Open Access

ARTICLE

Phase Field: A Variational Method for Structural Topology Optimization

by Michael Yu Wang¹, Shiwei Zhou²

1 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Tel.: +852-2609-8487; Fax: +852-2603-6002. E-mail: yuwang@acae.cuhk.edu.hk
2 Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Computer Modeling in Engineering & Sciences 2004, 6(6), 547-566. https://doi.org/10.3970/cmes.2004.006.547

Abstract

In this paper 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 Γ-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 wellposed problem 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 2D examples that have been extensively used in the recent literature of 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.

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