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

Atsushi Kawamoto*, Tadayoshi Matsumori, Shintaro Yamasaki, Tsuyoshi Nomura, Tsuguo Kondoh, Shinji Nishiwaki

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

177 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)19-24
Number of pages6
JournalStructural and Multidisciplinary Optimization
Issue number1
Publication statusPublished - 2011 Jul
Externally publishedYes


  • Filter
  • Heaviside projection method
  • Topology optimization

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Control and Optimization


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