Abstract
Multi-cell tubal structures have widely been used in the automobile industry due to proven superior crashworthiness performance than single-cell and foam-filled tubes. This superior performance is attributed to the number of corners within the cross-sectional profile of the tube. In this paper, a two-stage optimization design of a multi-cell tubal structure is presented to address an important design problem by combining a discrete and continuous optimization process into a sequential optimization that generates an overall optimum. The first stage entails a configurational optimization which is realized by formulating a discrete topological optimization problem where the webs within the tube configuration are taken as the topological design variables. Each topological configuration is represented using a binary scheme that shows the presence or not of an edge to create different combinations of the corners. The constraints in the first stage are connectivity, mass ratio, and peak crushing force (PCF). The binary genetic algorithm (BGA) is utilized in searching for the optimal configuration in the first stage. The second stage entails parameter optimization where the cell sizes are the design variables. The objective functions in the second stage are defined using meta-models. Multi-objective particle swarm optimization (MOPSO) is employed for Pareto searching and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is used to find the optimal point for each of the mass ratios considered. Compared with the baseline configuration, the optimized tubal structures demonstrated superior crashworthiness performance. The two-stage discrete and continuous optimization approach has demonstrated that it not only provides a systematic approach to searching optimal structure but also creates a series of novel multi-cell topological configurations with enhanced crashworthiness.
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All data generated or analyzed during this study are included in this published article and its Supplementary Information files
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The code developed during the current study is available from the corresponding author on reasonable request.
References
Abdullahi HS, Gao S (2020) A novel multi-cell square tubal structure based on Voronoi tessellation for enhanced crashworthiness. Thin-Walled Struct 150. https://doi.org/10.1016/j.tws.2020.106690
Abramowicz W, Jones N (1986) Dynamic progressive buckling of circular and square tubes. Int J Impact Eng 4(4):243–270. https://doi.org/10.1016/0734-743X(86)90017-5
Abramowicz W, Wierzbicki T (1989) Axial crushing of multicorner sheet metal columns. J Appl Mech 56(1):113. https://doi.org/10.1115/1.3176030
Afrousheh M, Marzbanrad J, Göhlich D (2019) Topology optimization of energy absorbers under crashworthiness using modified hybrid cellular automata (MHCA) algorithm. Struct Multidiscip Optim 60(3):1021–1034. https://doi.org/10.1007/s00158-019-02254-2
Bigdeli A, Nouri MD (2019) A crushing analysis and multi-objective optimization of thin-walled five-cell structures. Thin-Walled Struct 137:1–18. https://doi.org/10.1016/j.tws.2018.12.033
Cavazzuti M, Baldini A, Bertocchi E, Costi D, Torricelli E, Moruzzi P (2011) High performance automotive chassis design: a topology optimization based approach. Struct Multidisc Optim 44:45–56. https://doi.org/10.1007/s00158-010-0578-7
Chen MF, Tzeng GH (2004) Combining grey relation and TOPSIS concepts for selecting an expatriate host country. Mathematical and Computer Modelling. https://doi.org/10.1016/j.mcm.2005.01.006
Chen S, Yu H, Fang J (2018) A novel multi-cell tubal structure with circular corners for crashworthiness. Thin-Walled Struct 122:329–343. https://doi.org/10.1016/j.tws.2017.10.026
Chen W, Wierzbicki T (2001) Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption. Thin-Walled Struct 39:287–306. https://doi.org/10.1016/S0263-8231(01)00006-4
Christensen J, Bastien C, Blundell MV (2012) Effects of roof crush loading scenario upon body in white using topology optimisation. Int J Crashworthiness 17(1):29–38. https://doi.org/10.1080/13588265.2011.625640
Da DC, Chen JH, Cui XY, Li GY (2017) Design of materials using hybrid cellular automata. Struct Multidiscip Optim 56(1):131–137. https://doi.org/10.1007/s00158-017-1652-1
Duan L, Jiang H, Cheng A, Xue H, Geng G (2019a) Multi-objective reliability-based design optimization for the VRB-VCS FLB under front-impact collision. Struct Multidiscip Optim 59 (5):1835–1851. https://doi.org/10.1007/s00158-018-2142-9
Duan L, Jiang H, Geng G, Zhang X, Li Z (2019b) Parametric modeling and multiobjective crashworthiness design optimization of a new front longitudinal beam. Struct Multidiscip Optim 59 (5):1789–1812. https://doi.org/10.1007/s00158-018-2134-9
Duan L, Jiang H, Li H, Xiao N (2020) Crashworthiness optimization of VRB thin-walled structures under manufacturing constraints by the eHCA-VRB algorithm. Appl Math Model 80:126–150. https://doi.org/10.1016/j.apm.2019.11.030
Duddeck F, Hunkeler S, Lozano P, Wehrle E, Zeng D (2016) Topology optimization for crashworthiness of thin-walled structures under axial impact using hybrid cellular automata. Struct Multidisc Optim 54(3):415–428. https://doi.org/10.1007/s00158-016-1445-y
Fang J, Gao Y, Sun G, Qiu N, Li Q (2015) On design of multi-cell tubes under axial and oblique impact loads. Thin-Walled Struct 95:115–126. https://doi.org/10.1016/j.tws.2015.07.002
Fang J, Sun G, Qiu N, Kim NH, Li Q (2017a) On design optimization for structural crashworthiness and its state of the art. Struct Multidiscip Optim 55(3):1091–1119. https://doi.org/10.1007/s00158-016-1579-y
Fang J, Sun G, Qiu N, Steven GP, Li Q (2017b) Topology optimization of multicell tubes under out-of-plane crushing using a modified artificial bee colony algorithm. Journal of Mechanical Design, Transactions of the ASME 139(7):1–16. https://doi.org/10.1115/1.4036561
Fu J, Liu Q, Liufu K, Deng Y, Fang J, Li Q (2019) Design of bionic-bamboo thin-walled structures for energy absorption. Thin-Walled Struct 135:400–413. https://doi.org/10.1016/j.tws.2018.10.003
Haupt RL, Haupt SE (2004) The binary genetic algorithm. In: Practical genetic algorithms. https://doi.org/10.1002/0471671746.ch2. Wiley, Hoboken, pp 27–50
Hou S, Li Q, Long S, Yang X, Li W (2007) Design optimization of regular hexagonal thin-walled columns with crashworthiness criteria. Finite Elem Anal Des 43(6-7):555–565. https://doi.org/10.1016/J.FINEL.2006.12.008
Jang HH, Lee HA, Lee JY, Park GJ (2012) Dynamic response topology optimization in the time domain using equivalent static loads. AIAA Journal. https://doi.org/10.2514/1.J051256
Jeyakarthikeyan P, Yogeshwaran R, Abdullahi HS (2018) Time efficiency and error estimation in generating element stiffness matrices of plane triangular elements using Universal Matrix Method and Gauss-Quadrature. Ain Shams Eng J 9(4):965–972. https://doi.org/10.1016/j.asej.2016.05.002
Karagiozova D, Nurick GN, Chung Kim Yuen S (2005) Energy absorption of aluminium alloy circular and square tubes under an axial explosive load. Thin-Walled Struct 43:956–982. https://doi.org/10.1016/j.tws.2004.11.002
Kuri Morales AF, Quezada C (1998) A universal eclectic genetic algorithm for constrained optimization. In: Proceedings of the 6th European congress on intelligent techniques and soft computing, EUFIT’98
Li M, Tang W, Yuan M (2014) Structural dynamic topology optimization based on dynamic reliability using equivalent static loads. Struct Multidisc Optim 49:121–129. https://doi.org/10.1007/s00158-013-0965-y
Liu C, Zhu Y, Sun Z, Li D, Du Z, Zhang W, Guo X (2018) An efficient moving morphable component (MMC)-based approach for multi-resolution topology optimization. Struct Multidisc Optim 58:2455–2479. https://doi.org/10.1007/s00158-018-2114-0, 1805. 02008
Liu Y, Day ML (2007) Development of simplified finite element models for straight thin-walled tubes with octagonal cross section. Int J Crashworthiness 12(5):503–508. https://doi.org/10.1080/13588260701483557
Marzbanrad J, Ebrahimi MR (2011) Multi-objective optimization of aluminum hollow tubes for vehicle crash energy absorption using a genetic algorithm and neural networks. Thin-Walled Struct 49:1605–1615. https://doi.org/10.1016/j.tws.2011.08.009
Najafi A, Rais-Rohani M (2011) Mechanics of axial plastic collapse in multi-cell, multi-corner crush tubes. Thin-Walled Struct 49:1–12. https://doi.org/10.1016/j.tws.2010.07.002
Neubauer A (1997) The circular schema theorem for genetic algorithms and two-point crossover. In: Second international conference on genetic algorithms in engineering systems, IEE. https://doi.org/10.1049/cp:19971182, vol 1997, pp 209–214
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
Pang T, Zheng G, Fang J, Ruan D, Sun G (2019) Energy absorption mechanism of axially-varying thickness (AVT) multicell thin-walled structures under out-of-plane loading. Engineering Structures 196 (June):109,130. https://doi.org/10.1016/j.engstruct.2019.04.074
Patel NM, Kang BS, Renaud JE, Tovar A (2009) Crashworthiness design using topology optimization. In: Journal of Mechanical Design, Transactions of the ASME. https://doi.org/10.1115/1.3116256
Penninger CL, Watson LT, Tovar A, Renaud JE (2010) Convergence analysis of hybrid cellular automata for topology optimization. Struct Multidisc Optim 40:271–282. https://doi.org/10.1007/s00158-009-0360-x
Qiu N, Gao Y, Fang J, Feng Z, Sun G, Li Q (2015) Crashworthiness analysis and design of multi-cell hexagonal columns under multiple loading cases. Finite Elem Anal Des 104:89–101. https://doi.org/10.1016/j.finel.2015.06.004
Qiu N, Gao Y, Fang J, Sun G, Kim NH (2018) Topological design of multi-cell hexagonal tubes under axial and lateral loading cases using a modified particle swarm algorithm. Appl Math Model 53:567–583. https://doi.org/10.1016/j.apm.2017.08.017
Sun G, Liu T, Fang J, Steven GP, Li Q (2018a) Configurational optimization of multi-cell topologies for multiple oblique loads. Struct Multidiscip Optim 57(2):469–488. https://doi.org/10.1007/s00158-017-1839-5
Sun G, Liu T, Huang X, Zheng G, Li Q (2018b) Topological configuration analysis and design for foam filled multi-cell tubes. Eng Struct 155(October 2017):235–250. https://doi.org/10.1016/j.engstruct.2017.10.063
Tovar A, Patel N, Kaushik AK, Letona GA, Renaud JE, Sanders B (2004) Hybrid cellular automata: a biologically-inspired structural optimization technique. In: Collection of technical papers - 10th AIAA/ISSMO multidisciplinary analysis and optimization conference . https://doi.org/10.2514/6.2004-4558
Tran TN, Baroutaji A (2018) Crashworthiness optimal design of multi-cell triangular tubes under axial and oblique impact loading. Eng Fail Anal 93(May):241–256. https://doi.org/10.1016/j.engfailanal.2018.07.003
Van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidisc Optim 48:437–472. https://doi.org/10.1007/s00158-013-0912-y
Wang H, Xie H (2019) Multi-objective optimization of crashworthiness of vehicle front longitudinal beam. Struct Multidisc Optim 61:2111–2123. https://doi.org/10.1007/s00158-019-02459-5
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng. https://doi.org/10.1016/S0045-7825(02)00559-5
Wierzbicki T, Abramowicz W (1983) On the crushing mechanics of thin-walled structures. J Appl Mech 50(4a):727. https://doi.org/10.1115/1.3167137
Wu S, Zheng G, Sun G, Liu Q, Li G, Li Q (2016) On design of multi-cell thin-wall structures for crashworthiness. International Journal of Impact Engineering 88:102–117. https://doi.org/10.1016/j.ijimpeng.2015.09.003
Zeng D, Duddeck F (2017) Improved hybrid cellular automata for crashworthiness optimization of thin-walled structures. Struct Multidiscip Optim 56(1):101–115. https://doi.org/10.1007/s00158-017-1650-3
Zhang H, Sun G, Xiao Z, Li G, Li Q (2018a) Bending characteristics of top-hat structures through tailor rolled blank (TRB) process. Thin-Walled Struct 123:420–440. https://doi.org/10.1016/j.tws.2017.10.032
Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Struct Multidisc Optim 53:1243–1260. https://doi.org/10.1007/s00158-015-1372-3
Zhang X, Cheng G (2007) A comparative study of energy absorption characteristics of foam-filled and multi-cell square columns. Int J Impact Eng 34(11):1739–1752. https://doi.org/10.1016/J.IJIMPENG.2006.10.007
Zhang X, Cheng G, Zhang H (2006) Theoretical prediction and numerical simulation of multi-cell square thin-walled structures. Thin-Walled Struct 44:1185–1191. https://doi.org/10.1016/j.tws.2006.09.002
Zhang Y, Xu X, Sun G, Lai X, Li Q (2018b) Nondeterministic optimization of tapered sandwich column for crashworthiness. Thin-Walled Struct 122:193–207. https://doi.org/10.1016/j.tws.2017.09.028
Zhu B, Chen Q, Wang R, Zhang X (2018) Structural topology optimization using a moving morphable Component-Based method considering geometrical nonlinearity. Journal of Mechanical Design, Transactions of the ASME. https://doi.org/10.1115/1.4040547
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The authors are grateful for the financial support of the National Natural Science Foundation of China (61572432).
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All authors contributed to the study conception and design. Material preparation, experiment, analysis, and numerical simulation were performed by Hamza Sulayman Abdullahi and Shuming Gao. The first draft of the manuscript was written by Hamza Sulayman Abdullahi, and all authors commented on draft versions of the manuscript. All authors read and approved the final manuscript.
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The raw data for experimental results presented in Section 3 are given in ESM_1.mat. The raw data for results of the first and second optimization stage are given in ESM_2.xlsx and ESM_3.xlsx, respectively.
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Abdullahi, H.S., Gao, S. A two-stage approach to the optimization design of multi-cell square tubal structures. Struct Multidisc Optim 63, 897–913 (2021). https://doi.org/10.1007/s00158-020-02735-9
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DOI: https://doi.org/10.1007/s00158-020-02735-9