Structural and Multidisciplinary Optimization

, Volume 59, Issue 5, pp 1775–1788 | Cite as

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

  • Miche JansenEmail author
Research Paper


The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.


Topology optimization Level set XFEM Design regularization Minimum length scale 


Funding information

The work presented in this paper was performed in the framework of the Any-Shape 4.0 project supported by the Walloon Region (grant number 151066).


  1. Abdi M, Ashcroft I, Wildman R (2014) High resolution topology design with Iso-XFEM. In: Proceedings of the 25th annual international solid freeform fabrication symposium, pp 1288–1303Google Scholar
  2. Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefzbMATHGoogle Scholar
  3. Allaire G, Jouve F, Michailidis G (2016) Thickness control in structural optimization via a level set method. Struct Multidiscip Optim 53(6):1349–1382MathSciNetCrossRefGoogle Scholar
  4. Belytschko T, Parimi C, Moës N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56(4):609–635CrossRefzbMATHGoogle Scholar
  5. Bendsøe M (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1:193–202CrossRefGoogle Scholar
  6. Bendsøe M, Sigmund O (2004) Topology optimization: theory, methods and applications, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459CrossRefzbMATHGoogle Scholar
  9. Chen S, Wang M, Liu A (2008) Shape feature control in structural topology optimization. Comput Aided Des 40(9):951–962CrossRefGoogle Scholar
  10. Cook R, Malkus D, Plesha M, Witt R (2002) Concepts and applications of finite element analysis, 4th edn. Wiley, New YorkGoogle Scholar
  11. Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48(12):1741–1760CrossRefzbMATHGoogle Scholar
  12. Deaton J, Grandhi R (2013) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRefGoogle Scholar
  13. Dunning P (2018) Minimum length-scale constraints for parameterized implicit function based topology optimization. Struct Multidiscip Optim 58(1):155–169MathSciNetCrossRefGoogle Scholar
  14. Fries TP, Belytschko T (2010) The extended/generalized finite element method: An overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304MathSciNetzbMATHGoogle Scholar
  15. Geuzaine C, Remacle JF (2009) Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331MathSciNetCrossRefzbMATHGoogle Scholar
  16. Guest J, Prevost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254MathSciNetCrossRefzbMATHGoogle Scholar
  17. Guo X, Zhang W, Zhong W (2014) Explicit feature control in structural topology optimization via level set method. Comput Methods Appl Mech Eng 272:354–378MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jansen M, Lazarov B, Schevenels M, Sigmund O (2013) On the similarities between micro/nano lithography and topology optimization projection methods. Struct Multidiscip Optim 48(4):717–730CrossRefGoogle Scholar
  19. Jouve F, Mechkour H (2008) Level set based method for design of compliant mechanisms. Eur J Comput Mech 17(5–7):957–968CrossRefzbMATHGoogle Scholar
  20. Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 46(3):311–326MathSciNetCrossRefzbMATHGoogle Scholar
  21. Lazarov B, Schevenels M, Sigmund O (2012) Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct Multidiscip Optim 46(4):597–612MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lazarov B, Wang F, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86(1–2):189–218CrossRefGoogle Scholar
  23. Luo J, Luo Z, Chen S, Tong L, Wang M (2008) A new level set method for systematic design of hinge-free compliant mechanisms. Comput Methods Appl Mech Eng 198(2):318–331CrossRefzbMATHGoogle Scholar
  24. Makhija D, Maute K (2014) Numerical instabilities in level set topology optimization with the extended finite element method. Struct Multidiscip Optim 49(2):185–197MathSciNetCrossRefGoogle Scholar
  25. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150CrossRefzbMATHGoogle Scholar
  26. Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41 (8):1417–1434MathSciNetCrossRefzbMATHGoogle Scholar
  27. Rozvany G, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Multidiscip Optim 4:250–252CrossRefGoogle Scholar
  28. Schevenels M, Lazarov B, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49–52):3613– 3627CrossRefzbMATHGoogle Scholar
  29. Sethian J, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489–528MathSciNetCrossRefzbMATHGoogle Scholar
  30. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401– 424CrossRefGoogle Scholar
  31. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055MathSciNetCrossRefGoogle Scholar
  32. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidiscip Optim 16:68–75CrossRefGoogle Scholar
  33. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373MathSciNetCrossRefzbMATHGoogle Scholar
  34. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximation. SIAM J Optim 12:555–573MathSciNetCrossRefzbMATHGoogle Scholar
  35. van Dijk N, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472MathSciNetCrossRefGoogle Scholar
  36. Van Miegroet L (2012) Generalized shape optimization using XFEM and level set description. PhD thesis, University of Liege, Aerospace and Mechanical Engineering DepartmentGoogle Scholar
  37. Villanueva C, Maute K (2014) Density and level set-XFEM schemes for topology optimization of 3-D structures. Comput Mech 54(1):133–150MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246MathSciNetCrossRefzbMATHGoogle Scholar
  39. Wang F, Lazarov B, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43:767–784CrossRefzbMATHGoogle Scholar
  40. Wei P, Wang M, Xing X (2010) A study on X-FEM in continuum structural optimization using a level set model. Comput Aided Des 42(8):708–719CrossRefGoogle Scholar
  41. Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on Heaviside functions. Struct Multidiscip Optim 41:495–505MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhou M, Lazarov B, Sigmund O (2014) Topology optimization for optical projection lithography with manufacturing uncertainties. Appl Opt 53(12):2720–2729CrossRefGoogle Scholar
  43. Zhou M, Lazarov B, Wang F, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng 293:266–282MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CenaeroGosseliesBelgium

Personalised recommendations