An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases

Abstract

Finding the optimum distribution of material phases in a multi-material structure is a frequent and important problem in structural engineering which involves topology optimization. The Bi-directional Evolutionary Structural Optimization (BESO) method is now a well-known topology optimization method. In this paper an improved soft-kill BESO algorithm is introduced which can handle both single and multiple material distribution problems. A new filtering scheme and a gradual procedure inspired by the continuation approach are used in this algorithm. Capabilities of the proposed method are demonstrated using different examples. It is shown that the proposed method can result in considerable improvements compared to the normal BESO algorithm particularly when solving problems involving very soft material or void phase.

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Notes

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    Another conclusion is that the results of the variable thickness sheet problem will have less elements with intermediate thickness (grey elements) when the maximum and minimum allowed thicknesses are chosen close to each other. See Table 1.

References

  1. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Opt 1(4):193–202. doi:10.1007/BF01650949

    Article  Google Scholar 

  2. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. doi:10.1016/0045-7825(88)90086-2

    Article  Google Scholar 

  3. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654. doi:10.1007/s004190050248

    Article  Google Scholar 

  4. Blasques JP, Stolpe M (2012) Multi-material topology optimization of laminated composite beam cross sections. Compos Struct 94(11):3278–3289. doi:10.1016/j.compstruct.2012.05.002

    Article  Google Scholar 

  5. Bruyneel M (2011) SFP—a new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Opt 43(1):17–27. doi:10.1007/s00158-010-0548-0

    Article  Google Scholar 

  6. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Opt 49(1):1–38. doi:10.1007/s00158-013-0956-z

    MathSciNet  Article  Google Scholar 

  7. Gao T, Zhang W (2011) A mass constraint formulation for structural topology optimization with multiphase materials. Int J Numer Methods Eng 88(8):774–796. doi:10.1002/nme.3197

    MATH  Article  Google Scholar 

  8. Gao T, Zhang W, Duysinx P (2013) Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint. Structural and Multidisciplinary Optimization 48(6):1075–1088 . doi:10.1007/s00158-013-0948-z

    MathSciNet  Article  Google Scholar 

  9. Ghabraie K (2009) Exploring topology and shape optimisation techniques in underground excavations. PhD thesis, School of Civil, Environmental and Chemical Engineering Science. RMIT University, Melbourne, Australia

    Google Scholar 

  10. Ghabraie K (2014) The ESO method revisited. Structural and Multidisciplinary Optimization pp 1–12, doi:10.1007/s00158-014-1208-6

  11. Ghabraie K, Xie YM, Huang X (2010a) Using BESO method to optimize the shape and reinforcement of underground openings. In: Ghafoori N (ed) Challenges, Opportunities and Solutions in Structural Engineering and Construction: Proceedings of the 5th International Structural Engineering and ConstructiomConference (ISEC-5), vol 2009. Taylor and Francis, USA, London, pp 1001–1006

  12. Ghabraie K, Xie YM, Huang X, Ren G (2010b) Shape and reinforcement optimization of underground tunnels. J Comput Sci Technol 4(1):51–63. doi:10.1299/jcst.4.51

    Article  Google Scholar 

  13. Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43(14):1039–1049. doi:10.1016/j.finel.2007.06.006

    Article  Google Scholar 

  14. Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43(3):393–401. doi:10.1007/s00466-008-0312-0

    MATH  MathSciNet  Article  Google Scholar 

  15. Huang X, Xie YM (2010) Evolutionary topology optimization of continuum structures. Wiley

  16. Huang X, Xie YM, Burry MC (2006) A new algorithm for bi-directional evolutionary structural optimization. Japan Soc Mech Eng Int J Ser C 49(4):1091–1099. doi:10.1299/jsmec.49.1091

    Google Scholar 

  17. Liu Y, Jin F, Li Q, Zhou S (2008) A fixed-grid bidirectional evolutionary structural optimization method and its applications in tunnelling engineering. Int J Numer Methods Eng 73(12):1788–1810. doi:10.1002/nme.2145

    MATH  Article  Google Scholar 

  18. Lund E, Stegmann J (2006) Eigenfrequency and buckling optimization of laminated composite shell structures using discrete material optimization. In: e MPB ON, Sigmund O (eds) IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, vol 137. Springer, Netherlands, pp 147–156, DOI doi:10.1007/1-4020-4752-5_15

  19. Nguyen T, Ghabraie K, Tran-Cong T (2014) Applying bi-directional evolutionary structural optimisation method for tunnel reinforcement design considering nonlinear material behaviour. Comput Geotech 55:57–66. doi:10.1016/j.compgeo.2013.07.015

    Article  Google Scholar 

  20. Park J, Sutradhar A (2015) A multi-resolution method for 3d multi-material topology optimization. Comput Methods Appl Mech Eng 285:571–586. doi:10.1016/j.cma.2014.10.011

    MathSciNet  Article  Google Scholar 

  21. Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 15(8):1031–1048. doi:10.1108/02644409810244129

    MATH  Article  Google Scholar 

  22. Querin OM, Victoria M, Díaz C, Martí P (2015) Layout optimization of multi-material continuum structures with the isolines topology design method. Eng Opt 47(2):221–237. doi:10.1080/0305215X.2014.882332

    Article  Google Scholar 

  23. Rispler A, Steven G (1995) Shape optimisation of metallic inserts in composite bolted joints. In: International Aerospace Congress 1995: Proceedings; Second Pacific International Conference on Aerospace and Technology; Sixth Australian Aeronautical Conference, Institution of Engineers, Australia , pp 225–229

  24. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidiscip Opt 16(1):68–75. doi:10.1007/BF01214002

    Article  Google Scholar 

  25. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067. doi:10.1016/S0022-5096(96)00114-7

    MathSciNet  Article  Google Scholar 

  26. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027. doi:10.1002/nme.1259

    MATH  Article  Google Scholar 

  27. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Opt 22(2):116–124. doi:10.1007/s001580100129

    Article  Google Scholar 

  28. Tavakoli R (2014) Multimaterial topology optimization by volume constrained allen–cahn system and regularized projected steepest descent method. Comput Methods Appl Mech Eng 276:534–565. doi:10.1016/j.cma.2014.04.005

    MathSciNet  Article  Google Scholar 

  29. Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line matlab implementation. Struct Multidiscip Opt 49(4):621–642. doi:10.1007/s00158-013-0999-1

    MathSciNet  Article  Google Scholar 

  30. Thomsen J (1992) topology optimization of structures composed of one or two materials. Struct Opt 5 (1–2):108–115. doi:10.1007/BF01744703, presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Germany, Sept. 23 Oct. 4, 1991

    Article  Google Scholar 

  31. Wang MY, Wang X (2004) “color” level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6–8):469–496

    MATH  Article  Google Scholar 

  32. Wang Y, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586. doi:10.1016/j.cma.2014.11.002

    MathSciNet  Article  Google Scholar 

  33. Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896. doi:10.1016/0045-7949(93)90035-C

    Article  Google Scholar 

  34. Yang XY, Xie YM, Steven GP, Querin OM (1999) Bidirectional evolutionary method for stiffness optimization. Am Inst Aeronaut Astronaut J 37(11):1483–1488

    Article  Google Scholar 

  35. Zhou M, Rozvany GIN (2001) On the validity of ESO type methods in topology optimization. Struct Multidiscip Opt 21(1):80–83. doi:10.1007/s001580050170

    Article  Google Scholar 

  36. Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized cahn–hilliard model of multiphase transition. Struct Multidiscip Opt 33(2):89–111. doi:10.1007/s00158-006-0035-9

    MATH  Article  Google Scholar 

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Correspondence to Kazem Ghabraie.

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Ghabraie, K. An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases. Struct Multidisc Optim 52, 773–790 (2015). https://doi.org/10.1007/s00158-015-1268-2

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Keywords

  • Topology optimization
  • Gradual BESO
  • Multi-material design
  • Continuation approach
  • Soft-kill BESO