Structural and Multidisciplinary Optimization

, Volume 60, Issue 4, pp 1605–1618 | Cite as

Fail-safe truss topology optimization

  • Mathias StolpeEmail author
Research Paper


The classical minimum compliance problem for truss topology optimization is generalized to accommodate for fail-safe requirements. Failure is modeled as either a complete damage of some predefined number of members or by degradation of the member areas. The considered problem is modeled as convex conic optimization problems by enumerating all possible damage scenarios. This results in problems with a generally large number of variables and constraints. A working-set algorithm based on solving a sequence of convex relaxations is proposed. The relaxations are obtained by temporarily removing most of the complicating constraints. Some of the violated constraints are re-introduced, the relaxation is resolved, and the process is repeated. The problems and the associated algorithm are applied to optimal design of two-dimensional truss structures revealing several properties of both the algorithm and the optimal designs. The working-set approach requires only a few relaxations to be solved for the considered examples. The numerical results indicate that the optimal topology can change significantly even if the damage is not severe.


Fail-safe optimal design Truss topology optimization Minimum compliance Semidefinite programming Second-order cone programming 



The author would like to sincerely thank two anonymous reviewers for their knowledgeable comments and suggestions that significantly improved the manuscript in terms of contents and presentation.

Funding information

The research presented in this manuscript is funded by Independent Research Fund Denmark through the research project Fail-Safe Structural Optimization (SELMA) with grant no. 7017-00084B.

Compliance with ethical standards

Conflict of interest

The author declares no conflict of interest.


  1. Achtziger W (1996) Truss topology optimization including bar properties different for tension and compression. Structural Optimization 12(1):63–74CrossRefGoogle Scholar
  2. Achtziger W (1998) Multiple-load truss topology and sizing optimization: some properties of minimax compliance. J Optim Theory Appl 98(2):255–280MathSciNetCrossRefGoogle Scholar
  3. Achtziger W, Bendsøe M (1999) Optimal topology design of discrete structures resisting degradation effects. Structural Optimization 17(1):74–78CrossRefGoogle Scholar
  4. Achtziger W, Kanzow C (2008) Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math Program 114:69–99MathSciNetCrossRefGoogle Scholar
  5. Achtziger W, Kočvara M (2007) Structural topology optimization with eigenvalues. SIAM J Optim 18(4):1129–1164MathSciNetCrossRefGoogle Scholar
  6. Achtziger W, Bendsøe M, Ben-Tal A, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. Impact of Computing in Science and Engineering 4(4):315–345MathSciNetCrossRefGoogle Scholar
  7. Ben-Tal A, Nemirovski A (1997) Robust truss topology design via semidefinite programming. SIAM J Optim 7(4):991–1016MathSciNetCrossRefGoogle Scholar
  8. Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization: analysis, algorithms and engineering applications. SIAM, PhiladelphiaCrossRefGoogle Scholar
  9. Bendsøe M, Sigmund O (2003) Topology optimization - theory, methods, and applications. Springer VerlagGoogle Scholar
  10. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  11. Cheng G, Guo X (1997) ε-relaxed approach in structural topology optimization. Structural Optimization 13(4):258–266CrossRefGoogle Scholar
  12. Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis, 4th edn. Wiley, New YorkGoogle Scholar
  13. Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J de Mécanique 3:25–52Google Scholar
  14. Fujisawa K, Kojima M, Nakata K (1997) Exploiting sparsity in primal-dual interior-point methods for semidefinite programming. Math Program 79(1-3):235–253MathSciNetCrossRefGoogle Scholar
  15. Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(7-8):1044–1064CrossRefGoogle Scholar
  16. Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. In: Blondel V, Boyd S, Kimura H (eds) Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences. Springer-Verlag Limited, pp 95– 110Google Scholar
  17. Grant M, Boyd S (2014) CVX: Matlab software for disciplined convex programming, version 2.1.
  18. Hemp W (1973) Optimum structures. Clarendon PressGoogle Scholar
  19. Jansen M, Lombaert G, Schevenels M, Sigmund O (2014) Topology optimization of fail-safe structures using a simplified local damage model. Struct Multidiscip Optim 49(4):657–666MathSciNetCrossRefGoogle Scholar
  20. Kanno Y (2012) Worst scenario detection in limit analysis of trusses against deficiency of structural components. Eng Struct 42:33–42CrossRefGoogle Scholar
  21. Kanno Y (2016) Global optimization of trusses with constraints on number of different cross-sections: a mixed-integer second-order cone programming approach. Comput Optim Appl 63(1):203– 236MathSciNetCrossRefGoogle Scholar
  22. Kanno Y (2017) Redundancy optimization of finite-dimensional structures: concept and derivative-free algorithm. J Struct Eng 143(1):04016,151CrossRefGoogle Scholar
  23. Kanno Y, Ben-Haim Y (2011) Redundancy and robustness, or when is redundancy redundant? J Struct Eng 137(9):935–945CrossRefGoogle Scholar
  24. Kirsch U (1990) On singular topologies in optimum structural design. Structural Optimization 2(3):133–142MathSciNetCrossRefGoogle Scholar
  25. Kreisselmeier G, Steinhauser R (1983) Application of vector performance optimization to a robust control loop design for a fighter aircraft. Int J Control 37(2):251–284CrossRefGoogle Scholar
  26. Lobo M, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. Linear Algebra Appl 284(1–3):193–228MathSciNetCrossRefGoogle Scholar
  27. Mohr D, Stein I, Matzies T, Knapek C (2014) Redundant robust topology optimization of truss. Optim Eng 15(4):945–972MathSciNetCrossRefGoogle Scholar
  28. Rozvany G (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Structural Optimization 11:213–217CrossRefGoogle Scholar
  29. Rozvany G (2001) On design-dependent constraints and topologies. Struct Multidiscip Optim 21:164–172CrossRefGoogle Scholar
  30. Sturm J (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11-2(1-4):625–653MathSciNetCrossRefGoogle Scholar
  31. Sturm J (2002) Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optim Methods Softw 17(6):1105–1154MathSciNetCrossRefGoogle Scholar
  32. Sun P, Arora J, Haug E (1976) Fail-safe optimal design of structures. Eng Optim 2(1):43–53CrossRefGoogle Scholar
  33. Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefGoogle Scholar
  34. Sved G, Ginos Z (1968) Structural optimization under multiple loading. Int J Mech Sci 10(10):803–805CrossRefGoogle Scholar
  35. The Mathworks Inc (2018) MATLAB R2018a. Natick, MassachusettsGoogle Scholar
  36. Toh K, Todd M, Tütüncü R (1999) SDPT3 - A MATLAB software package for semidefinite programming, version 1.3. Optim Methods Softw 11(1-4):545–581MathSciNetCrossRefGoogle Scholar
  37. Tütüncü R, Toh K, Todd M (2003) Solving semidefinite-quadratic-linear programs using SDPT3. Math Program 95(2):189–217MathSciNetCrossRefGoogle Scholar
  38. Verbart A, Stolpe M (2018) A working-set approach for sizing optimization of frame-structures subjected to time-dependent constraints. Struct Multidiscip Optim 58(4):1367–1382MathSciNetCrossRefGoogle Scholar
  39. Yamashita M, Fujisawa K (2003) Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0). Optim Methods Softw 18(4):491–505MathSciNetCrossRefGoogle Scholar
  40. Zhou M, Fleury R (2016) Fail-safe topology optimization. Struct Multidiscip Optim 54(5):1225–1243CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.DTU Wind EnergyTechnical University of Denmark (DTU)RoskildeDenmark

Personalised recommendations