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Aerodynamic optimisation of a high-speed train head shape using an advanced hybrid surrogate-based nonlinear model representation method

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Abstract

A global optimisation strategy based on the hybrid surrogate model method and the competitive mechanism-based multi-objective particle swarm optimisation (CMOPSO) algorithm was developed to improve the accuracy of the aerodynamic performance optimisation of a high-speed train running in open air without a crosswind. Free-form deformation was used to improve the optimisation efficiency without remodelling or remeshing. The sample points and their responses were obtained via optimal Latin hypercube sampling and computational fluid dynamics (CFD) simulations. The hybrid surrogate model (HSM) was constructed by using the theory of optimal weighted surrogate to combine a polynomial response surface (PRS) model with a radial basis function (RBF) model. Comprehensive error evaluation results indicated that the prediction accuracy achieved with the HSM was higher than that achieved with either the PRS model or the RBF model; thus, the HSM was selected for use in each iteration to approximate the CFD simulation model of the high-speed train in subsequent optimisation. Then, the CMOPSO algorithm was selected as the optimisation algorithm. After optimisation, a series of Pareto-optimal solutions was obtained, and the optimal and original head shapes were compared. The use of the hybrid surrogate model and the CMOPSO algorithm greatly improved the optimisation efficiency.

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Acknowledgements

The research described in this paper was undertaken at Key Laboratory of Traffic Safety on the Track of Ministry of Education, China. The authors gratefully acknowledge the support from the Project of National Key R&D Program of China (Grant No. 2016YFB1200506-03).

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Authors and Affiliations

  1. Key Laboratory of Traffic Safety on the Track of Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha, 410075, China

    Zhao He, Xiaohui Xiong, Bo Yang & Haihong Li

  2. Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, China

    Zhao He, Xiaohui Xiong, Bo Yang & Haihong Li

  3. National and Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Changsha, 410075, China

    Zhao He, Xiaohui Xiong, Bo Yang & Haihong Li

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  1. Zhao He
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  2. Xiaohui Xiong
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  3. Bo Yang
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  4. Haihong Li
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Correspondence to Xiaohui Xiong.

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Appendix: The model expressions

Appendix: The model expressions

$$\begin{aligned} Cd{\text{-}}total & = 0.324\left( {0.30548 - 0.00751h_{1} + 0.00728h_{2} + 0.00134l_{1} - 0.01812l_{2} + 0.00365w} \right. \\ & \quad + 0.06868h_{1} h_{2} - 0.06372h_{1} l_{1} - 0.12739h_{1} l_{2} - 0.06221h_{1} w - 0.09942h_{2} l_{1} \\ & \quad - 0.28692h_{2} l_{2} + 0.12927h_{2} w - 0.10190l_{1} l_{2} + 0.16831l_{1} w + 0.09664l_{2} w \\ & \quad \left. { - 0.09330h_{1}^{2} + 0.04975h_{2}^{2} - 0.12480l_{1}^{2} - 0.24282l_{2}^{2} - 0.06386w^{2} } \right) \\ & \quad + 0.676\left( { - 95.15519\sqrt {(h_{1} + 0.0483)^{2} + (h_{2} + 0.0531)^{2} + (l_{1} + 0.0017)^{2} + (l_{2} + 0.1)^{2} + (w + 0.0238)^{2} + 1} } \right. \\ & \quad - 525.76639\sqrt {(h_{1} + 0.0017)^{2} + (h_{2} + 0.0262)^{2} + (l_{1} + 0.05)^{2} + (l_{2} + 0.0328)^{2} + (w - 0.021)^{2} + 1} \\ & \quad + 208.12898\sqrt {(h_{1} - 0.0138)^{2} + (h_{2} + 0.0666)^{2} + (l_{1} - 0.0948)^{2} + (l_{2} + 0.069)^{2} + (w - 0.0121)^{2} + 1} \\ & \quad - 452.74486\sqrt {(h_{1} + 0.0793)^{2} + (h_{2} + 0.0576)^{2} + (l_{1} - 0.1362)^{2} + (l_{2} + 0.0586)^{2} + (w + 0.0283)^{2} + 1} \\ & \quad + 75.31201\sqrt {(h_{1} + 0.0948)^{2} + (h_{2} + 0.0128)^{2} + (l_{1} + 0.0431)^{2} + (l_{2} + 0.0276)^{2} + (w + 0.0372)^{2} + 1} \\ & \quad + 54.91733\sqrt {(h_{1} - 0.0241)^{2} + (h_{2} - 0.0321)^{2} + (l_{1} - 0.0741)^{2} + (l_{2} + 0.0121)^{2} + (w - 0.0255)^{2} + 1} \\ & \quad - 179.07452\sqrt {(h_{1} + 0.0586)^{2} + (h_{2} - 0.0366)^{2} + (l_{1} - 0.019)^{2} + (l_{2} - 0.0138)^{2} + (w + 0.0821)^{2} + 1} \\ & \quad + 7.64420\sqrt {(h_{1} + 0.0845)^{2} + (h_{2} + 0.0083)^{2} + (l_{1} - 0.0466)^{2} + (l_{2} + 0.0534)^{2} + (w - 0.03)^{2} + 1} \\ & \quad + 545.24477\sqrt {(h_{1} - 0.0345)^{2} + (h_{2} - 0.0276)^{2} + (l_{1} + 0.0224)^{2} + (l_{2} - 0.0086)^{2} + (w + 0.0417)^{2} + 1} \\ & \quad - 177.83921\sqrt {(h_{1} - 0.0448)^{2} + (h_{2} + 0.0486)^{2} + (l_{1} - 0.0397)^{2} + (l_{2} - 0.0293)^{2} + (w + 0.0014)^{2} + 1} \\ & \quad - 82.53965\sqrt {(h_{1} + 0.069)^{2} + (h_{2} + 0.0621)^{2} + (l_{1} - 0.0328)^{2} + (l_{2} + 0.0379)^{2} + (w + 0.0955)^{2} + 1} \\ & \quad - 233.08150\sqrt {(h_{1} - 0.0086)^{2} + (h_{2} + 0.0441)^{2} + (l_{1} - 0.1017)^{2} + (l_{2} + 0.0897)^{2} + (w + 0.0776)^{2} + 1} \\ & \quad + 130.18649\sqrt {(h_{1} + 0.0897)^{2} + (h_{2} - 0.0186)^{2} + (l_{1} - 0.0672)^{2} + (l_{2} + 0.0793)^{2} + (w + 0.0641)^{2} + 1} \\ & \quad + 29.11423\sqrt {(h_{1} - 0.05)^{2} + (h_{2} - 0.0007)^{2} + (l_{1} - 0.0259)^{2} + (l_{2} + 0.0845)^{2} + (w + 0.0193)^{2} + 1} \\ & \quad + 48.65003\sqrt {(h_{1} + 0.0069)^{2} + (h_{2} + 0.0755)^{2} + (l_{1} - 0.1155)^{2} + (l_{2} - 0.0241)^{2} + (w + 0.0597)^{2} + 1} \\ & \quad + 79.95596\sqrt {(h_{1} - 0.0034)^{2} + (h_{2} - 0.0231)^{2} + (l_{1} - 0.1086)^{2} + (l_{2} - 0.05)^{2} + (w + 0.0462)^{2} + 1} \\ & \quad + 156.33029\sqrt {(h_{1} - 0.0293)^{2} + (h_{2} + 0.071)^{2} + (l_{1} + 0.0155)^{2} + (l_{2} + 0.0431)^{2} + (w + 0.0552)^{2} + 1} \\ & \quad + 0.30842\sqrt {(h_{1} - 0.0397)^{2} + (h_{2} + 0.0217)^{2} + (l_{1} - 0.0603)^{2} + (l_{2} - 0.0034)^{2} + (w + 0.1)^{2} + 1} \\ & \quad - 403.37828\sqrt {(h_{1} + 0.0276)^{2} + (h_{2} + 0.0397)^{2} + (l_{1} + 0.0362)^{2} + (l_{2} - 0.0397)^{2} + (w + 0.0686)^{2} + 1} \\ & \quad - 111.82187\sqrt {(h_{1} + 0.0379)^{2} + (h_{2} - 0.05)^{2} + (l_{1} + 0.0086)^{2} + (l_{2} + 0.0638)^{2} + (w + 0.0103)^{2} + 1} \\ & \quad - 135.10940\sqrt {(h_{1} + 0.0741)^{2} + (h_{2} - 0.041)^{2} + (l_{1} - 0.1224)^{2} + (l_{2} + 0.0069)^{2} + (w + 0.0148)^{2} + 1} \\ & \quad - 240.06943\sqrt {(h_{1} + 0.0328)^{2} + (h_{2} + 0.0352)^{2} + (l_{1} - 0.1431)^{2} + (l_{2} - 0.019)^{2} + (w - 0.0166)^{2} + 1} \\ & \quad + 556.35432\sqrt {(h_{1} + 0.0638)^{2} + (h_{2} + 0.08)^{2} + (l_{1} - 0.0121)^{2} + (l_{2} + 0.0017)^{2} + (w - 0.0031)^{2} + 1} \\ & \quad + 631.28340\sqrt {(h_{1} + 0.0431)^{2} + (h_{2} + 0.0038)^{2} + (l_{1} - 0.15)^{2} + (l_{2} + 0.0172)^{2} + (w + 0.091)^{2} + 1} \\ & \quad + 400.51037\sqrt {(h_{1} + 0.0172)^{2} + (h_{2} - 0.0141)^{2} + (l_{1} - 0.1293)^{2} + (l_{2} + 0.0948)^{2} + (w + 0.0059)^{2} + 1} \\ & \quad + 292.16546\sqrt {(h_{1} + 0.0121)^{2} + (h_{2} - 0.0052)^{2} + (l_{1} + 0.0293)^{2} + (l_{2} + 0.0741)^{2} + (w + 0.0866)^{2} + 1} \\ & \quad + 211.55698\sqrt {(h_{1} + 0.0534)^{2} + (h_{2} - 0.0097)^{2} + (l_{1} - 0.0052)^{2} + (l_{2} - 0.0448)^{2} + (w - 0.0076)^{2} + 1} \\ & \quad - 22.87909\sqrt {(h_{1} + 0.0224)^{2} + (h_{2} + 0.0172)^{2} + (l_{1} - 0.0534)^{2} + (l_{2} + 0.0224)^{2} + (w + 0.0328)^{2} + 1} \\ & \quad - 699.57531\sqrt {(h_{1} - 0.019)^{2} + (h_{2} - 0.0455)^{2} + (l_{1} - 0.0879)^{2} + (l_{2} + 0.0483)^{2} + (w + 0.0731)^{2} + 1} \\ & \left. {\quad - 68.70866\sqrt {(h_{1} + 0.1)^{2} + (h_{2} + 0.0307)^{2} + (l_{1} - 0.081)^{2} + (l_{2} - 0.0345)^{2} + (w + 0.0507)^{2} + 1} + 0.30415} \right). \\ \end{aligned}$$
(26)
$$\begin{aligned} Cl{\text{-}}tail & = 0.763\left( {0.04545 - 0.01538h_{1} - 0.02708h_{2} - 0.01180l_{1} + 0.00677l_{2} + 0.02729w} \right. \\ & \quad - 0.15703h_{1} h_{2} - 0.05448h_{1} l_{1} - 0.30762h_{1} l_{2} - 0.28262h_{1} w + 0.22535h_{2} l_{1} \\ & \quad + 0.49945h_{2} l_{2} - 0.20745h_{2} w + 0.04662l_{1} l_{2} - 0.42554l_{1} w + 0.06141l_{2} w \\ & \quad \left. { + 0.07543h_{1}^{2} - 0.05892h_{2}^{2} - 0.12885l_{1}^{2} - 0.14775l_{2}^{2} - 0.01507w^{2} } \right) \\ & \quad + 0.237\left( { - 0.06203\sqrt {(h_{1} + 0.0483)^{2} + (h_{2} + 0.0531)^{2} + (l_{1} + 0.0017)^{2} + (l_{2} + 0.1)^{2} + (w + 0.0238)^{2} + 0.00091809} } \right. \\ & \quad + 0.01052\sqrt {(h_{1} + 0.0017)^{2} + (h_{2} + 0.0262)^{2} + (l_{1} + 0.05)^{2} + (l_{2} + 0.0328)^{2} + (w - 0.021)^{2} + 0.00091809} \\ & \quad + 0.01721\sqrt {(h_{1} - 0.0138)^{2} + (h_{2} + 0.0666)^{2} + (l_{1} - 0.0948)^{2} + (l_{2} + 0.069)^{2} + (w - 0.0121)^{2} + 0.00091809} \\ & \quad + 0.04790\sqrt {(h_{1} + 0.0793)^{2} + (h_{2} + 0.0576)^{2} + (l_{1} - 0.1362)^{2} + (l_{2} + 0.0586)^{2} + (w + 0.0283)^{2} + 0.00091809} \\ & \quad - 0.00627\sqrt {(h_{1} + 0.0948)^{2} + (h_{2} + 0.0128)^{2} + (l_{1} + 0.0431)^{2} + (l_{2} + 0.0276)^{2} + (w + 0.0372)^{2} + 0.00091809} \\ & \quad + 0.01364\sqrt {(h_{1} - 0.0241)^{2} + (h_{2} - 0.0321)^{2} + (l_{1} - 0.0741)^{2} + (l_{2} + 0.0121)^{2} + (w - 0.0255)^{2} + 0.00091809} \\ & \quad - 0.05807\sqrt {(h_{1} + 0.0586)^{2} + (h_{2} - 0.0366)^{2} + (l_{1} - 0.019)^{2} + (l_{2} - 0.0138)^{2} + (w + 0.0821)^{2} + 0.00091809} \\ & \quad - 0.01921\sqrt {(h_{1} + 0.0845)^{2} + (h_{2} + 0.0083)^{2} + (l_{1} - 0.0466)^{2} + (l_{2} + 0.0534)^{2} + (w - 0.03)^{2} + 0.00091809} \\ & \quad - 0.02874\sqrt {(h_{1} - 0.0345)^{2} + (h_{2} - 0.0276)^{2} + (l_{1} + 0.0224)^{2} + (l_{2} - 0.0086)^{2} + (w + 0.0417)^{2} + 0.00091809} \\ & \quad - 0.02318\sqrt {(h_{1} - 0.0448)^{2} + (h_{2} + 0.0486)^{2} + (l_{1} - 0.0397)^{2} + (l_{2} - 0.0293)^{2} + (w + 0.0014)^{2} + 0.00091809} \\ & \quad + 0.04118\sqrt {(h_{1} + 0.069)^{2} + (h_{2} + 0.0621)^{2} + (l_{1} - 0.0328)^{2} + (l_{2} + 0.0379)^{2} + (w + 0.0955)^{2} + 0.00091809} \\ & \quad - 0.04826\sqrt {(h_{1} - 0.0086)^{2} + (h_{2} + 0.0441)^{2} + (l_{1} - 0.1017)^{2} + (l_{2} + 0.0897)^{2} + (w + 0.0776)^{2} + 0.00091809} \\ & \quad + 0.07559\sqrt {(h_{1} + 0.0897)^{2} + (h_{2} - 0.0186)^{2} + (l_{1} - 0.0672)^{2} + (l_{2} + 0.0793)^{2} + (w + 0.0641)^{2} + 0.00091809} \\ & \quad + 0.02055\sqrt {(h_{1} - 0.05)^{2} + (h_{2} - 0.0007)^{2} + (l_{1} - 0.0259)^{2} + (l_{2} + 0.0845)^{2} + (w + 0.0193)^{2} + 0.00091809} \\ & \quad + 0.07057\sqrt {(h_{1} + 0.0069)^{2} + (h_{2} + 0.0755)^{2} + (l_{1} - 0.1155)^{2} + (l_{2} - 0.0241)^{2} + (w + 0.0597)^{2} + 0.00091809} \\ & \quad + 0.03233\sqrt {(h_{1} - 0.0034)^{2} + (h_{2} - 0.0231)^{2} + (l_{1} - 0.1086)^{2} + (l_{2} - 0.05)^{2} + (w + 0.0462)^{2} + 0.00091809} \\ & \quad - 0.06121\sqrt {(h_{1} - 0.0293)^{2} + (h_{2} + 0.071)^{2} + (l_{1} + 0.0155)^{2} + (l_{2} + 0.0431)^{2} + (w + 0.0552)^{2} + 0.00091809} \\ & \quad - 0.02948\sqrt {(h_{1} - 0.0397)^{2} + (h_{2} + 0.0217)^{2} + (l_{1} - 0.0603)^{2} + (l_{2} - 0.0034)^{2} + (w + 0.1)^{2} + 0.00091809} \\ & \quad + 0.10118\sqrt {(h_{1} + 0.0276)^{2} + (h_{2} + 0.0397)^{2} + (l_{1} + 0.0362)^{2} + (l_{2} - 0.0397)^{2} + (w + 0.0686)^{2} + 0.00091809} \\ & \quad + 0.01717\sqrt {(h_{1} + 0.0379)^{2} + (h_{2} - 0.05)^{2} + (l_{1} + 0.0086)^{2} + (l_{2} + 0.0638)^{2} + (w + 0.0103)^{2} + 0.00091809} \\ & \quad - 0.01329\sqrt {(h_{1} + 0.0741)^{2} + (h_{2} - 0.041)^{2} + (l_{1} - 0.1224)^{2} + (l_{2} + 0.0069)^{2} + (w + 0.0148)^{2} + 0.00091809} \\ & \quad + 0.01664\sqrt {(h_{1} + 0.0328)^{2} + (h_{2} + 0.0352)^{2} + (l_{1} - 0.1431)^{2} + (l_{2} - 0.019)^{2} + (w - 0.0166)^{2} + 0.00091809} \\ & \quad - 0.04961\sqrt {(h_{1} + 0.0638)^{2} + (h_{2} + 0.08)^{2} + (l_{1} - 0.0121)^{2} + (l_{2} + 0.0017)^{2} + (w - 0.0031)^{2} + 0.00091809} \\ & \quad - 0.08451\sqrt {(h_{1} + 0.0431)^{2} + (h_{2} + 0.0038)^{2} + (l_{1} - 0.15)^{2} + (l_{2} + 0.0172)^{2} + (w + 0.091)^{2} + 0.00091809} \\ & \quad + 0.00608\sqrt {(h_{1} + 0.0172)^{2} + (h_{2} - 0.0141)^{2} + (l_{1} - 0.1293)^{2} + (l_{2} + 0.0948)^{2} + (w + 0.0059)^{2} + 0.00091809} \\ & \quad + 0.0517\sqrt {(h_{1} + 0.0121)^{2} + (h_{2} - 0.0052)^{2} + (l_{1} + 0.0293)^{2} + (l_{2} + 0.0741)^{2} + (w + 0.0866)^{2} + 0.00091809} \\ & \quad - 0.02626\sqrt {(h_{1} + 0.0534)^{2} + (h_{2} - 0.0097)^{2} + (l_{1} - 0.0052)^{2} + (l_{2} - 0.0448)^{2} + (w - 0.0076)^{2} + 0.00091809} \\ & \quad + 0.00487\sqrt {(h_{1} + 0.0224)^{2} + (h_{2} + 0.0172)^{2} + (l_{1} - 0.0534)^{2} + (l_{2} + 0.0224)^{2} + (w + 0.0328)^{2} + 0.00091809} \\ & \quad + 0.02216\sqrt {(h_{1} - 0.019)^{2} + (h_{2} - 0.0455)^{2} + (l_{1} - 0.0879)^{2} + (l_{2} + 0.0483)^{2} + (w + 0.0731)^{2} + 0.00091809} \\ & \quad \left. { - 0.04500\sqrt {(h_{1} + 0.1)^{2} + (h_{2} + 0.0307)^{2} + (l_{1} - 0.081)^{2} + (l_{2} - 0.0345)^{2} + (w + 0.0507)^{2} + 0.00091809} + 0.04367} \right). \\ \end{aligned}$$
(27)

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He, Z., Xiong, X., Yang, B. et al. Aerodynamic optimisation of a high-speed train head shape using an advanced hybrid surrogate-based nonlinear model representation method. Optim Eng 23, 59–84 (2022). https://doi.org/10.1007/s11081-020-09554-3

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  • DOI: https://doi.org/10.1007/s11081-020-09554-3

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