Structural and Multidisciplinary Optimization

, Volume 57, Issue 5, pp 2061–2073 | Cite as

Structural optimization with several discrete design variables per part by outer approximation

  • Mathias Stolpe
  • Kasper Sandal


The article proposes an optimal design approach to minimize the mass of load carrying structures with discrete design variables. The design variables are chosen from catalogues, and several variables are assigned to each part of the structure. This allows for more design freedom than only choosing parts from a catalogue. The problems are modelled as mixed 0–1 nonlinear problems with nonconvex continuous relaxations. An algorithm based on outer approximation is proposed to find optimized designs. The capabilities of the approach are demonstrated by optimal design of a space frame (jacket) structure for offshore wind turbines, with requirements on natural frequencies, strength, and fatigue lifetime.


Structural optimization Outer approximation Offshore wind turbines Jacket structures Discrete variables 



The research presented in this manuscript is part of the strategic research project ABYSS: Advancing BeYond Shallow waterS - Optimal design of offshore wind turbine support structures ( The project is funded by the Danish Council for Strategic Research. The funding is gratefully acknowledged. The authors would like to thank the anonymous reviewers for valuable inputs to the final manuscript. In particular, the suggested benchmark with the genetic algorithm provided new insights to the performance of the presented optimization approach.

We would like to thank our colleague Alexander Verbart for his contributions to JADOP.


  1. Achtziger W, Kanzow C (2008) Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math Program 114:69–99MathSciNetCrossRefzbMATHGoogle Scholar
  2. Achtziger W, Stolpe M (2007) Truss topology optimization with discrete design variables — guaranteed global optimality and benchmark examples. Struct Multidiscip Optim 34(1):1–20MathSciNetCrossRefzbMATHGoogle Scholar
  3. Achtziger W, Stolpe M (2009) Global optimization of truss topology with discrete bar areas-Part II: implementation and numerical results. Comput Optim Appl 44(2):315–341MathSciNetCrossRefzbMATHGoogle Scholar
  4. Amestoy PR, Duff IS, Koster J, L’Excellent J -Y (2001) A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J Matrix Anal Appl 23(1):15–41MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arora JS, Wang Q (2005) Review of formulations for structural and mechanical system optimization. Struct Multidiscip Optim 30(4):251–272MathSciNetCrossRefzbMATHGoogle Scholar
  6. Arora JS, Huang MW, Hsieh CC (1994) Methods for optimization of nonlinear problems with discrete variables - a review. Struct Optim 8(2–3):69–85CrossRefGoogle Scholar
  7. ASTM (2013) Standard practices for cycle conting in fatigue analysis. Technical Report E1049, ASTM,
  8. Bak C, Zahle F, Bitsche R, Kim T, Yde A, Henriksen LC, Natarajan A, Hansen M (2013) The DTU 10 MW reference wind turbine. Technical report. DTU Wind EnergyGoogle Scholar
  9. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654zbMATHGoogle Scholar
  10. Bendsøe MP, Sigmund O (2003) Topology optimization, theory, methods and applications. SpringerGoogle Scholar
  11. Borstel T (2013) Design report - reference jacket. Technical Report D4.3.1, Ramboll., Accessed 13 Feb 2017
  12. Cameron TM, Thirunavukarasu AC, El-Sayed MEM (2000) Optimization of frame structures with flexible joints. Struct Multidiscip Optim 19(3):204–213MathSciNetCrossRefGoogle Scholar
  13. Chang TC, Wysk RA (1997) Computer-aided manufacturing, 2nd edn. Prentice Hall PTR, Upper Saddle RiverGoogle Scholar
  14. Chew KH, Tai K, Ng EYK, Muskulus M (2016) Analytical gradient-based optimization of offshore wind turbine substructures under fatigue and extreme loads. Mar Struct 47:23–41CrossRefGoogle Scholar
  15. Chin CM, Fletcher R (2003) On the global convergence of an SLP-filter algorithm that takes EQP steps. Math Program 96(1):161– 177MathSciNetCrossRefzbMATHGoogle Scholar
  16. Christensen PW, Klarbring A (2008) An introduction to structural optimization. Solid mechanics and its applications. Springer, NetherlandsGoogle Scholar
  17. Cook RD, Malkus DS, Plesha ME, Witt RJ (2007) Concepts and applications of finite element analysis, 4th edn. WileyGoogle Scholar
  18. Damiani R (2016) JacketSE: an offshore wind turbine jacket sizing tool theory manual and sample usage with preliminary validation. Technical Report NREL/TP-5000-65417, National Renewable Energy Laboratory, NREL.
  19. Damiani R, Ning A, Maples B, Smith A, Dykes K (2017) Scenario analysis for techno-economic model development of U.S. offshore wind support structures. Wind Energy 20:731–747CrossRefGoogle Scholar
  20. DNVGL (2014) RP-C203 Fatigue design of offshore steel structures. Technical report DNVGLGoogle Scholar
  21. Duran MA, Grossmann IE (1986) An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math Program 36(3):307–339MathSciNetCrossRefzbMATHGoogle Scholar
  22. Faustino AM, Judice JJ, Ribeiro IA, Serra Neves A (2006) An integer programming model for truss topology optimization. Investigação Operacional 26:111–127Google Scholar
  23. Fischer T, de Vries W, Schmidt B (2010) Upwind design basis technical report FP6 upwind reportGoogle Scholar
  24. Fletcher R, Leyffer S (1994) Solving mixed integer nonlinear programs by outer approximation. Math Program 66(1–3):327–349MathSciNetCrossRefzbMATHGoogle Scholar
  25. Fredricson H, Johansen T, Klarbring A, Petersson J (2003) Topology optimization of frame structures with flexible joints. Struct Multidiscip Optim 25(3):199–214MathSciNetCrossRefzbMATHGoogle Scholar
  26. Grossmann IE (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3(3):227–252MathSciNetCrossRefzbMATHGoogle Scholar
  27. IBM Corporation (2014) IBM ILOG CPLEX optimization studio V12.6.0 documentation. Technical report. IBM CorporationGoogle Scholar
  28. Lund E, Stegmann J (2005) On structural optimization of composite shell structures using a discrete constitutive parametrization. Wind Energy 8(1):109–124CrossRefzbMATHGoogle Scholar
  29. Martens JH, Zwick D, Muskulus M (2015) Topology optimization of a jacket structure for an offshore wind turbine with a genetic algorithm. In: 11th World Congress on structural and multidisciplinary optimization. SydneyGoogle Scholar
  30. MathWorks Inc. (2016) MATLAB Primer. MathWorks Inc., R2016a edition.
  31. Mela K (2014) Resolving issues with member buckling in truss topology optimization using a mixed variable approach. Struct Multidiscip Optim 50(6):1037–1049MathSciNetCrossRefGoogle Scholar
  32. Muñoz E, Stolpe M (2011) Generalized Benders’ decomposition for topology optimization problems. J Glob Optim 51(1):149–183MathSciNetCrossRefzbMATHGoogle Scholar
  33. Muskulus M, Schafhirt S (2014) Design optimization of wind turbine support structures —a review. J Ocean Wind Energy 1(1):12–22Google Scholar
  34. Oest J, Sørensen R, Overgaard LCT, Lund E (2017) Structural optimization with fatigue and ultimate limit constraints of jacket structures for large offshore wind turbines. Struct Multidiscip Optim 55(3):779–793MathSciNetCrossRefGoogle Scholar
  35. Pasamontes L, Torres FG, Zwick D, Schafhirt S, Muskulus M (2014) Support structure optimization for offshore wind. In: Proceedings of the ASME 2014 33rd international conference on ocean, offshore and arctic engineering. The American Society of Mechanical Engineers (ASME), San Francisco, pp 1–7Google Scholar
  36. Pedersen CBW (2004) Crashworthiness design of transient frame structures using topology optimization. Comput Methods Appl Mech Eng 193(6-8):653–678CrossRefzbMATHGoogle Scholar
  37. Pedersen P, Jørgensen L (1984) Minimum mass design of elastic frames subjected to multiple load cases. Comput Struct 18(1):147–157CrossRefzbMATHGoogle Scholar
  38. Pedersen NL, Nielsen AK (2003) Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling. Struct Multidiscip Optim 25(5–6):436–445CrossRefGoogle Scholar
  39. Rozvany GIN (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 11:213–217CrossRefGoogle Scholar
  40. Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21:164–172CrossRefGoogle Scholar
  41. Schafhirt S, Zwick D, Muskulus M (2014) Reanalysis of jacket support structure for computer-aided optimization of offshore wind turbines with a genetic algorithm. J Ocean Wind Energy 1(4):209–216Google Scholar
  42. Seidel M, Voormeeren S, van der Steen JB (2016) State-of-the-art design processes for offshore wind turbine support structures. Stahlbau 85(9):583–590CrossRefGoogle Scholar
  43. Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8(4):207–227CrossRefGoogle Scholar
  44. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027CrossRefzbMATHGoogle Scholar
  45. Stolpe M (2007) On the reformulation of topology optimization problems as linear or convex quadratic mixed 0-1 programs. Optim Eng 8:163–192MathSciNetCrossRefzbMATHGoogle Scholar
  46. Stolpe M (2014) Truss topology optimization with discrete design variables by outer approximation. J Glob Optim 61(1):139–163MathSciNetCrossRefzbMATHGoogle Scholar
  47. Stolpe M, Svanberg K (2003) Modeling topology optimization problems as linear mixed 0–1 programs. Int J Numer Methods Eng 57(5):723–739CrossRefzbMATHGoogle Scholar
  48. Svanberg K (1981) Optimization of geometry in truss design. Comput Methods Appl Mech Eng 28(1):63–80CrossRefzbMATHGoogle Scholar
  49. Templeman AB (1988) Discrete optimum structural design. Comput Struct 30(3):511–518CrossRefGoogle Scholar
  50. Templeman AB, Yates DF (1983) A segmental method for the discrete optimum design of structures. Eng Optim 6(3):145–155CrossRefGoogle Scholar
  51. Wächter A, Biegler LT (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57MathSciNetCrossRefzbMATHGoogle Scholar
  52. Yates DF, Templeman AB, Boffey TB (1982) The complexity of procedures for determining minimum weight trusses with discrete member sizes. Int J Solids Struct 18(6):487–495CrossRefzbMATHGoogle Scholar
  53. Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4):595–622MathSciNetCrossRefzbMATHGoogle Scholar
  54. Zwick D, Muskulus M, Moe G (2012) Iterative optimization approach for the design of full-height lattice towers for offshore wind turbines. Energy Procedia 24:297–304CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.DTU Wind EnergyTechnical University of DenmarkRoskildeDenmark

Personalised recommendations