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Structural and Multidisciplinary Optimization

, Volume 59, Issue 4, pp 1181–1197 | Cite as

Non-probabilistic robust continuum topology optimization with stress constraints

  • Gustavo Assis da SilvaEmail author
  • Eduardo Lenz Cardoso
  • André Teófilo Beck
Research Paper
  • 178 Downloads

Abstract

This paper proposes a non-probabilistic robust design approach, based on optimization with anti-optimization, to handle unknown-but-bounded loading uncertainties in stress-constrained topology optimization. The objective of the proposed topology optimization problem is to find the lightest structure that respects the worst possible scenario of local stress constraints, given predefined bounds on magnitudes and directions of applied loads. A solution procedure based on the augmented Lagrangian method is proposed, where worst-case local stress constraints are handled without employing aggregation techniques. Results are post-processed, demonstrating that maximum stress of robust solutions is almost insensitive with respect to changes in loading scenarios. Numerical examples also demonstrate that obtained robust solutions satisfy the stress failure criterion for any load condition inside the predefined range of unknown-but-bounded uncertainties in applied loads.

Keywords

Topology optimization Stress constraints Uncertainties Non-probabilistic Robust Worst case 

Notes

Funding information

The authors received financial support from CNPq (National Council for Research and Development), grant number 306373/2016-5, FAPESP (São Paulo Research Foundation), grant number 2015/25199-0, and FAPESC, grant numbers 2017TR1747 and 2017TR784. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

References

  1. Amir O (2017) Stress-constrained continuum topology optimization: a new approach based on elasto-plasticity. Struct Multidiscip Optim 55(5):1797–1818.  https://doi.org/10.1007/s00158-016-1618-8 MathSciNetCrossRefGoogle Scholar
  2. Arora J S (2012) Introduction to optimum design, 3rd edn. Academic Press, BostonGoogle Scholar
  3. Barlow J (1976) Optimal stress locations in finite element models. Int J Numer Methods Eng 10(2):243–251.  https://doi.org/10.1002/nme.1620100202 CrossRefzbMATHGoogle Scholar
  4. Bathe K J (1996) Finite element procedures. Prentice Hall, Upper Sadle RiverzbMATHGoogle Scholar
  5. Beck AT, Gomes WJS (2012) A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty. Probab Eng Mech 28:18–29.  https://doi.org/10.1016/j.probengmech.2011.08.007 CrossRefGoogle Scholar
  6. Beck AT, Gomes WJS, Lopez RH, Miguel LFF (2015) A comparison between robust and risk-based optimization under uncertainty. Struct Multidiscip Optim 52(3):479–492.  https://doi.org/10.1007/s00158-015-1253-9 MathSciNetCrossRefGoogle Scholar
  7. Birgin E, Martínez J (2014) Practical augmented lagrangian methods for constrained optimization society for industrial and applied mathematics, Philadelphia, PA.  https://doi.org/10.1137/1.9781611973365
  8. Bruggi M, Duysinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscip Optim 46(3):369–384.  https://doi.org/10.1007/s00158-012-0759-7 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bucalem M L, Bathe K J (2011) The mechanics of solids and structures - hierarchical modeling and the finite element solution 1st edn. Computational Fluid and Solid Mechanics. Springer, Berlin.  https://doi.org/10.1007/978-3-540-26400-2 zbMATHGoogle Scholar
  10. Cheng GD, Guo X (1997) ε-relaxed approach in structural topology optimization. Struct Optim 13(4):258–266.  https://doi.org/10.1007/BF01197454 CrossRefGoogle Scholar
  11. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43 (8):1453–1478.  https://doi.org/10.1002/(SICI)1097-0207(19981230)43:8<1453::AID-NME480>3.0.CO;2-2 MathSciNetCrossRefzbMATHGoogle Scholar
  12. Elishakoff I, Ohsaki M (2010) Optimization and anti-optimization of structures under uncertainty. Imperial College Press, LondonCrossRefzbMATHGoogle Scholar
  13. Elishakoff I, Haftka R, Fang J (1994) Structural design under bounded uncertainty - optimization with anti-optimization. Comput Struct 53(6):1401–1405.  https://doi.org/10.1016/0045-7949(94)90405-7 CrossRefzbMATHGoogle Scholar
  14. Fancello E A, Pereira J T (2003) Structural topology optimization considering material failure constraints and multiple load conditions. Lat Am J Solids Struct 1(1):3–24Google Scholar
  15. Guest JK, Prvost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254.  https://doi.org/10.1002/nme.1064 MathSciNetCrossRefzbMATHGoogle Scholar
  16. Guest JK, Asadpoure A, Ha SH (2011) Eliminating beta-continuation from heaviside projection and density filter algorithms. Struct Multidiscip Optim 44(4):443–453.  https://doi.org/10.1007/s00158-011-0676-1 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Guo X, Bai W, Zhang W, Gao X (2009) Confidence structural robust design and optimization under stiffness and load uncertainties. Comput Methods Appl Mech Eng 198(41):3378–3399.  https://doi.org/10.1016/j.cma.2009.06.018 MathSciNetCrossRefzbMATHGoogle Scholar
  18. Guo X, Zhang W, Zhang L (2013) Robust structural topology optimization considering boundary uncertainties. Comput Methods Appl Mech Eng 253:356–368.  https://doi.org/10.1016/j.cma.2012.09.005 MathSciNetCrossRefzbMATHGoogle Scholar
  19. Guo X, Zhao X, Zhang W, Yan J, Sun G (2015) Multi-scale robust design and optimization considering load uncertainties. Comput Methods Appl Mech Eng 283:994–1009.  https://doi.org/10.1016/j.cma.2014.10.014 MathSciNetCrossRefzbMATHGoogle Scholar
  20. Gurav S, Goosen J, vanKeulen F (2005) Bounded-but-unknown uncertainty optimization using design sensitivities and parallel computing: Application to mems. Comput Struct 83(14):1134–1149.  https://doi.org/10.1016/j.compstruc.2004.11.021 CrossRefGoogle Scholar
  21. Holmberg E, Thore CJ, Klarbring A (2015) Worst-case topology optimization of self-weight loaded structures using semi-definite programming. Struct Multidiscip Optim 52(5):915–928.  https://doi.org/10.1007/s00158-015-1285-1 MathSciNetCrossRefGoogle Scholar
  22. Holmberg E, Thore CJ, Klarbring A (2017) Game theory approach to robust topology optimization with uncertain loading. Struct Multidiscip Optim 55(4):1383–1397.  https://doi.org/10.1007/s00158-016-1548-5 MathSciNetCrossRefGoogle Scholar
  23. Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620.  https://doi.org/10.1007/s00158-009-0440-y
  24. Liu J, Gea HC (2018) Robust topology optimization under multiple independent unknown-but-bounded loads. Comput Methods Appl Mech Eng 329(Supplement C):464–479.  https://doi.org/10.1016/j.cma.2017.09.033 MathSciNetCrossRefGoogle Scholar
  25. Lombardi M (1998) Optimization of uncertain structures using non-probabilistic models. Comput Struct 67 (1):99–103.  https://doi.org/10.1016/S0045-7949(97)00161-2 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Lombardi M, Haftka RT (1998) Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng 157(1):19–31.  https://doi.org/10.1016/S0045-7825(97)00148-5 CrossRefzbMATHGoogle Scholar
  27. Luo Y, Zhou M, Wang MY, Deng Z (2014) Reliability based topology optimization for continuum structures with local failure constraints. Comput Struct 143:73–84.  https://doi.org/10.1016/j.compstruc.2014.07.009 CrossRefGoogle Scholar
  28. Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidiscip Optim 26(1):50–66.  https://doi.org/10.1007/s00158-003-0301-z MathSciNetCrossRefzbMATHGoogle Scholar
  29. Qian X, Sigmund O (2013) Topological design of electromechanical actuators with robustness toward over- and under-etching. Comput Methods Appl Mech Eng 253:237–251.  https://doi.org/10.1016/j.cma.2012.08.020 MathSciNetCrossRefzbMATHGoogle Scholar
  30. Rao SS (2009) Engineering optimization: Theory and practice, 4th edn. Wiley, New JerseyCrossRefGoogle Scholar
  31. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424.  https://doi.org/10.1007/s00158-006-0087-x CrossRefGoogle Scholar
  32. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mechanica Sinica 25(2):227–239.  https://doi.org/10.1007/s10409-009-0240-z CrossRefzbMATHGoogle Scholar
  33. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055.  https://doi.org/10.1007/s00158-013-0978-6 MathSciNetCrossRefGoogle Scholar
  34. da Silva GA, Beck AT (2018) Reliability-based topology optimization of continuum structures subject to local stress constraints. Struct Multidiscip Optim 57(6):2339–2355.  https://doi.org/10.1007/s00158-017-1865-3 MathSciNetCrossRefGoogle Scholar
  35. da Silva GA, Cardoso EL (2017) Stress-based topology optimization of continuum structures under uncertainties. Comput Methods Appl Mech Eng 313:647–672.  https://doi.org/10.1016/j.cma.2016.09.049 MathSciNetCrossRefGoogle Scholar
  36. da Silva GA, Beck AT, Cardoso EL (2018) Topology optimization of continuum structures with stress constraints and uncertainties in loading. Int J Numer Methods Eng 113(1):153–178.  https://doi.org/10.1002/nme.5607 MathSciNetCrossRefGoogle Scholar
  37. Svärd H (2015) Interior value extrapolation: a new method for stress evaluation during topology optimization. Struct Multidiscip Optim 51(3):613–629.  https://doi.org/10.1007/s00158-014-1171-2 CrossRefGoogle Scholar
  38. Thore CJ (2016) On a nash game for topology optimization under load-uncertainty: Finding the worst load. In: VII European congress on computational methods in applied sciences and engineering, pp 5–10Google Scholar
  39. Thore CJ, Holmberg E, Klarbring A (2017) A general framework for robust topology optimization under load-uncertainty including stress constraints. Comput Methods Appl Mech Eng 319:1–18.  https://doi.org/10.1016/j.cma.2017.02.015 MathSciNetCrossRefGoogle Scholar
  40. Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty a polynomial chaos approach. Comput Methods Appl Mech Eng 201204:263–275.  https://doi.org/10.1016/j.cma.2011.09.009 MathSciNetCrossRefzbMATHGoogle Scholar
  41. Wang F, Jensen JS, Sigmund O (2011a) Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. J Opt Soc Am B 28(3):387–397.  https://doi.org/10.1364/JOSAB.28.000387
  42. Wang F, Lazarov BS, Sigmund O (2011b) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784.  https://doi.org/10.1007/s00158-010-0602-y

Copyright information

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

Authors and Affiliations

  1. 1.Department of Structural Engineering, São Carlos School of EngineeringUniversity of São PauloSão CarlosBrazil
  2. 2.Department of Mechanical EngineeringState University of Santa CatarinaJoinvilleBrazil

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