Structural and Multidisciplinary Optimization

, Volume 57, Issue 4, pp 1461–1474 | Cite as

Optimal truss and frame design from projected homogenization-based topology optimization

RESEARCH PAPER
  • 203 Downloads

Abstract

In this article, we propose a novel method to obtain a near-optimal frame structure, based on the solution of a homogenization-based topology optimization model. The presented approach exploits the equivalence between Michell’s problem of least-weight trusses and a compliance minimization problem using optimal rank-2 laminates in the low volume fraction limit. In a fully automated procedure, a discrete structure is extracted from the homogenization-based continuum model. This near-optimal structure is post-optimized as a frame, where the bending stiffness is continuously decreased, to allow for a final design that resembles a truss structure. Numerical experiments show excellent behavior of the method, where the final designs are close to analytical optima, and obtained in less than 10 minutes, for various levels of detail, on a standard PC.

Keywords

Optimal frame design Optimal truss design Michell theory Topology optimization 

Notes

Acknowledgements

The authors acknowledge the support of the Villum Fonden through the Villum investigator project InnoTop. The authors would also like to thank Andreas Bærentzen and Niels Aage for valuable discussions during the preparation of the work. Finally, the authors wish to thank Krister Svanberg for providing the MATLAB MMA code.

References

  1. Aage N, Nobel-jørgensen M, Andreasen CS, Sigmund O (2013) Interactive topology optimization on hand-held devices. Struct Multidiscip Optim 47(1):1–6.  https://doi.org/10.1007/s00158-012-0827-z CrossRefGoogle Scholar
  2. Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4):285–304.  https://doi.org/10.1007/s00158-006-0092-0 MathSciNetCrossRefMATHGoogle Scholar
  3. Bendsøe MP, Haber RB (1993) The michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study. Structural optimization 6(4):263–267.  https://doi.org/10.1007/BF01743385 CrossRefGoogle Scholar
  4. Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Structural optimization 7(3):141–159.  https://doi.org/10.1007/BF01742459 CrossRefGoogle Scholar
  5. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158.  https://doi.org/10.1002/nme.116 MathSciNetCrossRefMATHGoogle Scholar
  6. Bourdin B, Kohn R (2008) Optimization of structural topology in the high-porosity regime. J Mech Phys Solids 56(3):1043–1064.  https://doi.org/10.1016/j.jmps.2007.06.002 MathSciNetCrossRefMATHGoogle Scholar
  7. Bruns T (2006) Zero density lower bounds in topology optimization. Comput Methods Appl Mech Eng 196(1):566–578.  https://doi.org/10.1016/j.cma.2006.06.007 CrossRefMATHGoogle Scholar
  8. Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459.  https://doi.org/10.1016/S0045-7825(00)00278-4 CrossRefMATHGoogle Scholar
  9. Dobbs MW, Felton LP (1969) Optimization of truss geometry. J Struct Div 95(10):2105–2118Google Scholar
  10. Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. Journal de Mecanique 3:25–52Google Scholar
  11. Gao G, yu Liu Z, bin Li Y, feng Qiao Y (2017) A new method to generate the ground structure in truss topology optimization. Eng Optim 49(2):235–251.  https://doi.org/10.1080/0305215X.2016.1169050 MathSciNetCrossRefGoogle Scholar
  12. Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064.  https://doi.org/10.1108/02644400310503017 CrossRefMATHGoogle Scholar
  13. Graczykowski C, Lewiński T (2010) Michell cantilevers constructed within a half strip. tabulation of selected benchmark results. Struct Multidiscip Optim 42(6):869–877.  https://doi.org/10.1007/s00158-010-0525-7 CrossRefGoogle Scholar
  14. Groen JP, Sigmund O (2017) Homogenization-based topology optimization for high-resolution manufacturable micro-structures. Int J Numer Methods Eng :1–18.  https://doi.org/10.1002/nme.5575
  15. He L, Gilbert M (2015) Rationalization of trusses generated via layout optimization. Struct Multidiscip Optim 52(4):677–694.  https://doi.org/10.1007/s00158-015-1260-x MathSciNetCrossRefGoogle Scholar
  16. Hemp WS (1973) Optimum structures. Clarendon Press, OxfordGoogle Scholar
  17. Lewiński T, Rozvany GIN (2008) Exact analytical solutions for some popular benchmark problems in topology optimization iii: L-shaped domains. Struct Multidiscip Optim 35(2):165–174.  https://doi.org/10.1007/s00158-007-0157-8 MathSciNetCrossRefMATHGoogle Scholar
  18. Lewiński T, Zhou M, Rozvany G (1994a) Extended exact least-weight truss layouts—part ii: Unsymmetric cantilevers. Int J Mech Sci 36(5):399–419.  https://doi.org/10.1016/0020-7403(94)90044-2 CrossRefMATHGoogle Scholar
  19. Lewiński T, Zhou M, Rozvany G (1994b) Extended exact solutions for least-weight truss layouts—part i: Cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398.  https://doi.org/10.1016/0020-7403(94)90043-4 CrossRefMATHGoogle Scholar
  20. Martínez P, Martí P, Querin OM (2007) Growth method for size, topology, and geometry optimization of truss structures. Struct Multidiscip Optim 33(1):13–26.  https://doi.org/10.1007/s00158-006-0043-9 CrossRefGoogle Scholar
  21. Michell A (1904) The limits of economy of material in frame-structures. Phil Mag 8(47):589–597.  https://doi.org/10.1080/14786440409463229 CrossRefMATHGoogle Scholar
  22. 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 MathSciNetCrossRefMATHGoogle Scholar
  23. Pantz O, Trabelsi K (2010) Construction of minimization sequences for shape optimization. In: 15th international conference on methods and models in automation and robotics (MMAR), pp 278–283.  https://doi.org/10.1109/MMAR.2010.5587222
  24. Pedersen P (1969) On the minimum mass layout of trusses. In: AGARD conference proceedings no 36, symposium on structural optimization, pp 36–70Google Scholar
  25. Pedersen P (1989) On optimal orientation of orthotropic materials. Structural optimization 1(2):101–106.  https://doi.org/10.1007/BF01637666 CrossRefGoogle Scholar
  26. Pedersen P (1990) Bounds on elastic energy in solids of orthotropic materials. Structural optimization 2(1):55–63.  https://doi.org/10.1007/BF01743521 CrossRefGoogle Scholar
  27. Ramos JrAS, Paulino GH (2016) Filtering structures out of ground structures – a discrete filtering tool for structural design optimization. Struct Multidiscip Optim 54(1):95–116.  https://doi.org/10.1007/s00158-015-1390-1 MathSciNetCrossRefGoogle Scholar
  28. Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Structural optimization 15(1):42–48.  https://doi.org/10.1007/BF01197436 CrossRefMATHGoogle Scholar
  29. Rule W K (1994) Automatic truss design by optimized growth. J Struct Eng 120(10):3063–3070CrossRefGoogle Scholar
  30. Sokół T (2011) A 99 line code for discretized michell truss optimization written in mathematica. Struct Multidiscip Optim 43(2):181–190.  https://doi.org/10.1007/s00158-010-0557-z CrossRefMATHGoogle Scholar
  31. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373.  https://doi.org/10.1002/nme.1620240207 MathSciNetCrossRefMATHGoogle Scholar
  32. Washizawa T, Asai A, Yoshikawa N (2004) A new approach for solving singular systems in topology optimization using krylov subspace methods. Struct Multidiscip Optim 28(5):330–339.  https://doi.org/10.1007/s00158-004-0439-3 CrossRefGoogle Scholar
  33. Zegard T, Paulino GH (2014) Grand - ground structure based topology optimization for arbitrary 2d domains using matlab. Struct Multidiscip Optim 50(5):861–882.  https://doi.org/10.1007/s00158-014-1085-z MathSciNetCrossRefGoogle Scholar
  34. Zhou K, Li X (2008) Topology optimization for minimum compliance under multiple loads based on continuous distribution of members. Struct Multidiscip Optim 37(1):49–56.  https://doi.org/10.1007/s00158-007-0214-3 CrossRefGoogle Scholar
  35. Zhou K, Li X (2011) Topology optimization of truss-like continua with three families of members model under stress constraints. Struct Multidiscip Optim 43(4):487–493.  https://doi.org/10.1007/s00158-010-0584-9 CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations