The Generalized Cahn-Hilliard Equations of Multiphase Transition for Structural Topology Optimization

This paper describes a generalized Cahn-Hilliard model for the topology optimization of multi-material structure. Unlike the traditional Cahn-Hilliard model applied to spinodal separation which only has bulk energy and interface energy, the generalized model couples the elastic energy into the total free energy. As a result, the morphology of the small phase domain during phase separation and grain coarsening process is not random islands and zigzag web-like objects but regular truss structure. Although disturbed by elastic energy, the Cahn-Hilliard system still keeps its two most important properties: energy dissipation and mass conservation. Therefore, it is unnecessary to compute the Lagrange multipliers for the volume constraints and make great effort to minimize the elastic energy for the optimization of structural topology. Furthermore, this model also makes the simple interpolation of stiffness tensors reasonable for multi-material structure in real simulation. To resolve these fourth-order nonlinear parabolic Cahn-Hilliard equations coupled with elastic energy, we developed a powerful mutigrid algorithm. Finally, we demonstrate that this new method is effective in optimizing the topology of multi-material structure through several 2-D examples.