Structural and Multidisciplinary Optimization

, Volume 59, Issue 2, pp 633–645 | Cite as

Topology optimization of cellular materials with periodic microstructure under stress constraints

  • Pedro G. CoelhoEmail author
  • José M. Guedes
  • João B. Cardoso
Research Paper


Material design is a critical development area for industries dealing with lightweight construction. Trying to respond to these industrial needs topology optimization has been extended from structural optimization to the design of material microstructures to improve overall structural performance. Traditional formulations based on compliance and volume control result in stiffness-oriented optimal designs. However, strength-oriented designs are crucial in engineering practice. Topology optimization with stress control has been applied mainly to (macro) structures, but here it is applied to material microstructure design. Here, in the context of density-based topology optimization, well-established techniques and analyses are used to address known difficulties of stress control in optimization problems. A convergence analysis is performed and a density filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element meshes associated with highly non-linear stress behavior. A stress-constraint relaxation technique (qp-approach) is applied to overcome the singularity phenomenon. Parallel computing is used to minimize the impact of the local nature of the stress constraints and the finite difference design sensitivities on the overall computational cost of the problem. Finally, several examples test the developed model showing its inherent difficulties.


Topology Optimization Microstructures Convergence Stress Homogenization 



Authors wish to thank Professor Krister Svanberg (Royal Institute of Technology, Stockholm, Sweden) for the MMA optimization code and Professor Hélder C. Rodrigues (IDMEC, Instituto Superior Técnico, Portugal) for all the discussions on this work.

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Funding information

This work was partially supported by Fundação para a Ciência e a Tecnologia (Portugal) through the projects UID/EMS/00667/2013, UID/EMS/50022/2013, and PTDC/EMS-PRO/4732/2014.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UNIDEMI, Faculty of Sciences and TechnologyUniversidade Nova de Lisboa, FCTCaparicaPortugal
  2. 2.IDMEC, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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