Further elaborations on topology optimization via sequential integer programming and Canonical relaxation algorithm and 128-line MATLAB code

  • Yuan Liang
  • Gengdong ChengEmail author
Educational Paper


This paper provides further elaborations on discrete variable topology optimization via sequential integer programming and Canonical relaxation algorithm. Firstly, discrete variable topology optimization problem for minimum compliance subject to a material volume constraint is formulated and approximated by a sequence of discrete variable sub-programming with the discrete variable sensitivity. The differences between continuous variable sensitivity and discrete variable sensitivity are discussed. Secondly, the Canonical relaxation algorithm designed to solve this sub-programming is presented with a discussion on the move limit strategy. Based on the discussion above, a compact 128-line MATLAB code to implement the new method is included in Appendix 1. As shown by numerical experiments, the 128-line code can maintain black-white solutions during the optimization process. The code can be treated as the foundation for other problems with multiple constraints.


Discrete variable topology optimization Sequential approximate programming (SAP) Canonical relaxation algorithm MATLAB 


Funding information

This work is supported by the National Natural Science Foundation of China (grant number 11821202).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2396_MOESM1_ESM.m (5 kb)
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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Research Center for Computational MechanicsDalian University of TechnologyDalianPeople’s Republic of China

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