Structural and Multidisciplinary Optimization

, Volume 60, Issue 5, pp 2113–2130 | Cite as

An aggregation strategy of maximum size constraints in density-based topology optimization

  • Eduardo FernándezEmail author
  • Maxime Collet
  • Pablo Alarcón
  • Simon Bauduin
  • Pierre Duysinx
Research Paper


The maximum size constraint restricts the amount of material within a test region in each point of the design domain, leading to a highly constrained problem. In this work, the local constraints are gathered into a single one using aggregation functions. The challenge of this task is presented in detail, as well as the proposed strategy to address it. The latter is validated on different test problems as the compliance minimization, the minimum thermal compliance, and the compliant mechanism design. These are implemented in the MATLAB software for 2D design domains. As final validation, a 3D compliance minimization problem is also shown. The study includes two well-known aggregation functions, p-mean and p-norm. The comparison of these functions allows a deeper understanding about their behavior. For example, it is shown that they are strongly dependent on the distribution and amount of data. In addition, a new test region is proposed for the maximum size constraint which, in 2D, is a ring instead of a circle around the element under analysis. This slightly change reduces the introduction of holes in the optimized designs, which can contribute to improve manufacturability of maximum size–constrained components.


Length scale Constraints aggregation SIMP 



The work was supported by the project AERO+, funded by the Plan Marshall 4.0 and the Walloon Region of Belgium. Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Scientific Research Fund of Belgium (F.R.S.-FNRS).

Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical EngineeringUniversity of LiègeLiègeBelgium

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