Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition

  • Shiwei Zhou
  • Michael Yu WangEmail author
Research Paper


This paper describes a phase field method for the optimization of multimaterial structural topology with a generalized Cahn–Hilliard model. Similar to the well-known simple isotropic material with penalization method, the mass concentration of each material phase is considered as design variable. However, a variational approach is taken with the Cahn–Hilliard theory to define a thermodynamic model, taking into account of the bulk energy and interface energy of the phases and the elastic strain energy of the structure. As a result, the structural optimization problem is transformed into a phase transition problem defined by a set of nonlinear parabolic partial differential equations. The generalized Cahn–Hilliard model regularizes the original ill-posed topology optimization problem and provides flexibility of topology changes with interface coalescence and break-up due to phase separation and coarsening. We employ a powerful multigrid algorithm and extend it to include four material phases for numerical solution of the Cahn–Hilliard equations. We demonstrate our approach through several 2-D and 3-D examples to minimize mean compliance of the multimaterial structures.


Topology optimization Cahn–Hilliard equations Phase field Multimaterial Multigrid 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Automation and Computer-Aided EngineeringThe Chinese University of Hong KongShatin, NTPeople’s Republic of China

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