Topology optimization of energy absorbers under crashworthiness using modified hybrid cellular automata (MHCA) algorithm

  • Masoud Afrousheh
  • Javad MarzbanradEmail author
  • Dietmar Göhlich
Research Paper


Meta-heuristic and hyperheuristic algorithms are milestones that make the topology optimization practical for dynamic and nonlinear problems. However, researchers are continuing to improve these methods to get better results. Plastic behavior, complex deformation, and load-dependent material properties of the vehicle components during a crash event are challenging issues that are faced in this context. This research focuses on enhancing search efficiency of the hybrid cellular automata (HCA) algorithm with the aim of improving the energy absorption of vehicle structures exposed to high-impact collisions. An attempt is made to utilize an ideal amount of material in the structures to obtain a more uniform distribution of the plastic strain. Thus, the design is based on this criterion throughout the whole structure during the entire collision time. The variable neighborhood radius concept realizes an intelligent search strategy for the modified HCA (MHCA) algorithm. This innovation, applied here to topology optimization design, makes the MHCA algorithm more functional to controlling plastic strain energy. To confirm this, the crash analysis is performed using the finite-element software package LS-DYNA. An additional benefit of using this method is the quick and stable convergence while the energy absorption relative to the mass fraction is remarkably improved.


Crashworthiness Energy absorption Effective plastic strain Hybrid cellular automata Topology optimization Variable neighborhood radius 



  1. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43:1–16. CrossRefzbMATHGoogle Scholar
  2. Bandi P, Schmiedeler JP, Tovar A (2013) Design of crashworthy structures with controlled energy absorption in the hybrid cellular automaton framework. J Mech Des 135:091002. CrossRefGoogle Scholar
  3. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654. CrossRefzbMATHGoogle Scholar
  5. Bendsøe MP, Sigmund O (2004) Topology optimization; theory, methods and applications. Springer-Verlag, Berlin Heidelberg. zbMATHGoogle Scholar
  6. Bujny M, Aulig N, Olhofer M, Duddeck F (2016) Evolutionary level set method for crashworthiness topology optimization. In: ECCOMAS Congress 2016, Crete Island, GreeceGoogle Scholar
  7. Bujny M, Aulig N, Olhofer M, Duddeck F (2017) Identification of optimal topologies for crashworthiness with the evolutionary level set method. International Journal of Crashworthiness 1–22.
  8. Duddeck F, Volz K (2012) A new topology optimization approach for crashworthiness of passenger vehicles based on physically defined equivalent static loads. In: Proceedings ICRASH conference, Milano, 2012Google Scholar
  9. 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 Multidiscip Optim 54:415–428. CrossRefGoogle Scholar
  10. Duddeck F, Bujny M, Zeng D (2017) Topology optimization methods based on nonlinear and dynamic crash simulations. 11th Europ. LS-DYNA Conf., Salzburg, AustriaGoogle Scholar
  11. Fang J, Sun G, Qiu N, Kim NH, Li Q (2017) On design optimization for structural crashworthiness and its state of the art. Struct Multidiscip Optim 55:1091–1119. MathSciNetCrossRefGoogle Scholar
  12. Forsberg J, Nilsson L (2007) Topology optimization in crashworthiness design. Struct Multidiscip Optim 33:1–12. CrossRefGoogle Scholar
  13. Goel T, Roux W, Stander N (2009) A topology optimization tool for LS-Dyna users: LS-OPT/Topology. In: Proc., 7th European LS-DYNA Conf, Salzburg, AustriaGoogle Scholar
  14. Guo L, Tovar A, Penninger CL, Renaud JE (2011) Strain-based topology optimisation for crashworthiness using hybrid cellular automata. Int J Crashworthiness 16:239–252. CrossRefGoogle Scholar
  15. Guo L, Huang J, Zhou X, Tovar A (2012) The convergence and algorithm factors analysis of topology optimization for crashworthiness based on hybrid cellular automata. In: ASME 2012 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, Houston, Texas, USA. pp 131–139 doi:
  16. Gurdal Z, Tatting B (2000) Cellular automata for design of truss structures with linear and nonlinear response. In: 41st Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA, USA. p 1580. doi:
  17. Hallquist JO (2006) LS-DYNA. Theory manual, Livermore Software Technology Corporation, California, 3, 25–31Google Scholar
  18. Hassani B, Hinton E (1998) A review of homogenization and topology optimization I—homogenization theory for media with periodic structure. Comput Struct 69:707–717. CrossRefzbMATHGoogle Scholar
  19. Hassani B, Hinton E (2012) Homogenization and structural topology optimization: theory, practice and software. Springer Science & Business Media-Verlag, LondonGoogle Scholar
  20. Huang X, Xie M (2010) Evolutionary topology optimization of continuum structures: methods and applications. NY John Wiley & Sons, Chichester, New York. CrossRefzbMATHGoogle Scholar
  21. Hunkeler S (2014) Topology optimisation in crashworthiness design via hybrid cellular automata for thin walled structures. PhD thesis, Queen Mary University of London, UKGoogle Scholar
  22. Hunkeler S, Duddeck F, Rayamajhi M (2013) Topology optimisation method for crashworthiness design using hybrid cellular automata and thin-walled ground structures. In: 9th Europ LS-DYNA conf. Manchester, UKGoogle Scholar
  23. Kang B-S, Park G-J, Arora JS (2006) A review of optimization of structures subjected to transient loads. Struct Multidiscip Optim 31:81–95. MathSciNetCrossRefzbMATHGoogle Scholar
  24. Khandelwal K, Tovar A (2010) Hybrid cellular automaton: a novel framework for non-linear topology optimization. In: Structures Congress 2010: 19th Analysis and Computation Specialty Conference, Orlando, Florida, USA. pp 421–432. doi:
  25. Maleque MA, Salit MS (2013) Materials selection and design. SpringerBriefs in materials, Singapore. doi:
  26. Mozumder CK (2010) Topometry optimization of sheet metal structures for crashworthiness design using hybrid cellular automata. PhD thesis, University of Notre Dame, Indiana, USAGoogle Scholar
  27. Mozumder C, Bandi P, Patel N, Renaud J (2008) Thickness based topology optimization for crashworthiness design using hybrid cellular automata. In: 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, British Columbia. p 6046 doi:
  28. Mozumder C, Tovar A, Renaud JE (2009) Topology design of plastically deformable structures with a controlled energy absorption for prescribed force and displacement response. In: 8th World Congress on Structural and Multidisciplinary Optimization, Lisbon, PortugalGoogle Scholar
  29. Öchsner A (2014) Elasto-plasticity of frame structure elements. Heidelberg: Springer, Berlin doi:
  30. Patel N, Renaud J, Tovar A (2005) Compliant mechanism design using the hybrid cellular automaton method. In: 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Austin, Texas, USA. p 2276. doi:
  31. Patel NM, Kang B-S, Renaud JE, Tovar A (2009) Crashworthiness design using topology optimization. J Mech Des 131:061013. CrossRefGoogle Scholar
  32. Penninger CL, Tovar A, Watson LT, Renaud JE (2009) KKT conditions satisfied using adaptive neighboring in hybrid cellular automata for topology optimization. Computer Science Technical Reports 1030. Virginia Polytechnic Institute & State University, Blacksburg, Virginia, USAzbMATHGoogle Scholar
  33. Penninger CL, Watson LT, Tovar A, Renaud JE (2010) Convergence analysis of hybrid cellular automata for topology optimization. Struct Multidiscip Optim 40:271–282. MathSciNetCrossRefzbMATHGoogle Scholar
  34. Penninger CL, Patel NM, Tovar A (2012) A novel HCA framework for simulating the cellular mechanisms of bone remodeling. In: ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, Chicago, Illinois, USA. pp 1261–1270. doi:
  35. Raeisi S, Tapkir P, Tovar A, Mozumder C, Xu S (2019) Multimaterial topology optimization for crashworthiness using hybrid cellular automaton. SAE Technical Paper, No. 2019-01-0826Google Scholar
  36. Roux W (2016) The LS-TaSC™ tool topology and shape computations, user’s manual. Livermore Software Technology Corporation, CA, USA. Accessed: November 2016
  37. Rozvany GI, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4:250–252. CrossRefGoogle Scholar
  38. Schmit LA (1960) Structural design by systematic synthesis. In: Proceedings of the Second National Conference on Electronic Computation, ASCE Structural Division, Pittsburgh, Pennsylvania, USA. pp. 105–132Google Scholar
  39. Schwanitz P, Werner S, Zerbe J, Göhlich D (2014) Robust optimization of vehicle crashboxes. SAE Technical Paper, No. 2014-01-0397 doi:
  40. 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. MathSciNetCrossRefGoogle Scholar
  41. Sun G, Liu T, Huang X, Zhen G, Li Q (2018b) Topological configuration analysis and design for foam filled multi-cell tubes. Eng Struct 155:235–250. CrossRefGoogle Scholar
  42. Sun G, Tian J, Liu T, Yan X, Huang X (2018c) Crashworthiness optimization of automotive parts with tailor rolled blank. Eng Struct 169:201–215. CrossRefGoogle Scholar
  43. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373. MathSciNetCrossRefzbMATHGoogle Scholar
  44. Svanberg K (1993) The method of moving asymptotes (MMA) with some extensions. In: Optimization of large structural systems. Springer, Dordrecht, pp 555–566. CrossRefGoogle Scholar
  45. Taylor CE (2003) Dr. Daniel C. Drucker, 1918–2001, Graduate Research Professor Emeritus, University of Florida. J Appl Mech 70:158–159. CrossRefzbMATHGoogle Scholar
  46. Tenek LH, Hagiwara I (1993) Static and vibrational shape and topology optimization using homogenization and mathematical programming. Comput Methods Appl Mech Eng 109:143–154. CrossRefzbMATHGoogle Scholar
  47. Tovar A (2004) Bone remodeling as a hybrid cellular automaton optimization process. PhD thesis, University of Notre Dame, Indiana, USAGoogle Scholar
  48. Tovar A, Patel NM, Niebur GL, Sen M, Renaud JE (2006) Topology optimization using a hybrid cellular automaton method with local control rules. J Mech Des 128:1205–1216. CrossRefGoogle Scholar
  49. Tovar A, Patel NM, Kaushik AK, Renaud JE (2007) Optimality conditions of the hybrid cellular automata for structural optimization. AIAA J 45:673–683. CrossRefGoogle Scholar
  50. Ulam S (1950) Random processes and transformations. In: Proceedings of the International Congress on Mathematics, Cambridge, Massachusetts, USA. vol. 2, pp. 264–275Google Scholar
  51. Von Neumann J, Burks AW (1966) Theory of self-reproducing automata. IEEE Trans Neural Netw 5:3–14Google Scholar
  52. Wehrle E, Han Y, Duddeck F (2015) Topology optimization of transient nonlinear structures—a comparative assessment of methods. In: 10th European LS-DYNA Conference, Würzburg, GermanyGoogle Scholar
  53. Weider K, Schumacher A (2017) A topology optimization scheme for crash loaded structures using topological derivatives. In: World congress of structural and multidisciplinary optimisation. Springer, Cham, pp 1601–1614. Google Scholar
  54. Yamasaki S, Nishiwaki S, Yamada T, Izui K, Yoshimura M (2010) A structural optimization method based on the level set method using a new geometry-based re-initialization scheme. Int J Numer Methods Eng 83:1580–1624MathSciNetCrossRefzbMATHGoogle Scholar
  55. Zeng D, Duddeck F (2017) Improved hybrid cellular automata for crashworthiness optimization of thin-walled structures. Struct Multidiscip Optim 56:101–115. MathSciNetCrossRefGoogle Scholar
  56. Zuo K-T, Chen L-P, Zhang Y-Q, Yang J (2007) Study of key algorithms in topology optimization. Int J Adv Manuf Technol 32:787–796. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Automotive EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Technical University of Berlin, FG MPMBerlinGermany

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