Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

3D topology optimization for cost and time minimization in additive manufacturing


As the frontier of modern-day engineering challenges pushes forward, the integration of multiple strategies to reduce manufacturing cost and increase component performance has engineers turning to tools such as topology optimization (TO) and additive manufacturing (AM). Recent focus on these topics has led to the bridging of the gap between these two tools and the making of their integration in the conventional design cycle as seamless as possible. This paper expands upon existing mathematical constructs by providing an algorithm to minimize the cost and time associated with additively manufactured parts within a three-dimensional topology optimization framework. The formulation has been constructed in such a manner to accommodate large-scale topology optimization problems, including a filtering scheme requiring minimal storage of additional mesh information and an iterative finite element analysis solver. A rigorous trade-off analysis is conducted to determine the optimal contribution of additive manufacturing factors to minimize build time. A perimeter method-inspired approach for optimization of the surface area is explored, suggesting benefits for AM-specific process mechanics. Multiple academic example problems included in this work illustrate the applicability of this approach to three-dimensional geometries; physical models of these example problems created via fused filament fabrication serve to validate the numerical results obtained herein.

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
Fig. 17
Fig. 18
Fig. 19
Fig. 20


  1. Alund A, Lotstedt P, Sillen M (1997) Parallel single grid and multigrid solution of industrial compressible flow problems. Comput Fluids.

  2. Barrett R, Berry M, Chan TF et al (1995) Templates for the solution of linear systems: building blocks for iterative methods. Math Comput.

  3. Borrvall T, Petersson J (2001) Large-scale topology optimization in 3D using parallel computing. Comput Methods Appl Mech Eng.

  4. Brackett D, Ashcroft I, Hague R (2011a) Topology optimization for additive manufacturing. Solid Free Fabr Symp.

  5. Brackett D, Ashcroft I, Hague R (2011b) Topology optimization for additive manufacturing. Solid Free Fabr Symp:348–362.

  6. Caelers M (2017) Study of in - situ monitoring methods to create a robust SLM process. KTH Royal Institute Of Technology

  7. Calignano F (2014) Design optimization of supports for overhanging structures in aluminum and titanium alloys by selective laser melting. Mater Des.

  8. Clausen A, Andreassen E (2017) On filter boundary conditions in topology optimization. Struct Multidiscip Optim.

  9. Fayazfar H, Salarian M, Rogalsky A et al (2018) A critical review of powder-based additive manufacturing of ferrous alloys: process parameters, microstructure and mechanical properties. Mater Des.

  10. Gao W, Zhang Y, Ramanujan D et al (2015) The status, challenges, and future of additive manufacturing in engineering. Comput Des.

  11. Garaigordobil A, Ansola R, Santamaría J, Fernández de Bustos I (2018) A new overhang constraint for topology optimization of self-supporting structures in additive manufacturing. Struct Multidiscip Optim.

  12. Gaynor AT, Guest JK (2016) Topology optimization considering overhang constraints: eliminating sacrificial support material in additive manufacturing through design. Struct Multidiscip Optim 54:1157–1172.

  13. Guo X, Zhou J, Zhang W et al (2017) Self-supporting structure design in additive manufacturing through explicit topology optimization. Comput Methods Appl Mech Eng.

  14. Haber RB, Jog CS, Bendsoe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim.

  15. Hoffarth M, Gerzen N, Pedersen C (2017) ALM overhang constraint in topology optimization for industrial applications. In: World Congress on Structural and Multidisciplinary Optimization. p 10

  16. Kawamoto A, Matsumori T, Yamasaki S et al (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim 44:19–24.

  17. Langelaar M (2016) Topology optimization of 3D self-supporting structures for additive manufacturing. Addit Manuf 12:60–70.

  18. Langelaar M (2017) An additive manufacturing filter for topology optimization of print-ready designs. Struct Multidiscip Optim 55:871–883.

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

  20. Leary M, Merli L, Torti F, et al (2014) Optimal topology for additive manufacture: a method for enabling additive manufacture of support-free optimal structures

  21. Léonard F, Tammas-Williams S, Todd I (2016) CT for additive manufacturing process characterisation: assessment of melt strategies on defect population. In: 6th conference on industrial computed tomography

  22. Li C, Kim IY (2017) Multi-material topology optimization for automotive design problems. Proc Inst Mech Eng Part D J Automob Eng 0954407017

  23. Li D, Kim IY (2018) Multi-material topology optimization for practical lightweight design. Struct Multidiscip Optim.

  24. Li C, Kim IY, Jeswiet J (2015) Conceptual and detailed design of an automotive engine cradle by using topology, shape, and size optimization. Struct Multidiscip Optim 51:547–564.

  25. Liu J, Gaynor AT, Chen S et al (2018) Current and future trends in topology optimization for additive manufacturing. Struct Multidiscip Optim 57:2457–2483.

  26. Mirzendehdel AM, Suresh K (2016) Support structure constrained topology optimization for additive manufacturing. Comput Des 81:1–13.

  27. Moylan S, Slotwinski J, Cooke A, et al (2013) Lessons learned in establishing the NIST metal additive manufacturing laboratory

  28. Qian X (2017) Undercut and overhang angle control in topology optimization: a density gradient based integral approach. Int J Numer Methods Eng.

  29. Ranjan R, Samant R, Anand S (2017) Integration of design for manufacturing methods with topology optimization in additive manufacturing. J Manuf Sci Eng 139:061007.

  30. Ryan L, Kim IY (2019) A multiobjective topology optimization approach for cost and time minimization in additive manufacturing. Int J Numer Methods Eng.

  31. Sames WJ, List FA, Pannala S et al (2016) The metallurgy and processing science of metal additive manufacturing. Int Mater Rev

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

  33. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng.

  34. Thomas D (2010) The development of design rules for selective laser melting. Dissertation

  35. Wang D, Yang Y, Yi Z, Su X (2013) Research on the fabricating quality optimization of the overhanging surface in SLM process. Int J Adv Manuf Technol 65:1471–1484.

  36. Woischwill C, Kim IY (2018) Multimaterial multijoint topology optimization. Int J Numer Methods Eng

  37. Wong J, Ryan L, Kim IY (2018) Design optimization of aircraft landing gear assembly under dynamic loading. Struct Multidiscip Optim 57:1357–1375.

  38. Zegard T, Paulino GH (2016) Bridging topology optimization and additive manufacturing. Struct Multidiscip Optim.

  39. Zhou M, Lazarov B, Sigmund O (2014) Topology optimization for optical projection lithography with manufacturing uncertainties. Appl Opt.

Download references


This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Author information

Correspondence to Il Yong Kim.

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.

Responsible Editor: Xu Guo

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sabiston, G., Kim, I.Y. 3D topology optimization for cost and time minimization in additive manufacturing. Struct Multidisc Optim 61, 731–748 (2020).

Download citation


  • Topology optimization
  • Additive manufacturing
  • Support material
  • Surface area
  • Helmholtz PDE
  • Manufacturing constraint