Structural and Multidisciplinary Optimization

, Volume 55, Issue 2, pp 663–679 | Cite as

A unified aggregation and relaxation approach for stress-constrained topology optimization

  • Alexander Verbart
  • Matthijs Langelaar
  • Fred van Keulen
RESEARCH PAPER

Abstract

In this paper, we propose a unified aggregation and relaxation approach for topology optimization with stress constraints. Following this approach, we first reformulate the original optimization problem with a design-dependent set of constraints into an equivalent optimization problem with a fixed design-independent set of constraints. The next step is to perform constraint aggregation over the reformulated local constraints using a lower bound aggregation function. We demonstrate that this approach concurrently aggregates the constraints and relaxes the feasible domain, thereby making singular optima accessible. The main advantage is that no separate constraint relaxation techniques are necessary, which reduces the parameter dependence of the problem. Furthermore, there is a clear relationship between the original feasible domain and the perturbed feasible domain via this aggregation parameter.

Keywords

Stress constraints Singular optima Constraint aggregation Topology optimization 

Notes

Acknowledgments

The authors gratefully acknowledge the support of the Netherlands Aerospace Centre (NLR), Amsterdam for funding this research. We would also like to thank Krister Svanberg for providing his Matlab implementation of MMA.

References

  1. Achtziger W, Kanzow C (2008) Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math Program 114(1):69–99. doi: 10.1007/s10107-006-0083-3 MathSciNetCrossRefMATHGoogle Scholar
  2. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural optimization 1 (4):193–202. doi: 10.1007/BF01650949 CrossRefGoogle Scholar
  3. Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36(2):125–141. doi: 10.1007/s00158-007-0203-6 MathSciNetCrossRefMATHGoogle Scholar
  4. Bruggi M, Venini P (2008) A mixed fem approach to stress-constrained topology optimization. Int J Numer Methods Eng 73(12):1693–1714. doi: 10.1002/nme.2138 MathSciNetCrossRefMATHGoogle Scholar
  5. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459. doi: 10.1016/S0045-7825(00)00278-4 CrossRefMATHGoogle Scholar
  6. Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20(2):129–148. doi: 10.1080/03052159208941276 CrossRefGoogle Scholar
  7. Cheng GD, Guo X (1997) 𝜖-relaxed approach in structural topology optimization. Structural optimization 13(4):258–266. doi: 10.1007/BF01197454 CrossRefGoogle Scholar
  8. Duysinx P (1999) Topology optimization with different stress limit in tension and compression. In: Proceedings of the 3rd World Congress of Structural and Multidisciplinary Optimization WCSMO3Google Scholar
  9. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(8):1453–1478. doi: 10.1002/(SICI)1097-0207(19981230)43:8%3C1453::AID-NME480%3D3.0.CO;2-2 MathSciNetCrossRefMATHGoogle Scholar
  10. Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distributions. In: Proceedings of 7th AIAA/USAF/NASA/ISSMO symposium on Multidisciplinary Design Optimization, AIAAGoogle Scholar
  11. Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48(1):33–47. doi: 10.1007/s00158-012-0880-7 MathSciNetCrossRefMATHGoogle Scholar
  12. Kirsch U (1989) Optimal topologies of truss structures. Comput Methods Appl Mech Eng 72(1):15–28. doi: 10.1016/0045-7825(89)90119-9 MathSciNetCrossRefMATHGoogle Scholar
  13. Kirsch U (1990) On singular topologies in optimum structural design. Structural optimization 2(3):133–142. doi: 10.1007/BF01836562 CrossRefGoogle Scholar
  14. Kreisselmeier G (1979) Systematic control design by optimizing a vector performance index. In: International federation of active control symposium on computer-aided design of control systems, zurich, Switzerland, August 29-31, 1979Google Scholar
  15. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2009) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620. doi: 10.1007/s00158-009-0440-y CrossRefGoogle Scholar
  16. Luo Y, Wang M Y, Zhou M, Deng Z (2012) Optimal topology design of steel-concrete composite structures under stiffness and strength constraints. Comput Struct 112-113:433–444. doi: 10.1016/j.compstruc.2012.09.007 CrossRefGoogle Scholar
  17. Luo Y, Wang M Y, Kang Z (2013) An enhanced aggregation method for topology optimization with local stress constraints. Comput Methods Appl Mech Eng 254(0):31–41. doi: 10.1016/j.cma.2012.10.019 MathSciNetCrossRefMATHGoogle Scholar
  18. París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidiscip Optim 39(4):419–437MathSciNetCrossRefMATHGoogle Scholar
  19. París J, Navarrina F, Colominas I, Casteleiro M (2010) Improvements in the treatment of stress constraints in structural topology optimization problems. J Comput Appl Math 234(7):2231–2238. doi: 10.1016/j.cam.2009.08.080 MathSciNetCrossRefMATHGoogle Scholar
  20. Rozvany GIN (2001a) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21 (2):164–172. doi: 10.1007/s001580050181
  21. Rozvany GIN (2001b) Stress ratio and compliance based methods in topology optimization –a critical review. Struct Multidiscip Optim 21(2):109–119. doi: 10.1007/s001580050175
  22. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4):401–424. doi: 10.1007/s00158-006-0087-x MathSciNetCrossRefGoogle Scholar
  23. Stolpe M (2003) On models and methods for global optimization of structural topology. PhD thesis, KTH Royal Institute of TechnologyGoogle Scholar
  24. Stolpe M, Svanberg K (2001) On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct Multidiscip Optim 21(2):140–151. doi: 10.1007/s001580050178 CrossRefGoogle Scholar
  25. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. doi: 10.1002/nme.1620240207 MathSciNetCrossRefMATHGoogle Scholar
  26. Sved G, Ginos Z (1968) Structural optimization under multiple loading. Int J Mech Sci 10(10):803–805. doi: 10.1016/0020-7403(68)90021-0 CrossRefGoogle Scholar
  27. Verbart A, Langelaar M, van Keulen F (2015) Damage approach: A new method for topology optimization with local stress constraints. Struct Multidiscip Optim:1–18. doi: 10.1007/s00158-015-1318-9
  28. Yang R J, Chen C J (1996) Stress-based topology optimization. Struct Multidiscip Optim 12(2):98–105CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Alexander Verbart
    • 1
    • 3
  • Matthijs Langelaar
    • 2
  • Fred van Keulen
    • 2
  1. 1.Netherlands Aerospace Centre (NLR)AmsterdamThe Netherlands
  2. 2.Department of Precision and Microsystems EngineeringDelft University of TechnologyDelftThe Netherlands
  3. 3.Department of Wind EnergyTechnical University of Denmark (DTU)RoskildeDenmark

Personalised recommendations