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Multi-objective robust optimization of foam-filled tapered multi-cell thin-walled structures

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Abstract

Foam-filled multi-cell thin-walled structure has recently gained attentions for its excellent energy absorption capacity. Tapered thin-walled structure is less likely to fail by global buckling, and is more capable of bearing oblique impact loads. Thus, foam-filled tapered multi-cell thin-walled structure (FTMTS) may be an extremely excellent energy absorber candidate in future vehicle body. This paper focuses on the crashworthiness of four kinds of axisymmetric FTMTSs with different cell numbers. According to our study, we find that FTMTSs have very excellent energy absorption capacity as well as strong capacity of avoiding global buckling. According to our investigation, it was found that the crashworthiness of FTMTS was largely affected by design parameters such as geometric sizes and foam density. In order to find optimal designs of FTMTSs, it is very essential to carry out crashworthiness optimization for FTMTSs. However, the conventional deterministic design is likely to become less meaningful or even unacceptable when considering the uncertainties of design parameters due to the manufacturing or installation deviation. In order to overcome this drawback, a multi-objective robust optimization procedure which employs Kriging metamodels, multi-objective particle swarm optimization (MOPSO) algorithm, “k-sigma” robust design theory and Monte Carlo simulation (MCS) was developed. The comparison of the Pareto fronts obtained by the developed multi-objective robust optimization procedure and the traditional multi-objective deterministic optimization algorithm shows that the robust optimization result is more reliable than the deterministic optimization result. The robust optimal design of FTMTS not only has very excellent crashworthiness but also has very high reliability when considering the uncertainty of design parameters.

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References

  • Abramowicz W, Jones N (1986) Dynamic progressive buckling of circular and square tubes. Int J Impact Eng 4(4):243–270

    Article  Google Scholar 

  • Abramowicz W, Wierzbicki T (1988) Axial crushing of foam-filled columns. Int J Mech Sci 30(3/4):263–271

    Article  Google Scholar 

  • Ahmad Z, Thambiratnam DP (2009) Crushing response of foam-filled conical tubes under quasi-static axial loading. Mater Des 30(7):2393–2403

    Article  Google Scholar 

  • Aktay L, Toksoy AK, Güden M (2006) Quasi-static axial crushing of extruded polystyrene foam-filled thin-walled aluminum tubes: experimental and numerical analysis. Mater Des 27(7):556–565

    Article  Google Scholar 

  • Aktay L, Kröplin BH, Toksoy AK, Güden M (2008) Finite element and coupled finite element/smooth particle hydrodynamics modeling of the quasi-static crushing of empty and foam-filled single, bitubular and constraint hexagonal- and square-packed aluminum tubes. Mater Des 29(5):952–962

    Article  Google Scholar 

  • Alghamdi AAA (2001) Collapsible impact energy absorbers: an overview. Thin Wall Struct 39(2):189–213

    Article  MathSciNet  Google Scholar 

  • Bi J, Fang HB, Wang QA, Ren XC (2010) Modeling and optimization of foam-filled thin-walled columns for crashworthiness designs. Finite Elem Anal Des 46(9):698–709

    Article  Google Scholar 

  • Chen WG, Wierzbicki T (2001) Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption. Thin Wall Struct 39(4):287–306

    Article  Google Scholar 

  • Coello Coello C, Salazar Lechuga M (2002) MOPSO: A proposal for multiple objective particle swarm optimization. Congress on Evolutionary Computation (CEC), 1051–1056

  • Deshpande VS, Fleck NA (2000) Isotropic constitutive models for metallic foams. J Mech Phys Solids 48(6–7):1253–1283

    Article  MATH  Google Scholar 

  • Gu XG, Sun GY, Li GY, Mao LC, Li Q (2013) A comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure. Struct Multidisc Optim 48(3):669–684

    Article  Google Scholar 

  • Gümrük R, Karadeniz S (2008) A numerical study of the influence of bump type triggers on the axial crushing of top hat thin-walled sections. Thin Wall Struct 46(10):1094–1106

    Article  Google Scholar 

  • Hallquist JO (1998) LS-DYNA theoretical manual. Livemore Software Technology Corporation, California

    Google Scholar 

  • Hallquist JO (2003) LS-DYNA keyword user’s manual. Livemore Software Technology Corporation, California

    Google Scholar 

  • Hanssen AG, Langseth M, Hopperstad OS (2000a) Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. Int J Impact Eng 24(5):475–507

    Article  Google Scholar 

  • Hanssen AG, Langseth M, Hopperstad OS (2000b) Static and dynamic crushing of square aluminium extrusions with aluminium foam filler. Int J Impact Eng 24(4):347–383

    Article  Google Scholar 

  • Hanssen AG, Hopperstad OS, Langseth M, IIstad H (2002) Validation of constitutive models applicable to aluminium foams. Int J Mech Sci 44(2):359–406

    Article  Google Scholar 

  • Hou SJ, Li Q, Long SY, Yang XJ, Li W (2008) Multiobjective optimization of multi-cell sections for the crashworthiness design. Int J Impact Eng 35(11):1355–1367

    Article  Google Scholar 

  • Hou SJ, Li Q, Long SY, Yang XJ, Li W (2009) crashworthiness design for foam filled thin-walled structures. Mater Des 30(6):2024–2032

    Article  Google Scholar 

  • Hou SJ, Han X, Sun GY, Long SY, Li W, Yang XJ, Li Q (2011) Multiobjective optimization for tapered circular tubes. Thin Wall Struct 49(7):855–863

    Article  Google Scholar 

  • Kavi H, Toksoy AK, Guden M (2006) Predicting energy absorption in a foam-filled thin-walled aluminum tube based on experimentally determined strengthening coefficient. Mater Des 27(4):263–269

    Article  Google Scholar 

  • Kim HS (2002) New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency. Thin Wall Struct 40(4):311–327

    Article  Google Scholar 

  • Kleijnen JPC (2005) An overview of the design and analysis of simulation experiments for sensitivity analysis. Eur J Oper Res 164(2):287–300

    Article  MATH  Google Scholar 

  • Langseth M, Hopperstand OS (1996) Static and dynamic axial crushing of square thin-walled aluminum extrusions. Int J Impact Eng 18(7/8):949–968

    Article  Google Scholar 

  • Langseth M, Hopperstad OS, Hanssen AG (1998) Crash behaviour of thin-walled aluminium members. Thin Wall Struct 32(1–3):127–150

    Article  Google Scholar 

  • Liao XT, Li Q, Yang XJ, Zhang WG, Li W (2008) Multiobjective optimization for crash safety design of vehicles using stepwise regression model. Struct Multidisc Optim 35(6):561–569

    Article  Google Scholar 

  • Liu YC, Day ML (2008) Bending collapse of thin-walled circular tubes and computational application. Thin Wall Struct 46(4):442–450

    Article  Google Scholar 

  • Liu DS, Tan KC, Goh CK, Ho WK (2007) A multiobjective memetic algorithm based on particle swarm optimization. IEEE T Syst Man Cy B 37(1):42–50

    Article  Google Scholar 

  • Lu GX, Yu TX (2003) Energy absorption of structures and materials. CRC Press, Boca Raton

    Book  Google Scholar 

  • Mahmoudabadi MZ, Sadighi M (2011) A study on the static and dynamic loading of the foam filled metal hexagonal honeycomb—Theoretical and experimental. Mater Sci Eng A 530:333–343

    Article  Google Scholar 

  • Meguid SA, Attia MS, Monfort A (2004) On the crush behaviour of ultralight foam-filled structures. Mater Des 25(3):183–189

    Article  Google Scholar 

  • Mirfendereski L, Salimi M, Ziaei-Rad S (2008) Parametric study and numerical analysis of empty and foam-filled thin-walled tubes under static and dynamic loadings. Int J Mech Sci 50(6):1042–1057

    Article  Google Scholar 

  • Myers RH, Montgomery DC, Anderson-Cook CM (2009) Response surface methodology: Process and product optimization using designed experiments, 3rd edn. Wiley, Hoboken

    Google Scholar 

  • Paik JK, Kim BJ, Park DK, Jang BS (2011) On quasi-static crushing of thin-walled steel structures in cold temperature: experimental and numerical studies. Int J Impact Eng 38(1):13–28

    Article  Google Scholar 

  • Parsopoulos K, Vrahatis M (2002) Particle swarm optimization method in multiobjective problems. Proceedings of the ACM Symposium on Applied Computing (SAC), 603–607

  • Raquel C, Naval P (2005) An effective use of crowding distance in multiobjective particle swarm optimization. In: Proceedings of the 2005 conference on Genetic and evolutionary computation, Washington (DC), USA, 257–264

  • Reddy TY, Wall RJ (1988) Axial compression of foam-filled thin-walled circular tubes. Int J Impact Eng 7(2):151–166

    Article  Google Scholar 

  • Reid SR, Reddy TY (1986) Axial crushing of foam-filled tapered sheet metal tubes. Int J Mech Sci 28(10):643–656

    Article  Google Scholar 

  • Reid SR, Reddy TY, Gray MD (1986) Static and dynamic axial crushing of foam-filled sheet metal tubes. Int J Mech Sci 28(5):295–322

    Article  Google Scholar 

  • Reyes A, Langseth M, Hopperstad OS (2003a) Square aluminum tubes subjected to oblique loading. Int J Impact Eng 28(10):1077–1106

    Article  Google Scholar 

  • Reyes A, Hopperstad OS, Berstad T, Hanssen AG, Langseth M (2003b) Constitutive modeling of aluminum foam including fracture and statistical variation of density. Eur J Mech A-Solid 22(6):815–835

    Article  MATH  Google Scholar 

  • Rossi A, Fawaz Z, Behdinan K (2005) Numerical simulation of the axial collapse of thin-walled polygonal section tubes. Thin Wall Struct 43(10):1646–1661

    Article  Google Scholar 

  • Rubinstein RY (1981) Simulation and the Monte Carlo method. Wiley, New York

    Book  MATH  Google Scholar 

  • Rust W, Schweizerhof K (2003) Finite element limit load analysis of thin-walled structures by ANSYS (implicit), LS-DYNA (explicit) and in combination. Thin Wall Struct 41(2–3):227–244

    Article  Google Scholar 

  • Santosa SP, Wierzbicki T, Hanssen AG, Langseth M (2000) Experimental and numerical studies of foam-filled sections. Int J Impact Eng 24(5):509–534

    Article  Google Scholar 

  • Seitzberger M, Rammerstorfer FG, Gradinger R, Degischer HP, Blaimschein M, Walch C (2000) Experimental studies on the quasi-static axial crushing of steel columns filled with aluminium foam. Int J Solids Struct 37(30):4125–4147

    Article  Google Scholar 

  • Shahbeyk S, Petrinic N, Vafai A (2007) Numerical modelling of dynamically loaded metal foam-filled square columns. Int J Impact Eng 34(3):573–586

    Article  Google Scholar 

  • Song XG, Sun GY, Li GY, Gao WZ, Li Q (2013) Crashworthiness optimization design of foam-filled tapered thin-walled structures using multiple surrogate models. Struct Multidisc Optim 47(2):221–231

    Article  MathSciNet  MATH  Google Scholar 

  • Sun GY, Li GY, Hou SJ, Zhou SW, Li W, Li Q (2010) Crashworthiness design for functionally graded foam-filled thin-walled structures. Mater Sci Eng A Struct 527(7–8):1911–1919

    Article  Google Scholar 

  • Sun GY, Li GY, Zhou SW, Li HZ, Hou SJ, Li Q (2011) Crashworthiness design of vehicle by using multiobjective robust optimization. Struct Multidisc Optim 44(1):99–110

    Article  Google Scholar 

  • Sun GY, Song XG, Baek S, Li Q (2014) Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel. Struct Multidisc Optim 49(6):897–913

    Article  Google Scholar 

  • Tang ZL, Liu ST, Zhang ZH (2013) Analysis of energy absorption characteristics of cylindrical multi-cell columns. Thin Wall Struct 62:75–84

    Article  Google Scholar 

  • Tarigopula V, Langseth M, Hopperstad OS, Clausen AH (2006) Axial crushing of thin-walled high-strength steel sections. Int J Impact Eng 32(5):847–882

    Article  Google Scholar 

  • Vanderplaats GN, Moses F (1973) Structural optimization by methods of feasible directions. Comput Struct 3(4):739–755

    Article  Google Scholar 

  • Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. Int J Mech Des 129(4):370–380

    MathSciNet  Google Scholar 

  • Wang QC, Fan ZJ, Gui LJ (2006) A theoretical analysis for the dynamic axial crushing behaviour of aluminium foam-filled hat sections. Int J Solids Struct 43(7–8):2064–2075

    Article  MATH  Google Scholar 

  • Wang QC, Fan ZJ, Gui LJ (2007) Theoretical analysis for axial crushing behaviour of aluminium foam-filled hat sections. Int J Mech Sci 49(4):515–521

    Article  Google Scholar 

  • White MD, Jones N (1999) Experimental quasi-static axial crushing of top-hat and double-hat thin-walled sections. Int J Mech Sci 41:179–208

    Article  MATH  Google Scholar 

  • Wierzbicki T, Abramowicz W (1983) On the crushing mechanics of thin-walled structures. J Appl Mech 50(4):727–734

    Article  MATH  Google Scholar 

  • Williams CKI (1998) Prediction with Gaussian processes: from linear regression to linear prediction and beyond. Learning Graph Models 89:599–621

    Article  Google Scholar 

  • Xiang YJ, Wang Q, Fan ZJ, Fang HB (2006) Optimal crashworthiness design of a spot-welded thin-walled hat section. Finite Elem Anal Des 42(10):846–855

    Article  Google Scholar 

  • Yin HF, Wen GL, Gan NF (2011a) Crashworthiness design for honeycomb structures under axial dynamic loading. Int J Comput Methods 8(4):863–877

    Article  Google Scholar 

  • Yin HF, Wen GL, Hou SJ, Chen K (2011b) Crushing analysis and multiobjective crashworthiness optimization of honeycomb-filled single and bitubular polygonal tubes. Mater Des 32(8):4449–4460

    Article  Google Scholar 

  • Yin HF, Wen GL, Hou SJ, Qing QX (2013) Multiobjective crashworthiness optimization of functionally lateral graded foam-filled tubes. Mater Des 44:414–428

    Article  Google Scholar 

  • Yin HF, Wen GL, Wu X, Qing QX, Hou SJ (2014a) Crashworthiness design of functionally graded foam-filled multi-cell thin-walled structures. Thin Wall Struct 85:142–155

    Article  Google Scholar 

  • Yin HF, Wen GL, Liu ZB, Qing QX (2014b) Crashworthiness optimization design for foam-filled multi-cell thin-walled structures. Thin Wall Struct 75:8–17

    Article  Google Scholar 

  • Zarei HR, Kroger M (2007) Crashworthiness optimization of empty and filled aluminum crash boxes. Int J Crashworth 12(3):255–264

    Article  Google Scholar 

  • Zarei HR, Kroger M (2008) Optimization of the foam-filled aluminum tubes for crush box application. Thin Wall Struct 46(2):214–221

    Article  Google Scholar 

  • Zhang X, Huh H (2009) Energy absorption of longitudinally grooved square tubes under axial compression. Thin Wall Struct 47(12):1469–1477

    Article  Google Scholar 

  • Zhang X, Zhang H (2013a) Energy absorption limit of plates in thin-walled structures under compression. Int J Impact Eng 57:81–98

    Article  Google Scholar 

  • Zhang X, Zhang H (2013b) Energy absorption of multi-cell stub columns under axial compression. Thin Wall Struct 68:156–163

    Article  Google Scholar 

  • Zhang X, Zhang H (2014) Axial crushing of circular multi-cell columns. Int J Impact Eng 65:110–125

    Article  Google Scholar 

  • Zhang X, Cheng GD, Zhang H (2006) Theoretical prediction and numerical simulation of multi-cell square thin-walled structures. Thin Wall Struct 44(11):1185–1191

    Article  Google Scholar 

  • Zhang X, Cheng GD, You Z, Zhang H (2007a) Energy absorption of axially compressed thin-walled square tubes with patterns. Thin Wall Struct 45(9):737–746

    Article  Google Scholar 

  • Zhang Y, Zhu P, Chen GL (2007b) Lightweight design of automotive front side rail based on robust optimisation. Thin Wall Struct 45(7–8):670–676

    Article  Google Scholar 

  • Zhang X, Cheng GD, Zhang H (2009) Numerical investigations on a new type of energy-absorbing structure based on free inversion of tubes. Int J Mech Sci 51(1):64–76

    Article  MATH  Google Scholar 

  • Zhang ZH, Liu ST, Tang ZL (2011) Comparisons of honeycomb sandwich and foam-filled cylindrical columns. Thin Wall Struct 49(9):1071–1079

    Article  Google Scholar 

  • Zhang X, Wen ZZ, Zhang H (2014a) Axial crushing and optimal design of square tubes with graded thickness. Thin Wall Struct 84:263–274

    Article  Google Scholar 

  • Zhang Y, Sun GY, Xu XP, Li GY, Li Q (2014b) Multiobjective crashworthiness optimization of hollow and conical tubes for multiple load cases. Thin Wall Struct 82:331–342

    Article  Google Scholar 

  • Zhu P, Zhang Y, Chen GL (2009) Metamodel-based light weight design of an automotive front-body structure using robust optimization. Proc Inst Mech Eng Part D-J Automob Eng 223(D9):1133–1147

    Article  Google Scholar 

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Yin, H., Fang, H., Xiao, Y. et al. Multi-objective robust optimization of foam-filled tapered multi-cell thin-walled structures. Struct Multidisc Optim 52, 1051–1067 (2015). https://doi.org/10.1007/s00158-015-1299-8

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  • DOI: https://doi.org/10.1007/s00158-015-1299-8

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