Structural and Multidisciplinary Optimization

, Volume 46, Issue 5, pp 647–661

Stress-constrained topology optimization with design-dependent loading

  • Edmund Lee
  • Kai A. James
  • Joaquim R. R. A. Martins
Research Paper

Abstract

The purpose of this paper is to apply stress constraints to structural topology optimization problems with design-dependent loading. A comparison of mass-constrained compliance minimization solutions and stress-constrained mass minimization solutions is also provided. Although design-dependent loading has been the subject of previous research, only compliance minimization has been studied. Stress-constrained mass minimization problems are solved in this paper, and the results are compared with those of compliance minimization problems for the same geometries and loading. A stress-relaxation technique is used to avoid the singularity in the stress constraints, and these constraints are aggregated in blocks to reduce the total number of constraints in the optimization problem. The results show that these design-dependent loading problems may converge to a local minimum when the stress constraints are enforced. The use of a continuation method where the stress-constraint aggregation parameter is gradually increased typically leads to better convergence; however, this may not always be possible. The results also show that the topologies of compliance-minimization and stress-constrained solutions are usually vastly different, and the sizing optimization of a compliance solution may not lead to an optimum.

Keywords

Topology optimization Design dependent loads Stress constraints Block aggregated constraints Self-weight Pressure loading 

References

  1. Allaire G, Francfort G (1993) A numerical algorithm for topology and shape optimization. Kluwer, Dordrecht, pp 239–248Google Scholar
  2. Ansola R, Canales J, Tárrago JA (2006) An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads. Finite Elem Anal Des 42(14–15):1220–1230. doi:10.1016/j.finel.2006.06.001 CrossRefGoogle Scholar
  3. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidisc Optim 1(4):193–202. doi:10.1007/BF01650949 Google Scholar
  4. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. doi:10.1016/0045-7825(88)90086-2 CrossRefGoogle Scholar
  5. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, BerlinGoogle Scholar
  6. Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM: Control Optim Calc Var 9:19–48. doi:10.1051/cocv:2002070 MathSciNetMATHCrossRefGoogle Scholar
  7. Bruggi M, Cinquini C (2009) An alternative truly-mixed formulation to solve pressure load problems in topology optimization. Comput Methods Appl Mech Eng 198(17–20):1500–1512. doi:10.1016/j.cma.2008.12.009 MATHCrossRefGoogle Scholar
  8. 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 MathSciNetMATHCrossRefGoogle Scholar
  9. Bruyneel M, Duysinx P (2005) Note on topology optimization of continuum structures including self-weight. Struct Multidisc Optim 29:245–256. doi:10.1007/s00158-004-0484-y CrossRefGoogle Scholar
  10. Chen BC, Kikuchi N (2001) Topology optimization with design-dependent loads. Finite Elem Anal Des 37(1):57–70. doi:10.1016/S0168-874X(00)00021-4 MathSciNetMATHCrossRefGoogle Scholar
  11. Cheng GD, Guo X (1997) ε-relaxed approach in structural topology optimization. Struct Multidisc Optim 13:258–266. doi:10.1007/BF01197454 Google Scholar
  12. Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element analysis, 3rd edn. Wiley, New YorkMATHGoogle Scholar
  13. Du J, Olhoff N (2004) Topological optimization of continuum structures with design-dependent surface loading—part I: new computational approach for 2D problems. Struct Multidisc Optim 27(3):151–165. doi:10.1007/s00158-004-0379-y MathSciNetMATHCrossRefGoogle Scholar
  14. 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<1453::AID-NME480>3.0.CO;2-2 MATHCrossRefGoogle Scholar
  15. 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, AIAA, Saint Louis, Missouri, AIAA Paper 98–4906Google Scholar
  16. Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390. doi:10.1115/1.1388075 CrossRefGoogle Scholar
  17. Fuchs M, Shemesh N (2004) Density-based topological design of structures subjected to water pressure using a parametric loading surface. Struct Multidisc Optim 28:11–19. doi:10.1007/s00158-004-0406-z CrossRefGoogle Scholar
  18. Gill PE, Murray W, Saunders MA (2005) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev 47(1):99–131. doi:10.1137/S0036144504446096 MathSciNetMATHCrossRefGoogle Scholar
  19. Guilherme CEM, Fonseca JSO (2007) Topology optimization of continuum structures with ε-relaxed stress constraints. In: ABCM symposium series in solid mechanics, vol 1, pp 239–250Google Scholar
  20. Hammer V, Olhoff N (2000) Topology optimization of continuum structures subjected to pressure loading. Struct Multidisc Optim 19(2):85–92. doi:10.1007/s001580050088 CrossRefGoogle Scholar
  21. James KA, Hansen JS, Martins JRRA (2009) Structural topology optimization for multiple load cases using a dynamic aggregation technique. Eng Optim 41(12):1103–1118. doi:10.1080/03052150902926827 CrossRefGoogle Scholar
  22. Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226. doi:10.1016/0045-7825(95)00928-0 MathSciNetMATHCrossRefGoogle Scholar
  23. Kennedy GJ, Martins JRRA (2010) Parallel solution methods for aerostructural analysis and design optimization. In: 13th AIAA/ISSMO multidisciplinary analysis and optimization conference, Fort Worth, Texas, AIAA 2010–9308Google Scholar
  24. Kirsch U (1990) On singular topologies in optimum structural design. Struct Multidisc Optim 2:133–142. doi:10.1007/BF01836562 Google Scholar
  25. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. In: International federation of active controls syposium on computer aided design of control systems, Zurich, Switzerland, pp 113–117Google Scholar
  26. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidisc Optim 41:605–620. doi:10.1007/s00158-009-0440-y CrossRefGoogle Scholar
  27. Lee E, Martins JRRA (2011) Structural topology optimization with design-dependent pressure loads. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2012.04.007 Google Scholar
  28. Olhoff N (1989) Multicriterion structural optimization via bound formulation and mathematical programming. Struct Optim 1:11–17CrossRefGoogle Scholar
  29. París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidisc Optim 39:419–437. doi:10.1007/s00158-008-0336-2 CrossRefGoogle Scholar
  30. París J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41(3):433–441. doi:10.1016/j.advengsoft.2009.03.006 MATHCrossRefGoogle Scholar
  31. Pereira J, Fancello E, Barcellos C (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidisc Optim 26:50–66. doi:10.1007/s00158-003-0301-z MathSciNetMATHCrossRefGoogle Scholar
  32. Perez R, Jansen P, Martins JRRA (2011) pyOpt: a python-based object-oriented framework for nonlinear constrained optimization. Struct Multidisc Optim 45:101–118. doi:10.1007/s00158-011-0666-3 MathSciNetCrossRefGoogle Scholar
  33. Poon NMK, Martins JRRA (2007) An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Struct Multidisc Optim 30(1):61–73. doi:10.1007/s00158-006-0061-7 MathSciNetCrossRefGoogle Scholar
  34. Rahmatalla S, Swan C (2004) A Q4/Q4 continuum structural topology optimization implementation. Struct Multidisc Optim 27:130–135. doi:10.1007/s00158-003-0365-9 CrossRefGoogle Scholar
  35. Rozvany G (1989) Structural design via optimality criteria. Kluwer Academic Publishers, DordrechtMATHCrossRefGoogle Scholar
  36. Rozvany G, Prager W (1979) A new class of structural optimization problems: optimal archgrids. Comput Methods Appl Mech Eng 19(1):127–150. doi:10.1016/0045-7825(79)90038-0 MathSciNetMATHCrossRefGoogle Scholar
  37. Rozvany GIN, Bendsøe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48(2):41–119. doi:10.1115/1.3005097 CrossRefGoogle Scholar
  38. Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127. doi:10.1007/s001580050176 CrossRefGoogle Scholar
  39. Sigmund O, Clausen PM (2007) Topology optimization using a mixed formulation: an alternative way to solve pressure load problems. Comput Methods Appl Mech Eng 196(13–16):1874–1889. doi:10.1016/j.cma.2006.09.021 MathSciNetMATHCrossRefGoogle Scholar
  40. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75. doi:10.1007/BF01214002 CrossRefGoogle Scholar
  41. Stolpe M, Svanberg K (2001) On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct Multidisc Optim 21:140–151. doi:10.1007/s001580050178 CrossRefGoogle Scholar
  42. Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Methods Eng 24(3):359–373. doi:10.1002/nme.1620240207 MathSciNetMATHCrossRefGoogle Scholar
  43. 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
  44. Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Multidisc Optim 12:98–105. doi:10.1007/BF01196941 Google Scholar
  45. Zhang H, Zhang X, Liu S (2008) A new boundary search scheme for topology optimization of continuum structures with design-dependent loads. Struct Multidisc Optim 37:121–129. doi:10.1007/s00158-007-0221-4 CrossRefGoogle Scholar
  46. Zheng B, Chang CJ, Gea HC (2009) Topology optimization with design-dependent pressure loading. Struct Multidisc Optim 38(6):535–543. doi:10.1007/s00158-008-0317-5 CrossRefGoogle Scholar
  47. Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336. doi:10.1016/0045-7825(91)90046-9 CrossRefGoogle Scholar
  48. Zuo KT, Chen LP, Zhang YQ, Yang J (2007) Study of key algorithms in topology optimization. Int J Adv Manuf Technol 32:787–796. doi:10.1007/s00170-005-0387-0 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Edmund Lee
    • 1
  • Kai A. James
    • 1
  • Joaquim R. R. A. Martins
    • 2
  1. 1.Institute for Aerospace StudiesUniversity of TorontoTorontoCanada
  2. 2.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA

Personalised recommendations