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Analytical relationships for imposing minimum length scale in the robust topology optimization formulation

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Abstract

When using the robust topology optimization formulation in the density framework, the minimum size of the solid and void phases must be imposed implicitly through the parameters that define the density filter and the smoothed Heaviside projection. Finding these parameters can be time consuming and cumbersome, hindering a general code implementation of the robust formulation. Motivated by this issue, in this article, we provide analytical expressions that explicitly relate the minimum length scale and the parameters that define it. The expressions are validated on a density-based framework. To facilitate the reproduction of results, MATLAB codes are provided.

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Notes

  1. For the sake of clarity, this manuscript changes two notations with respect to Qian and Sigmund (2013). Here, \(2r_{\text {min.Solid}}^{\text {int}}\) and \(r_{\min \limits }\) respectively represent b and R in the cited article.

  2. The maximum size is actually imposed using an annular region (Fernández et al. 2020), but for illustrative purposes, a circular region is drawn.

  3. https://github.com/DenisTri/Analytical_Min_Size

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Funding

The authors acknowledge the research project FAFil (Fabrication Additive laser par dépôt de Fil), funded by INTERREG V A Grande Région and the European Regional Development Fund (ERDF).

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Liège, 4000, Liège, Belgium

    Denis Trillet, Pierre Duysinx & Eduardo Fernández

Authors
  1. Denis Trillet
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  2. Pierre Duysinx
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  3. Eduardo Fernández
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Corresponding author

Correspondence to Denis Trillet.

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The authors declare that they have no conflict of interest.

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Responsible Editor: Gengdong Cheng

Replication of results

This manuscript contains two MATLAB codes as supplementary material that can be found on GitHubsFootnote 3. The first is called SizeSolution.m and provides a list of filter and projection parameters that impose user defined minimum length scales. The second is called NumericalSolution.m and builds the graphs in Fig. 6 using the numerical method proposed by Wang et al. (2011).

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Appendix

Appendix

As we show the contour plot of the minimum sizes ratio, it is possible to produce the same kind of figure for the offset distances. These are plotted in Fig. 19 and illustrate again the symmetric behavior of the sizes with the threshold values for a case where ηint = 0.5.

Fig. 19
figure 19

Contour plot of the offset distances in the domain (ηero,ηdil)

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Trillet, D., Duysinx, P. & Fernández, E. Analytical relationships for imposing minimum length scale in the robust topology optimization formulation. Struct Multidisc Optim 64, 2429–2448 (2021). https://doi.org/10.1007/s00158-021-02998-w

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  • DOI: https://doi.org/10.1007/s00158-021-02998-w

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