Identification of Optimal Topologies for Continuum Structures Using Metaheuristics: A Comparative Study

Abstract

The development of low dimensional explicit based topology optimization approaches such as moving morphable components method increased the hopes to develop and expand evolutionary based solutions in the topology optimization of continuum structures. Despite the low dimensionality of the parametrization which helps to increase the efficiency, due to the multimodal behavior of the objective function and the correlation between the design variables more researches should be done to improve the efficiency. This paper is dedicated to comparing nine non-gradient approach based approaches based on the moving morphable parameterization. The algorithms are compared by the convergence speed, the quality of final designs, and the abilities to explore and exploit based on a diversity index. It is demonstrated that only some of these algorithms can lead to globally optimal solutions. This research clarifies the ability of the aforementioned algorithms to solve the topology optimization problem which can help future researchers to develop more suitable and efficient algorithms for this problem.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

References

  1. 1.

    Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654. https://doi.org/10.1007/s004190050248

    Article  MATH  Google Scholar 

  2. 2.

    Wang X, Wang MY, Guo D (2004) Structural shape and topology optimization in a level-set-based framework of region representation. Struct Multidiscip Optim 27(1–2):1–19. https://doi.org/10.1007/s00158-003-0363-y

    Article  Google Scholar 

  3. 3.

    Apte AP, Wang BP (2008) Topology optimization using hyper radial basis function network. AIAA J 46(9):2211–2218. https://doi.org/10.2514/1.28723

    Article  Google Scholar 

  4. 4.

    Overvelde, J. T. (2012). The moving node approach in topology optimization [Master Thesis].

  5. 5.

    Lin J, Guan Y, Zhao G, Naceur H, Lu P (2017) Topology optimization of plane structures using smoothed particle hydrodynamics method. Int J Numer Meth Eng 110(8):726–744. https://doi.org/10.1002/nme.5427

    MathSciNet  Article  Google Scholar 

  6. 6.

    Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically: a new moving morphable components based framework. J Appl Mech. https://doi.org/10.1115/1.4027609

    Article  Google Scholar 

  7. 7.

    Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306–327. https://doi.org/10.1016/j.cma.2015.05.005

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Smith H, Norato JA (2020) A MATLAB code for topology optimization using the geometry projection method. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-020-02552-0

    MathSciNet  Article  Google Scholar 

  9. 9.

    Zhang W, Chen J, Zhu X, Zhou J, Xue D, Lei X, Guo X (2017) Explicit three dimensional topology optimization via Moving Morphable Void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614. https://doi.org/10.1016/j.cma.2017.05.002

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413. https://doi.org/10.1016/j.cma.2018.01.050

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Gai Y, Zhu X, Zhang YJ, Hou W, Hu P (2020) Explicit isogeometric topology optimization based on moving morphable voids with closed B-spline boundary curves. Struct Multidiscip Optim 61(3):963–982. https://doi.org/10.1007/s00158-019-02398-1

    MathSciNet  Article  Google Scholar 

  12. 12.

    Xue R, Liu C, Zhang W, Zhu Y, Tang S, Du Z, Guo X (2019) Explicit structural topology optimization under finite deformation via Moving Morphable Void (MMV) approach. Comput Methods Appl Mech Eng 344:798–818. https://doi.org/10.1016/j.cma.2018.10.011

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Zhang W, Li D, Kang P, Guo X, Youn SK (2020) Explicit topology optimization using IGA-based moving morphable void (MMV) approach. Comput Methods Appl Mech Eng 360:112685. https://doi.org/10.1016/j.cma.2019.112685

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    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 Multidiscip Optim 53(6):1243–1260. https://doi.org/10.1007/s00158-015-1372-3

    MathSciNet  Article  Google Scholar 

  15. 15.

    Zhang W, Zhang J, Guo X (2016) Lagrangian description based topology optimization: a revival of shape optimization. J Appl Mech. https://doi.org/10.1115/1.4032432

    Article  Google Scholar 

  16. 16.

    Yang H, Huang J (2020) An explicit structural topology optimization method based on the descriptions of areas. Struct Multidiscip Optim 61(3):1123–1156. https://doi.org/10.1007/s00158-019-02414-4

    MathSciNet  Article  Google Scholar 

  17. 17.

    Wang R, Zhang X, Zhu B (2019) Imposing minimum length scale in moving morphable component (MMC)-based topology optimization using an effective connection status (ECS) control method. Comput Methods Appl Mech Eng 351:667–693. https://doi.org/10.1016/j.cma.2019.04.007

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Bai J, Zuo W (2020) Hollow structural design in topology optimization via moving morphable component method. Struct Multidiscip Optim 61(1):187–205. https://doi.org/10.1007/s00158-019-02353-0

    Article  Google Scholar 

  19. 19.

    Niu B, Wadbro E (2019) On equal-width length-scale control in topology optimization. Struct Multidiscip Optim 59(4):1321–1334. https://doi.org/10.1007/s00158-018-2131-z

    Article  Google Scholar 

  20. 20.

    Hoang VN, Nguyen NL, Nguyen-Xuan H (2020) Topology optimization of coated structure using moving morphable sandwich bars. Struct Multidiscip Optim 61(2):491–506

    Article  Google Scholar 

  21. 21.

    Cui T, Sun Z, Liu C et al (2020) Topology optimization of plate structures using plate element-based moving morphable component (MMC) approach. Acta Mech Sin. https://doi.org/10.1007/s10409-020-00944-5

    MathSciNet  Article  Google Scholar 

  22. 22.

    Sun Z, Cui R, Cui T et al (2020) An optimization approach for stiffener layout of composite stiffened panels based on moving morphable components (MMCs). Acta Mech Solida Sin. https://doi.org/10.1007/s10338-020-00161-4

    Article  Google Scholar 

  23. 23.

    Marzbanrad, J., & Rostami, P. (2020, January). Weight optimization of thick plate structures using radial basis functions parameterization. In IOP Conference Series: Materials Science and Engineering (Vol. 671, No. 1, p. 012011). IOP Publishing. Doi: https://doi.org/10.1007/s10409-020-00942-7.

  24. 24.

    Lei X, Liu C, Du Z, Zhang W, Guo X (2019) Machine learning-driven real-time topology optimization under moving morphable component-based framework. J Appl Mech. https://doi.org/10.1115/1.4041319

    Article  Google Scholar 

  25. 25.

    Yu M, Ruan S, Wang X, Li Z, Shen C (2019) Topology optimization of thermal–fluid problem using the MMC-based approach. Struct Multidiscip Optim 60(1):151–165. https://doi.org/10.1007/s00158-019-02206-w

    MathSciNet  Article  Google Scholar 

  26. 26.

    Liu D, Du J (2019) A moving morphable components based shape reconstruction framework for electrical impedance tomography. IEEE Trans Med Imag 38(12):2937–2948. https://doi.org/10.1109/TMI.2019.2918566

    Article  Google Scholar 

  27. 27.

    Sun J, Tian Q, Hu H, Pedersen NL (2018) Topology optimization of a flexible multibody system with variable-length bodies described by ALE–ANCF. Nonlinear Dyn 93(2):413–441. https://doi.org/10.1007/s11071-018-4201-6

    Article  MATH  Google Scholar 

  28. 28.

    Sun J, Tian Q, Hu H, Pedersen NL (2019) Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components. J Sound Vib 448:83–107. https://doi.org/10.1016/j.jsv.2019.01.054

    Article  Google Scholar 

  29. 29.

    Guo X, Zhou J, Zhang W, Du Z, Liu C, Liu Y (2017) Self-supporting structure design in additive manufacturing through explicit topology optimization. Comput Methods Appl Mech Eng 323:27–63. https://doi.org/10.1016/j.cma.2017.05.003

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Takalloozadeh M, Yoon GH (2017) Implementation of topological derivative in the moving morphable components approach. Finite Elem Anal Des 134:16–26. https://doi.org/10.1016/j.finel.2017.05.008

    Article  Google Scholar 

  31. 31.

    Wang, S. Y., & Tai, K. (2003, December). A bit-array representation GA for structural topology optimization. In The 2003 Congress on Evolutionary Computation, 2003. CEC'03. (Vol. 1, pp. 671–677). IEEE. Doi: https://doi.org/10.1109/CEC.2003.1299640.

  32. 32.

    Chapman CD (1996) Genetic algorithm-based structural topology design with compliance and manufacturability considerations. J Mech Design 118:89–98

  33. 33.

    Kita E, Tanie H (1999) Topology and shape optimization of continuum structures using GA and BEM. Struct Optim 17(2–3):130–139. https://doi.org/10.1007/BF01195937

    Article  Google Scholar 

  34. 34.

    Tai K, Chee TH (2000) Design of structures and compliant mechanisms by evolutionary optimization of morphological representations of topology. J Mech Des 122(4):560–566. https://doi.org/10.1115/1.1319158

    Article  Google Scholar 

  35. 35.

    Tai K, Akhtar S (2005) Structural topology optimization using a genetic algorithm with a morphological geometric representation scheme. Struct Multidiscip Optim 30(2):113–127. https://doi.org/10.1007/s00158-004-0504-y

    Article  Google Scholar 

  36. 36.

    Cappello F, Mancuso A (2003) A genetic algorithm for combined topology and shape optimisations. Comput Aided Des 35(8):761–769. https://doi.org/10.1016/S0010-4485(03)00007-1

    Article  Google Scholar 

  37. 37.

    Wang SY, Tai K (2004) Graph representation for structural topology optimization using genetic algorithms. Comput Struct 82(20–21):1609–1622. https://doi.org/10.1016/j.compstruc.2004.05.005

    MathSciNet  Article  Google Scholar 

  38. 38.

    Bureerat S, Kunakote T (2006) Topological design of structures using population-based optimization methods. Inverse Probl Sci Eng 14(6):589–607. https://doi.org/10.1080/17415970600573437

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Bureerat S, Limtragool J (2006) Performance enhancement of evolutionary search for structural topology optimisation. Finite Elem Anal Des 42(6):547–566. https://doi.org/10.1016/j.finel.2005.10.011

    MathSciNet  Article  Google Scholar 

  40. 40.

    Kaveh A, Hassani B, Shojaee S, Tavakkoli SM (2008) Structural topology optimization using ant colony methodology. Eng Struct 30(9):2559–2565. https://doi.org/10.1016/j.engstruct.2008.02.012

    Article  Google Scholar 

  41. 41.

    Luh GC, Lin CY (2009) Structural topology optimization using ant colony optimization algorithm. Appl Soft Comput 9(4):1343–1353. https://doi.org/10.1016/j.asoc.2009.06.001

    Article  Google Scholar 

  42. 42.

    Luh GC, Lin CY, Lin YS (2011) A binary particle swarm optimization for continuum structural topology optimization. Appl Soft Comput 11(2):2833–2844. https://doi.org/10.1016/j.asoc.2010.11.013

    Article  Google Scholar 

  43. 43.

    Bureerat S, Limtragool J (2008) Structural topology optimisation using simulated annealing with multiresolution design variables. Finite Elem Anal Des 44(12–13):738–747. https://doi.org/10.1016/j.finel.2008.04.002

    Article  Google Scholar 

  44. 44.

    Balamurugan R, Ramakrishnan CV, Singh N (2008) Performance evaluation of a two stage adaptive genetic algorithm (TSAGA) in structural topology optimization. Appl Soft Comput 8(4):1607–1624. https://doi.org/10.1016/j.asoc.2007.10.022

    Article  Google Scholar 

  45. 45.

    Garcia-Lopez NP, Sanchez-Silva M, Medaglia AL, Chateauneuf A (2011) A hybrid topology optimization methodology combining simulated annealing and SIMP. Comput Struct 89(15–16):1512–1522. https://doi.org/10.1016/j.compstruc.2011.04.008

    Article  Google Scholar 

  46. 46.

    Cardillo A, Cascini G, Frillici FS, Rotini F (2013) Multi-objective topology optimization through GA-based hybridization of partial solutions. Eng Comput 29(3):287–306. https://doi.org/10.1007/s00366-012-0272-z

    Article  Google Scholar 

  47. 47.

    Garcia-Lopez NP, Sanchez-Silva M, Medaglia AL, Chateauneuf A (2013) An improved robust topology optimization approach using multiobjective evolutionary algorithms. Comput Struct 125:1–10. https://doi.org/10.1016/j.compstruc.2013.04.025

    Article  Google Scholar 

  48. 48.

    Ahmed F, Deb K, Bhattacharya B (2016) Structural topology optimization using multi-objective genetic algorithm with constructive solid geometry representation. Appl Soft Comput 39:240–250. https://doi.org/10.1016/j.asoc.2015.10.063

    Article  Google Scholar 

  49. 49.

    Pandey A, Datta R, Bhattacharya B (2017) Topology optimization of compliant structures and mechanisms using constructive solid geometry for 2-d and 3-d applications. Soft Comput 21(5):1157–1179. https://doi.org/10.1007/s00500-015-1845-8

    Article  MATH  Google Scholar 

  50. 50.

    Valdez SI, Marroquín JL, Botello S, Faurrieta N (2018) A meta-heuristic for topology optimization using probabilistic learning. Appl Intell 48(11):4267–4287. https://doi.org/10.1007/s10489-018-1215-1

    Article  Google Scholar 

  51. 51.

    Li B, Xuan C, Tang W, Zhu Y, Yan K (2019) Topology optimization of plate/shell structures with respect to eigenfrequencies using a biologically inspired algorithm. Eng Optim 51(11):1829–1844. https://doi.org/10.1080/0305215X.2018.1552952

    MathSciNet  Article  Google Scholar 

  52. 52.

    Bielefeldt BR, Reich GW, Beran PS, Hartl DJ (2019) Development and validation of a genetic L-System programming framework for topology optimization of multifunctional structures. Comput Struct 218:152–169. https://doi.org/10.1016/j.compstruc.2019.02.005

    Article  Google Scholar 

  53. 53.

    Salajegheh F, Kamalodini M, Salajegheh E (2020) Momentum method powered by swarm approaches for topology optimization. Appl Soft Comput 90:106174. https://doi.org/10.1016/j.asoc.2020.106174

    Article  Google Scholar 

  54. 54.

    Ram L, Sharma D (2017) Evolutionary and GPU computing for topology optimization of structures. Swarm and evolutionary computation 35:1–13. https://doi.org/10.1016/j.swevo.2016.08.004

    Article  Google Scholar 

  55. 55.

    Jaafer AA, Al-Bazoon M, Dawood AO (2020) Structural topology design optimization using the binary bat algorithm. Appl Sci 10(4):1481. https://doi.org/10.3390/app10041481

    Article  Google Scholar 

  56. 56.

    Aulig, N. (2017). Generic topology optimization based on local state features (Vol. 468). PhD Dissertation, Tu-Darmstadt, VDI Verlag

  57. 57.

    Aulig, N., & Olhofer, M. (2016, July). Evolutionary computation for topology optimization of mechanical structures: An overview of representations. In 2016 IEEE Congress on Evolutionary Computation (CEC) (pp. 1948–1955). IEEE. Doi: https://doi.org/10.1109/CEC.2016.7744026.

  58. 58.

    Guirguis D, Aulig N, Picelli R, Zhu B, Zhou Y, Vicente W, Saitou K (2019) Evolutionary black-box topology optimization: challenges and promises. IEEE Trans Evol Comput. https://doi.org/10.1109/tevc.2019.2954411

    Article  Google Scholar 

  59. 59.

    Biyikli E, To AC (2015) Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PloS one. https://doi.org/10.1371/journal.pone.0145041

    Article  Google Scholar 

  60. 60.

    Wang H, Cheng W, Du R et al (2020) Improved proportional topology optimization algorithm for solving minimum compliance problem. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-020-02504-8

    MathSciNet  Article  Google Scholar 

  61. 61.

    Tovar, A. (2004). Bone remodeling as a hybrid cellular automaton optimization process [Doctoral dissertation].

  62. 62.

    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

    MathSciNet  Article  Google Scholar 

  63. 63.

    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

    MathSciNet  Article  Google Scholar 

  64. 64.

    Bujny, M., Aulig, N., Olhofer, M., & Duddeck, F. (2016, June). Evolutionary level set method for crashworthiness topology optimization. In ECCOMAS Congress. Doi: https://doi.org/10.7712/100016.1814.11054.

  65. 65.

    Bujny M, Aulig N, Olhofer M, Duddeck F (2016) Evolutionary crashworthiness topology optimization of thin-walled structures. ASMO UK, Munich, Germany

    Google Scholar 

  66. 66.

    Bujny M, Aulig N, Olhofer M, Duddeck F (2018) Identification of optimal topologies for crashworthiness with the evolutionary level set method. Int J Crashworthiness 23(4):395–416. https://doi.org/10.1080/13588265.2017.1331493

    Article  Google Scholar 

  67. 67.

    Bujny, M., Aulig, N., Olhofer, M., & Duddeck, F. (2016, July). Hybrid evolutionary approach for level set topology optimization. In 2016 IEEE Congress on Evolutionary Computation (CEC) (pp. 5092–5099). IEEE. DOI: https://doi.org/10.1109/CEC.2016.7748335.

  68. 68.

    Rostami P, Marzbanrad J (2020) Hybrid algorithms for handling the numerical noise in topology optimization. Acta Mech Sin. https://doi.org/10.1007/s10409-020-00942-7

    MathSciNet  Article  Google Scholar 

  69. 69.

    Marzbanrad J, Varnousfaderani PR (2019) A new hybrid differential evolution with gradient search for level set topology optimization. ZANCO J Pure Appl Sci 31(s3):329–334. https://doi.org/10.21271/ZJPAS.31.s3.46

    Article  Google Scholar 

  70. 70.

    Rostami P, Marzbanrad J (2020) Cooperative coevolutionary topology optimization using moving morphable components. Eng Optim. https://doi.org/10.1080/0305215X.2020.1759579

    Article  Google Scholar 

  71. 71.

    Raponi E, Bujny M, Olhofer M, Aulig N, Boria S, Duddeck F (2017) Kriging-guided level set method for crash topology optimization. GACM, Stuttgart, Germany

    Google Scholar 

  72. 72.

    Raponi E, Bujny M, Olhofer M, Aulig N, Boria S, Duddeck F (2019) Kriging-assisted topology optimization of crash structures. Comput Methods Appl Mech Eng 348:730–752. https://doi.org/10.1016/j.cma.2019.02.002

    MathSciNet  Article  MATH  Google Scholar 

  73. 73.

    Raponi, E., Bujny, M., Olhofer, M., Boria, S., & Duddeck, F. (2019). Hybrid Kriging-assisted Level Set Method for Structural Topology Optimization. Doi: https://doi.org/10.5220/0008067800700081

  74. 74.

    Storn R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359. https://doi.org/10.1023/A:1008202821328

    MathSciNet  Article  MATH  Google Scholar 

  75. 75.

    Hansen N (2006) The CMA evolution strategy: a comparing review. In Towards a new evolutionary computation. Springer, Berlin, Heidelberg, pp 75–102

    Google Scholar 

  76. 76.

    Rao RV, Savsani VJ, Vakharia DP (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315. https://doi.org/10.1016/j.cad.2010.12.015

    Article  Google Scholar 

  77. 77.

    Rao RV, Savsani VJ, Vakharia DP (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183(1):1–15. https://doi.org/10.1016/j.ins.2011.08.006

    MathSciNet  Article  Google Scholar 

  78. 78.

    Reynolds, R. G. (1994, February). An introduction to cultural algorithms. In Proceedings of the third annual conference on evolutionary programming (pp. 131–139). River Edge, NJ: World Scientific. Doi: https://doi.org/10.1142/9789814534116.

  79. 79.

    Engelbrecht AP (2007) Computational intelligence: an introduction. John Wiley & Sons, Chichester

    Google Scholar 

  80. 80.

    Karaboga D, Akay B (2009) A comparative study of artificial bee colony algorithm. Appl Math Comput 214(1):108–132. https://doi.org/10.1016/j.amc.2009.03.090

    MathSciNet  Article  MATH  Google Scholar 

  81. 81.

    Socha K, Dorigo M (2008) Ant colony optimization for continuous domains. Eur J Oper Res 185(3):1155–1173. https://doi.org/10.1016/j.ejor.2006.06.046

    MathSciNet  Article  MATH  Google Scholar 

  82. 82.

    Geem, Z. W. (2007, September). Harmony search algorithm for solving sudoku. In International Conference on Knowledge-Based and Intelligent Information and Engineering Systems (pp. 371–378). Springer, Berlin, Heidelberg. Doi: https://doi.org/10.1007/978-3-540-74819-9_46.

  83. 83.

    Yang, X. S. (2010). Nature-inspired metaheuristic algorithms. Book Luniver press

  84. 84.

    Simon D (2008) Biogeography-based optimization. IEEE Trans Evol Comput 12(6):702–713. https://doi.org/10.1109/TEVC.2008.919004

    Article  Google Scholar 

  85. 85.

    Kaveh A, Zolghadr A (2014) Comparison of nine meta-heuristic algorithms for optimal design of truss structures with frequency constraints. Adv Eng Softw 76:9–30. https://doi.org/10.1016/j.advengsoft.2014.05.012

    Article  Google Scholar 

Download references

Funding

This research is not supported by any institute of grant.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Javad Marzbanrad.

Ethics declarations

Conflict of interest

The authors have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rostami, P., Marzbanrad, J. Identification of Optimal Topologies for Continuum Structures Using Metaheuristics: A Comparative Study. Arch Computat Methods Eng (2021). https://doi.org/10.1007/s11831-021-09546-1

Download citation