An efficient 3D homogenization-based topology optimization methodology

Abstract

Homogenization theory forms the basis for solving the topology optimization problem (TOP) formulated for designing composite materials. Homogenization is proved to be an efficient approach to effectively determine the equivalent macroscopic properties of the composite material. It relies on the assumption that the composite material presents a periodic pattern on a microstructural level; the simplest repeating unit of the microstructure, that if isolated represents exactly the macroscopic behaviour of the material, is called the \(unit\ cell\). Scope of homogenization is to determine the macroscopic (or else effective) properties of the non-homogeneous unit cell, that is, determine the properties of the unit cell as if it was composed by homogeneous material. In this study, a simple methodology is proposed where homogenization is implemented on a 3D lattice unit cell, with its radius being considered as the alternating parameter for the homogenization procedure; different values of the radius result to different unit cell configurations and hence, to different equivalent properties. A fitting process takes place in order to appropriately model the variations of the obtained effective properties w.r.t the design parameter. The corresponding, homogenization-based TOP is formed and the accuracy of the proposed methodology is assessed on several case studies.

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
Fig. 18

References

  1. 1.

    Lagaros ND (2018) The environmental and economic impact of structural optimization. Struct Multidiscip Optim 58(4):1751–1768. https://doi.org/10.1007/s00158-018-1998-z

    Article  Google Scholar 

  2. 2.

    Kanellopoulos I, Sotiropoulos S, Kazakis G, Lagaros ND (2017) Topology optimization aided structural design: interpretation, computational aspects and 3d printing. Heliyon. https://doi.org/10.1016/j.heliyon.2017.e00431

    Article  Google Scholar 

  3. 3.

    Sotiropoulos S, Kazakis G, Lagaros ND (2020) Conceptual design of structural systems based on topology optimization and prefabricated components. Comput Struct. https://doi.org/10.1016/j.compstruc.2019.106136

    Article  Google Scholar 

  4. 4.

    Christensen PW, Klarbring A (2009) An introduction to structural optimization, volume 153 of Solid mechanics and its applications. Springer, Netherlands. ISBN 978-1-4020-8665-6. https://doi.org/10.1007/978-1-4020-8666-3

  5. 5.

    Bendsøe MP, Lund E, Olhoff N, Sigmund O (2005) Topology optimization-broadening the areas of application. Control Cybern 34:7–35

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Sigmund O(2019) Topology optimization state-of-the-art and future perspective, (last accessed November ). https://goo.gl/PCqrgk

  7. 7.

    Paulino GH (2013) Where are we in topology optimization? In: 10th World Congress on structural and multidisciplinary optimization. Orlando, FL, USA

  8. 8.

    Rozvany GIN, Olhoff N (2001) Topology optimization of structures and composite continua, vol 7, 1st edn. Nato Science Series II. Springer, Dordrecht

    Google Scholar 

  9. 9.

    Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202. https://doi.org/10.1007/BF01650949

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zhou M, Rozvany GIN (1991) The coc algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336. https://doi.org/10.1016/0045-7825(91)90046-9

    Article  Google Scholar 

  11. 11.

    Mlejnek HP (1992) Some aspects of the genesis of structures. Struct Multidiscip Optim 5(1–2):64–69. https://doi.org/10.1007/BF01744697

    Article  Google Scholar 

  12. 12.

    Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246. https://doi.org/10.1016/S0045-7825(02)00559-5

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and level-set method. J Comput Phys 194:363–393. https://doi.org/10.1016/j.jcp.2003.09.032

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Xie Y, Steven GP(1992) Shape and layout optimization via an evolutionary procedure. In: Proceedings of the international conference computational engineering science, volume 194. Hong Kong University, Hong Kong, 17–22 December, p 363

  15. 15.

    Xie Y, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896. https://doi.org/10.1016/0045-7949(93)90035-C

    Article  Google Scholar 

  16. 16.

    Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimization using a bidirectional algorithm. Eng Comput 15(8):1031–1048. https://doi.org/10.1108/02644409810244129

    Article  MATH  Google Scholar 

  17. 17.

    Fujii D, Chen BC, Kikuchi N (2001) Composite material design of two-dimensional structures using the homogenization design method. Int J Numer Meth Eng 50:2031–2051. https://doi.org/10.1002/nme.105

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Liu L, Yan J, Cheng G (2008) Optimum structure with homogeneous optimum truss-like material. Comput Struct 86:1417–1425. https://doi.org/10.1016/j.compstruc.2007.04.030

    Article  Google Scholar 

  19. 19.

    Wang Y, Wang MY, Chen F (2016) Structure-material integrated design by level sets. Struct Multidiscip Optim 54:1145–1156. https://doi.org/10.1007/s00158-016-1430-5

    MathSciNet  Article  Google Scholar 

  20. 20.

    Chen W, Tong L, Liu S (2017) Concurrent topology design of structure and material using a two-scale topology optimization. Comput Struct 178:119–128. https://doi.org/10.1016/j.compstruc.2016.10.013

    Article  Google Scholar 

  21. 21.

    Rodrigues H, Guedes J, Bendsøe MP (2002) Hierachical optimization of material and structure. Struct Multidiscip Optim 24(1):1–10. https://doi.org/10.1007/s00158-002-0209-z

    Article  Google Scholar 

  22. 22.

    Lazarov B(2013) Topology optimization using multiscale finite element method for high-contrast media. In: International conference on large-scale scientific computing-LSSC 2013, Sozopol, Bulgaria

  23. 23.

    Alexandersen J, Lazarov B (2015) Topology optimization of manufacturable microstructural details without length scale separation using a spectral coarse basis precoditioner. Comput Methods Appl Mech Eng 290:156–182. https://doi.org/10.1016/j.cma.2015.02.028

    Article  MATH  Google Scholar 

  24. 24.

    Pantz O, Trabelsi K (2008) A post-treatment of the homogenization method for shape optimization. SIAM J Control Optim 47(3):1380–1398. https://doi.org/10.1137/070688900

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Pantz O, Trabelsi K(2010) Construction of minimization sequences for shape optimization. In: 15th International conference on methods and models in automation and robotics, Miedzyzdroje, Poland, pp 278–283. https://doi.org/10.1109/MMAR.2010.5587222

  26. 26.

    Groen JP, Sigmund O (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int J Numer Methods Eng. https://doi.org/10.1002/nme.5575

    MathSciNet  Article  Google Scholar 

  27. 27.

    Westermann N, Sigmund R, Wu O, Aage J (2018) Infill optimization for additive manufacturing-approaching bone-like porous structures. IEE Trans Visual Comput Graph. https://doi.org/10.1109/TVCG.2017.2655523

    Article  Google Scholar 

  28. 28.

    Sigmund J, Groen O, Wu JP (2019) Homogenization-based stiffness optimization and projection of 2d coated structures with orthotropic infill. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2019.02.031

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Pantz G, Geoffroy-Bonders O, Allaire P (2020) 3-d topology optimization of modulated ad oriented periodic microstructures by the homogenization method. J Comput Phys. https://doi.org/10.1016/j.jcp.2019.108994

    Article  MATH  Google Scholar 

  30. 30.

    Cheng X, Yan G, Guo J (2016) Multi-scale concurrent material and structural design under mechanical and thermal loads. Comput Mech. https://doi.org/10.1007/s00466-015-1255-x

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Shukla A, Misra A (2013) Review of optimality criterion approach scope, limitation and development in topology optimization. Int J Adv Eng Technol 6(4):1886–1889

    Google Scholar 

  32. 32.

    Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Meth Eng 24(2):359–373. https://doi.org/10.1002/nme.1620240207

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Bendsøe MP, Sigmund O (2004) Topology optimization: theory, 2nd edn. Methods and applications. Springer, Berlin Heidelberg. ISBN 978-3-540-42992-0. https://doi.org/10.1007/978-3-662-05086-6

  34. 34.

    Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures, vol 7, 1st edn. North Holland, New York. ISBN 978-0-8218-5324-5

  35. 35.

    Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. https://doi.org/10.1016/0045-7825(88)90086-2

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Hassani B, Hinton E (1998) A review of homogenization and topology optimization i-homogenization theory for media with periodic structure. Comput Struct 69(6):707–717. https://doi.org/10.1016/S0045-7949(98)00131-X

    Article  MATH  Google Scholar 

  37. 37.

    Hassani B, Hinton E (1998) A review of homogenization and topology optimization ii analytical and numerical solution of homogenization equations. Comput Struct 69(6):719–738. https://doi.org/10.1016/S0045-7949(98)00132-1

    Article  MATH  Google Scholar 

  38. 38.

    Hassani B, Hinton E (1998) A review of homogenization and topology optimization i-topology optimization using optimality criteria. Comput Struct 69(6):739–756. https://doi.org/10.1016/S0045-7949(98)00133-3

    Article  MATH  Google Scholar 

  39. 39.

    Groen J, Stutz F, Aage N , Bærentzen JA, Sigmund O. De-homogenization of optimal multi-scale 3d topologies. https://arxiv.org/abs/1910.13002

  40. 40.

    Gao J, Luo Z, Xia L, Gao L (2019) Concurrent topology optimization of multiscale composite structures in matlab. Struct Multidiscip Optim 60:2621–2651. https://doi.org/10.1007/s00158-019-02323-6

    MathSciNet  Article  Google Scholar 

  41. 41.

    Allaire G, Geoffroy-Donders P, Pantz O (2019) Topology optimization of modulated and oriented periodic microstructures by the homogenization method. Comput Math Appl 78(7):2197–2229. https://doi.org/10.1016/j.camwa.2018.08.007

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Groen JP, Wu J, Sigmund O (2019) Homogenization-based stiffness optimization and projection of 2d coated structures with orthotropic infill. Comput Methods Appl Mech Eng 349:722–742. https://doi.org/10.1016/j.cma.2019.02.031

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Monteiro A (2017) Topology optimization of microstructures with constraints on average stress and material properties. Master’s thesis, Instituto Superior Técnico, Universidade de Lisboa, Portugal

  44. 44.

    Wu J, Wang W, Gao X (2019) Design and optimization of conforming lattice structures. IEEE Trans Visual Comput Graph. https://doi.org/10.1109/TVCG.2019.2938946

    Article  Google Scholar 

  45. 45.

    Andreassen E, Andreasen CS (2014) How to determine composite material properties using numerical homogenization. Comput Mater Sci 83:488–495. https://doi.org/10.1016/j.commatsci.2013.09.006

    Article  Google Scholar 

  46. 46.

    Guoying D, Yunlong T, Yaoyao FZ (2018) A 149 line homogenization code for three-dimensional cellular materials written in matlab. J Eng Mater Technol 141(1):488–495. https://doi.org/10.1115/1.4040555

    Article  Google Scholar 

  47. 47.

    Liu K, Tovar A (2014) An efficient 3d topology optimization code written in matlab. Struct Multidiscip Optim 50(6):1175–1196. https://doi.org/10.1007/s00158-014-1107-x

    MathSciNet  Article  Google Scholar 

  48. 48.

    Gill P, Murray W, Saunders M (2002) SNOPT: an SQP algorithm for large-scale constrained optimization. Soc Ind Appl Math 12(4):979–1006

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Bourdin B (2001) Filters in topology optimization. Int J Numer Meth Eng 50(9):2143–2158. https://doi.org/10.1002/nme.116

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Pedersen CG, Lund JJ, Damkilde L, Kristensen AS (2006) Topology optimization—improved checker-board filtering with sharp contours. In: Proceedings of the 19th nordic seminar on computational mechanics, Lund, Sweden, 20–21 October, p 182

  51. 51.

    Wang SY, Lim KM, Khoo BC, Wang MY (2008) A hybrid sensitivity filtering for topology optimization. Comput Model Eng Sci 24(1):21–50. https://doi.org/10.3970/cmes.2008.024.021

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21(2):120–127. https://doi.org/10.1007/s001580050176

    Article  Google Scholar 

  53. 53.

    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):1–16. https://doi.org/10.1007/s00158-010-0594-7

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This research has been co-financed by the European Union and Greek national funds through the Operational Program Competitiveness, Entrepreneurship and Innovation, under the call \({\textit{RESEARCH}}-{\textit{CREATE}}-{\textit{INNOVATE}}\) (project code: T1EDK-05603). Konstantinos Iason Ypsilantis acknowledges the support of the Research Foundation Flanders (FWO) under grant number 1SA8721N.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nikos D. Lagaros.

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

Ypsilantis, KI., Kazakis, G. & Lagaros, N.D. An efficient 3D homogenization-based topology optimization methodology. Comput Mech (2021). https://doi.org/10.1007/s00466-020-01943-w

Download citation

Keywords

  • Design of composite materials
  • Topology optimization
  • Homogenization approach
  • 3D lattice unit cell
  • Fitting process