Topology optimization by a time-dependent diffusion equation

A. Kawamoto*, T. Matsumori, T. Nomura, T. Kondoh, S. Yamasaki, S. Nishiwaki

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


Most topology optimization problems are formulated as constrained optimization problems; thus, mathematical programming has been the mainstream. On the other hand, solving topology optimization problems using time evolution equations, seen in the level set-based and the phase field-based methods, is yet another approach. One issue is the treatment of multiple constraints, which is difficult to incorporate within time evolution equations. Another issue is the extra re-initialization steps that interrupt the time integration from time to time. This paper proposes a way to describe, using a Heaviside projection-based representation, a time-dependent diffusion equation that addresses these two issues. The constraints are treated using a modified augmented Lagrangian approach in which the Lagrange multipliers are updated by simple ordinary differential equations. The proposed method is easy to implement using a high-level finite element code. Also, it is very practical in the sense that one can fully utilize the existing framework of the code: GUI, parallelized solvers, animations, data imports/exports, and so on. The effectiveness of the proposed method is demonstrated through numerical examples in both the planar and spatial cases.

Original languageEnglish
Pages (from-to)795-817
Number of pages23
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
Publication statusPublished - 2013 Feb 24
Externally publishedYes


  • Heaviside projection method
  • Time-dependent diffusion equation
  • Topology optimization

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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