The Cahn-Hilliard Phase-Field Model for Topology Optimization of Solids

Abstract

In this paper we present a method to optimize the topology of a solid structure based on the theory of phase transition with diffuse interface. A well-known physical model describing phase separation, the Cahn-Hilliard model is modified to reflect the objective of the design optimization. The topology optimization is formulated with a phase parameter that varies continuously at any position within the specified design domain. The relaxation of the parameter is driven by local minimization of the free energy of the system, including the design objective, the bulk energy, and the interface energy. The bulk energy plays a role to draw the design variable to its two distinct material phases (solid and void), while the interface energy smoothes the structure boundary to its minimum perimeter. With properties of mass conservation and energy dissipation of the model, complex morphological and topological material transitions such as coalescence and break-up of the boundary can be naturally captured. A thermodynamic equation for the phase parameter is obtained for the creation, evolution, and dissolution of controlled phase interfaces. We use a multigrid method to solve the resulting fourth-order Cahn-Hilliard equation. The capabilities of the method are demonstrated with three examples in 2-D for the mean-compliance minimization of structures. 1. The Problem of Solids Optimization Let us consider the minimum compliance optimization problem of a statically loaded linear elastic structure under a single loading case (Bendsoe and Kikuchi 1988; Rozvany 1989; Haber et al. 1996). Let d R ⊆ Ω ) 3 or 2 ( = d be an open and bounded set occupied by the linear isotropic elastic structure. The boundary of Ω consists of three parts: 2 1 0 Γ ∪ Γ ∪ Γ = Ω ∂ = Γ , with Dirichlet boundary conditions on 1 Γ and Neumann boundary conditions on 2 Γ . It is assumed that the boundary segment 0 Γ is