Skip to main content
SpringerLink
Log in

Filter the shape sensitivity in level set-based topology optimization methods

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

Filtering methods are widely used in density-based topology optimization methods, such as the SIMP method, to prevent checkerboards and mesh dependency due to their ease of implementation and their efficiency. In this study, several filtering schemes are presented to filter the shape sensitivity in level set-based structural topology optimization methods. It is revealed that filtering of the shape sensitivity can yield convergence with less iterations without considerably increasing the computing time of each iteration step. Thus, it can improve the overall computational efficiency. The validity of the method is tested on both the mean compliance minimization problem and the compliant mechanisms design problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  • Allaire G (2012) A 2-d scilab code for shape and topology optimization by the level set method. http://www.cmap.polytechnique.fr/allaire/levelset_en.html

  • Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. C R Math 334(12):1125–1130

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Amir O, Sigmund O (2011) On reducing computational effort in topology optimization: how far can we go?. Struct Multidiscip Optim 44(1):25–29

    Article  MATH  Google Scholar 

  • Ansola R, Veguería E, Canales J, Tárrago JA (2007) Asimple evolutionary topology optimization procedure for compliant mechanism design. Finite Elem in Anal Des 44(1):53–62

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

  • 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 

  • Burger M (2003) A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Boundaries 5(3):301–330

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Challis VJ, Guest JK (2009) Level set topology optimization of fluids in stokes flow. Int J Numer Methods Eng 79(10):1284–1308

    Article  MATH  MathSciNet  Google Scholar 

  • Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1: linear systems. Springer

  • Deepak SR, Dinesh M, Sahu DK et al (2009) A comparative study of the formulations and benchmark problems for the topology optimization of compliant mechanisms. J Mech Robot 1(1):1–8

    Article  Google Scholar 

  • van Dijk NP, Langelaar M, van Keulen F (2012) Explicit level-set-based topology optimization using an exact heaviside function and consistent sensitivity analysis. Int J Numer Methods Eng 91(1):67–97

    Article  MATH  MathSciNet  Google Scholar 

  • van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level set methods for structural topology optimization: a review, vol 48, pp 437–472

  • Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–389

    Article  Google Scholar 

  • de Gournay F (2006) Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J Control Optim 45(1):343–367

    Article  MATH  MathSciNet  Google Scholar 

  • Hamza K, Saitou K (2012) A co-evolutionary approach for design optimization via ensembles of surrogates with application to vehicle crashworthiness. Trans ASME, J Mech Des 134(1):01100,101–01100,110

    Article  Google Scholar 

  • Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors considering design-dependent effects including, heat conduction and convection. Int J Heat Mass Trans 52(11-12):2721–2732

    Article  MATH  Google Scholar 

  • Liu Z, Korvink JG, Huang R (2011) Structure topology optimization: fully coupled level set method via femlab. Struct Multidiscip Optim 29(6):407–417

    Article  MathSciNet  Google Scholar 

  • Luo J, Luo Z, Chen L, Tong L, Wang MY (2008a) A semi-implicit level set method for structural shape and topology optimization. J Comput Phys 227(11):5561–5581

    Article  MATH  MathSciNet  Google Scholar 

  • Luo Z, Tong L, Wang MY, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227(1):680–705

    Article  MATH  MathSciNet  Google Scholar 

  • Luo Z, Wang M Y, Wang S, Wei P (2008b) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76(1):1–26

    Article  MATH  MathSciNet  Google Scholar 

  • Mei Y, Wang X (2004) A level set method for structural topology optimization and its applications advances in engineering software 35(7):415–441

  • Mohammadi B, Pironneau O, Mohammadi B et al (2001) Applied shape optimization for fluids. Oxford University Press, Oxford

  • Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer, New York

    Google Scholar 

  • Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. J Comput Phys 79(1):12–49

    Article  MATH  MathSciNet  Google Scholar 

  • Osher SJ, Santosa F (2001) Level set methods for optimization problems involving geometry and constraints, I frequencies of a two-density inhomogeneous drum. J Comput Phys 171(1):272–288

    Article  MATH  MathSciNet  Google Scholar 

  • Rong JH, Liang QQ (2008) A level set method for topology optimization of continuum structures with bounded design domains. Comput Methods Appl Mech Eng 197(17):1447–1465

    Article  MATH  MathSciNet  Google Scholar 

  • Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37(3):217–237

    Article  MATH  MathSciNet  Google Scholar 

  • Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4(3-4):250– 252

    Article  Google Scholar 

  • Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, omputer version, and material science. Cambridge University Press

  • 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. Technical University of Denmark, Denmark

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

    Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip 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 Multidiscip Optim 16(1):68–75

    Article  Google Scholar 

  • Sokolowski J, Zolesio JP (1992) Introduction to shape optimization Shape sensitivity analysis. Springer

  • Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146–159

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Wang MY (2005) Toplsm 199-line version. http://spring.mae.cuhk.edu.hk/cmdl/download.htm

  • Wang MY (2009) A kinetoelastic formulation of compliant mechanism optimization. J Mech Robot 1(2):02101,101–02101,110

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Wang X, Wang M, Guo D (2004) Structural shape and topology optimization in a level-set-based framework of region representation, vol 27, pp 1–19

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

    Article  MATH  Google Scholar 

  • Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  • Yamasaki S, Nomura T, Kawamoto A, Sato K, Izui K, Nishiwaki S (2010) A level set based topology optimization method using the discretized signed distance function as the design variables. Struct Multidiscip Optim 41(5):685–698

    Article  MATH  MathSciNet  Google Scholar 

  • Yildiz AR, Saitou K (2011) Topology synthesis of multicomponent structural assemblies in continuum domains. Trans ASME, J Mech Des 133(1):01100,801–01100,809

    Article  Google Scholar 

  • Zhou M, Wang MY (2012) A semi-lagrangian level set method for structural optimization. Struct Multidiscip Optim 46(4):487–501

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu B, Zhang X, Wang N (2013) Topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method. Struct Multidiscip Optim 47(5):659–672

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu B, Zhang X, Fatikow S (2014a) Level set-based topology optimization of hinge-free compliant mechanisms using a two-step elastic modeling method. J Mech Des 136(3):031,007

    Article  MathSciNet  Google Scholar 

  • Zhu B, Zhang X, Fatikow S (2014b) A velocity predictor-corrector scheme in level-set based topology optimization to improve the computational efficiency Journal of Mechanical Design(In press)

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant No. 91223201), the Natural Science Foundation of Guangdong Province (Grant No. S2013030013355), Project GDUPS (2010), and the Fundamental Research Funds for the Central Universities (2012ZP0004). This support is greatly acknowledged. The authors would like to thank Professor M. Y. Wang and Professor G. Allaire for sharing their computer codes on the internet. It has been helpful for demonstrating our method. The first author also would like to thank the support of the Sino-German(CSC-DAAD) Postdoc Scholarship.

Author information

Authors and Affiliations

  1. Key Laboratory of Precision Equipment and Manufacturing Technology of Guangdong Province, South China University of Technology, Guangzhou, Guangdong, 510640, China

    Benliang Zhu, Xianmin Zhang & Sergej Fatikow

Authors
  1. Benliang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianmin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergej Fatikow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianmin Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, B., Zhang, X. & Fatikow, S. Filter the shape sensitivity in level set-based topology optimization methods. Struct Multidisc Optim 51, 1035–1049 (2015). https://doi.org/10.1007/s00158-014-1194-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-014-1194-8

Keywords

Search

Navigation