Advertisement

Engineering with Computers

, Volume 34, Issue 2, pp 287–305 | Cite as

Multi-material proportional topology optimization based on the modified interpolation scheme

  • Mingtao CuiEmail author
  • Yifei Zhang
  • Xinfeng Yang
  • Chenchun Luo
Original Article

Abstract

A multi-material proportional topology optimization (PTO) method based on the modified material interpolation scheme is proposed in this work. PTO method is a highly heuristic algorithm by which satisfactory results are obtained. When the proposed method is used to solve the minimum compliance problem, the design variables are assigned to elements proportionally by the value of compliance during the optimization process. It is worth mentioning that PTO algorithm does not incorporate sensitivities. Accordingly, there is nothing about sensitivity calculation but just a weighted density used as filtering in the proposed method. Hence, non-sensitivity is also one of the salient features of this method. According to the characteristics, a density interpolation approach based on the logistic function is introduced in the present study. This approach cannot only establish the relationship between the material densities and Young’s modulus more reasonably, but also effectively realize the polarization of the intermediate-density elements. The complication associated with sensitivities can be avoided by the complicated interpolation scheme in conjunction with PTO algorithm. The multi-material interpolation scheme is modified from the extended SIMP interpolation approach in three-phase topology optimization. A density-filter-based Heaviside threshold function combined with the modified interpolation is introduced in this work to obtain clear 0/1 optimal topology design. The effectiveness and feasibility of the proposed method are demonstrated by several typical numerical examples of multi-material topology optimization, in which the optimal design with distinct boundaries can be obtained.

Keywords

Multi-material topology optimization PTO algorithm Logistic function Modified material interpolation scheme Heaviside threshold function 

Notes

Acknowledgements

This work was supported by the Project of China Scholarship Council (201506965015). The authors are also grateful to the anonymous reviewers for their valuable suggestions for improving the manuscript.

References

  1. 1.
    Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202CrossRefGoogle Scholar
  3. 3.
    Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16CrossRefzbMATHGoogle Scholar
  4. 4.
    Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Allaire G, Gournay FD, Jouve F, Toader AM (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybernet 34(1):59–80MathSciNetzbMATHGoogle Scholar
  6. 6.
    Luo Z, Tong LY, Kang Z (2009) A level set method for structural shape and topology optimization using radial basis functions. Comput Struct 87(7–8):425–434CrossRefGoogle Scholar
  7. 7.
    Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654zbMATHGoogle Scholar
  8. 8.
    Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4):401–424CrossRefGoogle Scholar
  9. 9.
    Tavakoli R (2014) Multimaterial topology optimization by volume constrained Allen–Cahn system and regularized projected steepest descent method. Comput Methods Appl Mech Eng 276:534–565MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fuchs MB, Jiny S, Peleg N (2005) The SRV constraint for 0/1 topological design. Struct Multidiscip Optim 30(4):320–326CrossRefGoogle Scholar
  11. 11.
    Du YX, Yan SQ, Zhang Y, Xie HH, Tian QH (2015) A modified interpolation approach for topology optimization. Acta Mech Solida Sin 28(4):420–430CrossRefGoogle Scholar
  12. 12.
    Svanberg K, Werme M (2007) Sequential integer programming methods for stress constrained topology optimization. Struct Multidiscip Optim 34:277–299MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Munk DJ, Vio GA, Steven GP (2015) Topology and shape optimization methods using evolutionary algorithms: a review. Struct Multidiscip Optim 52:613–631MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tai K, Akhtar S (2005) Structural topology optimization using a genetic algorithm with a morphological geometric representation scheme. Struct Multidiscip Optim 30:113–127CrossRefGoogle Scholar
  15. 15.
    Prager W (1968) Optimality criteria in structural design. Proc Natl Acad Sci USA 61(3):794–796CrossRefGoogle Scholar
  16. 16.
    Rozvany GIN. (1988) Optimality criteria and layout theory in structural design: recent developments and applications. In: Rozvany GIN, Karihaloo BL (eds) Structural optimization. Springer, DordrechtCrossRefGoogle Scholar
  17. 17.
    Rozvany GIN, Zhou M, Rotthaus M, Gollub W, Spengemann F (1989) Continuum-type optimality criteria methods for large finite element systems with a displacement constraint-Part I. Struct Optim 1(1):47–72CrossRefGoogle Scholar
  18. 18.
    Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21(2):120–127CrossRefGoogle Scholar
  19. 19.
    Gill PE, Murray W, Saunders MA (2002) SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fleury C, Braibant V (1986) Structural optimization: A new dual method using mixed variables. Int J Numer Methods Eng 23(3):409–428MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Biyikli E, To AC (2015) Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLoS One 10(12):1–23CrossRefGoogle Scholar
  23. 23.
    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):1037–1067MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825CrossRefzbMATHGoogle Scholar
  25. 25.
    Yin L, Ananthasuresh GK (2001) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidiscip Optim 23(1):49–62CrossRefGoogle Scholar
  26. 26.
    Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49(4):621–642MathSciNetCrossRefGoogle Scholar
  27. 27.
    Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computat Mech 43:393–401MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gao T, Zhang WH (2011) A mass constraint formulation for structural topology optimization with multiphase materials. Int J Numer Methods Eng 88(8):774–796CrossRefzbMATHGoogle Scholar
  29. 29.
    Wang MY, Zhou S (2004) Synthesis of shape and topology of multi-material structures with a phase-field method. J Comput Aided Mater Des 11(2–3):117–138CrossRefGoogle Scholar
  30. 30.
    Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct Multidiscip Optim 33(2):89–111MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45CrossRefGoogle Scholar
  32. 32.
    Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459CrossRefzbMATHGoogle Scholar
  34. 34.
    Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Guest JK, Prevost 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):238–254MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidiscip Optim 41(4):495–505MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang FW, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2017

Authors and Affiliations

  • Mingtao Cui
    • 1
    • 2
    Email author
  • Yifei Zhang
    • 1
  • Xinfeng Yang
    • 1
  • Chenchun Luo
    • 1
  1. 1.School of Mechano-electronic EngineeringXidian UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontrealCanada

Personalised recommendations