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Crashworthiness optimisation of a composite energy-absorbing structure for subway vehicles based on hybrid particle swarm optimisation

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Abstract

To improve the crashworthiness of subway vehicles, a composite energy-absorbing structure (EAS) is designed by coupling a thin-walled metal tube and aluminium honeycomb structures. On this basis, the effectiveness of the composite structure and aluminium honeycomb finite element model (FEM) is validated by conducting trolley impact tests and quasi-static compression experiments. Based on the verified FEM, the surrogate models, including polynomial response surface (PRS), Kriging, radial basis function (RBF) and supported vector regression (SVR), are established to get the relationship between crashworthiness indexes and design variables. Then, the most accurate model is employed for crashworthiness optimisation through comparing the accuracies of these four models. Owing to the mutually effects among components of the composite EAS and the relatively complex mathematical formulae acquired by the high-precision surrogate model, a hybrid particle swarm optimisation (HPSO) algorithm is put forward. The performance of the HPSO is tested by a typical engineering optimisation case. The results indicate that the HPSO algorithm presents various advantages (e.g. strong global search capability, high optimisation accuracy, etc.). By optimising the mathematical model of the composite EAS, the optimal configurations of the structure are obtained, which confirms that the HPSO algorithm has favourable applicability and performance in the crashworthiness optimisation of the EASs for subway vehicles.

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Acknowledgments

This research was undertaken at Key Laboratory of Traffic Safety on Track (Central South University), Ministry of Education, China. The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant no. 51775558), the supported by Innovation-Driven Project of Central South University (No. 2018CX023) and the support from the Shenghua Yu-ying Talents Program of the Central South University (Principle Investigator: Prof. Suchao Xie). The corresponding author would like to acknowledge the China Scholarship Council/University of Manchester Joint Scholarship for PhD study (201706370205).

Author information

Authors and Affiliations

  1. Ministry of Education, Key Laboratory of Traffic Safety on Track (Central South University), Changsha, 410075, China

    Suchao Xie, Haihong Li & Shuguang Yao

  2. School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Pariser Building, Manchester, M13 9PL, UK

    Chengxing Yang

  3. School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, China

    Suchao Xie, Haihong Li & Shuguang Yao

Authors
  1. Suchao Xie
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  2. Haihong Li
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  3. Chengxing Yang
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  4. Shuguang Yao
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Corresponding author

Correspondence to Chengxing Yang.

Appendix: The response model expressions

Appendix: The response model expressions

$$ {\displaystyle \begin{array}{l} SEA=\hbox{-} 949.2562{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+701.7790{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}+598.7812{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 1149.4894{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+1035.1016{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+2291.0571{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.75em \hbox{-} 2297.4196{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+56.7215{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+1241.1849{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 2598.5402{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 1452.4172{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+2625.0468{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 3235.2708{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+2703.6099{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+1913.4500{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 1667.6270{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+2171.7826{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+363.8051{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 2427.5173{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 774.1893{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+2154.2208{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 4115.0621{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+3274.2119{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 1048.0831{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em +639.4714{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\hbox{-} 0.00015694\end{array}} $$
(20)
$$ {\displaystyle \begin{array}{l} CFE=\hbox{-} 11.3532{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+14.8120{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 0.7069{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 7.2826{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+5.9566{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+26.8433{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.75em \hbox{-} 50.7002{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+21.9695{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+4.5661{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 12.5050{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 8.7201{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+11.5366{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\\ {}\kern1.5em +6.2832{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\hbox{-} 3.9383{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+8.9278{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 21.1077{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+73.9078{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\hbox{-} 73.7803{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\\ {}\kern1.5em +17.3019{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 0.7784{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+18.9962{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 60.2296{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+57.0276{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 16.0722{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 0.1643{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+0.000070585\end{array}} $$
(21)
$$ {\displaystyle \begin{array}{l}{F}_{ip}=6835.8680{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 50974.6406{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_3-0.07\right)}^2\right)/8\right]}+43255.6059{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 29657.5978{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+4255.0217{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+3399.7037{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.75em +116438.4361{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\hbox{-} 70852.9481{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+38845.0350{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em +13541.1670{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 27917.4190{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\hbox{-} 88906.4254{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\\ {}\kern1.5em +6229.7421{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+16230.2918{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 39395.7577{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.5em +12260.7972{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+48688.5739{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+8657.6748{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 20686.4121{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}+20295.9443{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+15418.0245{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 47073.2019{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+40112.7689{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 24575.1546{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em +9918.08293{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\hbox{-} 0.0027\end{array}} $$
(22)
$$ {\displaystyle \begin{array}{l}{F}_{avg}=\hbox{-} 90049.9564{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+5072.6960{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}+151010.6483{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 186903.5460{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+105716.6446{e}^{\left[-\left({\left({t}_1-1.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+267167.6072{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.75em \hbox{-} 161084.3139{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\hbox{-} 103470.2327{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+257555.9588{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 257887.7089{e}^{\left[-\left({\left({t}_1-2\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\hbox{-} 197905.1519{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+305740.2096{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 481509.8579{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+323578.3181{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+137802.4456{e}^{\left[-\left({\left({t}_1-2.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 246955.6966{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}+301241.4774{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}+187071.3895{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 376267.3601{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 17173.4284{e}^{\left[-\left({\left({t}_1-3\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}+320346.5287{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-1\right)}^2+{\left({t}_2-0.16\right)}^2\right)/8\right]}\\ {}\kern1.5em \hbox{-} 573877.7356{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-2\right)}^2+{\left({t}_2-0.04\right)}^2\right)/8\right]}+383242.3155{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-3\right)}^2+{\left({t}_2-0.07\right)}^2\right)/8\right]}\hbox{-} 102457.5937{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-4\right)}^2+{\left({t}_2-0.1\right)}^2\right)/8\right]}\\ {}\kern1.5em +57075.2454{e}^{\left[-\left({\left({t}_1-3.5\right)}^2+{\left(l-5\right)}^2+{\left({t}_2-0.13\right)}^2\right)/8\right]}\hbox{-} 0.0198\end{array}} $$
(23)

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Xie, S., Li, H., Yang, C. et al. Crashworthiness optimisation of a composite energy-absorbing structure for subway vehicles based on hybrid particle swarm optimisation. Struct Multidisc Optim 58, 2291–2308 (2018). https://doi.org/10.1007/s00158-018-2022-3

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