Skip to main content
SpringerLink
Log in

Canonical Duality Theory for Topology Optimization

  • Chapter
  • First Online:
Canonical Duality Theory

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 37))

Abstract

This paper presents a canonical duality approach for solving a general topology optimization problem of nonlinear elastic structures. Based on the principle of minimum total potential energy, this most challenging problem can be formulated as a bi-level mixed integer nonlinear programming problem (MINLP), i.e., for a given deformation, the first-level optimization is a typical linear constrained 0–1 programming problem, while for a given structure, the second-level optimization is a general nonlinear continuous minimization problem in computational nonlinear elasticity. It is discovered that for linear elastic structures, first-level optimization is a typical Knapsack problem , which is considered to be NP-complete in computer science. However, by using canonical duality theory, this well-known problem can be solved analytically to obtain exact integer solution. A perturbed canonical dual algorithm (CDT) is proposed and illustrated by benchmark problems in topology optimization. Numerical results show that the proposed CDT method produces desired optimal structure without any gray elements. The checkerboard issue in traditional methods is much reduced. Additionally, an open problem on NP-hardness of the Knapsack problem is proposed.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ali, E.J., Gao, D.Y.: Improved canonical dual finite element method and algorithm for post buckling analysis of nonlinear gao beam. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality-Triality: Unified Theory and Methodology for Multidisciplinary Study. Springer, Berlin (2016)

    Google Scholar 

  2. Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S., Sigmund, O.: Efficient topology optimization in MATLAB using 88 lines of code. Struct. Multidiscip. Optim. 43(1), 1–16 (2011)

    Article  MATH  Google Scholar 

  3. Bendsoe, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193C202 (1989)

    Google Scholar 

  4. Bendsoe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 72(2), 197–224 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, D.Y.: Panpenalty finite element programming for limit analysis. Comput. Struct. 28, 749–755 (1988)

    Article  MATH  Google Scholar 

  6. Gao, D.Y.: Complementary finite element method for finite deformation nonsmooth mechanics. J. Eng. Math. 30, 339–353 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, D.Y.: Canonical duality theory: unified understanding and generalized solutions for global optimization. Comput. Chem. Eng. 33, 1964–1972 (2009)

    Article  Google Scholar 

  8. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications, pp. xviii + 454. Springer, New York (2000)

    Google Scholar 

  9. Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Indust. Manage. Optim. 3(2), 293–304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, D.Y.: On unified modeling, theory, and method for solving multi-scale global optimization problems. AIP Conf. Proc. 1776, 020005 (2016). doi:10.1063/1.4965311

    Article  Google Scholar 

  11. Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47, 463–484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, D.Y., Strang, G.: Geometric nonlinearity: Potential energy, complementary energy, and the gap function. Quart. Appl. Math. 47(3), 487–504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karp, R.K.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Chapter  Google Scholar 

  14. Moreau, J.J.: La notion de sur-potentiel et les liaisons unilatérales en élastostatique. C.R. Acad. Sci. Paris 267 A, 954–957 (1968)

    Google Scholar 

  15. Santos, H.A.F.A., Gao, D.Y.: Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam. Int. J. Nonlinear Mech. 7, 240–247 (2011)

    Google Scholar 

  16. Sigmund, O.: A 99 line topology optimization code written in matlab. Struct. Multidiscip. Optim. 21(2), 120–127 (2001)

    Article  MathSciNet  Google Scholar 

  17. Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Optim. 16(1), 68–75 (1998)

    Article  Google Scholar 

  18. Sigmund, O., Maute, K.: Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013)

    Article  MathSciNet  Google Scholar 

  19. Sokolowski, J., Zochowski, A.: On the topological derivative in shape optimization. Struct. Optim. 37, 1251–1272 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Stolpe, M., Bendsoe, M.P.: Global optima for the Zhou-Rozvany problem. Struct. Multidiscip. Optim. 43(2), 151–164 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. van Dijk, N.P., Maute, K., Langelaar, M., van Keulen, F.: Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48(3), 437–472 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Matlab code for the CDT algorithm was helped by Professor M. Li from Zhejiang University. The research is supported by US Air Force Office of Scientific Research under grants FA2386-16-1-4082 and FA9550-17-1-0151.

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Mount Helen, VIC, Australia

    David Yang Gao

Authors
  1. David Yang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Yang Gao .

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    David Yang Gao

  2. Department of Computer, Control, and Managment Engineering, Sapienza University of Rome, Rome, Roma, Italy

    Vittorio Latorre

  3. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    Ning Ruan

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Gao, D.Y. (2017). Canonical Duality Theory for Topology Optimization . In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_13

Download citation

Publish with us

Policies and ethics

Search

Navigation