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Structural and Multidisciplinary Optimization

, Volume 42, Issue 5, pp 665–679 | Cite as

A 199-line Matlab code for Pareto-optimal tracing in topology optimization

  • Krishnan SureshEmail author
Educational Article

Abstract

The paper ‘A 99-line topology optimization code written in Matlab’ by Sigmund (Struct Multidisc Optim 21(2):120–127, 2001) demonstrated that SIMP-based topology optimization can be easily implemented in less than hundred lines of Matlab code. The published method and code has been used even since by numerous researchers to advance the field of topology optimization. Inspired by the above paper, we demonstrate here that, by exploiting the notion of topological-sensitivity (an alternate to SIMP), one can generate Pareto-optimal topologies in about twice the number of lines of Matlab code. In other words, optimal topologies for various volume fractions can be generated in a highly efficient manner, by directly tracing the Pareto-optimal curve.

Keywords

Pareto-optimal Topological sensitivity Topology optimization 

References

  1. Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393zbMATHCrossRefMathSciNetGoogle Scholar
  2. Amstutz S (2006) Sensitivity analysis with respect to a local perturbation of the material property. Asymptot Anal 49(1–2):87–108zbMATHMathSciNetGoogle Scholar
  3. Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57(8):1177–1196zbMATHCrossRefGoogle Scholar
  4. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  5. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in optimal design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224CrossRefGoogle Scholar
  6. Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362zbMATHCrossRefMathSciNetGoogle Scholar
  7. Céa J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimizations connection. Comput Methods Appl Mech Eng 188:713–726zbMATHCrossRefGoogle Scholar
  8. Chen TY, Wu S-C (1998) Multiobjective optimal topology design of structures. Comput Mech 21:483–492zbMATHCrossRefGoogle Scholar
  9. Cohon JL (1978) Multiobjective programming and planning, vol 140. Academic, LondonGoogle Scholar
  10. Dambrine M, Vial G (2005) Influence of a boundary perforation on the Dirichlet energy. Control Cybern 34(1):117–136zbMATHMathSciNetGoogle Scholar
  11. Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69CrossRefGoogle Scholar
  12. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, ChichesterzbMATHGoogle Scholar
  13. Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–389CrossRefGoogle Scholar
  14. Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51CrossRefGoogle Scholar
  15. Feijóo RA, Novotny AA, Taroco E, Padra C (2003) The topological derivative for the Poisson’s problem. Math Models Methods Appl Sci 13(12):1825–1844zbMATHCrossRefMathSciNetGoogle Scholar
  16. Feijóo RA, Novotny AA, Taroco E, Padra C (2005) The topological-shape sensitivity method in two-dimensional linear elasticity topology design. In: Idelsohn VSSR (ed) Applications of computational mechanics in structures and fluids. CIMNE, BarcelonaGoogle Scholar
  17. Gopalakrishnan SH, Suresh K (2008) Feature sensitivity: a generalization of topological sensitivity. Finite Elem Anal Des 44(11):696–704CrossRefMathSciNetGoogle Scholar
  18. Hamda H, Roudenko O, Schoenauer M (2002) Application of a multi-objective evolutionary algorithm to topology optimum design. In: Fifth international conference on adaptive computing in design and manufactureGoogle Scholar
  19. Lin J, Luo Z, Tong L (2010) A new multi-objective programming scheme for topology optimization of compliant mechanisms. Struct Multidisc Optim 30:241–255CrossRefGoogle Scholar
  20. Luo Z, Chen L, Yang J, Zhang Y, Abdel-Malek K (2005) Compliant mechanism design using multi-objective topology optimization scheme of continuum structures. Struct Multidisc Optim 30:142–154CrossRefGoogle Scholar
  21. Madeira JFA, Rodrigues H, Pina H (2005) Multi-objective optimization of structures topology by genetic algorithms. Adv Eng Softw 36:21–28zbMATHCrossRefGoogle Scholar
  22. Messac A, Ismail-Yahaya A (2001) Required relationship between objective function and Pareto frontier orders: practical implications. AIAA J 39(11):2168–2174CrossRefGoogle Scholar
  23. Messac A, Sundararaj GJ, Tappeta RV, Renaud JE (2000) Ability of objective functions to generate points on non-convex Pareto frontiers. AIAA J 38(6):1084–1091CrossRefGoogle Scholar
  24. Norato JA, Bendsøe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33:375–386CrossRefGoogle Scholar
  25. Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological-shape sensitivity analysis. Comput Methods Appl Mech Eng 192(7):803–829zbMATHCrossRefGoogle Scholar
  26. Novotny AA, Feijóo RA, Taroco E, Padra C (2005) Topological sensitivity analysis for three-dimensional linear elasticity problem. In: 6th world congress on structural and multidisciplinary optimization, Rio de JaneiroGoogle Scholar
  27. Novotny AA, Feijóo RA, Taroco E, Padra C (2006) Topological-shape sensitivity method: theory and applications. Solid Mech Appl 137:469–478CrossRefGoogle Scholar
  28. Padhye N (2008) Topology optimization of compliant mechanism using multi-objective particle swarm optimization. In: GECCO’08, July 12–16. ACM, AtlantaGoogle Scholar
  29. Papalambros PY (2002) The optimization paradigm in engineering design: promises and challenges. Comput Aided Des 34(12):939–951CrossRefGoogle Scholar
  30. Rozvany GIN (2001a) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidisc Optim 21(2):90–108CrossRefGoogle Scholar
  31. Rozvany GIN (2001b) Stress ratio and compliance based methods in topology optimization—a critical review. Struct Multidisc Optim 21(2):109–119CrossRefGoogle Scholar
  32. Samet B (2003) The topological asymptotic with respect to a singular boundary perturbation. Comptes Rendus Mathematique 336(12):1033–1038zbMATHCrossRefMathSciNetGoogle Scholar
  33. Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127CrossRefGoogle Scholar
  34. Sokolowski J, Zochowski A (1999) On topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272zbMATHCrossRefMathSciNetGoogle Scholar
  35. Sokolowski J, Zochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42(4):1198–1221zbMATHCrossRefMathSciNetGoogle Scholar
  36. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246zbMATHCrossRefMathSciNetGoogle Scholar
  37. Zhang WH, Yang HC (2002) Efficient gradient calculation of the Pareto optimal curve in multicriteria optimization. Struct Multidisc Optim 23:311–319zbMATHCrossRefGoogle Scholar
  38. Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89:197–224CrossRefGoogle Scholar
  39. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier, AmsterdamzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.University of WisconsinMadisonUSA

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