A computational study of symmetry and well-posedness of structural topology optimization

Abstract

We are concerned with the computational topological optimization of elastic structures, in particular minimization of compliance subject to a constraint on the mass. Through computational experiments, it is discovered that even very simple optimization problems can exhibit complex behavior such as critical points and bifurcation. In the vicinity of critical points, structural topology optimization problems are not well-posed since infinitesimally small perturbations lead to distinct topologies.

This is a preview of subscription content, log in to check access.

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

References

  1. Baier H (1994) Ill posed problems in structural optimization and their practical consequences. Struct Optim 7:184–190

    Article  Google Scholar 

  2. Bendsoe MP, Sigmund O (2003) Topology Optimization Theory, Methods, and Applications. Springer, Berlin

    Google Scholar 

  3. Blanchard P, Devaney RL, Hall GR (2006) Differential Equations. Thomson Brooks/Cole, USA

    Google Scholar 

  4. Cheng G, Lui X (2011) Discussion on symmetry of optimum topology design. Struct Multidisc Optim 44:713–717

    Article  MATH  Google Scholar 

  5. Guo X, Ni C, Cheng G, Du Z (2012) Some symmetry results for optimal solutions in structural optimization. Struct Multidisc Optim 46:631–645

    MathSciNet  Article  MATH  Google Scholar 

  6. Kosakam I, Swan CC (1999) A symmetry reduction method for continuum structural topology optimization. Comp Struct 70:47–61

    MathSciNet  Article  MATH  Google Scholar 

  7. Lazarov B, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Meth Eng 86:765–781

    MathSciNet  Article  MATH  Google Scholar 

  8. Richardson JN, Adriaenssens S, Bouillard P, Coelho RF (2013) Symmetry and asymmetry of solutions in discrete variable structural optimization. Struct Multidisc Optim 47:631–643

    MathSciNet  Article  MATH  Google Scholar 

  9. Rozvany G (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Struct Optim 15:42–48

    Article  MATH  Google Scholar 

  10. Rozvany GIN (2011) On symmetry and non-uniqueness in exact topology optimization. Struct Multidisc Optim 43:297–317

    MathSciNet  Article  MATH  Google Scholar 

  11. Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidis Optimiz 21:120–127

    Article  Google Scholar 

  12. Stolpe M (2010) On some fundamental properties of structural topology optimization problems. Struct Multidisp Optim 41:661–670

    Article  MATH  Google Scholar 

  13. Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106(1):5–57

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Daniel A. White.

Additional information

Publisher’s Note

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

Prepared by LLNL under Contract DE-AC52-07NA27344

Responsible Editor: Fred van Keulen

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

White, D.A., Voronin, A. A computational study of symmetry and well-posedness of structural topology optimization. Struct Multidisc Optim 59, 759–766 (2019). https://doi.org/10.1007/s00158-018-2098-9

Download citation

Keywords

  • Structures
  • Topology
  • Optimization
  • Bifurcation
  • Well-posed