted from the multi-objective optimization process. Finally, Nearest to Ideal Point (NIP) method and Technique for Ordering Preferences by Similarity to Ideal Solution (TOPSIS) method are employed to find trade-off optimum design points out of all non-dominated individuals compromising all three objective functions together.

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Correspondence to Abolfazl Khalkhali.

Appendix A

Appendix A

Corresponding polynomial representation for absorbed energy of types A, B, C and D obtained using GEvoM are as follows:

For Type A:

$$ {Y}_1= 4849.3294- 37.2842D+ 0.42835{D}^2{Y}_2= 4922.798- 543.668n+ 91.682{n}^2{Y}_3=- 0.0058+ 16.5079{Y}_1- 16.629{Y}_2- 0.000984{Y}_1^2+ 0.002858{Y}_2^2- 0.00161{1}_2{Y}_1{Y}_4= 0.000394+ 0.0055219D+ 0.848535{Y}_2+ 0.3681567{D}^2+ 6.8241156{Y}_2^2- 0.008267{Y}_2D Energy= 0.005755- 24.66255{Y}_3+ 25.44598{Y}_4+ 0.01772{Y}_3^2+ 0.0119459{Y}_4^2- 0.0296227{\mathrm{Y}}_3{\mathrm{Y}}_4 $$

For Type B:

$$ {Y}_1= 4189.0752- 195.03608n+ 36.69127D+ 83.207066{n}^2- 0.34526{D}^2- 14.05867nD{Y}_2= 4321.49518- 7.42745D- 0.02677{D}^2{Y}_3=- 0.46187- 91.45453D+ 1.94702{Y}_1+ 0.10591{D}^2- 0.00022{Y_1}^2+ 0.0205698D{Y}_1{Y}_4=- 0.003539+ 10.14823{Y}_1- 8.94379{Y}_2- 0.000487{Y_1}^2+ 0.0017609{Y_2}^2- 0.001321{Y}_1{Y}_2 Energy = - 0.03759- 14.38065{Y}_3+ 15.49129{Y}_4- 0.03292{Y_3}^2- 0.03677{Y_4}^2+ 0.06967{Y}_3{Y}_4 $$

For Type C:

$$ {Y}_1= 5778.8423- 811.9339n+ 60.38783D+ 142.6619{n}^2- 1.38307{D}^2- 5.90438892781643nD{Y}_2= 4787.22183+ 39.71079D- 1.1467{D}^2{Y}_3= 5974.887- 1188.5924n+ 227.629{n}^2{Y}_4=- 0.00215+ 5.2949{Y}_1- 4.32067{Y}_2- 0.000182{Y_1}^2+ 0.00075{Y_2}^2- 0.000568{Y}_1{Y}_2{Y}_5= 4.579* 1{0}^{\mathit{\hbox{-}} 5}+ 0.986081{Y}_2+ 0.004101+ 2.8974* 1{0}^{\mathit{\hbox{-}} 6}{Y_2}^2{Y}_5= 0.007725- 0.421966n+ 0.96043{Y}_2+105.5038{n}^2+ 5.60010* 1{0}^{\mathit{\hbox{-}} 5}{Y_2}^2- 0.15632n{Y}_2{Y}_6= 0.000523+ 0.008127D+ 1.2338{Y}_3- 1.1458{D}^2- 4.4885* 1{0}^{\mathit{\hbox{-}} 5}Y 32+ 0.008275D{Y}_3{Y}_7=- 0.00091+ 3.00512{Y}_4- 1.87748{Y}_5- 0.00035{Y_4}^2+ 0.0001{Y_5}^2+ 0.00023{Y}_4{Y}_5{Y}_8= 0.0025106- 9.26457{Y}_5+ 10.43103{Y}_6+ 0.00177{Y_5}^2- 0.00029{Y_6}^2- 0.001524{Y}_5{Y}_6 Energy = - 0.00504+ 8.22627{Y}_7- 6.74763{Y}_8+ 0.00338{Y_7}^2+ 0.004716{Y_8}^2- 0.008209{Y}_7{Y}_7 $$

For Type D:

$$ {Y}_1= 6860.736- 980.45n- 14.57D+ 179.605{n}^2- 0.1042{D}^2- 10.9524nD{Y}_2= 6051.591- 1054.599n+ 139.517{n}^2{Y}_3= 5931.642+ 56.764D+ 0.44536{D}^2{Y}_4= 0.14737+ 0.9183{Y}_1+ 10.1842D+ 1.4308{Y_1}^2- 0.0636{D}^2- 0.0013{Y}_1D{Y}_5=- 0.001627- 6.610366{Y}_2+ 6.45366{Y}_3+ 0.00097{Y_2}^2- 0.00044{Y_3}^2- 0.000278{Y}_2{Y}_3{Y}_6= 9.6249+ 0.7868{Y}_1+ 0.21856{Y}_2- 4.341* 1{0}^{\mathit{\hbox{-}} 5}{Y_1}^2- 9.0711* 1{0}^{\mathit{\hbox{-}} 5}{Y_2}^2+ 0.0001338{Y}_1{Y}_2{Y}_7= 0.004549* 1{0}^{- 5}+ 9.6103{Y}_4- 8.568{Y}_5+ 0.00247{Y_4}^2+ 0.00465{Y_5}^2- 0.0071{Y}_4{Y}_6{Y}_8= 0.0024+ 11.954{Y}_7- 10.825{Y}_5- 0.001643{Y_7}^2+ 0.00106{Y_6}^2+ 0.000551{Y}_7{Y}_6 Energy = - 0.04637- 9.368{Y}_7- 9.368{Y}_8+ 10.2556{Y_7}^2+ 0.05517{Y_8}^2+ 0.05395{Y}_7{Y}_8 $$

Similarly, the corresponding GEvoM polynomial representations to model the peak crushing force of types A, B, C and D are in the form of:

For Type A:

$$ {Y}_1= 71238.1224- 870.4868n+ 454.9678{n}^2{Y}_2= 84266.5655- 2926.3874n+- 376.9989D+ 518.2299{n}^2- 0.259431{D}^2+ 29.1475nD{Y}_3= 1.3662* 1{0}^{- 5}+ 0.499{Y}_2+ 0.49771{Y}_1+ 1.16274* 1{0}^{\mathit{\hbox{-}} 6}{Y_2}^2- 5. 7945* 1{0}^{\mathit{\hbox{-}} 6}{Y_1}^2+ 4.668* 1{0}^{\mathit{\hbox{-}} 6}{Y}_2{Y}_1{F}_{max}= 1.38108* 1{0}^{- 5}+ 0.48753{Y}_3+ 0.48719{Y}_2+ 0.122235{Y_3}^2+ 0.122339{Y_2}^2- 0.24457{Y}_3{Y}_2 $$

For Type B:

$$ \begin{array}{l}\begin{array}{l}{Y}_1= 85895.284- 3899.7748n+- 5 82.285D+ 1253.2236{n}^2+ 4.76153{D}^2- 125.8719nD\hfill \\ {}{Y}_2= 3.05887* 1{0}^{- 5}+ 1.00757{Y}_1+ 3.6393* 1{0}^{\mathit{\hbox{-}} 5}n- 1.00799* 1{0}^{- 7}{Y_1}^2+ 2.24438* 1{0}^{- 5}{n}^2- 0.00032{Y}_1n\hfill \end{array}\\ {}{F}_{max}= 1.24071* 1{0}^{- 5}+ 0.5155{Y}_1+ 0.51372{Y}_2- 0.16627{Y_1}^2- 0.16633{Y_2}^2+ 0.33261{Y}_2{Y}_1\end{array} $$

For Type C:

$$ {Y}_1= 66993.5678- 4 235.856\ 6n+ 1348.3893{n}^2{Y}_2= 108903.8198- 14710.046n- 1185.1545D+ 2066.7706{n}^2+ 3.29294{D}^2+ 131.38259nD{Y}_3= 84253.48327- 5 97.72D- 1.352117{D}^2{Y}_4= 2.70313* 1{0}^{- 4}+ 1.00426{Y}_1- 6. 42003* 1{0}^{- 8}{Y_1}^2{Y}_5= 1.58153* 1{0}^{- 5}+ 0.50779{Y}_2+ 0.5082{Y}_3+ 0.0004234{Y_2}^2+ 0.0004294{Y_3}^2- 0.0008536{Y}_2{Y}_3{F}_{max}= 9.97831* 1{0}^{- 6}+ 0.3374849{Y}_4+ 0.33079{Y}_5+ 2.156436* 1{0}^{- 6}{Y_4}^2+ 7.03301{Y_5}^2- 4. 36705{Y}_4{Y}_5 $$

For Type D:

$$ {Y}_1= 99723.8754- 9784.638n- 1747.028D+ 1608.228{n}^2+ 16.2803{D}^2+ 52.97698nD{Y}_2= 61963.2877- 2577.7168n+ 310.175{n}^2{Y}_3=- 1.50565* 1{0}^{- 5}+ 0.962219{Y}_1- 0.00595D+ 3.348628* 1{0}^{- 7}{Y_1}^2- 0.841921{D}^2+ 0.0012934{Y}_1D{Y}_4= 1.3662* 1{0}^{- 5}+ 0.4074{Y}_1+ 0.39829{Y}_2- 8.0952* 1{0}^{- 6}{Y_1}^2- 1.58136* 1{0}^{- 5}{Y_2}^2+ 2.75687* 1{0}^{- 5}{Y}_1{Y}_2{Y}_5= 1.3662* 1{0}^{- 5}+ 0.39831{Y}_2+ 0.4074 7{Y}_1- 1.58138- 05{Y_2}^2- 8.0952* 1{0}^{- 6}{Y_1}^2+ 2.75686* 1{0}^{- 5}{Y}_2{Y}_1{Y}_6= 1.62774* 1{0}^{- 5}+ 0.49318{Y}_3+ 0.489156{Y}_4+ 0.004105{Y_3}^2+ 0.004102{Y_4}^2- 0.008209{Y}_3{Y}_4{Y}_7= 1.68271* 1{0}^{- 5}+ 0.49943{Y}_5+ 0.50486{Y}_1+ 0.0038 2{Y_5}^2+ 0.0038{Y_1}^2- 0.007657{Y}_5{Y}_1{F}_{max}= 1.76673* 1{0}^{- 5}+ 0.532{Y}_6+ 0.531{Y}_7- 0.00276{Y_6}^2- 0.00276{Y_7}^2+ 0.005484{Y}_6{Y}_7 $$

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Khalkhali, A., Mostafapour, M., Tabatabaie, S.M. et al. Multi-objective crashworthiness optimization of perforated square tubes using modified NSGAII and MOPSO. Struct Multidisc Optim 54, 45–61 (2016).

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