Abstract
The crashworthiness optimization of the subway end structure is essentially a topology–shape–size collaborative optimization problem of a composite structure under collision conditions. To this end, a rapid multivariate co-optimization method based on weighted graph model and differential evolution algorithm is proposed. The underframe configuration is mapped as a weighted graph and then represented by mathematical matrices. By parameter inversion, a basic parametric simplified finite element model is established and validated by the detailed finite element model and the impact test. Three different configuration optimizations are accomplished and compared to prove the efficiency of the proposed method. The research results of this paper provide a reliable method and a reference for the further development of the crashworthiness optimization for complex structures.
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References
Abramowicz W, Jones N (1984) Dynamic axial crushing of square tubes. Int J Impact Eng 2:179–208
Afrousheh M, Marzbanrad J, Göhlich D (2019) Topology optimization of energy absorbers under crashworthiness using modified hybrid cellular automata (MHCA) algorithm. Struct Multidisc Optim 60:1021–1034. https://doi.org/10.1007/s00158-019-02254-2
Albooyeh A, Soleymani P, Taghipoor H (2022) Evaluation of the mechanical properties of hydroxyapatite-silica aerogel/epoxy nanocomposites: optimizing by response surface approach. J Mech Behav Biomed Mater 136:105513. https://doi.org/10.1016/j.jmbbm.2022.105513
Bahramian N, Khalkhali A (2020) Crashworthiness topology optimization of thin-walled square tubes, using modified bidirectional evolutionary structural optimization approach. Thin-Walled Struct. https://doi.org/10.1016/j.tws.2019.106524
Bai YC, Zhou HS, Lei F, Lei HS (2019) An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads. Struct Multidisc Optim 59:117–130. https://doi.org/10.1007/s00158-018-2054-8
Bakhtiarinejad M, Lee S, Joo J (2017) Component allocation and supporting frame topology optimization using global search algorithm and morphing mesh. Struct Multidisc Optim 55:297–315. https://doi.org/10.1007/s00158-016-1468-4
Chen M, Xiao X, Tong J et al (2018) Optimization of loading path in hydroforming of parallel double branched tube through response surface methodology. Adv Eng Softw 115:429–438. https://doi.org/10.1016/j.advengsoft.2017.11.003
Chia CM, Rongong JA, Worden K (2006) Structural optimisation using a hybrid cellular automata (HCA) algorithm. Appl Mech Mater 56:93–100
Cui L, Li G, Lin Q et al (2016) Adaptive differential evolution algorithm with novel mutation strategies in multiple sub-populations. Comput Oper Res 67:155–173. https://doi.org/10.1016/j.cor.2015.09.006
Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186:311–338. https://doi.org/10.1016/S0045-7825(99)00389-8
Duddeck F, Hunkeler S, Lozano P et al (2016) Topology optimization for crashworthiness of thin-walled structures under axial impact using hybrid cellular automata. Struct Multidisc Optim 54:415–428. https://doi.org/10.1007/s00158-016-1445-y
Eyvazian A, Taghipoor H, Tran T (2022) Analytical and experimental investigations on axial crushing of aluminum tube with vertically corrugated. Int J Crashworthiness 27(4):1032–1045. https://doi.org/10.1080/13588265.2021.1892954
Forsberg J, Nilsson L (2005) On polynomial response surfaces and Kriging for use in structural optimization of crashworthiness. Struct Multidisc Optim 29:232–243. https://doi.org/10.1007/s00158-004-0487-8
Gao GJ, Zhuo TY, Guan WY (2020) Recent research development of energy-absorption structure and application for railway vehicles. J Cent South Univ 27:1012–1038. https://doi.org/10.1007/s11771-020-4349-3
Giger M, Ermanni P (2006) Evolutionary truss topology optimization using a graph-based parameterization concept. Struct Multidisc Optim 32:313–326. https://doi.org/10.1007/s00158-006-0028-8
Giger M, Keller D, Ermanni P et al (2008) A graph-based parameterization concept for global laminate optimization ETH Library A graph-based parameterization concept for global laminate optimization. Struct Multidisc Optim. https://doi.org/10.3929/ethz-b-000015175
Gustafsson E, Strömberg N (2008) Shape optimization of castings by using successive response surface methodology. Struct Multidisc Optim 35:11–28. https://doi.org/10.1007/s00158-007-0114-6
Herskovits J, Dias G, Santos G, Mota Soares C (2000) Shape structural optimization with an interior point nonlinear programming algorithm. Struct Multidisc Optim 20:107–115
Islam SM, Das S, Ghosh S et al (2012) An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization. IEEE Trans Syst Man Cybern B 42:482–500. https://doi.org/10.1109/TSMCB.2011.2167966
Karev A, Harzheim L, Immel R, Erzgräber M (2019) Free sizing optimization of a front hood using the ESL method: overcoming challenges and traps. Struct Multidisc Optim 60:1687–1707. https://doi.org/10.1007/s00158-019-02285-9
Kawamoto A, Bendsøe MP, Sigmund O (2004) Planar articulated mechanism design by graph theoretical enumeration. Struct Multidisc Optim 27:295–299. https://doi.org/10.1007/s00158-004-0409-9
Lee J, Cho M (2018) Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load. Struct Multidisc Optim 58:1489–1504. https://doi.org/10.1007/s00158-018-1976-5
Liu Y, Liu Z, Qin H et al (2018) An efficient structural optimization approach for the modular automotive body conceptual design. Struct Multidisc Optim 58:1275–1289. https://doi.org/10.1007/s00158-018-1949-8
Lu Z, Li B, Yang C et al (2017) Numerical and experimental study on the design strategy of a new collapse zone structure for railway vehicles. Int J Crashworthiness 22:488–502. https://doi.org/10.1080/13588265.2017.1281080
Opara KR, Arabas J (2019) Differential evolution: a survey of theoretical analyses. Swarm Evol Comput 44:546–558. https://doi.org/10.1016/j.swevo.2018.06.010
Ortmann C, Schumacher A (2013) Graph and heuristic based topology optimization of crash loaded structures. Struct Multidisc Optim 47:839–854. https://doi.org/10.1007/s00158-012-0872-7
Padhye N, Bhardawaj P, Deb K (2013) Improving differential evolution through a unified approach. J Glob Optim 55:771–799. https://doi.org/10.1007/s10898-012-9897-0
Passos AG, Luersen MA (2018) Multiobjective optimization of laminated composite parts with curvilinear fibers using Kriging-based approaches. Struct Multidisc Optim 57:1115–1127. https://doi.org/10.1007/s00158-017-1800-7
Sauter M, Kress G, Giger M, Ermanni P (2008) Complex-shaped beam element and graph-based optimization of compliant mechanisms. Struct Multidisc Optim 36:429–442. https://doi.org/10.1007/s00158-007-0182-7
Shu W, Qiu B, Yang J et al (2020) Loading path optimization of shaft clinching forming assembly using finite element simulation and response surface methodology. Proc Inst Mech Eng C 234:734–745. https://doi.org/10.1177/0954406219887770
Storn R (1996) On the usage of differential evolution for function optimization. In: Proceedings of North American fuzzy information processing. IEEE, pp 519–523
Taghipoor H, Eyvazian A (2022) Quasi-static axial crush response and energy absorption of composite wrapped metallic thin-walled tube. J Braz Soc Mech Sci Eng 44:158. https://doi.org/10.1007/s40430-022-03449-3
Taghipoor H, Nouri MD (2018) Experimental and numerical investigation of lattice core sandwich beams under low-velocity bending impact. J Sandwich Struct Mater 21:6. https://doi.org/10.1177/1099636218761315
Taghipoor H, Nouri MD (2018) Axial crushing and transverse bending responses of sandwich structures with lattice core. J Sandwich Struct Mater 22:3. https://doi.org/10.1177/1099636218761321
Taghipoor H, Sefidi M (2022) Energy absorption of foam-filled corrugated core sandwich panels under quasi-static loading. Inst Mech Eng 237:1. https://doi.org/10.1177/14644207221110483
Taghipoor H, Eyvazian A, Musharavati F et al (2020) Experimental investigation of the three-point bending properties of sandwich beams with polyurethane foam-filled lattice cores. Structures 28:424–432. https://doi.org/10.1016/j.istruc.2020.08.082
Taghipoor H, Ghiaskar A, Shavalipour A (2022) Crashworthiness performance of thin-walled, square tubes with circular hole discontinuities under high-speed impact loading. Int J Crashworthiness 27(6):1622–1634. https://doi.org/10.1080/13588265.2021.1981125
Tyrell DC (2002) US rail equipment crashworthiness standards. Proc Inst Mech Eng F 216:123–130. https://doi.org/10.1243/09544090260082362
Wang J, Sun Z (2018) The stepwise accuracy-improvement strategy based on the Kriging model for structural reliability analysis. Struct Multidisc Optim 58:595–612. https://doi.org/10.1007/s00158-018-1911-9
Weider K, Schumacher A (2018) A topology optimization scheme for crash loaded structures using topological derivatives. In: Schumacher A, Vietor T, Fiebig S, Bletzinger K-U, Maute K (eds) Advances in structural and multidisciplinary optimization. Springer, Cham, pp 1601–1614
Wierzbicki T, Abramowicz W (1983) On the crushing mechanics of thin-walled structures. J Appl Mech 50:727–734
Xie S, Zhou H (2014) Impact characteristics of a composite energy absorbing bearing structure for railway vehicles. Composites B 67:455–463. https://doi.org/10.1016/j.compositesb.2014.08.019
Xie S, Liang X, Zhou H, Li J (2016) Crashworthiness optimisation of the front-end structure of the lead car of a high-speed train. Struct Multidisc Optim 53:339–347. https://doi.org/10.1007/s00158-015-1332-y
Xing J, Xu P, Yao S et al (2020) Crashworthiness optimisation of a step-like bi-tubular energy absorber for subway vehicles. Int J Crashworthiness 25:252–262. https://doi.org/10.1080/13588265.2019.1577522
Xing J, Xu P, Yao S et al (2021) A novel weighted graph representation-based method for structural topology optimization. Adv Eng Softw. https://doi.org/10.1016/j.advengsoft.2021.102977
Xu P, Yang C, Peng Y et al (2016) Crash performance and multi-objective optimization of a gradual energy-absorbing structure for subway vehicles. Int J Mech Sci 107:1–12. https://doi.org/10.1016/j.ijmecsci.2016.01.001
Xu P, Yang C, Peng Y et al (2016) Cut-out grooves optimization to improve crashworthiness of a gradual energy-absorbing structure for subway vehicles. Mater Des 103:132–143. https://doi.org/10.1016/j.matdes.2016.04.059
Xu P, Xing J, Yao S et al (2017) Energy distribution analysis and multi-objective optimization of a gradual energy-absorbing structure for subway vehicles. Thin-Walled Struct 115:255–263. https://doi.org/10.1016/j.tws.2017.02.033
Yang C, Xu P, Yao S et al (2018) Optimization of honeycomb strength assignment for a composite energy-absorbing structure. Thin-Walled Struct 127:741–755. https://doi.org/10.1016/j.tws.2018.03.014
Yao S, Zhang P, Xing J et al (2022) Application of a weighted graph representation method to the crashworthiness optimization of subway collision frame structures. Structures 45:1095–1109. https://doi.org/10.1016/j.istruc.2022.09.065
Yoshimura M, Shimoyama K, Misaka T, Obayashi S (2019) Optimization of passive grooved micromixers based on genetic algorithm and graph theory. Microfluid Nanofluidics 23:30. https://doi.org/10.1007/s10404-019-2201-6
Yu X, Cai M, Cao J (2015) A novel mutation differential evolution for global optimization. J Intell Fuzzy Syst 28:1047–1060. https://doi.org/10.3233/IFS-141388
Zhang J, Sanderson AC (2009) JADE: Adaptive differential evolution with optional external archive. IEEE Trans Evol Comput 13:945–958. https://doi.org/10.1109/TEVC.2009.2014613
Acknowledgements
The research presented in this paper was conducted with the support of the National Key Research and Development Program of China (Grant No. 2021YFB3703801), the Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ30853), and the Leading Talents of Science and Technology of Hunan Province (Grant No. 2019RS3018).
Funding
Funding was provided by the National Key Research and Development Program of China (Grant No. 2021YFB3703801), Natural Science Foundation of Hunan Province (Grant No. 2021JJ30853), and the Leading Talents of Science and Technology of Hunan Province (Grant No. 2019RS3018).
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Yao, S., Zhou, Y., Xing, J. et al. A crashworthiness optimization method of subway underframe structures based on the differential evolution of the weighted graph representation. Struct Multidisc Optim 67, 62 (2024). https://doi.org/10.1007/s00158-024-03780-4
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DOI: https://doi.org/10.1007/s00158-024-03780-4