Structural and Multidisciplinary Optimization

, Volume 58, Issue 4, pp 1395–1410 | Cite as

Design of buckling-induced mechanical metamaterials for energy absorption using topology optimization

  • Qi Chen
  • Xianmin ZhangEmail author
  • Benliang Zhu


A novel design concept for buckling-induced mechanical metamaterials for energy absorption is presented. The force-displacement curves of the mechanical metamaterials are analyzed according to the curves of their unit cells, and the energy-absorbing characteristics of mechanical metamaterials are evaluated. Two topology optimization models are proposed. One maximizes the buckling-induced dissipated energy to facilitate the design of metamaterials with high energy absorption and low elastic strain energy. The other maximizes the dissipated energy with a constraint that the mechanical metamaterials should be self-recoverable. An energy interpolation scheme is employed to avoid numerical instabilities in the geometric nonlinear finite element analysis. A two-phase algorithm is proposed to find the optimized result from a uniform initial guess, and sensitivity analysis is performed. The optimized design has a larger amount of buckling-induced dissipated energy than the previously proposed structural prototypes. Moreover, the self-recoverable mechanical metamaterial is successfully designed by topology optimization.


Mechanical metamaterial Energy absorption Topology optimization Buckling 



This work was supported by the National Natural Science Foundation of China (Grant Nos. U1501247 and U1609206). This support is greatly appreciated. Additionally, the authors thank Dr. K. Svanberg at KTH (Stockholm, Sweden) for providing the MMA code for academic research.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflicts of interest.

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  1. Bochenek B, Tajs-Zielińska K (2015) Minimal compliance topologies for maximal buckling load of columns. Struct Multidiscip Optim 51(5):1149–1157MathSciNetCrossRefGoogle Scholar
  2. Bruns TE, Sigmund O (2004) Toward the topology design of mechanisms that exhibit snap-through behavior. Comput Methods Appl Mech Eng 193(36):3973–4000MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bruns TE, Sigmund O, Tortorelli DA (2002) Numerical methods for the topology optimization of structures that exhibit snap-through. Int J Numer Methods Eng 55(10):1215–1237CrossRefzbMATHGoogle Scholar
  4. Bruyneel M, Duysinx P, Fleury C (2002) A family of mma approximations for structural optimization. Struct Multidiscip Optim 24(4):263–276CrossRefGoogle Scholar
  5. Che K, Yuan C, Wu J, Qi HJ, Meaud J (2017) Three-dimensional-printed multistable mechanical metamaterials with a deterministic deformation sequence. J Appl Mech 84(1):011004CrossRefGoogle Scholar
  6. Correa DM, Klatt T, Cortes S, Haberman M, Kovar D, Seepersad C (2015) Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyp J 21(2):193–200CrossRefGoogle Scholar
  7. Costas M, Díaz J, Romera L, Hernández S (2014) A multi-objective surrogate-based optimization of the crashworthiness of a hybrid impact absorber. Int J Mech Sci 88:46–54CrossRefGoogle Scholar
  8. Deepak SR, Dinesh M, Sahu DK, Ananthasuresh GK (2009) A comparative study of the formulations and benchmark problems for the topology optimization of compliant mechanisms. J Mech Robot 1(1):011003CrossRefGoogle Scholar
  9. Dunning PD, Ovtchinnikov E, Scott J, Alicia Kim H (2016) Level-set topology optimization with many linear buckling constraints using an efficient and robust eigensolver. Int J Numer Methods Eng 107:1029–1053MathSciNetCrossRefzbMATHGoogle Scholar
  10. Evans AG, He MY, Deshpande VS, Hutchinson JW, Jacobsen AJ, Carter WB (2010) Concepts for enhanced energy absorption using hollow micro-lattices. Int J Impact Eng 37(9):947–959CrossRefGoogle Scholar
  11. Fang J, Sun G, Na Q, Kim NH, Li Q (2016) On design optimization for structural crashworthiness and its state of the art. Struct Multidiscip Optim 55:1–29MathSciNetGoogle Scholar
  12. Findeisen C, Hohe J, Kadic M, Gumbsch P (2017) Characteristics of mechanical metamaterials based on buckling elements. J Mech Phys Solids 102:151–164MathSciNetCrossRefGoogle Scholar
  13. Forsberg J, Nilsson L (2007) Topology optimization in crashworthiness design. Struct Multidiscip Optim 33(1):1–12CrossRefGoogle Scholar
  14. Frenzel T, Findeisen C, Kadic M, Gumbsch P, Wegener M (2016) Tailored buckling microlattices as reusable light-weight shock absorbers. Adv Mater 28(28):5865–5870CrossRefGoogle Scholar
  15. Gao X, Ma H (2015) Topology optimization of continuum structures under buckling constraints. Comput Struct 157:142–152CrossRefGoogle Scholar
  16. Gatt R, Mizzi L, Azzopardi JI, Azzopardi KM, Attard D, Casha A, Briffa J, Grima JN (2015) Hierarchical auxetic mechanical metamaterials. Sci Rep 5:8395CrossRefGoogle Scholar
  17. Haghpanah B, Salari-Sharif L, Pourrajab P, Hopkins J, Valdevit L (2016) Multistable shape-reconfigurable architected materials. Adv Mater 28(36):7915–7920CrossRefGoogle Scholar
  18. He G, Huang X, Hu W, Li G (2016) Topology optimization of periodic structures using beso based on unstructured design points. Struct Multidiscip Optim 53(2):271–275MathSciNetCrossRefGoogle Scholar
  19. Hu N, Burgueño R (2015) Buckling-induced smart applications: recent advances and trends. Smart Mater Struct 24(6):063001CrossRefGoogle Scholar
  20. James KA, Waisman H (2016) Layout design of a bi-stable cardiovascular stent using topology optimization. Comput Methods Appl Mech Eng 305:869–890MathSciNetCrossRefGoogle Scholar
  21. Kawamoto A (2009) Stabilization of geometrically nonlinear topology optimization by the levenberg–marquardt method. Struct Multidiscip Optim 37(4):429–433CrossRefGoogle Scholar
  22. Kiani M, Motoyama K, Rais-Rohani M, Shiozaki H (2014) Joint stiffness analysis and optimization as a mechanism for improving the structural design and performance of a vehicle. Proceedings of the Institution of Mechanical Engineers Part D Journal of Automobile Engineering 228(6):689–700CrossRefGoogle Scholar
  23. Kim JJ, Jang IG (2016) Image resolution enhancement for healthy weight-bearing bones based on topology optimization. J Biomech 49(13):3035–3040CrossRefGoogle Scholar
  24. Lahuerta RD, Simões ET, Campello EMB, Pimenta PM, Silva ECN (2013) Towards the stabilization of the low density elements in topology optimization with large deformation. Comput Mech 52(4):779–797MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lee H-A, Park G-J (2012) Topology optimization for structures with nonlinear behavior using the equivalent static loads method. J Mech Des 134(3):031004CrossRefGoogle Scholar
  26. Lee J-H, Wang L, Kooi S, Boyce MC, Thomas EL (2010) Enhanced energy dissipation in periodic epoxy nanoframes. Nano Lett 10(7):2592–2597CrossRefGoogle Scholar
  27. Lee J-H, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials. Adv Mater 24(36):4782–4810CrossRefGoogle Scholar
  28. Li L, Zhang G, Khandelwal K (2017) Topology optimization of energy absorbing structures with maximum damage constraint. Int J Numer Methods Eng 112:737–775MathSciNetCrossRefGoogle Scholar
  29. Lindgaard E, Dahl J (2013) On compliance and buckling objective functions in topology optimization of snap-through problems. Struct Multidiscip Optim 47(3):409–421MathSciNetCrossRefzbMATHGoogle Scholar
  30. Liu M, Zhang X, Fatikow S (2017) Design and analysis of a multi-notched flexure hinge for compliant mechanisms. Precis Eng 48:292–304CrossRefGoogle Scholar
  31. Luo Q, Tong L (2015) Structural topology optimization for maximum linear buckling loads by using a moving iso-surface threshold method. Struct Multidiscip Optim 52(1):71–90MathSciNetCrossRefGoogle Scholar
  32. Luo Q, Tong L (2016) An algorithm for eradicating the effects of void elements on structural topology optimization for nonlinear compliance. Struct Multidiscip Optim 53(4):695–714MathSciNetCrossRefGoogle Scholar
  33. Mayer RR, Kikuchi N, Scott RA (1996) Application of topological optimization techniques to structural crashworthiness. Int J Numer Methods Eng 39(8):1383–1403CrossRefzbMATHGoogle Scholar
  34. Mohammadiha O, Beheshti H (2014) Optimization of functionally graded foam-filled conical tubes under axial impact loading. J Mech Sci Technol 28(5):1741–1752CrossRefGoogle Scholar
  35. Nicolaou ZG, Motter AE (2012) Mechanical metamaterials with negative compressibility transitions. Nat Mater 11(7):608–613CrossRefGoogle Scholar
  36. Patel NM, Kang B-S, Renaud JE, Tovar A (2009) Crashworthiness design using topology optimization. J Mech Des 131(6):061013CrossRefGoogle Scholar
  37. Puglisi G, Truskinovsky L (2000) Mechanics of a discrete chain with bi-stable elements. J Mech Phys Solids 48(1):1–27MathSciNetCrossRefzbMATHGoogle Scholar
  38. Rafsanjani A, Akbarzadeh A, Pasini D (2015) Snapping mechanical metamaterials under tension. Adv Mater 27(39):5931–5935CrossRefGoogle Scholar
  39. Rojas-Labanda S, Stolpe M (2015) Automatic penalty continuation in structural topology optimization. Struct Multidiscip Optim 52(6):1205–1221MathSciNetCrossRefGoogle Scholar
  40. Shan S, Kang SH, Raney JR, Wang P, Fang L, Candido F, Lewis JA, Bertoldi K (2015) Multistable architected materials for trapping elastic strain energy. Adv Mater 27(29):4296–4301CrossRefGoogle Scholar
  41. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055MathSciNetCrossRefGoogle Scholar
  42. Tran AV, Zhang X, Zhu B (2017) The development of a new piezoresistive pressure sensor for low pressure. IEEE Trans Ind Electron PP(99):1–1Google Scholar
  43. van Dijk NP, Langelaar M, van Keulen F (2014) Element deformation scaling for robust geometrically nonlinear analyses in topology optimization. Struct Multidiscip Optim 50(4):537–560MathSciNetCrossRefGoogle Scholar
  44. Wang R, Zhang X (2018) Parameters optimization and experiment of a planar parallel 3-dof nanopositioning system. IEEE Trans Ind Electron 65(3):2388–2397CrossRefGoogle Scholar
  45. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784CrossRefzbMATHGoogle Scholar
  46. Wang F, Sigmund O, Jensen JS (2014a) Design of materials with prescribed nonlinear properties. J Mech Phys Solids 69:156– 174MathSciNetCrossRefGoogle Scholar
  47. Wang F, Lazarov BS, Sigmund O, Jensen JS (2014b) Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Comput Methods Appl Mech Eng 276:453–472MathSciNetCrossRefzbMATHGoogle Scholar
  48. Yoon GH, Kim YY (2005) Element connectivity parameterization for topology optimization of geometrically nonlinear structures. Int J Solids Struct 42(7):1983–2009MathSciNetCrossRefzbMATHGoogle Scholar
  49. Zheng X, Lee H, Weisgraber TH, Shusteff M, DeOtte J, Duoss EB, Kuntz JD, Biener MM, Ge Q, Jackson JA et al (2014) Ultralight, ultrastiff mechanical metamaterials. Science 344(6190):1373–1377CrossRefGoogle Scholar
  50. Zheng X, Smith W, Jackson J, Moran B, Cui H, Chen D, Ye J, Fang N, Rodriguez N, Weisgraber T et al (2016) Multiscale metallic metamaterials. Nat Mater 15:1100–1106CrossRefGoogle Scholar
  51. Zhu B, Zhang X, Fatikow S (2015) Structural topology and shape optimization using a level set method with distance-suppression scheme. Comput Methods Appl Mech Eng 283:1214–1239MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Guangdong Key Laboratory of Precision Equipment and Manufacturing TechnologySouth China University of TechnologyGuangzhouChina

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