Structural and Multidisciplinary Optimization

, Volume 59, Issue 4, pp 1021–1031 | Cite as

Simple single-scale microstructures based on optimal rank-3 laminates

  • E. Träff
  • O. Sigmund
  • J. P. GroenEmail author
Research Paper


With the goal of identifying optimal elastic single-scale microstructures for multiple loading situations, the paper shows that qualified starting guesses, based on knowledge of optimal rank-3 laminates, significantly improves chances of convergence to near optimal designs. Rank-3 laminates, optimal for a given set of anisotropic loading conditions, are approximated on a single scale using a simple mapping approach. We demonstrate that these mapped microstructures perform relatively close to theoretical energy bounds. Microstructures with a performance even closer to the bounds can be obtained by using the approximated rank-3 structures in a further step as starting guesses for inverse homogenization problems. Due to the nonconvex nature of inverse homogenization problems, the starting guesses based on rank-3 laminates outperform classical starting guesses with homogeneous or random material distributions. Furthermore, the obtained single-scale microstructures are relatively simple, which enhances manufacturability. Results, obtained for a wide range of loading cases, indicate that microstructures with performance within 5–8% of the theoretical optima can be guarantied, as long as feature sizes are not limited by minimium size constraints.


Inverse homogenization Optimal microstructures Starting guess 



The authors acknowledge the support of the Villum Fonden through the Villum investigator project InnoTop. The authors would also like to thank Krister Svanberg for providing the MATLAB MMA code.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkLyngbyDenmark

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