Advertisement

Structural and Multidisciplinary Optimization

, Volume 53, Issue 1, pp 175–192 | Cite as

Bridging topology optimization and additive manufacturing

  • Tomás Zegard
  • Glaucio H. PaulinoEmail author
EDUCATIONAL ARTICLE
First Online:

Abstract

Topology optimization is a technique that allows for increasingly efficient designs with minimal a priori decisions. Because of the complexity and intricacy of the solutions obtained, topology optimization was often constrained to research and theoretical studies. Additive manufacturing, a rapidly evolving field, fills the gap between topology optimization and application. Additive manufacturing has minimal limitations on the shape and complexity of the design, and is currently evolving towards new materials, higher precision and larger build sizes. Two topology optimization methods are addressed: the ground structure method and density-based topology optimization. The results obtained from these topology optimization methods require some degree of post-processing before they can be manufactured. A simple procedure is described by which output suitable for additive manufacturing can be generated. In this process, some inherent issues of the optimization technique may be magnified resulting in an unfeasible or bad product. In addition, this work aims to address some of these issues and propose methodologies by which they may be alleviated. The proposed framework has applications in a number of fields, with specific examples given from the fields of health, architecture and engineering. In addition, the generated output allows for simple communication, editing, and combination of the results into more complex designs. For the specific case of three-dimensional density-based topology optimization, a tool suitable for result inspection and generation of additive manufacturing output is also provided.

Keywords

Additive manufacturing Ground structure method Density-based topology optimization Three-dimensional optimal structures Structural manufacturing 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The authors appreciate constructive comments and insightful suggestions from the anonymous reviewers. We are thankful to the support from the US National Science Foundation under grant CMMI #1335160. We also acknowledge the support from SOM (Skidmore, Owings and Merrill LLP) and from the Donald B. and Elizabeth M. Willett endowment at the University of Illinois at Urbana–Champaign. Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.

Supplementary material

158_2015_1274_MOESM1_ESM.zip (8.2 mb)
(ZIP 8.21 MB)

References

  1. Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4-5):285–304CrossRefMathSciNetzbMATHGoogle Scholar
  2. Allaire G, Francfort G (1993) A numerical algorithm for topology and shape optimization. In: Bendsøe M P, Mota Soares C A (eds) Topology design of structures. Springer Netherlands, Sesimbra, pp 239–248Google Scholar
  3. Allaire G, Kohn R (1993) Topology optimization and optimal shape design using homogenization. In: Bendsøe M P, Mota Soares CA (eds) Topology design of structures. Springer Netherlands, Sesimbra, pp 207–218Google Scholar
  4. Almeida SRM, Paulino GH, Silva ECN (2009) A simple and effective inverse projection scheme for void distribution control in topology optimization. Struct Multidiscip Optim 39(4):359–371CrossRefMathSciNetGoogle Scholar
  5. Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var Partial Diff Equ 1(1):55–69CrossRefMathSciNetzbMATHGoogle Scholar
  6. Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optimi 43(1):1–16CrossRefzbMATHGoogle Scholar
  7. ASTM ISO (2013) ASTM52915-13, Standard specification for additive manufacturing file format (AMF) Version 1.1. ASTM International, West Conshohocken, PAGoogle Scholar
  8. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202CrossRefGoogle Scholar
  9. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224CrossRefGoogle Scholar
  10. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Engineering Online Library, 2nd edn. Springer, BerlinGoogle Scholar
  11. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158CrossRefMathSciNetzbMATHGoogle Scholar
  12. Brackett D, Ashcroft I, Hague R (2011) Topology optimization for additive manufacturing. In: 22nd annual solid freeform fabrication symposium, pp 348–362Google Scholar
  13. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26-27):3443–3459CrossRefzbMATHGoogle Scholar
  14. Brutzman D, Daly L (2010) X3D: extensible 3D graphics for Web authors, 1st edn. Morgan Kaufmann, San FranciscoGoogle Scholar
  15. Chen Y (2006) A mesh-based geometric modeling method for general structures. In: ASME 2006 international design engineering technical conferences and computers and information in engineering conference. PhiladelphiaGoogle Scholar
  16. Christensen P, Klarbring A (2009) An introduction to structural optimization, 1st edn. Springer, BerlinzbMATHGoogle Scholar
  17. Crump S (1992) Apparatus and method for creating three-dimensional objects. US Patent 5,121,329Google Scholar
  18. Deaton JD, Grandhi RV (2013) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38CrossRefMathSciNetGoogle Scholar
  19. Deckard C (1989) Method and apparatus for producing parts by selective sintering. US Patent 4,863,538Google Scholar
  20. Dewhurst P, Srithongchai S (2005) An investigaton of minimum-weight dual-material symmetrically loaded wheels and torsion arms. J Appl Mech 72(2):196–202CrossRefzbMATHGoogle Scholar
  21. Dewhurst P, Taggart D (2009) Three-dimensional cylindrical truss structures: a case study for topological optimization. In: Hernández S, Brebbia C A (eds) Computer aided optimum design in engineering XI. WIT Press, pp 83–94Google Scholar
  22. Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45CrossRefGoogle Scholar
  23. Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J Mecanique 3(1):25–52Google Scholar
  24. EOS GmbH (2014) Electro optical systems: orthopaedic technology. http://www.eos.info/industries_markets/medical/orthopaedic_technology
  25. France AK (2013) Make: 3D printing - the essential guide to 3D printers. Maker Media, SebastopolGoogle Scholar
  26. Gaynor AT, Meisel Na, Williams CB, Guest JK (2014) Multiple-material topology optimization of compliant mechanisms created via PolyJet three-dimensional printing. J Manuf Sci Engi 136(6):061015CrossRefGoogle Scholar
  27. Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064CrossRefzbMATHGoogle Scholar
  28. Graczykowski C, Lewiński T (2005) The lightest plane structures of a bounded stress level transmitting a point load to a circular support. Control Cybern 34(1):227–253zbMATHGoogle Scholar
  29. Graczykowski C, Lewiński T (2006a) Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets. Struct Multidiscip Optim 32(5):347–368CrossRefMathSciNetzbMATHGoogle Scholar
  30. Graczykowski C, Lewiński T (2006b) Michell cantilevers constructed within trapezoidal domains—Part II: virtual displacement fields. Struct Multidiscip Optim 32(6):463–471CrossRefMathSciNetzbMATHGoogle Scholar
  31. Graczykowski C, Lewiński T (2006c) Michell cantilevers constructed within trapezoidal domains—Part III: force fields. Struct Multidiscip Optim 33(1):1–19CrossRefGoogle Scholar
  32. Graczykowski C, Lewiński T (2007) Michell cantilevers constructed within trapezoidal domains—Part IV: complete exact solutions of selected optimal designs and their approximations by trusses of finite number of joints. Struct Multidiscip Optim 33(2): 113–129CrossRefMathSciNetzbMATHGoogle Scholar
  33. Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254CrossRefzbMATHGoogle Scholar
  34. Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Opt 11(1-2):1–12CrossRefGoogle Scholar
  35. Hegemier G, Prager W (1969) On Michell trusses. Int J Mech Sci 11(2):209–215CrossRefzbMATHGoogle Scholar
  36. Hemp WS (1973) Optimum structures, 1st edn. Oxford University Press, OxfordGoogle Scholar
  37. Hull C (1986) Apparatus for production of three-dimensional objects by stereolithography. US Patent 4,575,330Google Scholar
  38. Jones R, Haufe P, Sells E, Iravani P, Olliver V, Palmer C, Bowyer A (2011) RepRap — the replicating rapid prototyper. Robotica 29(1):177–191CrossRefGoogle Scholar
  39. Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395CrossRefMathSciNetzbMATHGoogle Scholar
  40. Lewiński T (2004) Michell structures formed on surfaces of revolution. Struct Multidiscip Optim 28(1):20–30CrossRefMathSciNetGoogle Scholar
  41. Lewiński T, Rozvany GIN (2007) Exact analytical solutions for some popular benchmark problems in topology optimization—Part II: three-sided polygonal supports. Struct Multidiscip Optim 33(4-5):337–349CrossRefMathSciNetGoogle Scholar
  42. Lewiński T, Rozvany GIN (2008a) Analytical benchmarks for topological optimization—Part IV: square-shaped line support. Struct Multidiscip Optim 36(2):143–158CrossRefMathSciNetGoogle Scholar
  43. Lewiński T, Rozvany GIN (2008b) Exact analytical solutions for some popular benchmark problems in topology optimization—Part III: L-shaped domains. Struct Multidiscip Optim 35(2):165–174CrossRefMathSciNetGoogle Scholar
  44. Lewiński T, Rozvany GIN, Sokół T, Bołbotowski K (2013) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited. Struct Multidiscip Optim 47 (6):937–942CrossRefMathSciNetGoogle Scholar
  45. Lewiński T, Zhou M, Rozvany GIN (1994a) Extended exact least-weight truss layouts?–Part II: unsymmetric cantilevers. Int J Mech Sci 36(5):399–419CrossRefGoogle Scholar
  46. Lewiński T, Zhou M, Rozvany GIN (1994b) Extended exact solutions for least-weight truss layouts?–Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398CrossRefzbMATHGoogle Scholar
  47. Lipson H, Kurman M (2013) Fabricated: the new world of 3D printing, 1st edn. Wiley, IndianapolisGoogle Scholar
  48. Liu K, Tovar A (2014) An efficient 3D topology optimization code written in Matlab. Struct Multidiscip Optim. doi: 10.1007/s00158-014-1107-x
  49. Meiners W, Wissenbach K, Gasser A (1998) Shaped body especially prototype or replacement part production. DE Patent 19,649,865Google Scholar
  50. Meisel NA, Gaynor A, Williams CB, Guest JK (2013) Multiple-material topology optimization of compliant mechanisms created via polyjet 3d printing. In: 24rd annual international solid freeform fabrication symposium, pp. 980–997Google Scholar
  51. Michell AGM (1904) The limits of economy of material in frame-structures. 6 8(47):589–597zbMATHGoogle Scholar
  52. Ohsaki M (2010) Optimization of finite dimensional structures, 1st edn. CRC Press, BocaCrossRefGoogle Scholar
  53. Petersson J (1999) A finite element analysis of optimal variable thickness sheets. SIAM J Numer Anal 36(6):1759–1778CrossRefMathSciNetzbMATHGoogle Scholar
  54. Ramos AS, Paulino GH (2014) Convex topology optimization for hyperelastic trusses based on the ground-structure approach. Struct Multidiscip OptimGoogle Scholar
  55. Reinhart G, Teufelhart S (2011) Load-adapted design of generative manufactured lattice structures. Phys Procedia 12(Part A):385–392CrossRefGoogle Scholar
  56. Rezaie R, Badrossamay M, Ghaie a, Moosavi H (2013) Topology optimization for fused deposition modeling process. Procedia CIRP 6:521–526CrossRefGoogle Scholar
  57. Rozvany G, Gollub W (1990) Michell layouts for various combinations of line supports—I. Int J Mech Sci 32(12):1021– 1043CrossRefzbMATHGoogle Scholar
  58. Rozvany G, Gollub W, Zhou M (1997) Exact Michell layouts for various combinations of line supports—Part II. Struct Optim 14(2-3):138–149CrossRefGoogle Scholar
  59. Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Struct Optim 15(1):42–48CrossRefzbMATHGoogle Scholar
  60. Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37(3):217–237CrossRefMathSciNetzbMATHGoogle Scholar
  61. Salmi M (2013) Medical applications of additive manufacturing in surgery and dental care. Phd thesis, Aalto University, Helsinki, FinlandGoogle Scholar
  62. Salmi M, Tuomi J, Paloheimo K-S, Björkstrand R, Paloheimo M, Salo J, Kontio R, Mesimäki K, Mäkitie AA (2012) Patient-specific reconstruction with 3D modeling and DMLS additive manufacturing. Rapid Prototyp J 18(3):209–214CrossRefGoogle Scholar
  63. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524CrossRefGoogle Scholar
  64. Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21 (2):120–127CrossRefGoogle Scholar
  65. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424CrossRefGoogle Scholar
  66. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055CrossRefMathSciNetGoogle Scholar
  67. 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(1):68–75CrossRefGoogle Scholar
  68. Sokół T (2011) A 99 line code for discretized Michell truss optimization written in mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefzbMATHGoogle Scholar
  69. Sundararajan V (2011) Topology optimization for additive manufacturing of customized meso-structures using homogenization and parametric smoothing functions. Msc thesis, University of Texas at AustinGoogle Scholar
  70. Sutradhar A, Park J, Carrau D, Miller MJ (2014) Experimental validation of 3D printed patient-specific implants using digital image correlation and finite element analysis. Comput Biol Med 52:8–17CrossRefGoogle Scholar
  71. Sutradhar A, Paulino GH, Miller MJ, Nguyen TH (2010) Topological optimization for designing patient-specific large craniofacial segmental bone replacements. Proc Nat Acad Sci USA 107(30):13222–13227CrossRefGoogle Scholar
  72. Topping BHV (1983) Shape optimization of skeletal structures: A review. J Struct Eng 109(8):1933–1951CrossRefGoogle Scholar
  73. Villanueva CH, Maute K (2014) Density and level set-XFEM schemes for topology optimization of 3-D structures. Comput Mech 54(1):133–150CrossRefMathSciNetzbMATHGoogle Scholar
  74. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784CrossRefzbMATHGoogle Scholar
  75. Wittbrodt B, Glover aG, Laureto J, Anzalone G, Oppliger D, Irwin J, Pearce J (2013) Life-cycle economic analysis of distributed manufacturing with open-source 3-D printers. Mechatronics 23(6):713–726CrossRefGoogle Scholar
  76. Wright MH (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Amer Math Soc 42(1):39–56CrossRefGoogle Scholar
  77. Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidiscip Optim 41(4):495–505CrossRefMathSciNetzbMATHGoogle Scholar
  78. Zegard T. (2014) Structural optimization: from continuum and ground structures to additive manufacturing. Phd thesis, University of Illinois at Urbana–Champaign, Urbana, ILGoogle Scholar
  79. Zegard T, Paulino GH (2014a) GRAND – Ground structure based topology optimization on arbitrary 2D domains using MATLAB. Struct Multidiscip Optim 50(5):861–882CrossRefMathSciNetGoogle Scholar
  80. Zegard T, Paulino GH (2014b) GRAND3 – Ground structure based topology optimization on arbitrary 3D domains using MATLAB. Struct Multidiscip Optim. doi: 10.1007/s00158-015-1284-2
  81. Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1-3):309–336CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  2. 2.School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations

Cite article

Buy options