Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 133–146 | Cite as

Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures

  • Narindra Ranaivomiarana
  • François-Xavier Irisarri
  • Dimitri Bettebghor
  • Boris DesmoratEmail author
Original Article


An optimization methodology to find concurrently material spatial distribution and material anisotropy repartition is proposed for orthotropic, linear and elastic two-dimensional membrane structures. The shape of the structure is parameterized by a density variable that determines the presence or absence of material. The polar method is used to parameterize a general orthotropic material by its elasticity tensor invariants by change of frame. A global structural stiffness maximization problem written as a compliance minimization problem is treated, and a volume constraint is applied. The compliance minimization can be put into a double minimization of complementary energy. An extension of the alternate directions algorithm is proposed to solve the double minimization problem. The algorithm iterates between local minimizations in each element of the structure and global minimizations. Thanks to the polar method, the local minimizations are solved explicitly providing analytical solutions. The global minimizations are performed with finite element calculations. The method is shown to be straightforward and efficient. Concurrent optimization of density and anisotropy distribution of a cantilever beam and a bridge are presented.


Topology optimization SIMP Distributed orthotropy Polar method Material design 


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Copyright information

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

Authors and Affiliations

  • Narindra Ranaivomiarana
    • 1
  • François-Xavier Irisarri
    • 1
  • Dimitri Bettebghor
    • 1
  • Boris Desmorat
    • 2
    Email author
  1. 1.DMAS, ONERAUniversité Paris SaclayChâtillonFrance
  2. 2.Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert, UMR 7190ParisFrance

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