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ARTICLE

3D Multi-Material Structural Topology Optimization with the Generalized Cahn-Hilliard Equations

Shiwei Zhou¹, Michael Yu Wang²

1 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China. E-mail address: swzhou@acae.cuhk.edu.hk
2 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China. E-mail address: yuwang@acae.cuhk.edu.hk

Computer Modeling in Engineering & Sciences 2006, 16(2), 83-102. https://doi.org/10.3970/cmes.2006.016.083

Abstract

This paper describes a self-mass-conservative Cahn-Hilliard (C-H) model with elastic strain energy (mean compliance) for the optimization of multi-material structure topology. The total free energy of the generalized C-H system can be represented as a Lyapunov functional so that the elastic strain energy, as a part of the total free energy, decreases gradually to attain optimal material distribution. The interface energy relating to phase gradient in the total free energy plays an important role in regularizing the original ill-posed problem by restricting the structure's boundaries. On the other hand, interface coalescence and break-up due to phase separation and grain coarsening result in spontaneous and flexible topology changes. To solve the generalized C-H system with elasticity in 3D, which is described by a set of fourth-order nonlinear parabolic PDEs, we develop a powerful implicit multigrid algorithm and prove non-increasing in total free energy and the convergence. Our results are compared with the results of the well-known SIMP model for single material structural topology in 3D. Furthermore, applications to several well-known 3D examples for multi-material structural topology optimization are presented.

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