Skip to main content
SpringerLink
Log in

Heaviside projection based topology optimization by a PDE-filtered scalar function

  • Brief Notes
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

This paper deals with topology optimization based on the Heaviside projection method using a scalar function as design variables. The scalar function is then regularized by a PDE based filter. Several image-processing based filtering techniques have so far been proposed for regularization or restricting the minimum length scale. They are conventionally applied to the design sensitivities rather than the design variables themselves. However, it causes discrepancies between the filtered sensitivities and the actual sensitivities that may confuse the optimization process and disturb the convergence. In this paper, we propose a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation. Therefore, the optimality can be strictly discussed in terms of the KKT condition. In order to demonstrate the effectiveness of the proposed method, a minimum compliance problem is solved.

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

References

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

    Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Topology optimization of non-linear elastic structures and compliant mechanisms. Int J Numer Methods Eng 50(9):2143–2158

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidisc Optim 230(3):1615–1488

    MathSciNet  Google Scholar 

  • Gill PE, Murray W, Saunders MA (2007) User’s guide for SNOPT version 7: software for large-scale nonlinear programming. Department of Mathematics, University of California

    Google Scholar 

  • 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):238–254

    Article  MathSciNet  MATH  Google Scholar 

  • Lazarov B, Sigmund O (2009) Sensitivity filters in topology optimisation as a solution to Helmholtz type differential equation. In: 8th world congress on structural and multidisciplinary optimization, vol 1370

  • Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state Navier–Stokes flow. Int J Numer Methods Eng 265(7):975–1001

    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 

Download references

Acknowledgement

The authors gratefully acknowledge the encouragement by Ole Sigmund at Technical University of Denmark.

Author information

Authors and Affiliations

  1. Toyota Central R&D Labs., Inc., Nagakute Aichi, 480-1192, Japan

    Atsushi Kawamoto, Tadayoshi Matsumori, Tsuyoshi Nomura & Tsuguo Kondoh

  2. Shibaura Institute of Technology, 307 Fukasaku, Minuma, Saitama, Saitama, 337-8570, Japan

    Shintaro Yamasaki

  3. Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Kyoto, 606-8501, Japan

    Shinji Nishiwaki

Authors
  1. Atsushi Kawamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tadayoshi Matsumori
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shintaro Yamasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tsuyoshi Nomura
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tsuguo Kondoh
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shinji Nishiwaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atsushi Kawamoto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kawamoto, A., Matsumori, T., Yamasaki, S. et al. Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidisc Optim 44, 19–24 (2011). https://doi.org/10.1007/s00158-010-0562-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-010-0562-2

Keywords

Search

Navigation