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Multi-material topology optimization using ordered SIMP interpolation

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Abstract

In this paper an ordered multi-material SIMP (solid isotropic material with penalization) interpolation is proposed to solve multi-material topology optimization problems without introducing any new variables. Power functions with scaling and translation coefficients are introduced to interpolate the elastic modulus and the cost properties for multiple materials with respect to the normalized density variables. Besides a mass constraint, a cost constraint is also considered in compliance minimization problems. A heuristic updating scheme of the design variables is derived from the Kuhn-Tucker optimality condition (OC). Since the proposed method does not rely on additional variables to represent material selection, the computational cost is independent of the number of materials considered. The iteration scheme is designed to jump across the discontinuous point of interpolation derivatives to make stable transition from one material phase to another. Numerical examples are included to demonstrate the proposed method. Because of its conceptual simplicity, the proposed ordered multi-material SIMP interpolation can be easily embedded into any existing single material SIMP topology optimization codes.

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References

  • Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidisc Optim 43(1):1–16

    Article  MATH  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202

    Article  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenisation method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MATH  Google Scholar 

  • Bendsøe M, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654

    Article  MATH  Google Scholar 

  • Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin

    MATH  Google Scholar 

  • Blasques JP (2014) Multi-material topology optimization of laminated composite beams with eigenfrequency constraints. Compos Struct 111:45–55

    Article  Google Scholar 

  • Blasques JP, Stolpe M (2012) Multi-material topology optimization of laminated composite beam cross sections. Compos Struct 94(11):3278–3289

    Article  Google Scholar 

  • Challis VJ (2010) A discrete level-set topology optimization code written in Matlab. Struct Multidisc Optim 41(3):453–464

    Article  MathSciNet  MATH  Google Scholar 

  • Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45

    Article  Google Scholar 

  • Fredricson H (2005) Topology optimization of frame structures-joint penalty and material selection. Struct Multidisc Optim 30(3):193–200

    Article  MathSciNet  Google Scholar 

  • Guest J, Asadpoure A, Ha S (2011) Eliminating beta-continuation from Heaviside projection and density filter algorithms. Struct Multidisc Optim 44(4):443–453

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Cheng G-D (2010) Recent development in structural design and optimization. Acta Mech Sinica 26(6):807–823

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Zhang WS, Wang MY, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47-48):3439–3452

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014a) Doing topology optimization explicitly and geometrically-a new moving morphable components based framework. J Appl Mech 81(8):081009

    Article  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014b) Explicit feature control in structural topology optimization via level set method. Comput Methods Appl Mech Eng 272:354–378

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014c) Stress-related topology optimization of continuum structures involving multi-phase materials. Comput Methods Appl Mech Eng 268:632–655

    Article  MathSciNet  MATH  Google Scholar 

  • Hvejsel C, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidisc Optim 43(6):811–825

    Article  MATH  Google Scholar 

  • Liu K, Tovar A (2014) An efficient 3D topology optimization code written in Matlab. Struct Multidisc Optim 50(6):1175–1196

    Article  MathSciNet  Google Scholar 

  • Luo Z, Tong L, Luo J (2009) Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J Comput Phys 228(7):2643–2659

    Article  MathSciNet  MATH  Google Scholar 

  • Ramani A (2009) A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials. Struct Multidisc Optim 41:913–934

    Article  Google Scholar 

  • Ramani A (2011) Multi-material topology optimization with strength constraints. Struct Multidisc Optim 43(5):597–615

    Article  Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524

    Article  Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127

    Article  MathSciNet  Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4-5):401–424

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75

    Article  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Tavakoli R, Mohseni S (2013) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidisc Optim 49(4):621–642

    Article  MathSciNet  Google Scholar 

  • Wang M, 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

    Article  MathSciNet  MATH  Google Scholar 

  • Wang M, Zhou S (2004) Phase field: a variational method for structural topology optimization. Comput Model Eng Sci 6(6):547–566

    MathSciNet  MATH  Google Scholar 

  • Wang MY, Wang MY, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246

    Article  MathSciNet  MATH  Google Scholar 

  • Wei P, Wang M (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Meth Eng 78(4):379–402

    Article  MathSciNet  MATH  Google Scholar 

  • Xie Y, Steven G (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896

    Article  Google Scholar 

  • Yin L, Ananthasuresh GK (2001) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidisc Optim 23(1):49–62

    Article  Google Scholar 

  • Zhang W, Yuan J, Zhang J, Guo X (2016a) A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Struct Multidisc Optim:1–18. doi:10.1007/s00158-015-1372-3

  • Zhang W, Zhang J, Guo X (2016a) Explicit structural topology optimization via moving morphable components-a revival of shape optimization. J Appl Mech 83:041010

    Article  Google Scholar 

  • Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89(1):309–336

    Article  Google Scholar 

  • Zhou S, Wang M (2007) Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct Multidisc Optim 33(2):89–111

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou M, YK S, HL T (2001) Checkerboard and minimum member size control in topology optimization. Struct Multidisc Optim 21(2):152–158

    Article  Google Scholar 

Download references

Acknowledgments

This work was conducted during the first author’s visit to the University of Michigan, which was supported by the National Natural Science Foundation of China (Grant No. 51575226) and the Plan for Scientific and Technological Development of Jilin Province (No. 20140101071JC).

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Authors and Affiliations

  1. State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, 130025, People’s Republic of China

    Wenjie Zuo

  2. School of Mechanical Science and Engineering, Jilin University, Changchun, 130025, People’s Republic of China

    Wenjie Zuo

  3. Department of Mechanical Engineering, University of Michigan, Ann Arbor, 48109, USA

    Kazuhiro Saitou

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  1. Wenjie Zuo
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  2. Kazuhiro Saitou
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Correspondence to Wenjie Zuo.

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Zuo, W., Saitou, K. Multi-material topology optimization using ordered SIMP interpolation. Struct Multidisc Optim 55, 477–491 (2017). https://doi.org/10.1007/s00158-016-1513-3

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  • DOI: https://doi.org/10.1007/s00158-016-1513-3

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