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Morphology-based black and white filters for topology optimization

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Abstract

To ensure manufacturability and mesh independence in density-based topology optimization schemes, it is imperative to use restriction methods. This paper introduces a new class of morphology-based restriction schemes that work as density filters; that is, the physical stiffness of an element is based on a function of the design variables of the neighboring elements. The new filters have the advantage that they eliminate grey scale transitions between solid and void regions. Using different test examples, it is shown that the schemes, in general, provide black and white designs with minimum length-scale constraints on either or both minimum hole sizes and minimum structural feature sizes. The new schemes are compared with methods and modified methods found in the literature.

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References

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MATH  MathSciNet  Google Scholar 

  • Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var 1:55–69

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Bendsøe MP (1995) Optimization of structural topology, shape and material. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods and applications. Springer, Berlin Heidelberg New York, XIV+370 pp.

    Google Scholar 

  • Borel PI, Frandsen LH, Harpøth A, Kristensen M, Jensen JS, Sigmund O (2005) Topology optimised broadband photonic crystal Y-splitter. Electron Lett 41(2):69–71

    Article  Google Scholar 

  • Borrvall T (2001) Topology optimization of elastic continua using restriction. Arch Comput Methods Eng 8(4):351–385

    MATH  MathSciNet  Google Scholar 

  • Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4928

    Article  MATH  MathSciNet  Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158

    Article  MATH  MathSciNet  Google Scholar 

  • Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM COCV Abr 9:19–48

    MATH  MathSciNet  Google Scholar 

  • Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26-27):3443–3459

    Article  MATH  Google Scholar 

  • Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57(10):1413–1430

    Article  MATH  Google Scholar 

  • Cardoso EL, Fonseca JSO (2003) Complexity control in the topology optimization of continuum structures. J Braz Soc Mech Sci Eng (3):293–301

  • Guest J, Prevost J, 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

    Article  MATH  MathSciNet  Google Scholar 

  • Guo X, Gu YX (2004) A new density-stiffness interpolation scheme for topology optimization of continuum structures. Eng Comput Int J Comput Aided Eng 21(1):9–22

    Article  MATH  Google Scholar 

  • Haber RB, Jog CS, Bendsøe MP (1994) Variable-topology shape optimization with a constraint on perimeter. In: Gilmore B, et al. (eds) Advances in design automation, vol DE 69-2. ASME, New York, pp 261–272

    Google Scholar 

  • Jensen JS, Sigmund O (2004) Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends. Appl Phys Lett 84(12):2022–2024

    Article  Google Scholar 

  • Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J Opt Soc Am B: Opt Phys 22(6):1191–1198

    Article  Google Scholar 

  • Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization—a new paradigm. Int J Solids Struct 37:5529–5559

    Article  MATH  MathSciNet  Google Scholar 

  • Kreisselmeier G, Steinhauser R (1983) Application of vector performance optimization to a robust control loop design for a fighter aircraft. Int J Control 37(2):251–84

    MATH  Google Scholar 

  • Niordson FI (1983) Optimal design of plates with a constraint on the slope of the thickness function. Int J Solids Struct 19:141–151

    Article  MATH  Google Scholar 

  • Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41(8):1417– 1434

    Article  MATH  MathSciNet  Google Scholar 

  • Poulsen TA (2002) Topology optimization in wavelet space. Int J Numer Methods Eng 53:567–582

    Article  MATH  MathSciNet  Google Scholar 

  • Poulsen TA (2003) A new scheme for imposing a minimum length scale in topology optimization. Int J Numer Methods Eng 57(6):741–760

    Article  MATH  MathSciNet  Google Scholar 

  • Pratt WK (1991) Digital image processing. Wiley, New York

    MATH  Google Scholar 

  • Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528

    Article  MATH  MathSciNet  Google Scholar 

  • Sigmund O (1994) Design of material structures using topology optimization. PhD thesis, Department of Solid Mechanics, Technical University of Denmark, DK-2800 Lyngby, the Danish Center for Applied Mathematics and Mechanics, DCAMM Special Report No. S69

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

    Google Scholar 

  • Sigmund O (2000) Topology optimization: a tool for the tailoring of structures and materials. Philos Trans R Soc Math Phys Eng Sci 358(1765):211–228

    Article  MATH  MathSciNet  Google Scholar 

  • Sigmund O (2001a) A 99 line topology optimization code written in MATLAB. Struct Multidisc Optim 21:120–127. DOI 10.1007/s001580050176, MATLAB code available online at www.topopt.dtu.dk

  • Sigmund O (2001b) Design of multiphysics actuators using topology optimization - Part II: two-material structures. Comput Methods Appl Mech Eng 190(49-50):6605–6627. DOI 10.1016/S0045-7825(01)00252-3

  • Sigmund O, Jensen JS (2003) Systematic design of phononic band gap materials and structures by topology optimization. Philos Trans R Soc A Math Phys Eng Sci 361:1001–1019

    Article  MATH  MathSciNet  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 

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1-2):227–246

    Article  MATH  MathSciNet  Google Scholar 

  • Wang MY, Wang S (2005) Bilateral filtering for structural topology optimization. Int J Numer Methods Eng 63(13):1911–1938

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Alle, B. 404, DK-2800, Lyngby, Denmark

    Ole Sigmund

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  1. Ole Sigmund
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Correspondence to Ole Sigmund.

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Sigmund, O. Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33, 401–424 (2007). https://doi.org/10.1007/s00158-006-0087-x

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  • DOI: https://doi.org/10.1007/s00158-006-0087-x

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