Structural and Multidisciplinary Optimization

, Volume 45, Issue 3, pp 329–357 | Cite as

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

  • Cameron Talischi
  • Glaucio H. PaulinoEmail author
  • Anderson Pereira
  • Ivan F. M. Menezes
Educational Article


We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.


Topology optimization Unstructured meshes Polygonal finite elements Matlab software 



The first two authors acknowledge the support by the Department of Energy Computational Science Graduate Fellowship Program of the Office of Science and National Nuclear Security Administration in the Department of Energy under contract DE-FG02-97ER25308. The last two authors acknowledge the financial support by Tecgraf (Group of Technology in Computer Graphics), PUC-Rio, Rio de Janeiro, Brazil.

Supplementary material (15 kb)
(ZIP 19.8 KB)


  1. Allaire G (2001) Shape optimization by the homogenization method. Springer, BerlinGoogle Scholar
  2. Allaire G, Francfort GA (1998) Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann Inst Henri Poincare Anal 15(3):301–339MathSciNetzbMATHCrossRefGoogle Scholar
  3. Allaire G, Jouve F (2005) A level-set method for vibration and multiple loads structural optimization. Comput Methods Appl Mech Eng 194(30–33):3269–3290. doi: 10.1016/j.cma.2004.12.018 MathSciNetzbMATHCrossRefGoogle Scholar
  4. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393. doi: 10.1016/ MathSciNetzbMATHCrossRefGoogle Scholar
  5. Almeida SRM, Paulino GH, Silva ECN (2010) Layout and material gradation in topology optimization of functionally graded structures: a global-local approach. Struct Multidisc Optim 42(6):885–868. doi: 10.1007/s00158-010-0514-x MathSciNetCrossRefGoogle Scholar
  6. Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var Partial Differ Equ 1(1):55–69MathSciNetzbMATHCrossRefGoogle Scholar
  7. Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2010) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidisc Optim. doi: 10.1007/s00158-010-0594-7 zbMATHGoogle Scholar
  8. Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57(8):1177–1196. doi: 10.1002/nme.824 zbMATHCrossRefGoogle Scholar
  9. Bendsoe MP (1989) Optimal design as material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  10. Bendsoe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654Google Scholar
  11. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, BerlinGoogle Scholar
  12. Borrvall T (2001) Topology optimization of elastic continua using restriction. Arch Comput Methods Eng 8(4):251–285MathSciNetCrossRefGoogle Scholar
  13. Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190(37–38):4911–4928MathSciNetzbMATHCrossRefGoogle Scholar
  14. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetzbMATHCrossRefGoogle Scholar
  15. Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM, Controle Optim Calc Var 9(2):19–48MathSciNetzbMATHCrossRefGoogle Scholar
  16. Bruns TE (2005) A reevaluation of the simp method with filtering and an alternative formulation for solid-void topology optimization. Struct Multidisc Optim 30(6):428–436. doi: 10.1007/s00158-005-0537-x MathSciNetCrossRefGoogle Scholar
  17. Bruyneel M, Duysinx P (2005) Note on topology optimization of continuum structures including self-weight. Struct Multidisc Optim 29(4):245–256. doi: 10.1007/s00158-004-0484-y CrossRefGoogle Scholar
  18. Cherkaev A (2000) Variational methods for structural optimization. Springer, New YorkzbMATHCrossRefGoogle Scholar
  19. Dambrine M, Kateb D (2009) On the Ersatz material approximation in level-set methods. ESAIM, Controle Optim Calc Var 16(3):618–634. doi: 10.1051/cocv/2009023 MathSciNetCrossRefGoogle Scholar
  20. de Ruiter MJ, Van Keulen F (2004) Topology optimization using a topology description function. Struct Multidisc Optim 26(6):406–416. doi: 10.1007/s00158-003-0375-7 CrossRefGoogle Scholar
  21. Delfour MC, Zolésio JP (2001) Shapes and geometries: analysis, differential calculus, and optimization. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar
  22. Ghosh S (2010) Micromechanical analysis and multi-scale modeling using the Voronoi cell finite element method. In: Computational mechanics and applied analysis. CRC Press, Boca RatonGoogle Scholar
  23. Groenwold AA, Etman LFP (2008) On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem. Int J Numer Methods Eng 73(3):297–316. doi: 10.1002/nme.2071 MathSciNetzbMATHCrossRefGoogle Scholar
  24. Guest JK, Prevost 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):238–254. doi: 10.1002/nmc.1064 MathSciNetzbMATHCrossRefGoogle Scholar
  25. Haber RB, Jog CS, Bendsoe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1):1–12CrossRefGoogle Scholar
  26. Hughes TJR (2000) The finite element method: linear static and dynamic finite elemnt analysis. Dover, New YorkGoogle Scholar
  27. Kohn RV, Strang G (1986a) Optimal design and relaxation of variational problems I. Commun Pure Appl Math 39(1):113–137MathSciNetzbMATHCrossRefGoogle Scholar
  28. Kohn RV, Strang G (1986b) Optimal design and relaxation of variational problems. II. Commun Pure Appl Math 38(1):139–182CrossRefGoogle Scholar
  29. Kohn RV, Strang G (1986c) Optimal design and relaxation of variational problems. III. Commun Pure Appl Math 39:353–377MathSciNetzbMATHCrossRefGoogle Scholar
  30. Kosaka I, Swan CC (1999) A symmetry reduction method for continuum structural topology optimization. Comput Struct 70(1):47–61MathSciNetzbMATHCrossRefGoogle Scholar
  31. Langelaar M (2007) The use of convex uniform honeycomb tessellations in structural topology optimization. In: 7th world congress on structural and multidisciplinary optimization, Seoul, South Korea, May 21–25Google Scholar
  32. Martinez JM (2005) A note on the theoretical convergence properties of the SIMP method. Struct Multidisc Optim 29(4):319–323. doi: 10.1007/s00158-004-0479-8 CrossRefGoogle Scholar
  33. Mousavi SE, Xiao H, Sukumar N (2009) Generalized gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng. doi: 10.1002/nme.2759 Google Scholar
  34. Natarajan S, Bordas SPA, Mahapatra DR (2009) Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping. Int J Numer Methods Eng. doi: 10.1002/nme.2589 MathSciNetGoogle Scholar
  35. Olhoff N, Bendsoe MP, Rasmussen J (1991) On cad-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89(1–3):259–279CrossRefGoogle Scholar
  36. Persson P, Strang G (2004) A simple mesh generator in MATLAB. Siam Rev 46(2):329–345. doi: 10.1137/S0036144503429121 MathSciNetzbMATHCrossRefGoogle Scholar
  37. Petersson J (1999) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171(1–2):123–140MathSciNetzbMATHCrossRefGoogle Scholar
  38. Rietz A (2001) Sufficiency of a finite exponent in SIMP (power law) methods. Struct Multidisc Optim 21:159–163CrossRefGoogle Scholar
  39. Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidisc Optim 37(3):217–237. doi: 10.1007/s00158-007-0217-0 MathSciNetCrossRefGoogle Scholar
  40. Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4(3–4):250–252CrossRefGoogle Scholar
  41. Saxena A (2008) A material-mask overlay strategy for continuum topology optimization of compliant mechanisms using honeycomb discretization. J Mech Des 130(8):2304-1-9. doi: 10.1115/1.2936891 CrossRefGoogle Scholar
  42. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424. doi: 10.1007/s00158-006-0087-x CrossRefGoogle Scholar
  43. 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
  44. Stolpe M, Svanberg K (2001a) On the trajectories of penalization methods for topology optimization. Struct Multidisc Optim 21(2):128–139CrossRefGoogle Scholar
  45. Stolpe M, Svanberg K (2001b) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidisc Optim 22(2):116–124CrossRefGoogle Scholar
  46. Stromberg LL, Beghini A, Baker WF, Paulino GH (2011) Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidisc Optim 43(2):165–180. doi: 10.1007/s00158-010-0563-1 CrossRefGoogle Scholar
  47. Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12):2045–2066. doi: 10.1002/nme.1141 MathSciNetzbMATHCrossRefGoogle Scholar
  48. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetzbMATHCrossRefGoogle Scholar
  49. Tabarraei A, Sukumar N (2006) Application of polygonal finite elements in linear elasticity. Int J Comput Methods 3(4):503–520. doi: 10.1142/S021987620600117X MathSciNetzbMATHCrossRefGoogle Scholar
  50. Talischi C, Paulino GH, Le CH (2009) Honeycomb wachspress finite elements for structural topology optimization. Struct Multidisc Optim 37(6):569–583. doi: 10.1007/s00158-008-0261-4 MathSciNetCrossRefGoogle Scholar
  51. Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: a unifying paradigm. Int J Numer Methods Eng 82(6):671–698. doi: 10.1002/nme.2763 zbMATHGoogle Scholar
  52. Talischi C, Paulino GH, Pereira A, Menezes IFM (2011) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidisc Optim, acceptedGoogle Scholar
  53. Tartar L (2000) An introduction to the homogenization method in optimal design. In: Optimal shape design: lecture notes in mathematics, no. 1740. Springer, Berlin, pp 47–156Google Scholar
  54. Van Dijk NP, Langelaar M, Van Keulen F (2009) A discrete formulation of a discrete level-set method treating multiple constraints. In: 8th World Congress on Structural and Multidisciplinary Optimization, June 1-5, 2009, Lisbon, PortugalGoogle Scholar
  55. Wang MY, Wang XM, Guo DM (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246zbMATHCrossRefGoogle Scholar
  56. 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 2011

Authors and Affiliations

  • Cameron Talischi
    • 1
  • Glaucio H. Paulino
    • 1
    Email author
  • Anderson Pereira
    • 2
  • Ivan F. M. Menezes
    • 2
  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-ChampaignChampaignUSA
  2. 2.TecgrafPontifical Catholic University of Rio de Janeiro (PUC-Rio)Rio de JaneiroBrazil

Personalised recommendations