Discrete topology optimization in augmented space: integrated element removal for minimum size and mesh sensitivity control


A computational strategy is proposed to circumvent some of the major issues that arise in the classical threshold-based approach to discrete topology optimization. These include the lack of an integrated element removal strategy to prevent the emergence of hair-like elements, the inability to effectively enforce a minimum member size of arbitrary magnitude, and high sensitivity of the final solution to the choice of ground structure. The proposed strategy draws upon the ideas used to arrive at mesh-independent solutions in continuum topology optimization and enables efficient imposition of a minimum size constraint onto the set of non vanishing elements. This is achieved via augmenting the design variables by a set of auxiliary variables, called existence variables, that not only prove very effective in addressing the aforementioned issues but also bring in a set of added benefits such as better convergence and complexity control. 2D and 3D examples from truss-like structures are presented to demonstrate the superiority of the proposed approach over the classical approach to discrete topology optimization.

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


  1. Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4-5):285–304. https://doi.org/10.1007/s00158-006-0092-0

    MathSciNet  Article  Google Scholar 

  2. Amir O (2013) A topology optimization procedure for reinforced concrete structures. Comput Struct 114-115:46–58, https://doi.org/10.1016/j.compstruc.2012.10.011

  3. Asadpoure A, Valdevit L (2015a) Topology optimization of lightweight periodic lattices under simultaneous compressive and shear stiffness constraints. Int J Solids Struct 60-61:1–16. https://doi.org/10.1016/j.ijsolstr.2015.01.016

    Article  Google Scholar 

  4. Asadpoure A, Guest JK, Valdevit L (2015b) Incorporating fabrication cost into topology optimization of discrete structures and lattices. Struct Multidiscip Optim 51(2):385–396. https://doi.org/10.1007/s00158-014-1133-8

    Article  Google Scholar 

  5. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202. https://doi.org/10.1007/BF01650949

    Article  Google Scholar 

  6. Bojczuk D, Mroz Z (1999) Optimal topology and configuration design of trusses with stress and buckling constraints. Struct Optim 17(1):25–35. https://doi.org/10.1007/BF01197710

    Article  Google Scholar 

  7. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38. https://doi.org/10.1007/s00158-013-0956-z

    MathSciNet  Article  Google Scholar 

  8. Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J Mecan 3(1):25–52

    Google Scholar 

  9. Gilbert M, Tyas A (2003) Layout optimization of large?scale pin?jointed frames. Eng Comput 20(8):1044–1064. https://doi.org/10.1108/02644400310503017

    Article  Google Scholar 

  10. Guest JK (2009) Imposing maximum length scale in topology optimization. Struct Multidiscip Optim 37(5):463–473. https://doi.org/10.1007/s00158-008-0250-7

    MathSciNet  Article  Google Scholar 

  11. 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. https://doi.org/10.1002/nme.1064

    MathSciNet  Article  Google Scholar 

  12. Guest JK, Asadpoure A, Ha SH (2011) Eliminating beta-continuation from Heaviside projection and density filter algorithms. Structural and Multidisciplinary Optimization. https://doi.org/10.1007/s00158-011-0676-1

  13. Hagishita T, Ohsaki M (2009) Topology optimization of trusses by growing ground structure method. Struct Multidiscip Optim 37(4):377–393. https://doi.org/10.1007/s00158-008-0237-4

    Article  Google Scholar 

  14. Havelia P (2016) A ground structure method to optimize topology and sizing of steel frame structures to minimize material, fabrication and erection cost. Stanford University

  15. He L, Gilbert M (2015) Rationalization of trusses generated via layout optimization. Struct Multidiscip Optim 52(4):677–694. https://doi.org/10.1007/s00158-015-1260-x

    MathSciNet  Article  Google Scholar 

  16. Kanno Y (2018a) Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach. Comput Optim Appl 71(2):403–433. https://doi.org/10.1007/s10589-018-0013-3

    MathSciNet  Article  Google Scholar 

  17. Kanno Y, Fujita S (2018b) Alternating direction method of multipliers for truss topology optimization with limited number of nodes: a cardinality-constrained second-order cone programming approach. Optim Eng 19(2):327–358. https://doi.org/10.1007/s11081-017-9372-3,1712.03385

  18. Kirsch1 U (1996) Integration of reduction and expansion processes in layout optimization. Struct Optim 11(1-2):13–18. https://doi.org/10.1007/BF01279649

    Article  Google Scholar 

  19. Michell A (1904) The limits of economy of material in frame-structures. Lond Edinb Dublin Philos Mag J Sci 8(47):589–597. https://doi.org/10.1080/14786440409463229

    Article  Google Scholar 

  20. Ohsaki M (1998) Simultaneous optimization of topology and geometry of a regular plane truss. Comput Struct 66(1):69–77. https://doi.org/10.1016/S0045-7949(97)00050-3

    Article  Google Scholar 

  21. Parkes EW (1975) Joints in optimum frameworks. Int J Solids Struct 11(9):1017–1022. https://doi.org/10.1016/0020-7683(75)90044-X

    Article  Google Scholar 

  22. Pritchard T, Gilbert M, Tyas A (2005) Plastic layout optimization of large-scale frameworks subject to multiple load cases, member self-weight and with joint length penalties. In: Proceedings of the 6th World Congress of Structural and Multidisciplinary Optimization, Rio de Janeiro

  23. Rozvany GIN (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 11(3-4):213–217. https://doi.org/10.1007/BF01197036

    Article  Google Scholar 

  24. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424. https://doi.org/10.1007/s00158-006-0087-x

    Article  Google Scholar 

  25. Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48(6):1031–1055. https://doi.org/10.1007/s00158-013-0978-6

    MathSciNet  Article  Google Scholar 

  26. Sigmund O, Aage N, Andreassen E (2016) On the (non-)optimality of Michell structures. Struct Multidiscip Optim 54(2):361–373. https://doi.org/10.1007/s00158-016-1420-7

    MathSciNet  Article  Google Scholar 

  27. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124. https://doi.org/10.1007/s001580100129

    Article  Google Scholar 

  28. Svanberg K (1987) The method of moving asymptotes–a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. https://doi.org/10.1002/nme.1620240207

    MathSciNet  Article  Google Scholar 

  29. Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Olhoff N, Rozvany GIN (eds) Proceedings of the First World Congress of Structural and Multidisciplinary Optimization, Pergamon Press. NY, Elmsford, pp 6–16

  30. Tin-Loi F (1999) A smoothing scheme for a minimum weight problem in structural plasticity. Struct Optim 17(4):279–285. https://doi.org/10.1007/BF01207004

    Article  Google Scholar 

  31. Torii AJ, Lopez RH, Miguel LFF (2016) Complexity control in truss optimization. Struct Multidiscip Optim 54(2):289–299. https://doi.org/10.1007/s00158-016-1403-8

  32. Yunkang S, Deqing Y (1998) A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech Sinica 14(2):179–185. https://doi.org/10.1007/BF02487752

    Article  Google Scholar 

  33. 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. https://doi.org/10.1016/0045-7825(91)90046-9

    Article  Google Scholar 

Download references


MT received financial support from the National Science Foundation under grants CMMI-1401575 and CMMI-1351742. AA also received support from UMass Dartmouth College of Engineering.

Author information



Corresponding author

Correspondence to Alireza Asadpoure.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

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

Replication of results

All the necessary data to reproduce the results reported here are provided in Sections 3 and 4.

Responsible Editor: James K Guest

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Asadpoure, A., Harati, M. & Tootkaboni, M. Discrete topology optimization in augmented space: integrated element removal for minimum size and mesh sensitivity control. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158-020-02630-3

Download citation


  • Topology optimization
  • Discrete design
  • Mesh sensitivity
  • Minimum size constraint
  • Hair-like elements
  • Complexity control