DMTO – a method for Discrete Material and Thickness Optimization of laminated composite structures

  • Søren N. Sørensen
  • René Sørensen
  • Erik Lund


This paper presents a gradient based topology optimization method for Discrete Material and Thickness Optimization of laminated composite structures, labelled the DMTO method. The capabilities of the proposed method are demonstrated on mass minimization, subject to constraints on the structural criteria; buckling load factors, eigenfrequencies, and limited displacements. Furthermore, common design guidelines or rules, referred to as manufacturing constraints, are included explicitly in the optimization problem as series of linear inequalities. The material selection and thickness variation are optimized simultaneously through interpolation functions with penalization. Numerical results for several parameterizations of a finite element model of a generic main spar from a wind turbine blade are presented. The different parameterizations represent different levels of complexity with respect to manufacturability. The results will thus give insight into the relation between potential weight saving and design complexity. The results show that the DMTO method is capable of solving the problems robustly with only few intermediate valued design variables.


Multi-material topology optimization Mass minimization Laminated composites Variable thickness Manufacturing constraints 



This work was supported by the Danish Centre for Composite Structures and Materials for Wind Turbines (DCCSM), grant no. 09-067212 from the Danish Strategic Research Council (DSF), and by the Danish Research Council for Technology and Production Sciences (FTP), grant no. 10-082695. This support is gratefully acknowledged. The authors thank Mathias Stolpe, Technical University of Denmark, Department of Wind Energy, for fruitful discussions and inputs for this research.


  1. Ahmad S, Irons BM, Zienkiewicz OC (1970) Analysis of thick and thin shell structures by curved elements. Int J NumerMethods Eng 2:419–451CrossRefGoogle Scholar
  2. Arora JS (2004) Introduction to optimum design, 2nd edn. Elsevier Google Scholar
  3. Arora J, Wang Q (2005) Review of formulations for structural and mechanical system optimization. Struct Multidiscip Optim and mechanical system optimization. Struct Multidiscip Optim 30(4):251–272CrossRefMATHMathSciNetGoogle Scholar
  4. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202Google Scholar
  5. Bruggi M, Duysinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscip Optim 46(3):369–384CrossRefMATHMathSciNetGoogle Scholar
  6. Bruyneel M (2011) SFP-a new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Optim 43(1):17–27CrossRefGoogle Scholar
  7. Bruyneel M, Beghin C, Craveur G, Grihon S, Sosonkina M (2012) Stacking sequence optimization for constant stiffness laminates based on a continuous optimization approach. Struct Multidiscip Optim 46(6):783–794CrossRefGoogle Scholar
  8. Bruyneel M, Duysinx P, Fleury C, Gao T (2011) Extensions of the shape functions with penalization parameterization for compositeply optimization. AIAA J 49(10):2325–2329CrossRefGoogle Scholar
  9. Costin DP, Wang BP (1993) Optimum design of a composite structure with manufacturing constraints. Thin-Walled Struct 17(3):185–202CrossRefGoogle Scholar
  10. Duysinx P, Sigmund O (1998) New development in handling stress constraints in optimal material distribution. In: 7th Symposium on multidisciplinary analysis and optimization, AIAA-98-4906. pp 1501–1509Google Scholar
  11. Gao T, Zhang W, Duysinx P (2012) A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate. Int J Numer Methods Eng 91(1):98–114CrossRefMATHGoogle Scholar
  12. Ghiasi H, Pasini D, Lessard L (2009) Optimum stacking sequence design of composite materials part I: constant stiffness design. Compos Struct 90(1):–11CrossRefGoogle Scholar
  13. Ghiasi H, Fayazbakhsh K, Pasini D, Lessard L (2010) Optimum stacking sequence design of composite materials part II: variable stiffness design. Compo Struct 93(1):1–13CrossRefGoogle Scholar
  14. Gill P, Murray W, Saunders M (2005) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev 47(1):99–131CrossRefMATHMathSciNetGoogle Scholar
  15. Groenwold A, Haftka R (2006) Optimization with non-homogeneous failure criteria like tsai-wu for composite laminates. Struct Multidiscip Optim 32(3):183–190CrossRefGoogle Scholar
  16. Hvejsel C, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825CrossRefMATHGoogle Scholar
  17. Hvejsel C, Lund E, Stolpe M (2011) Optimization strategies for discrete multi-material stiffness optimization. Struct Multidiscip Optim 44(2):149–163CrossRefGoogle Scholar
  18. Kennedy GJ, Martins JRRA (2013) A laminate parametrization technique for discrete ply-angle problems with manufacturing constraints. Struct Multidiscip OptimGoogle Scholar
  19. Kim C, Song S, Hwang W, Park H, Han K (1994) On the failure indices of quadratic failure criteria for optimal stacking sequence design of laminated plate. Appl Compos Mater 1(1):81–85CrossRefGoogle Scholar
  20. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620CrossRefGoogle Scholar
  21. Le Riche R, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA 31(5):951–956CrossRefMATHGoogle Scholar
  22. Lindgaard E, Dahl J (2012) On compliance and buckling objective functions in topology optimization of snap-through problems. Struct Multidiscip Optim 47(3):409–421CrossRefMathSciNetGoogle Scholar
  23. Lindgaard E, Lund E (2010) Nonlinear buckling optimization of composite structures. Comput Methods Appl Mech Eng 199(37-40):2319–2330CrossRefMATHMathSciNetGoogle Scholar
  24. Liu B, Haftka RT, Akgn MA, Todoroki A (1999) Permutation geneticalgorithm for stacking sequence design of composite laminates. Comput Methods Appl Mech Eng 186:357–372CrossRefGoogle Scholar
  25. Liu D, Toropov VV, Querin OM, Barton DC (2011) Bilevel optimization of blended composite wing panels. J Aircr 48(1):107–118CrossRefGoogle Scholar
  26. Lund E (2009) Buckling topology optimization of laminated multimaterial composite shell structures. Compos Struct 91(2):158–167CrossRefGoogle Scholar
  27. Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parametrization. Wind Energy 8(1):109–124CrossRefGoogle Scholar
  28. Lund E, Sørensen R, Sørensen SN (2013) Multi-criteriamulti-material topology optimization of laminated composite structures including local constraints. In: Book of abstracts, 10th world congress on structural and multidisciplinary optimization. Orlando, FloridaGoogle Scholar
  29. Manne PM, Tsai SW (1998) Design optimization of composite plates: part II–structural optimization by plydrop tapering. J Compos Mater 32(6):572–598CrossRefGoogle Scholar
  30. Neves M, Rodrigues H, Guedes J (1995) Generalized topology design of structures with a buckling load criterion. Struct Optim 10:71–78CrossRefGoogle Scholar
  31. Niu B, Olhoff N, Lund E, Cheng G (2010) Discrete material optimiza tion of vibrating laminated composite plates for minimum sound radiation. Int J Solids Struct 47(16):2097–2114CrossRefMATHGoogle Scholar
  32. Olhoff N, Bendsøe MP, Rasmussen J (1991) On cad-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89(1):259–279CrossRefGoogle Scholar
  33. Overgaard L, Lund E, Thomsen O (2010) Structural collapse of a wind turbine blade. Part A: static test and equivalent single layered models. Compos A: Appl Sci Manuf 41(2):257–270CrossRefGoogle Scholar
  34. Panda S, Natarajan R (1981) Analysis of laminated composite shell structures by finite element method. Comput Struct 14(3–4):225–230CrossRefMATHGoogle Scholar
  35. París J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41:433–441CrossRefMATHGoogle Scholar
  36. Rietz A (2001) Sufficiency of a finite exponent in simp (power law) methods. Struct Multidiscip Optim 21(2):159–163CrossRefGoogle Scholar
  37. Rodrigues H, Guedes J (1995) Necessary conditions for optimal design of structures with a nonsmooth eigenvalue based criterion. Struct Optim 9:52–56CrossRefGoogle Scholar
  38. Seresta O, Grdal Z, Adams DB, Watson LT (2007) Optimal design of composite wing structures with blended laminates. Compos Part B: Eng 38(4):469–480CrossRefGoogle Scholar
  39. Seyranian A, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8:207–227CrossRefGoogle Scholar
  40. Sørensen SN, Lund E (2013) Topology and thickness optimization of laminated composites including manufacturing constraints. Struct Multidiciplinary Optim 48:249–265CrossRefGoogle Scholar
  41. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027CrossRefMATHGoogle Scholar
  42. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124CrossRefGoogle Scholar
  43. Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373CrossRefMATHMathSciNetGoogle Scholar
  44. Svanberg K (2004) Some modelling aspects for the fortran implementation of MMA. Technical reportGoogle Scholar
  45. Toropov VV, Jones R, Willment T, Funnell M (2005) Weight and manufacturability optimization of composite aircraft components based on a genetic algorithm. In: Proceedings of 6th world congress of structural and multidisciplinary optimization (WCSMO). Rio de JaneiroGoogle Scholar
  46. Tsai SW, Wu EM (1971) A general theory of strength for anisotropic materials. J Compos Mater 5(1):58–80CrossRefGoogle Scholar
  47. Wittrick W (1962) Rates of change of eigenvalues, with reference to buckling and vibration problems. J Royal Aeronaut Soc 66:590–591Google Scholar
  48. Zein S, Colson B, Grihon S (2012) A primal-dual backtracking optimization method for blended composite structures. Struct Multidiscip Optim 45(5):669–680CrossRefMATHMathSciNetGoogle Scholar
  49. Zhou M, Fleury R (2012) Composite optimization - ply drop-rate constraints for concepts and detailed design. In: proceedings of the 23rd international congress of theoretical and applied mechanics (ICTAM). BeijingGoogle Scholar
  50. Zhou M, Fleury R, KempM(2011) Optimization of composits - recent advances and application. In: The 7th Altair CAE technology conference. AltairGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Manufacturing EngineeringAalborg UniversityAalborg EastDenmark

Personalised recommendations