An Adaptive Phase-Field Method for Structural Topology Optimization

TL;DR

The paper develops an adaptive phase-field approach for the volume-constrained minimum compliance problem in topology optimization, coupling a phase-field density to linear elasticity with $\mathcal{C}=\rho^p\mathcal{C}_0$ and regularizing the design via a ModicaMortola-type functional $\mathcal{F}_\gamma$. An AFEM loop driven by two residual-based a posteriori estimators solves the relaxed problem on progressively refined meshes, with an augmented Lagrangian enforcing the volume constraint and a gradient-flow update for the density. The authors prove that the adaptively generated sequence contains a subsequence converging to a solution of the continuous first-order optimality system (under an integer $p>1$), and demonstrate on six benchmarks that the adaptive method achieves comparable objectives to uniform refinement while producing sharper interfaces at lower computational cost. The work provides the first provably convergent adaptive phase-field topology optimization framework and offers practical guidance on estimator behavior and efficiency.

Abstract

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

Figures (10)

• Figure 1: The schematic illustration of the geometry, loading and boundary conditions of the examples.
• Figure 2: The initial phase--field functions $\rho_0$ for Examples (a)-(c).
• Figure 3: The evolution of the mesh during the adaptive process, from $k=0$ (initial) to 5 for Example (a), with the number of vertices of each mesh being 1718, 2782, 4039, 6464, 10684 and 16687. The second, third and last columns show the optimized design $\rho_k^*$, the error indicators $\eta_{k,1}$ and $\eta_{k,2}$ respectively.
• Figure 4: The evolution of the mesh during the adaptive procedure, from $k=0$ (initial) to 5 for Example (b) (MBB), with the number of vertices of each mesh being 2230, 3338, 5237, 8525, 14084 and 23314. The second, third, and fourth columns show the optimized design $\rho_k^*$, error indicators $\eta_{k,1}$ and $\eta_{k,2}$ respectively.
• Figure 5: The evolution of the mesh during the adaptive refinement procedure from $k=0$ (initial) to 5 for Example (c), with the number of vertices of each mesh being 1767, 2269, 3094, 4473, 6572 and 9959. The second, third and fourth columns show the optimized design $\rho_k^*$, error indicators $\eta_{k,1}$ and $\eta_{k,2}$ respectively.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.1
• proof
• Theorem 3.1
• Lemma 5.1
• proof
• Theorem 5.1
• proof