Structural and Multidisciplinary Optimization

, Volume 51, Issue 2, pp 385–396 | Cite as

Incorporating fabrication cost into topology optimization of discrete structures and lattices

  • Alireza Asadpoure
  • James K. Guest
  • Lorenzo ValdevitEmail author


In this article, we propose a method to incorporate fabrication cost in the topology optimization of light and stiff truss structures and periodic lattices. The fabrication cost of a design is estimated by assigning a unit cost to each truss element, meant to approximate the cost of element placement and associated connections. A regularized Heaviside step function is utilized to estimate the number of elements existing in the design domain. This makes the cost function smooth and differentiable, thus enabling the application of gradient-based optimization schemes. We demonstrate the proposed method with classic examples in structural engineering and in the design of a material lattice, illustrating the effect of the fabrication unit cost on the optimal topologies. We also show that the proposed method can be efficiently used to impose an upper bound on the allowed number of elements in the optimal design of a truss system. Importantly, compared to traditional approaches in structural topology optimization, the proposed algorithm reduces the computational time and reduces the dependency on the threshold used for element removal.


Fabrication cost Material cost Minimum weight Topology optimization Lattices 



This work was financially supported by the Office of Naval Research under Grant No. N00014-11-1-0884 (program manager: D. Shifler). This support is gratefully acknowledged. The authors are also thankful to Krister Svanberg for providing the MMA optimizer subroutine.


  1. Achtziger W, Stolpe M (2006) Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples. Struct Multidiscip Optim 34(1):1–20. doi: 10.1007/s00158-006-0074-2 CrossRefMathSciNetGoogle Scholar
  2. Achtziger W, Stolpe M (2007a) Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems. Comput Optim Appl 40(2):247–280. doi: 10.1007/s10589-007-9138-5 CrossRefMathSciNetGoogle Scholar
  3. Achtziger W, Stolpe M (2007b) Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results. Comput Optim Appl 44(2):315–341. doi: 10.1007/s10589-007-9152-7 CrossRefMathSciNetGoogle Scholar
  4. Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var 1(1):55–69. doi: 10.1007/BF02163264 CrossRefzbMATHMathSciNetGoogle Scholar
  5. Amir O, Sigmund O (2013) Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization. Struct Multidiscip Optim 47(2):157–174. doi: 10.1007/s00158-012-0817-1 CrossRefzbMATHMathSciNetGoogle Scholar
  6. Asadpoure A, Tootkaboni MP, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness—Application to truss structures. Comput Struct 89(11–12):1131–1141. doi: 10.1016/j.compstruc.2010.11.004 CrossRefGoogle Scholar
  7. Ben-Tal A, Bendsøe MP (1993) A new method for optimal truss topology design. SIAM J Optim 2:322–358. doi: 10.1137/0803015 CrossRefGoogle Scholar
  8. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202. doi: 10.1007/BF01650949 CrossRefGoogle Scholar
  9. Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Structural Optimization 7(3):141–159. doi: 10.1007/BF01742459 CrossRefGoogle Scholar
  10. Bendsøe MP, Sigmund O (2003) Topology Optimization: Theory, Methods and Applications. SpringerGoogle Scholar
  11. Dorn William S, Gomory Ralph E, Greenberg Herbert J (1964) Automatic design of optimal structures. Journal de Mecanique 3(1):25–52Google Scholar
  12. Groenwold AA, Stander N, Snyman JA (1996) A pseudo-discrete rounding method for structural optimization. Struct Optim 11(3–4):218–227. doi: 10.1007/BF01197037 CrossRefGoogle Scholar
  13. Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83(2):143–198. doi: 10.1016/0045-7825(90)90148-F CrossRefzbMATHMathSciNetGoogle Scholar
  14. Guest JK (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199(1–4):123–135. doi: 10.1016/j.cma.2009.09.023 CrossRefzbMATHMathSciNetGoogle Scholar
  15. Guest JK, Asadpoure A, Ha S-H (2011) Eliminating beta-continuation from Heaviside projection and density filter algorithms. Struct Multidiscip Optim. doi: 10.1007/s00158-011-0676-1
  16. Guest JK, Prévost JH, Belytschko TB (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254. doi: 10.1002/nme.1064 CrossRefzbMATHGoogle Scholar
  17. Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1–2):1–12. doi: 10.1007/BF01279647 CrossRefGoogle Scholar
  18. Jalalpour M, Igusa T , Guest JK (2011) Optimal design of trusses with geometric imperfections: Accounting for global instability. Int J Solids Struct 48(21):3011–3019. doi: 10.1016/j.ijsolstr.2011.06.020 CrossRefGoogle Scholar
  19. Jog CS, Haber RB, Bendsøe MP (1994) Topology design with optimized, self-adaptive materials. Int J Numer Methods Eng 37(8):1323–1350. doi: 10.1002/nme.1620370805 CrossRefzbMATHGoogle Scholar
  20. Parkes EW (1975) Joints in optimum frameworks. Int J Solids Struct 11(9):1017–1022. doi: 10.1016/0020-7683(75)90044-X CrossRefzbMATHGoogle Scholar
  21. Poulsen TA (2003) A new scheme for imposing a minimum length scale in topology optimization. Int J Numer Methods Eng 57(6):741–760. doi: 10.1002/nme.694 CrossRefzbMATHMathSciNetGoogle Scholar
  22. Rozvany GIN (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 11(3–4):213–217. doi: 10.1007/BF01197036 CrossRefGoogle Scholar
  23. Rozvany GIN (2011) On symmetry and non-uniqueness in exact topology optimization. Struct Multidiscip Optim 43(3):297–317. doi: 10.1007/s00158-010-0564-0 CrossRefzbMATHMathSciNetGoogle Scholar
  24. Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20(4):351–368. doi: 10.1016/0167-6636(94)00069-7 CrossRefMathSciNetGoogle Scholar
  25. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424. doi: 10.1007/s00158-006-0087-x CrossRefGoogle Scholar
  26. Stolpe M (2004) Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound. Int J Numer Methods Eng 61(8):1270–1309. doi: 10.1002/nme.1112 CrossRefzbMATHMathSciNetGoogle Scholar
  27. Svanberg K (1987) The method of moving asymptotesa new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. doi: 10.1002/nme.1620240207 CrossRefzbMATHMathSciNetGoogle Scholar
  28. Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Olhoff N, Rozvany G I N (eds) Proceedings of the First World Congress of Structural and Multidisciplinary Optimization. Pergamon Press, Elmsford, pp 6–16Google Scholar
  29. Zhou M, Rozvany GIN (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
  30. Mu Z, Yang Y, Gaynor AT, Guest JK (2014) Considering Constructability in Structural Topology Optimization. In: Structures Congress 2014. American Society of Civil Engineers, Reston, pp 2754–2764. doi: 10.1061/9780784413357.241 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alireza Asadpoure
    • 1
  • James K. Guest
    • 2
  • Lorenzo Valdevit
    • 1
    Email author
  1. 1.Mechanical and Aerospace EngineeringUniversity of California IrvineIrvineUSA
  2. 2.Department of Civil EngineeringJohns Hopkins UniversityBaltimoreUSA

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