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Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1213–1232 | Cite as

Including global stability in truss layout optimization for the conceptual design of large-scale applications

  • Alexis Tugilimana
  • Rajan Filomeno Coelho
  • Ashley P. Thrall
RESEARCH PAPER
  • 204 Downloads

Abstract

Including stability in truss topology optimization is critical to avoid unstable optimized designs in practical applications. While prior research addresses this challenge by implementing local buckling and linear prebuckling, numerical difficulties remain due to the global stability singularity phenomenon. Therefore, the goal of this paper is to develop an optimization formulation for truss topology optimization including global stability without numerical singularities, within the framework of the preliminary design of large-scale structures. This task is performed by considering an appropriate simultaneous analysis and design formulation, in which the use of a disaggregated form for the equilibrium equations alleviates the singularities inherent to global stability. By implementing a local buckling criterion for hollow truss elements, the resulting formulation is well-suited for the preliminary design of large-scale trusses in civil engineering applications. Three applications illustrate the efficiency of the proposed approach, including a benchmark truss structure and the preliminary design of a footbridge and a dome. The results demonstrate that including local buckling and global stability can considerably affect the optimized design, while offering a systematic means of avoiding unstable solutions. It is also shown that the proposed approach is in a good agreement with linear prebuckling assumptions.

Keywords

Truss topology optimization Local buckling Global stability Nonlinear semidefinite programming Simultaneous analysis and design 

Notes

Acknowledgements

The authors would like to thank the Fond National de la Recherche Scientifique (F.R.S-FNRS, Belgium) for its financial support.

References

  1. Achtziger W (1999) Local stability of trusses in the context of topology optimization. Part i: Exact modelling. Struct Optim 17:235–246Google Scholar
  2. ArcelorMittal (2016) Cold-finished round hollow structural sectionsGoogle Scholar
  3. Aroztegui M, Herskovits J, Roche JR, Bazán E (2014) A feasible direction interior point algorithm for nonlinear semidefinite programming. Struct Multidisc Optim 50:1019–1035MathSciNetCrossRefGoogle Scholar
  4. Barret B, Berry M, Chan T, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, Van der Vorst H (1993) Templates for the solution of linear systems: building blocks for iterative methods, 2nd edn. SIAM, PhiladephiaMATHGoogle Scholar
  5. Bathe KJ (1982) Finite element procedures in engineering analysis. Springer, chap 6, Finite Element Nonlinear Analysis in Solid and Structural Mechanics, pp 628–637Google Scholar
  6. Ben-Tal A, Jarre F, Kočvara M, Nemirovski A, Zowe J (2000) Optimal design of trusses under a noncovex global buckling constraint. Optim Eng 1:189–213MathSciNetCrossRefMATHGoogle Scholar
  7. Bendsøe M, Sigmund O (2013) Topology optimization: theory, methods and applications. Springer Science & Business Media, BerlinMATHGoogle Scholar
  8. Bendsøe M P, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Struct Optim 7:141–159CrossRefGoogle Scholar
  9. Benson HY, Vanderbei RJ (2003) Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Math Program 95:279–302MathSciNetCrossRefMATHGoogle Scholar
  10. Byrd RH, Gilbert JC, Nocedal J (2000) A trust region method based on interior point techniques for nonlinear programming. Math Program 89:149–185MathSciNetCrossRefMATHGoogle Scholar
  11. Byrd RH, Marazzi M, Nocedal J (2004) On the convergence of newton iterations to non-stationary points. Math Program 99 :127–148MathSciNetCrossRefMATHGoogle Scholar
  12. Changizi N, Jalalpour M (2017) Stress-based topology optimization of steel-frame structures using members with standard cross sections: Gradient-based approach. J Struct Eng 143(8).  https://doi.org/10.1061/(ASCE)ST.1943-541X.0001807
  13. Changizi N, Kaboodanian H, Jalalpour M (2017) Stress-based topology optimization of frame structures under geometric uncertainty. Comput Methods Appl Mech Engrg 315:121–140MathSciNetCrossRefGoogle Scholar
  14. Cheng GD, Guo X (1997) Epsilon-relaxed approach in structural topology optimization. Struct Optim 13:258–266CrossRefGoogle Scholar
  15. Cook RD (1974) Concepts and applications in finite element analysis. Wiley, chap 14, Stress Stiffening and Buckling, pp 429–447Google Scholar
  16. Descamps B, Filomeno Coelho R (2013) R Metaheuristic Applications in Structures and Infrastructures, Elsevier B. V., chap Graph theory in evolutionary truss design optimization, pp 241–268Google Scholar
  17. Descamps B, Filomeno Coelho R (2014) The nominal force method for truss geometry and topology optimization incorporating stability considerations. International Journal of Solids and Structures 51:2390–2399CrossRefGoogle Scholar
  18. Dorn WS, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J Mech 3:25–52Google Scholar
  19. Du J, Olhoff N (2007) Topoogical design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidisc Optim 34:91–110CrossRefMATHGoogle Scholar
  20. Ertel S, Schittkowski K, Zillober C (2008) Sequential convex programming for free material optimization with displacement and stress constraints. Technical report, Department of Comptuer Science. University of Bayreuth, BayreuthGoogle Scholar
  21. Eurocode (2005) NBN EN 1990: Eurocode: Basis of structural design. European Committee for Standardization (CEN)Google Scholar
  22. Evgrafov A (2005) On globally stable singular truss topologies. Struct Multidisc Optim 29:170–177MathSciNetCrossRefMATHGoogle Scholar
  23. Evgrafov A, Patriksson M (2005) On the convergence of stationary sequences in topology optimization. Int J Numer Methods Eng 64:17–44MathSciNetCrossRefMATHGoogle Scholar
  24. Fredricson H, Johansen T, Klarbring A, Petersson J (2003) Topology optimization of frame structures with flexible joints. Struct Multidisc Optim 25:199–214MathSciNetCrossRefMATHGoogle Scholar
  25. Fuhry M, Reichel L (2012) A new tikhonov regularization method. Numer Algor 59:433–445MathSciNetCrossRefMATHGoogle Scholar
  26. Fujisawa K, Fukuda M, Kobayashi K, Kojima M, Nakata K, Nakata M, Yamashita M (2003) Sdpa (semidefinite programming algorithm in matlab) and sdpa-gmp. Technical report, Operation Research Department of Mathematical and Computing Sciences. Tokyo Institute of Technology, TokyoGoogle Scholar
  27. Gilbert M, Darwich W, Tyas A, Shepherd P (2005) Application of large-scale layout optimization techniques in structural engineering practice. In: 6th world congresses of structural and multidisciplinary optimizationGoogle Scholar
  28. Gravesen J, Evgrafov A, Nguyen DM (2011) On the sensitivities of multiple eigenvalues. Struct Multidisc Optim 44:583–587MathSciNetCrossRefMATHGoogle Scholar
  29. Guo X, Cheng G, Yamazaki K (2001) A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints. Struct Multidisc Optim 22:364–372CrossRefGoogle Scholar
  30. Guo X, Cheng GD, Olhoff N (2005) Optimum design of truss topology under buckling constraints. Struct Multidisc Optim 30 :169–180CrossRefGoogle Scholar
  31. Hansen PC (1987) The truncated svd as a method for regularization. BIT Numer Math 27:534–553MathSciNetCrossRefMATHGoogle Scholar
  32. Haug EJ, Choi KK (1982) Systematic occurence of repeated eigenvalues in structural optimization. J Optim Theory Appl 38:251– 274MathSciNetCrossRefMATHGoogle Scholar
  33. Holmberg E, Thore CJ, Klarbring A (2015) Worst-case topology optimization of self-weight loaded structures using semi-definite programming. Struct Multidisc Optim 52:915–928MathSciNetCrossRefGoogle Scholar
  34. Kanno Y (2016) Mixed-integer second-order cone programming for global optimization of compliance of frame structure with discrete design variables. Struct Multidisc Optim 54:301–316MathSciNetCrossRefGoogle Scholar
  35. Kanno Y, Yamada H (2017) A note on truss topology optimization under self-weight load: mixed-integer second-order cone programming approach. Struct Multidisc Optim 56:221–226MathSciNetCrossRefGoogle Scholar
  36. Kanno Y, Ohsaki M, Katoh N (2001) Sequential semidefinite programming for optimization of framed structures under multimodal buckling constraints. Int J Struct Stab Dyn 1:585–602CrossRefMATHGoogle Scholar
  37. Kanzow C, Nagel C, Kato H, Fukushima M (2005) Successive linearization methods for nonlinear semidefinite programs. Comput Optim Appl 31:251–273MathSciNetCrossRefMATHGoogle Scholar
  38. Kondoh K, Atluri SN (1985) Influence of local buckling on global instability: simplified, large deformation, post-buckling analysis of plane trusses. Comput Struct 21:613–627CrossRefMATHGoogle Scholar
  39. Kočvara M (2002) On the modelling and solving of the truss design problem with global stability constraints. Struct Multidiscip Optim 23:189–203CrossRefGoogle Scholar
  40. Kočvara M, Stingl M (2003) Pennon: A code for convex nonlinear and semidefinite programming. Optim Methods Softw 18:317–333MathSciNetCrossRefMATHGoogle Scholar
  41. Kočvara M, Stingl M (2004) Solving nonconvex sdp problems of structural optimization with stability control. Optim Methods Softw 19:595–609MathSciNetCrossRefMATHGoogle Scholar
  42. Li L, Khandelwal K (2017) Topology optimization of geometrically nonlinear trusses with spurious eigenmodes control. Eng Struct 131:324–344CrossRefGoogle Scholar
  43. Lindgaard E, Lund E (2010) Nonlinear buckling optimization of composite structures. Comput Methods Appl Mech Eng 199:2319–2330MathSciNetCrossRefMATHGoogle Scholar
  44. Luebekman C, Shea K (2005) Cdo: Computational design + optimization in buidling practice. The Arup JournalGoogle Scholar
  45. MathWorks (2015) Optimization toolbox tm: User’s guide (r2015b). http://www.mathworks.com/help/pdfdoc/optim/optimtb.pdf
  46. Mela K (2014) Resolving issues with member buckling in truss topology optimization using a mixed variable approach. Struct Multidisc Optim 50:1037–1049MathSciNetCrossRefGoogle Scholar
  47. Neumaier A (2006) Solving ill-conditioned and singular linear systems: A tutorial on regularization. SIAM Rev 40(3):636–666MathSciNetCrossRefMATHGoogle Scholar
  48. Neves MM, Rodrigues H, Guedes JM (1995) Generalized topology design of structures with a bucklin load criterion. Struct Optim 10:71–78CrossRefGoogle Scholar
  49. Neves MM, Sigmund O, Bendsøe MP (2002) Topology optimization of periodic microstructures with a penalization of highly localized buckling modes. Int J Numer Methods Eng 54:809–834MathSciNetCrossRefMATHGoogle Scholar
  50. Ogita T, Oishi S (2012) Accurate and robust inverse cholesky factorization. Nonlinear Theory Appl IEICE 3:103–111CrossRefGoogle Scholar
  51. Olhoff N, Rasmussen SH (1977) On single and bimodal optimum buckling loads of clamped columns. Int J Solids Struct 13:605–614CrossRefMATHGoogle Scholar
  52. Pedersen NL, Nielsen AK (2003) Optimization of practical trusses with constraints on eigenfrequencies, displacements, stresses, and buckling. Struct Multidisc Optim 25:436–445CrossRefGoogle Scholar
  53. Rodrigues HC, Guedes JM, Bendsøe M P (1995) Necessary conditions for optimal design of structures with a nonsmooth eigenvalue based criterion. Struct Optim 9:52–56CrossRefGoogle Scholar
  54. Rozvany GIN (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 11:213–217CrossRefGoogle Scholar
  55. Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8:207–227CrossRefGoogle Scholar
  56. Stolpe M, Svanberg K (2001) On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct Multidiscip Optim 21:140–151CrossRefGoogle Scholar
  57. Stolpe M, Svanberg K (2003) A note on stress-constrained truss topology optimization. Struct Multidisc Optim 25:62–64CrossRefMATHGoogle Scholar
  58. Suleman A, Sedaghati R (2005) Benchmark case studies in optimization of geometrically nonlinear structures. Struct Multidiscip Optim 30:273–296CrossRefGoogle Scholar
  59. Thore CJ (2013) Fminsdp-A code for solving optimization problems with matrix inequality constraints. http://www.mathworks.com/matlabcentral/fileexchange/43643-fminsdp
  60. Vanderbei R, Benson HY (2000) On formulating semidefinite programming problems as smooth convex nonlinear optimization problems. Technical report, Center for Discrete Mathematics and Theoretical COmputer ScienceGoogle Scholar
  61. Yanagisawa Y, Ogita T, Oishi S (2014) Convergence analysis of an algorithm for accurate inverse cholesky factorization. Japan J Indust Appl Math 31:461–482MathSciNetCrossRefMATHGoogle Scholar
  62. Zhou M (1996) Difficulties in truss topology optimization with stress and local buckling constraints. Struct Optim 11:134– 136CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.BATir DepartmentUniversité libre de BruxellesBrusselsBelgium
  2. 2.Department of Civil & Environmental, Earth SciencesUniversity of Notre DameNotre DameUSA

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