Skip to main content
SpringerLink
Log in

Length scale and manufacturability in density-based topology optimization

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Since its original introduction in structural design, density-based topology optimization has been applied to a number of other fields such as microelectromechanical systems, photonics, acoustics and fluid mechanics. The methodology has been well accepted in industrial design processes where it can provide competitive designs in terms of cost, materials and functionality under a wide set of constraints. However, the optimized topologies are often considered as conceptual due to loosely defined topologies and the need of postprocessing. Subsequent amendments can affect the optimized design performance and in many cases can completely destroy the optimality of the solution. Therefore, the goal of this paper is to review recent advancements in obtaining manufacturable topology-optimized designs. The focus is on methods for imposing minimum and maximum length scales, and ensuring manufacturable, well-defined designs with robust performances. The overview discusses the limitations, the advantages and the associated computational costs. The review is completed with optimized designs for minimum compliance, mechanism design and heat transfer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Aage, N., Andreassen, E., Lazarov, B.: Topology optimization using petsc: an easy-to-use, fully parallel, open source topology optimization framework. Struct. Multidiscip. Optim. 51(3), 565–572 (2015). doi:10.1007/s00158-014-1157-0

    Article  MathSciNet  Google Scholar 

  2. Aage, N., Lazarov, B.: Parallel framework for topology optimization using the method of moving asymptotes. Struct. Multidiscip. Optim. 47(4), 493–505 (2013). doi:10.1007/s00158-012-0869-2

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexandersen, J., Lazarov, B.: Tailoring macroscale response of mechanical and heat transfer systems by topology optimization of microstructural details. In: Lagaros, N.D., Papadrakakis, M. (eds.) Engineering and Applied Sciences Optimization, Computational Methods in Applied Sciences, vol. 38, pp. 267–288. Springer, Berlin (2015). doi:10.1007/978-3-319-18320-6_15

    Chapter  Google Scholar 

  4. Alexandersen, J., Lazarov, B.S.: Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput. Methods Appl. Mech. Eng. 290, 156–182 (2015). doi:10.1016/j.cma.2015.02.028

    Article  MathSciNet  Google Scholar 

  5. Allaire, G., Francfort, G.: A numerical algorithm for topology and shape optimization. In: Bendse, M., Soares, C. (eds.) Topology Design of Structures, NATO ASI Series, vol. 227, pp. 239–248. Springer, Netherlands (1993). doi:10.1007/978-94-011-1804-0_16

    Chapter  Google Scholar 

  6. Allaire, G., Jouve, F., Michailidis, G.: Thickness control in structural optimization via a level set method (2014). https://hal.archives-ouvertes.fr/hal-00985000

  7. Allaire, G., Jouve, F., Toader, A.M.: A level-set method for shape optimization. Comptes Rendus Mathematique 334(12), 1125–1130 (2002). doi:10.1016/S1631-073X(02)02412-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004). doi:10.1016/j.jcp.2003.09.032

    Article  MathSciNet  MATH  Google Scholar 

  9. Allaire, G., Kohn, R.: Topology optimization and optimal shape design using homogenization. In: Bendse, M., Soares, C. (eds.) Topology Design of Structures, NATO ASI Series, vol. 227, pp. 207–218. Springer, Netherlands (1993). doi:10.1007/978-94-011-1804-0_14

    Chapter  Google Scholar 

  10. Amir, O.: Revisiting approximate reanalysis in topology optimization: on the advantages of recycled preconditioning in a minimum weight procedure. Struct. Multidiscip. Optim. 51(1), 41–57 (2015). doi:10.1007/s00158-014-1098-7

    Article  Google Scholar 

  11. Amir, O., Aage, N., Lazarov, B.: On multigrid-CG for efficient topology optimization. Struct. Multidiscip. Optim. 49(5), 815–829 (2014). doi:10.1007/s00158-013-1015-5

    Article  MathSciNet  Google Scholar 

  12. Amir, O., Sigmund, O.: On reducing computational effort in topology optimization: how far can we go? Struct. Multidiscip. Optim. 44(1), 25–29 (2011). doi:10.1007/s00158-010-0586-7

    Article  MATH  Google Scholar 

  13. Amir, O., Sigmund, O., Lazarov, B.S., Schevenels, M.: Efficient reanalysis techniques for robust topology optimization. Comput. Methods Appl. Mech. Eng. 245246, 217–231 (2012). doi:10.1016/j.cma.2012.07.008

    Article  MathSciNet  Google Scholar 

  14. Andreassen, E., Lazarov, B.S., Sigmund, O.: Design of manufacturable 3D extremal elastic microstructure. Mech. Mater. 69(1), 1–10 (2014). doi:10.1016/j.mechmat.2013.09.018

    Article  Google Scholar 

  15. Arora, J., Wang, Q.: Review of formulations for structural and mechanical system optimization. Struct. Multidiscip. Optim. 30(4), 251–272 (2005). doi:10.1007/s00158-004-0509-6

    Article  MathSciNet  MATH  Google Scholar 

  16. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  17. Bendsøe, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1(4), 193–202 (1989). doi:10.1007/BF01650949

    Article  Google Scholar 

  18. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988). doi:10.1016/0045-7825(88)90086-2, http://www.sciencedirect.com/science/article/pii/0045782588900862

  19. Bendsøe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69, 635–654 (1999)

    Article  Google Scholar 

  20. Bendsøe, M.P., Sigmund, O.: Topology Optimization—Theory, Methods and Applications. Springer, Berlin (2003)

    Google Scholar 

  21. Beyer, H.G., Sendhoff, B.: Robust optimization—a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196(33–34), 3190–3218 (2007). doi:10.1016/j.cma.2007.03.003

    Article  MathSciNet  MATH  Google Scholar 

  22. Borrvall, T., Petersson, J.: Topology optimization using regularized intermediate density control. Comput. Methods Appl. Mech. Eng. 190(37–38), 4911–4928 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bourdin, B.: Filters in topology optimization. Int. J. Numer. Methods Eng. 50, 2143–2158 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. ESAIM: Control Optim. Calc. Var. 9, 19–48 (2003). doi:10.1051/cocv:2002070

    Article  MathSciNet  MATH  Google Scholar 

  25. Brackett, D., Ashcroft, I., Hague, R.: Topology optimization for additive manufacturing. In: Proceedings of the 22nd Annual International Solid Freeform Fabrication Symposium (2011)

  26. Brunner, T.A., Ferguson, R.A.: Approximate models for resist processing effects. Proc. SPIE 2726, 198–207 (1996). doi:10.1117/12.240906

    Article  Google Scholar 

  27. Bruns, T.E., Tortorelli, D.A.: Topology optimization of non-linear elastic structures and compliant mechanisms. Comput. Methods Appl. Mech. Eng. 190, 3443–3459 (2001)

    Article  MATH  Google Scholar 

  28. Bückmann, T., Stenger, N., Kadic, M., Kaschke, J., Frölich, A., Kennerknecht, T., Eberl, C., Thiel, M., Wegener, M.: Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv. Mater. 24(20), 2710–2714 (2012). doi:10.1002/adma.201200584

    Article  Google Scholar 

  29. Chang, T.H.P.: Proximity effect in electronbeam lithography. J. Vac. Sci. Technol. 12(6), 1271–1275 (1975). doi:10.1116/1.568515, http://scitation.aip.org/content/avs/journal/jvst/12/6/10.1116/1.568515

  30. Chen, S., Chen, W.: A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct. Multidiscip. Optim. 44(1), 1–18 (2011). doi:10.1007/s00158-011-0660-9

    Article  MathSciNet  MATH  Google Scholar 

  31. Chen, S., Chen, W., Lee, S.: Level set based robust shape and topology optimization under random field uncertainties. Struct. Multidiscip. Optim. 41, 507–524 (2010). doi:10.1007/s00158-009-0449-2

    Article  MathSciNet  MATH  Google Scholar 

  32. Christiansen, R.E., Lazarov, B.S., Jensen, J.S., Sigmund, O.: Creating geometrically robust designs for highly sensitive problems using topology optimization. Struct. Multidiscipl. Optim. 1–18 (2015). doi:10.1007/s00158-015-1265-5

  33. Clausen, A., Aage, N., Sigmund, O.: Topology optimization with flexible void area. Struct. Multidiscip. Optim. 50(6), 927–943 (2014). doi:10.1007/s00158-014-1109-8

    Article  MathSciNet  Google Scholar 

  34. Clausen, A., Aage, N., Sigmund, O.: Topology optimization of coated structures and material interface problems. Comput. Methods Appl. Mech. Eng. 290, 524–541 (2015). doi:10.1016/j.cma.2015.02.011

    Article  MathSciNet  Google Scholar 

  35. Coelho, P., Fernandes, P., Rodrigues, H., Cardoso, J., Guedes, J.: Numerical modeling of bone tissue adaptational hierarchical approach for bone apparent density and trabecular structure. J. Biomech. 42(7), 830–837 (2009). doi:10.1016/j.jbiomech.2009.01.020

  36. Cui, Z.: Nanofabrication: Principles, Capabilities and Limits. Springer, US (2008). doi:10.1007/978-0-387-75577-9_2

    Book  Google Scholar 

  37. Deaton, J.D., Grandhi, R.V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49(1), 1–38 (2014). doi:10.1007/s00158-013-0956-z

    Article  MathSciNet  Google Scholar 

  38. Diaz, A., Sigmund, O.: Checkerboard patterns in layout optimization. Struct. Multidiscip. Optim. 10, 40–45 (1995)

    Article  Google Scholar 

  39. Dill, F., Neureuther, A.R., Tuttle, J., Walker, E.: Modeling projection printing of positive photoresists. IEEE Trans. Electron Devices 22(7), 456–464 (1975). doi:10.1109/T-ED.1975.18161

    Article  Google Scholar 

  40. Donoso, A., Sigmund, O.: Topology optimization of piezo modal transducers with null-polarity phases. Struct. Multidiscip. Optim. (2015). doi:10.1007/s00158-015-1330-0

  41. Doyle, K.: Bioprinting: from patches to parts. Genet. Eng. Biotechnol. News 34, 34–35 (2014). doi:10.1089/gen.34.10.02

    Google Scholar 

  42. Dunning, P.D., Kim, H.A.: Robust topology optimization: minimization of expected and variance of compliance. AIAA J. 51(11), 2656–2664 (2013). doi:10.2514/1.J052183

    Article  Google Scholar 

  43. Elesin, Y., Lazarov, B., Jensen, J., Sigmund, O.: Design of robust and efficient photonic switches using topology optimization. Photonics Nanostruct. Fundam. Appl. 10(1), 153–165 (2012). doi:10.1016/j.photonics.2011.10.003

    Article  Google Scholar 

  44. Elesin, Y., Lazarov, B., Jensen, J., Sigmund, O.: Time domain topology optimization of 3D nanophotonic devices. Photonics Nanostruct. Fundam. Appl. 12(1), 23–33 (2014). doi:10.1016/j.photonics.2013.07.008

    Article  Google Scholar 

  45. Elishakoff, I., Ohsaki, M.: Optimization and Anti-optimization of Structures Under Uncertainty. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  46. Evgrafov, A.: State space newton’s method for topology optimization. Comput. Methods Appl. Mech. Eng. 278, 272–290 (2014). doi:10.1016/j.cma.2014.06.005

    Article  MathSciNet  Google Scholar 

  47. Evgrafov, A.: On chebyshevs method for topology optimization of stokes flows. Struct. Multidiscip. Optim. 1–11 (2015). doi:10.1007/s00158-014-1176-x

  48. Evgrafov, A., Rupp, C., Maute, K., Dunn, M.: Large-scale parallel topology optimization using a dual-primal substructuring solver. Struct. Multidiscip. Optim. 36(4), 329–345 (2008). doi:10.1007/s00158-007-0190-7

    Article  MathSciNet  MATH  Google Scholar 

  49. Frangopol, D.M., Maute, K.: Life-cycle reliability-based optimization of civil and aerospace structures. Comput. Struct. 81(7), 397–410 (2003). doi:10.1016/S0045-7949(03)00020-8

    Article  Google Scholar 

  50. Gan, Z., Cao, Y., Evans, R.A., Gu, M.: Three-dimensional deep sub-diffraction optical beam lithography with 9nm feature size. Nat. Commun. 4 (2013). doi:10.1038/ncomms3061

  51. Gaynor, A.T., Meisel, N.A., Williams, C.B., Guest, J.K.: Multiple-material topology optimization of compliant mechanisms created via polyjet three-dimensional printing. J. Manuf. Sci. Eng. 136, 061015 (2014). doi:10.1115/1.4028439

    Article  Google Scholar 

  52. Gaynor, A.T., Meisel, N.A., Williams, C.B., Guest, J.K.: Topology optimization for additive manufacturing: Considering maximum overhang constraint. In: AIAA Aviation, pp. –. American Institute of Aeronautics and Astronautics (2014). doi:10.2514/6.2014-2036

  53. Gersborg, A.R., Andreasen, C.S.: An explicit parameterization for casting constraints in gradient driven topology optimization. Struct. Multidiscip. Optim. 44(6), 875–881 (2011). doi:10.1007/s00158-011-0632-0

    Article  Google Scholar 

  54. Ghanem, R., Spanos, P.: Stochastic Finite Elements—A Spectral Approach. Dover Publications Inc, Mineola (2003)

    Google Scholar 

  55. Gibson, I., Rosen, D.W., Stucker, B.: Additive Manufacturing Technologies. Springer, Berlin (2010)

    Book  Google Scholar 

  56. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987). doi:10.1016/0021-9991(87)90140-9

    Article  MathSciNet  MATH  Google Scholar 

  57. Groover, M.P.: Fundamentals of Modern Manufacturing: Materials, Processes, and Systems. Wiley, New York (2010)

    Google Scholar 

  58. Guest, J.: Imposing maximum length scale in topology optimization. Struct. Multidiscip. Optim. 37, 463–473 (2009). doi:10.1007/s00158-008-0250-7

    Article  MathSciNet  MATH  Google Scholar 

  59. Guest, J., Asadpoure, A., Ha, S.H.: Eliminating beta-continuation from heaviside projection and density filter algorithms. Struct. Multidiscip. Optim. 44(4), 443–453 (2011). doi:10.1007/s00158-011-0676-1

    Article  MathSciNet  MATH  Google Scholar 

  60. Guest, J., Prevost, J., Belytschko, T.: Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int. J. Numer. Methods Eng. 61(2), 238–254 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  61. Guest, J.K.: Topology optimization with multiple phase projection. Comput. Methods Appl. Mech. Eng. 199(1–4), 123–135 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. Guest, J.K.: Optimizing the layout of discrete objects in structures and materials: a projection-based topology optimization approach. Comput. Methods Appl. Mech. Eng. 283, 330–351 (2015). doi:10.1016/j.cma.2014.09.006

    Article  Google Scholar 

  63. Guest, J.K., Ha, S.H.: Design of Multifunctional Material Structures Using Topology Optimization with Feature Control, pp. 129–134. Wiley, New York (2011). doi:10.1002/9781118147726.ch17

    Google Scholar 

  64. Guest, J.K., Prvost, J.H.: Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int. J. Solids Struct. 43(2223), 7028–7047 (2006). doi:10.1016/j.ijsolstr.2006.03.001

    Article  MATH  Google Scholar 

  65. Guest, J.K., Smith Genut, L.C.: Reducing dimensionality in topology optimization using adaptive design variable fields. Int. J. Numer. Methods Eng. 81(8), 1019–1045 (2010). doi:10.1002/nme.2724

    MATH  Google Scholar 

  66. Guest, J.K., Zhu, M.: Casting and milling restrictions in topology optimization via projection-based algorithms. In: ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference Volume 3: 38th Design Automation Conference, Parts A and B Chicago, Illinois, USA, August, p. 1215 (2012)

  67. Guo, X., Zhang, W., Zhong, W.: Explicit feature control in structural topology optimization via level set method. Comput. Methods Appl. Mech. Eng. 272, 354–378 (2014). doi:10.1016/j.cma.2014.01.010

    Article  MathSciNet  MATH  Google Scholar 

  68. Ha, S.H., Guest, J.: Optimizing inclusion shapes and patterns in periodic materials using discrete object projection. Struct. Multidiscip. Optim. 50(1), 65–80 (2014). doi:10.1007/s00158-013-1026-2

    Article  Google Scholar 

  69. Haber, R., Jog, C., Bendsoe, M.: A new approach to variable-topology shape design using a constraint on perimeter. Struct. Optim. 11(1), 1–12 (1996)

    Article  Google Scholar 

  70. Hannon, A.F., Ding, Y., Bai, W., Ross, C.A., Alexander-Katz, A.: Optimizing topographical templates for directed self-assembly of block copolymers via inverse design simulations. Nano Lett. 14(1), 318–325 (2014). doi:10.1021/nl404067s

    Article  Google Scholar 

  71. Harzheim, L., Graf, G.: A review of optimization of cast parts using topology optimization. Struct. Multidiscip. Optim. 31(5), 388–399 (2006). doi:10.1007/s00158-005-0554-9

    Article  Google Scholar 

  72. Jansen, M., Lazarov, B.S., Schevenels, M., Sigmund, O.: On the similarities between micro/nano lithography and topology optimization projection methods. Struct. Multidiscip. Optim. 48(4), 717–730 (2013). doi:10.1007/s00158-013-0941-6

    Article  MathSciNet  Google Scholar 

  73. Jansen, M., Lombaert, G., Diehl, M., Lazarov, B.S., Sigmund, O., Schevenels, M.: Robust topology optimization accounting for misplacement of material. Struct. Multidiscip. Optim. 47, 317–333 (2013). doi:10.1007/s00158-012-0835-z

    Article  MathSciNet  MATH  Google Scholar 

  74. Jansen, M., Lombaert, G., Schevenels, M.: Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput. Methods Appl. Mech. Eng. 285, 452–467 (2015). doi:10.1016/j.cma.2014.11.028

    Article  MathSciNet  Google Scholar 

  75. Jansen, M., Lombaert, G., Schevenels, M., Sigmund, O.: Topology optimization of fail-safe structures using a simplified local damage model. Struct. Multidiscip. Optim. 49(4), 657–666 (2014). doi:10.1007/s00158-013-1001-y

    Article  MathSciNet  Google Scholar 

  76. Jog, C.S., Haber, R.B.: Stability of finite element models for distributed-parameter optimization and topology design. Comput. Methods Appl. Mech. Eng. 130(3–4), 203–226 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  77. Kawamoto, A., Matsumori, T., Kondoh, T., Nomura, T., Yamasaki, S., Nishiwaki, S.: Topology optimization by using a time-dependent diffusion equation. In: Proceedings of 9th World Congress on Structural and Multidisciplinary Optimization, Shizuoka, Japan (2011)

  78. Kawamoto, A., Matsumori, T., Yamasaki, S., Nomura, T., Kondoh, T., Nishiwaki, S.: Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct. Multidiscip. Optim. 44, 19–24 (2011). doi:10.1007/s00158-010-0562-2

    Article  MATH  Google Scholar 

  79. Kawata, S., Sun, H.B., Tanaka, T., Takada, K.: Finer features for functional microdevices. Nature 412, 697–698 (2001)

    Article  Google Scholar 

  80. Kharmanda, G., Olhoff, N., Mohamed, A., Lemaire, M.: Reliability-based topology optimization. Struct. Multidiscip. Optim. 26, 295–307 (2004)

    Article  Google Scholar 

  81. Kim, T.S., Kim, J.E., Jeong, J.H., Kim, Y.Y.: Filtering technique to control member size in topology design optimization. KSME Int. J. 18(2), 253–261 (2004). doi:10.1007/BF03184735

    Google Scholar 

  82. Kogiso, N., Ahn, W., Nishiwaki, S., Izui, K., Yoshimura, M.: Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J. Adv. Mech. Des. Syst. Manif. 2(1), 96–107 (2008). doi:10.1299/jamdsm.2.96

    Google Scholar 

  83. Kreisselmeier, G., Steinhauser, R.: Application of vector performance optimization to a robust control loop design for a fighter aircraft. Int. J. Control 37(2), 251–284 (1983). doi:10.1080/00207179.1983.9753066

    Article  MATH  Google Scholar 

  84. Landis, S. (ed.): Nano-Lithography. Wiley, New York (2013)

    Google Scholar 

  85. Lazarov, B., Matzen, R., Elesin, Y.: Topology optimization of pulse shaping filters using the hilbert transform envelope extraction. Struct. Multidiscip. Optim. 44, 409–419 (2011). doi:10.1007/s00158-011-0642-y

    Article  MathSciNet  MATH  Google Scholar 

  86. Lazarov, B., Sigmund, O.: Sensitivity filters in topology optimisation as a solution to Helmholtz type differential equation. In: Proceedings of the 8th World Congress on Structural and Multidisciplinary Optimization (2009)

  87. Lazarov, B., Wang, F.: Maximum length scale in density based topology optimization (2015) (in review)

  88. Lazarov, B.S.: Topology optimization using multiscale finite element method for high-contrast media. In: Lirkov, I., Margenov, S., Waniewski, J. (eds.) Large-Scale Scientific Computing, Lecture Notes in Computer Science, pp. 339–346. Springer, Berlin (2014). doi:10.1007/978-3-662-43880-0_38

    Google Scholar 

  89. Lazarov, B.S., Schevenels, M., Sigmund, O.: Robust design of large-displacement compliant mechanisms. Mech. Sci. 2(2), 175–182 (2011). doi:10.5194/ms-2-175-2011

    Article  Google Scholar 

  90. Lazarov, B.S., Schevenels, M., Sigmund, O.: Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct. Multidiscip. Optim. 46, 597–612 (2012). doi:10.1007/s00158-012-0791-7

    Article  MathSciNet  MATH  Google Scholar 

  91. Lazarov, B.S., Schevenels, M., Sigmund, O.: Topology optimization with geometric uncertainties by perturbation techniques. Int. J. Numer. Methods Eng. 90(11), 1321–1336 (2012). doi:10.1002/nme.3361

    Article  MATH  Google Scholar 

  92. Lazarov, B.S., Sigmund, O.: Filters in topology optimization based on Helmholtz-type differential equations. Int. J. Numer. Methods Eng. 86(6), 765–781 (2011). doi:10.1002/nme.3072

    Article  MathSciNet  MATH  Google Scholar 

  93. Le, C., Norato, J., Bruns, T., Ha, C., Tortorelli, D.: Stress-based topology optimization for continua. Struct. Multidiscip. Optim. 41(4), 605–620 (2010). doi:10.1007/s00158-009-0440-y

    Article  Google Scholar 

  94. Leary, M., Merli, L., Torti, F., Mazur, M., Brandt, M.: Optimal topology for additive manufacture: a method for enabling additive manufacture of support-free optimal structures. Mater. Des. 63, 678–690 (2014). doi:10.1016/j.matdes.2014.06.015

    Article  Google Scholar 

  95. Liu, J., Ma, Y.S.: 3d level-set topology optimization: a machining feature-based approach. Struct. Multidiscip. Optim. 1–20 (2015). doi:10.1007/s00158-015-1263-7

  96. Liu, S., Li, Q., Chen, W., Hu, R., Tong, L.: H-DGTP-a heaviside-function based directional growth topology parameterization for design optimization of stiffener layout and height of thin-walled structures. Struct. Multidiscip. Optim. 1–11 (2015). doi:10.1007/s00158-015-1281-5

  97. Lógó, J., Ghaemi, M., Rad, M.: Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech. Based Des. Struct. Mach. 37(3), 327–348 (2009)

    Article  Google Scholar 

  98. Mack, C.: Fundamental Principles of Optical Lithography: The Science of Microfabrication. Wiley, New York (2007)

    Book  Google Scholar 

  99. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952). doi:10.1111/j.1540-6261.1952.tb01525.x

    Google Scholar 

  100. Maruo, S., Nakamura, O., Kawata, S.: Three-dimensional microfabrication with two-photon-absorbed photopolymerization. Opt. Lett. 22(2), 132–134 (1997). doi:10.1364/OL.22.000132

    Article  Google Scholar 

  101. Maute, K.: Topology optimization under uncertainty. In: Rozvany, G., Lewiski, T. (eds.) Topology Optimization in Structural and Continuum Mechanics, CISM International Centre for Mechanical Sciences, vol. 549, pp. 457–471. Springer, Vienna (2014). doi:10.1007/978-3-7091-1643-2_20

    Chapter  Google Scholar 

  102. Maute, K., Frangopol, D.: Reliability-based design of mems mechanisms by topology optimization. Comput. Struct. 81, 813–824 (2003)

    Article  Google Scholar 

  103. Mei, Y., Wang, X., Cheng, G.: A feature-based topological optimization for structure design. Adv. Eng. Softw. 39(2), 71–87 (2008). doi:10.1016/j.advengsoft.2007.01.023

    Article  Google Scholar 

  104. Men, H., Freund, R.M., Nguyen, N.C., Saa-Seoane, J., Peraire, J.: Fabrication-adaptive optimization with an application to photonic crystal design. Oper. Res. 62(2), 418–434 (2014). doi:10.1287/opre.2013.1252

    Article  MathSciNet  MATH  Google Scholar 

  105. Michailidis, G.: Manufacturing constraints and multi-phase shape and topology optimization via a level-set method. Ph.D. thesis, Ecole Polytechnique (2014). https://pastel.archives-ouvertes.fr/pastel-00937306

  106. Mlejnek, H.: Some aspects of the genesis of structures. Struct. Optim. 5(1–2), 64–69 (1992). doi:10.1007/BF01744697

    Article  Google Scholar 

  107. Mohammad, M., Muhammad, M., Dew, S.K., Stepanova, M.: Fundamentals of electron beam exposure and development. In: Stepanova, M., Dew, S. (eds.) Nanofabrication, pp. 11–41. Springer, Vienna (2012). doi:10.1007/978-3-7091-0424-8_2

    Chapter  Google Scholar 

  108. Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley, New York (2013). doi:10.1002/9781118600788

    Book  Google Scholar 

  109. Nomura, T., Sato, K., Taguchi, K., Kashiwa, T., Nishiwaki, S.: Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique. Int. J. Numer. Methods Eng. 71(11), 1261–1296 (2007). doi:10.1002/nme.1974

    Article  MATH  Google Scholar 

  110. Norato, J., Bell, B., Tortorelli, D.: A geometry projection method for continuum-based topology optimization with discrete elements. Comput. Methods Appl. Mech. Eng. (2015). doi:10.1016/j.cma.2015.05.005, http://www.sciencedirect.com/science/article/pii/S0045782515001711

  111. Oropallo, W., Piegl, L.: Ten challenges in 3D printing. Eng. Comput. 1–14 (2015). doi:10.1007/s00366-015-0407-0

  112. Poonawala, A., Milanfar, P.: Mask design for optical microlithography—an inverse imaging problem. IEEE Trans. Image Process. 16(3), 774–788 (2007). doi:10.1109/TIP.2006.891332

    Article  MathSciNet  Google Scholar 

  113. Poulsen, T.A.: A new scheme for imposing a minimum length scale in topology optimization. Int. J. Numer. Methods Eng. 57, 741–760 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  114. Qian, X., Sigmund, O.: Topological design of electromechanical actuators with robustness toward over- and under-etching. Comput. Methods Appl. Mech. Eng. 253, 237–251 (2013). doi:10.1016/j.cma.2012.08.020

    Article  MathSciNet  MATH  Google Scholar 

  115. Rehman, S., Langelaar, M., van Keulen, F.: Robust optimization of 2x2 multimode interference couplers with fabrication uncertainties. Proc. SPIE 8627, 862713 (2013). doi:10.1117/12.2000647

    Article  Google Scholar 

  116. Rehman, S., Langelaar, M., van Keulen, F.: Efficient kriging-based robust optimization of unconstrained problems. J. Comput. Sci. 5(6), 872–881 (2014). doi:10.1016/j.jocs.2014.04.005, http://www.sciencedirect.com/science/article/pii/S1877750314000477

  117. Rodrigues, H., Guedes, J., Bendsoe, M.: Hierarchical optimization of material and structure. Struct. Multidiscip. Optim. 24(1), 1–10 (2002). doi:10.1007/s00158-002-0209-z

    Article  Google Scholar 

  118. Rojas-Labanda, S., Stolpe, M.: Benchmarking optimization solvers for structural topology optimization. Struct. Multidiscip. Optim. 1–21 (2015). doi:10.1007/s00158-015-1250-z

  119. Ruszczynski, A., Shapiro, A.: Optimization of convex risk functions. Mathe. Oper. Res. 31(3), 433–452 (2006). doi:10.1287/moor.1050.0186

    Article  MathSciNet  MATH  Google Scholar 

  120. Saxena, A.: Topology design with negative masks using gradient search. Struct. Multidiscip. Optim. 44(5), 629–649 (2011). doi:10.1007/s00158-011-0649-4

    Article  Google Scholar 

  121. Schevenels, M., Lazarov, B., Sigmund, O.: Robust topology optimization accounting for spatially varying manufacturing errors. Comput. Methods Appl. Mech. Eng. 200(49–52), 3613–3627 (2011). doi:10.1016/j.cma.2011.08.006

    Article  MATH  Google Scholar 

  122. Sigmund, O.: On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25, 493–524 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  124. Sigmund, O.: Manufacturing tolerant topology optimization. Acta Mech. Sin. 25, 227–239 (2009)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  126. Sokolowski, J., Zochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999). doi:10.1137/S0363012997323230

    Article  MathSciNet  MATH  Google Scholar 

  127. Solomon, C., Breckon, T.: Fundamentals of Digital Image Processing: A Practical Approach with Examples in Matlab. Wiley, New York (2010)

    Book  Google Scholar 

  128. Sørensen, R., Lund, E.: Thickness filters for gradient based multi-material and thickness optimization of laminated composite structures. Struct. Multidiscip. Optim. 1–24 (2015). doi:10.1007/s00158-015-1230-3

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

    Article  Google Scholar 

  130. Stolpe, M., Svanberg, K.: On the trajectories of penalization methods for topology optimization. Struct. Multidiscip. Optim. 21(2), 128–139 (2001). doi:10.1007/s001580050177

    Article  Google Scholar 

  131. Suzuki, K.: Electron beam lithography systems. In: Microlithography, pp. 329–360. CRC Press (2007). doi:10.1201/9781420051537.ch6

  132. Svanberg, K.: The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  133. Svanberg, K.: A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12, 555–573 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  134. Svanberg, K., Svärd, H.: Density filters for topology optimization based on the pythagorean means. Struct. Multidiscip. Optim. 48(5), 859–875 (2013). doi:10.1007/s00158-013-0938-1

    Article  MathSciNet  Google Scholar 

  135. Takezawa, A., Kobashi, M., Kitamura, M.: Porous composite with negative thermal expansion obtained by photopolymer additive manufacturing (2015). http://arxiv.org/abs/1504.07724v1

  136. Tomlin, M., Meyer, J.: Topology optimization of an additive layer manufactured (ALM) aerospace part. In: The 7th Altair CAE Technology Conference (2011)

  137. Tootkaboni, M., Asadpoure, A., Guest, J.K.: Topology optimization of continuum structures under uncertainty a polynomial chaos approach. Comput. Methods Appl. Mech. Eng. 201–204, 263–275 (2012). doi:10.1016/j.cma.2011.09.009

    Article  MathSciNet  Google Scholar 

  138. Travitzky, N., Bonet, A., Dermeik, B., Fey, T., Filbert-Demut, I., Schlier, L., Schlordt, T., Greil, P.: Additive manufacturing of ceramic-based materials. Adv. Eng. Mater. 16(6), 729–754 (2014). doi:10.1002/adem.201400097

    Article  Google Scholar 

  139. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  140. Tumbleston, J.R., Shirvanyants, D., Ermoshkin, N., Janusziewicz, R., Johnson, A.R., Kelly, D., Chen, K., Pinschmidt, R., Rolland, J.P., Ermoshkin, A., Samulski, E.T., DeSimone, J.M.: Continuous liquid interface production of 3D objects. Science 347(6228), 1349–1352 (2015). doi:10.1126/science.aaa2397

    Article  Google Scholar 

  141. van Dijk, N., Maute, K., Langelaar, M., van Keulen, F.: Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48(3), 437–472 (2013). doi:10.1007/s00158-013-0912-y

    Article  MathSciNet  Google Scholar 

  142. Vassilevski, P.S.: Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York (2008)

    Google Scholar 

  143. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006). doi:10.1007/s10107-004-0559-y

    Article  MathSciNet  MATH  Google Scholar 

  144. Wadbro, E., Udawalpola, R., Berggren, M.: Shape and topology optimization of an acoustic horn–lens combination. J. Comput. Appl. Math. 234(6), 1781–1787 (2010). doi:10.1016/j.cam.2009.08.028

    Article  MATH  Google Scholar 

  145. Wang, F., Jensen, J.S., Sigmund, O.: Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. J. Opt. Soc. Am. B 28(3), 387–397 (2011). doi:10.1364/JOSAB.28.000387

    Article  Google Scholar 

  146. Wang, F., Jensen, J.S., Sigmund, O.: High-performance slow light photonic crystal waveguides with topology optimized or circular-hole based material layouts. Photonics Nanostruct. Fundam. Appl. 10(4), 378–388 (2012). doi:10.1016/j.photonics.2012.04.004 (TaCoNa-Photonics 2011)

  147. Wang, F., Lazarov, B., Sigmund, O.: On projection methods, convergence and robust formulations in topology optimization. Struct. Multidiscip. Optim. 43(6), 767–784 (2011). doi:10.1007/s00158-010-0602-y

    Article  MATH  Google Scholar 

  148. Wang, F., Lazarov, B.S., Sigmund, O., Jensen, J.S.: Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Comput. Methods Appl. Mech. Eng. 276, 453–472 (2014). doi:10.1016/j.cma.2014.03.021

    Article  MathSciNet  Google Scholar 

  149. Wang, F., Sigmund, O., Jensen, J.S.: Design of materials with prescribed nonlinear properties. J. Mech. Phys. Solids 69(1), 156–174 (2014). doi:10.1016/j.jmps.2014.05.003

    Article  MathSciNet  Google Scholar 

  150. Wang, M., Zhou, S., Ding, H.: Nonlinear diffusions in topology optimization. Struct. Multidiscip. Optim. 28, 262–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  151. Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(12), 227–246 (2003). doi:10.1016/S0045-7825(02)00559-5

    Article  MATH  Google Scholar 

  152. Wu, S., Serbin, J., Gu, M.: Two-photon polymerisation for three-dimensional micro-fabrication. J. Photochem. Photobiol. A Chem. 181(1), 1–11 (2006). doi:10.1016/j.jphotochem.2006.03.004

    Article  Google Scholar 

  153. Xia, Q., Shi, T.: Constraints of distance from boundary to skeleton: for the control of length scale in level set based structural topology optimization. Comput. Methods Appl. Mech. Eng. 295, 525–542 (2015). doi:10.1016/j.cma.2015.07.015

    Article  MathSciNet  Google Scholar 

  154. Xia, Q., Shi, T., Wang, M., Liu, S.: A level set based method for the optimization of cast part. Struct. Multidiscip. Optim. 41(5), 735–747 (2010). doi:10.1007/s00158-009-0444-7

    Article  MathSciNet  Google Scholar 

  155. Xia, Q., Shi, T., Wang, M., Liu, S.: Simultaneous optimization of cast part and parting direction using level set method. Struct. Multidiscip. Optim. 44(6), 751–759 (2011). doi:10.1007/s00158-011-0690-3

    Article  MathSciNet  MATH  Google Scholar 

  156. Xie, Y., Steven, G.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49(5), 885–896 (1993). doi:10.1016/0045-7949(93)90035-C

    Article  Google Scholar 

  157. Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)

    Google Scholar 

  158. Xu, S., Cai, Y., Cheng, G.: Volume preserving nonlinear density filter based on heaviside functions. Struct. Multidiscip. Optim. 41, 495–505 (2010)

  159. Zauderer, E.: Partial Differential Equations of Applied Mathematics, 2 edn. Wiley, New York (1998)

  160. Zhang, W., Xia, L., Zhu, J., Zhang, Q.: Some recent advances in the integrated layout design of multicomponent systems. J. Mech. Des. 133, 15 (2011)

    Google Scholar 

  161. Zhang, W., Zhong, W., Guo, X.: An explicit length scale control approach in SIMP-based topology optimization. Comput. Methods Appl. Mech. Eng. 282, 71–86 (2014). doi:10.1016/j.cma.2014.08.027

    Article  MathSciNet  Google Scholar 

  162. Zhao, J., Wang, C.: Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput. Methods Appl. Mech. Eng. 273, 204–218 (2014). doi:10.1016/j.cma.2014.01.018

    Article  MATH  Google Scholar 

  163. Zhao, L., Ha, S., Sharp, K.W., Geltmacher, A.B., Fonda, R.W., Kinsey, A.H., Zhang, Y., Ryan, S.M., Erdeniz, D., Dunand, D.C., Hemker, K.J., Guest, J.K., Weihs, T.P.: Permeability measurements and modeling of topology-optimized metallic 3-D woven lattices. Acta Mater. 81, 326–336 (2014). doi:10.1016/j.actamat.2014.08.037

    Article  Google Scholar 

  164. Zhou, M., Fleury, R., Shyy, Y.K., Thomas, H., Brennan, J.: Progress in topology optimization with manufacturing constraints. In: 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization (2002). doi:10.2514/6.2002-5614

  165. Zhou, M., Lazarov, B.S., Sigmund, O.: Topology optimization for optical projection lithography with manufacturing uncertainties. Appl. Opt. 53(12), 2720–2729 (2014). doi:10.1364/AO.53.002720

    Article  Google Scholar 

  166. Zhou, M., Lazarov, B.S., Wang, F., Sigmund, O.: Minimum length scale in topology optimization by geometric constraints. Comput. Methods Appl. Mech. Eng. 293, 266–282 (2015). doi:10.1016/j.cma.2015.05.003

  167. Zhou, M., Rozvany, G.: The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89(13), 309–336 (1991). doi:10.1016/0045-7825(91)90046-9

    Article  Google Scholar 

  168. Zhou, M., Wang, M.Y.: Engineering feature design for level set based structural optimization. Comput. Aided Des. 45(12), 1524–1537 (2013). doi:10.1016/j.cad.2013.06.016

    Article  MathSciNet  Google Scholar 

  169. Zhu, J.H., Gu, X.J., Zhang, W.H., Beckers, P.: Structural design of aircraft skin stretch-forming die using topology optimization. J. Comput. Appl. Math. 246, 278–288 (2013). doi:10.1016/j.cam.2012.09.001

    Article  MathSciNet  MATH  Google Scholar 

  170. Zhu, J.H., Zhang, W.H., Xia, L.: Topology optimization in aircraft and aerospace structures design. Arch. Comput. Methods Eng. 1–28 (2015). doi:10.1007/s11831-015-9151-2

  171. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.: The Finite Element Method. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors acknowledge the financial support received from the NextTop project, sponsored by the Villum foundation and the Hypercool project J. Nr. 117-2014-1 sponsored by the Innovation Fund Denmark.

Author information

Authors and Affiliations

  1. Technical University of Denmark, Nils Koppels Allé, Building 404, 2800, Kgs. Lyngby, Denmark

    Boyan S. Lazarov, Fengwen Wang & Ole Sigmund

Authors
  1. Boyan S. Lazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fengwen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ole Sigmund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boyan S. Lazarov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lazarov, B.S., Wang, F. & Sigmund, O. Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86, 189–218 (2016). https://doi.org/10.1007/s00419-015-1106-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-015-1106-4

Keywords

Search

Navigation