Phase field: a variational method for structural topology optimization.

*(English)*Zbl 1152.74382Summary: We present a variational method to address the topology optimization problem the phase transition method. A phase-field model is employed based on the phase-transition theory in the fields of mechanics and material sciences. The topology optimization is formulated as a continuous problem with the phase-field as design variables within a fixed reference domain. All regions are described in terms of the phase field which makes no distinction between the solid, void and their interface. The van der Waals-Cahn-Hilliard theory is applied to define the variational topology optimization as a dynamic process of phase transition. The \(\Gamma\)-convergence theory is then adapted for an approximate solution to this free-discontinuity problem. As a result, a two-step, alternating numerical procedure is developed which treats the whole design domain simultaneously without any explicit tracking of the interface. Within this variational framework, we show that a regularization theory can be incorporated to lead to a well-posed formulation. We also show that the phase-field model has a close relationship with the general Mumford-Shah model of image segmentation in computer vision. The proposed variational method is illustrated with several two-dimensional examples that have been extensively used in the recent literature on topology optimization, especially in the homogenization-based methods. Extension of the proposed method to the general problems of multiple material phases other than just solid and void is discussed, and it is further suggested that such a variational approach may represent a promising alternative to the widely-used material distribution model for the future development in topology optimization.