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Revisiting topology optimization with buckling constraints

  • Federico FerrariEmail author
  • Ole Sigmund
Research Paper
  • 100 Downloads

Abstract

We review some features of topology optimization with a lower bound on the critical load factor, as computed by linearized buckling analysis. The change of the optimized design, the competition between stiffness and stability requirements and the activation of several buckling modes, depending on the value of such lower bound, are studied. We also discuss some specific issues which are of particular interest for this problem, as the use of non-conforming finite elements for the analysis, the use of inconsistent sensitivity that may lead to wrong signs of sensitivities and the replacement of the single eigenvalue constraints with an aggregated measure. We discuss the influence of these practices on the optimization result, giving some recommendations.

Keywords

Topology optimization Eigenvalue optimization Linearized buckling Aggregation functions Finite elements Sensitivity analysis 

Notes

Acknowledgements

The current project is supported by the Villum Fonden through the Villum Investigator Project “InnoTop.” The authors are grateful to Prof. Pauli Pedersen for several fruitful discussions on the topic of the paper.

References

  1. Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga–voxel computational morphogenesis for structural design. Nature 550(7674):84–86Google Scholar
  2. Achtziger W (1999) Local stability of trusses in the context of topology optimization, part I: exact modelling. Struct Optim 17:235–246Google Scholar
  3. Armand JL, Lodier B (1978) Optimal design of bending elements. Int J Numer Methods Eng 13:373–384zbMATHGoogle Scholar
  4. Bathe KJ, Dvorkin E (1983) On the automatic solution of nonlinear finite element equations. Comput Struct 17(5–6):871–879Google Scholar
  5. Bendsøe MP, Sigmund O (2004) Topology optimization: theory methods and applications. Springer, BerlinzbMATHGoogle Scholar
  6. Berke L (1970) An efficient approach to the minimum weight design of deflection limited structures. AFFDDL-TM-70-4Google Scholar
  7. Bian X, Feng Y (2017) Large–scale buckling–constrained topology optimization based on assembly–free finite element analysis. Adv Mech Eng 9(9):1–12Google Scholar
  8. Bochenek B, Tajs-Zieliṅska K (2015) Minimal compliance topologies for maximal buckling load of columns. Struct Multidiscip Optim 51(5):1149–1157MathSciNetGoogle Scholar
  9. Brantman R (1977) On the use of linearized instability analyses to investigate the buckling of nonsymmetrical systems. Acta Mech 26(1):75–89MathSciNetGoogle Scholar
  10. Bruyneel M, Colson B, Remouchamps A (2008) Discussion on some convergence problems in buckling optimisation. Struct Multidiscip Optim 35(2):181–186Google Scholar
  11. Chen X, Qi H, Qi L, Teo K L (2004) Smooth convex approximation to the maximum eigenvalue function. J Glob Optim 30(2):253–270MathSciNetzbMATHGoogle Scholar
  12. Cheng G, Xu L (2016) Two–scale topology design optimization of stiffened or porous plate subject to out–of–plane buckling constraint. Struct Multidiscip Optim 54(5):1283–1296MathSciNetGoogle Scholar
  13. Chin TW, Kennedy GJ (2016) Large–scale compliance–minimization and buckling topology optimization of the undeformed common research model wing. In: AIAA SciTechForumGoogle Scholar
  14. Cook RD, Malkus DS, Plesha ME, Witt RJ (2001) Concepts and applications of finite element analysis, 4th edn. Wiley, New YorkGoogle Scholar
  15. Cox S, McCarthy C (1998) The shape of the tallest column. SIAM J Math Anal 29:547–554MathSciNetzbMATHGoogle Scholar
  16. Cox S, Overton M (1992) On the optimal design of columns against buckling. SIAM J Math Anal 23:287–325MathSciNetzbMATHGoogle Scholar
  17. de Borst R, Crisfield MA, Remmers JJC, Verhoosel CV (2012) Non-linear finite element analysis of solids and structures, 2nd edn. WileyGoogle Scholar
  18. Dunning PD, Ovtchinnikov E, Scott J, Kim A (2016) Level–set topology optimization with many linear buckling constraints using and efficient and robust eigensolver. International Journal for Numerical Methods, in EngineeringGoogle Scholar
  19. Duysinx P, Sigmund O (1998) New developments in handling optimal stress constraints in optimal material distributions, pp 1501–1509Google Scholar
  20. Ferrari F, Lazarov BS, Sigmund O (2018) Eigenvalue topology optimization via efficient multilevel solution of the Frequency Response. Int J Numer Methods Eng 115(7):872–892MathSciNetGoogle Scholar
  21. Folgado J, Rodrigues H (1998) Structural optimization with a non-smooth buckling load criterion. Control Cybern 27:235–253MathSciNetzbMATHGoogle Scholar
  22. Frauenthal JC (1972) Constrained optimal design of circular plates against buckling. J Struct Mech 1:159–186Google Scholar
  23. Fröier M, Nilsson L, Samuelsson A (1974) The rectangular plane stress element by Turner, Pian and Wilson. Int J Numer Methods Eng 8:433–437Google Scholar
  24. Gao X, Ma H (2015) Topology optimization of continuum structures under buckling constraints. Comput Struct 157:142–152Google Scholar
  25. Gao X, Li L, Ma H (2017) An adaptive continuation method for topology optimization of continuum structures considering buckling constraints. Int J Appl Math 9(7):24Google Scholar
  26. Gravesen J, Evgrafov A, Nguyen DM (2011) On the sensitivities of multiple eigenvalues. Struct Multidiscip Optim 44(4):583–587MathSciNetzbMATHGoogle Scholar
  27. Haftka R, Gurdal Z (2012) Elements of structural optimization. Solid mechanics and its applications. Springer, NetherlandsGoogle Scholar
  28. Haftka R, Prasad B (1981) Optimum structural with plate bending elements – a survey. AIAA J 19:517–522zbMATHGoogle Scholar
  29. Hall SK, Cameron GE, Grierson DE (1988) Least–weight design of steel frameframe accounting for P −Δ effects. Struct Eng ASCE 115(6):1463–1475Google Scholar
  30. Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3):203–226MathSciNetzbMATHGoogle Scholar
  31. Kemmler R, Lipka A, Ramm E (2005) Large deformations and stability in topology optimization. Struct Multidiscip Optim 30:459–476MathSciNetzbMATHGoogle Scholar
  32. Kennedy GJ, Hicken JE (2015) Improved constraint–aggregation methods. Comput Methods Appl Mech Eng 289(Supplement C):332–354MathSciNetGoogle Scholar
  33. Kerr ADT, Soifer MT (1968) The linearization of the prebuckling state and its effect on the determined instability loads. J Appl Mech 36:775–783Google Scholar
  34. Khot NS, Venkayya VB, Berke L (1976) Optimum structural design with stability constraints. Int J Numer Methods Eng 10(5):1097–1114zbMATHGoogle Scholar
  35. Kirmser PG, Hu KK (1995) The shape of the ideal column reconsidered. Math Intell 15(3):62–67MathSciNetzbMATHGoogle Scholar
  36. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. IFAC Proceedings 12(7):113–117. iFAC Symposium on Computer Aided Design of Control Systems, Zurich, Switzerland, 29-31 AugustzbMATHGoogle Scholar
  37. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620Google Scholar
  38. Lee E, James K, Martins J (2012) Stress–constrained topology optimization with design dependent loads. Struct Multidiscip Optim 46(5):647–661MathSciNetzbMATHGoogle Scholar
  39. Lee K, Ahn K, Yoo J (2016) A novel p–norm correction method for lightweight topology optimization under maximum stress constraints. Comput Struct 171(Supplement C):18–30Google Scholar
  40. Lindgaard E, Dahl J (2013) On compliance and buckling objective functions in topology optimization of snap–through problems. Struct Multidiscip Optim 47:409–421MathSciNetzbMATHGoogle Scholar
  41. Lund E (2009) Buckling topology optimization of laminated multi–material composite shell structures. Compos Struct 91(2):158– 167Google Scholar
  42. Manh ND, Evgrafov A, Gersborg AR, Gravesen J (2011) Isogeometric shape optimization of vibrating membranes. Comput Methods Appl Mech Eng 200(13):1343–1353MathSciNetzbMATHGoogle Scholar
  43. Munk DJ, Vio GA, Steven GP (2017) A simple alternative formulation for structural optimisation with dynamic and buckling objectives. Struct Multidiscip Optim 55(3):969–986MathSciNetGoogle Scholar
  44. Neves MM, Rodrigues H, Guedes JM (1995) Generalized topology design of structures with a buckling load criterion. Struct Optim 10(2):71–78Google Scholar
  45. 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(6):809– 834MathSciNetzbMATHGoogle Scholar
  46. Ohsaki M, Ikeda K (2007) Stability and optimization of structures: generalized sensitivity analysis. Mechanical Engineering Series. Springer, BerlinzbMATHGoogle Scholar
  47. Olhoff N, Rasmussen SH (1977) On single and bimodal optimum buckling loads of clamped columns. Int J Solids Struct 13(7):605–614zbMATHGoogle Scholar
  48. Pedersen NL, Pedersen P (2018) Buckling load optimization for 2D continuum models with alternative formulation for buckling load estimation. Struct Multidiscip Optim 58(5):2163–2172MathSciNetGoogle Scholar
  49. Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA J 2:1333–1336Google Scholar
  50. Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Methods Eng 20(9):1685– 1695zbMATHGoogle Scholar
  51. Poon NMK, Martins JRA (2007) An adaptive approach to contstraint aggregation using adjoint sensitivity analysis. Struct Multidiscip Optim 34:61–73Google Scholar
  52. Rahmatalla S, Swan C (2003) Continuum topology optimization of buckling–sensitive structures. AIAA J 41(6):1180–1189Google Scholar
  53. Raspanti CG, Bandoni JA, Biegler LT (2000) New strategies for flexibility analysis and design under uncertainties. Comput Chem Eng 24:2193–2209Google Scholar
  54. Reitinger R, Ramm E (1995) Buckling and imperfection sensitivity in the optimization of shell structures. Thin-Walled Struct 23:159–177Google Scholar
  55. Rodrigues HC, Guedes JM, Bendsøe MP (1995) Necessary conditions for optimal design of structures with a nonsmooth eigenvalue based criterion. Struct Optim 9:52–56Google Scholar
  56. Rojas-Labanda S, Stolpe M (2015) Automatic penalty continuation in structural topology optimization. Struct Multidiscip Optim 52:1205–1221MathSciNetGoogle Scholar
  57. Rozvany G (1996) Difficulties in topology optimization with stress, local buckling and system stability constraints. Struct Optim 11:213–217Google Scholar
  58. Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8(4):207–227Google Scholar
  59. Sigmund O (2007) Morphology–based black and white filters for topology optimization. Struct Multidiscip Optim 33(4):401– 424Google Scholar
  60. Simitses GJ (1973) Optimal versus the stiffened circular plate. AIAA J 11:1409–1412Google Scholar
  61. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638MathSciNetzbMATHGoogle Scholar
  62. Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetzbMATHGoogle Scholar
  63. Szyszkowski W, Watson L (1988) Optimization of the buckling load of columns and frames. Eng Struct 10 (4):249–256Google Scholar
  64. Thomsen CR, Wang F, Sigmund O (2018) Buckling strength topology optimization of 2D periodic materials based on linearized bifurcation analysis. Comput Methods Appl Mech Eng 339:115–136MathSciNetGoogle Scholar
  65. Torii AJ, Faria JR (2017) Structural optimization considering smallest magnitude eigenvalues: a smooth approximation. J Braz Soc Mech Sci Eng 39(5):1745–1754Google Scholar
  66. Turner MJ, Clough RJ, Martin HC, Topp LJ (1956) Stiffness and deflection analysis of complex structures. J Aerosol Sci 23:805–823zbMATHGoogle Scholar
  67. Verbart A, Langelaar M, van Keulen F (2017) A unified aggregation and relaxation approach for stress–constrained topology optimization. Struct Multidiscip Optim 55:663–679MathSciNetGoogle Scholar
  68. Wang F, Lazarov B, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784zbMATHGoogle Scholar
  69. Wilson EL, Taylor RL, Doherty W, Glaboussi J (1973) Incompatible displacement models, Academic Press, pp 41–57Google Scholar
  70. Wu CC, Arora JS (1988) Design sensitivity analysis of non–linear buckling load. Comput Mech 3:129–140zbMATHGoogle Scholar
  71. Yang RJ, Chen CJ (1996) Stress–based topology optimization. Struct Optim 12(2):98–105Google Scholar
  72. Ye HL, Wang WW, Chen N, Sui YK (2016) Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering functions. Acta Mech Sinica 32(4):649–658MathSciNetzbMATHGoogle Scholar
  73. Zhou M (2004) Topology optimization of shell structures with linear buckling responses. In: Iakkis O (ed) WCCM VI in Beijing, China, pp 795–800Google Scholar
  74. Zhou M, Sigmund O (2017) On fully stressed design and p–norm measures in structural optimization. Struct Multidiscip Optim 56(731–736)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTechnical University of DenmarkKongens LyngbyDenmark

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