A Unified Phase-Field Fourier Neural Network Framework for Topology Optimization

TL;DR

This paper presents a unified and physics-driven framework of alternating phase-field Fourier neural networks (APF-FNNs) for topology optimization, establishing a powerful and versatile foundation for physics-driven computational design.

Abstract

This paper presents a unified and physics-driven framework of alternating phase-field Fourier neural networks (APF-FNNs) for topology optimization. At its core, an alternating architecture decouples the optimization by parameterizing the state, adjoint and topology fields with three distinct Fourier Neural Networks (FNNs). These networks are trained through a collaborative and stable alternating optimization scheme applicable to both self-adjoint and non-self-adjoint systems. The Ginzburg-Landau energy functional is incorporated into the topology network's loss function, acting as an intrinsic regularizer that promotes well-defined designs with smooth and distinct interfaces. By employing physics-informed losses derived from either variational principles or strong-form PDE residuals, the broad applicability of the APF-FNNs is demonstrated across a spectrum of 2D and 3D multi-physics benchmarks, including compliance minimization, eigenvalue maximization, and Stokes/Navier-Stokes flow optimization. The proposed APF-FNNs consistently yield high-performance and high-resolution topologies, establishing a powerful and versatile foundation for physics-driven computational design.

Figures (21)

• Figure 1: Schematic overview of the Alternating Phase-Field Fourier Neural Networks (APF-FNNs). The process begins with the physics settings (left panel). The core framework (middle panel) features an alternating architecture with three distinct FNNs parameterizing the state $\bm{u}$, adjoint $\bm{w}$ and topology $\phi$ fields. These networks are trained collaboratively through a stable alternating optimization scheme. The versatility of the framework (right panel) is demonstrated by solving a range of benchmark problems, encompassing both self-adjoint (e.g., linear elasticity and Stokes flow) and non-self-adjoint (e.g., Navier-Stokes flow) systems.
• Figure 2: Flowchart of the nested iterative scheme for conventional phase-field topology optimization.
• Figure 3: Flowchart of the proposed APF-FNNs. The process begins with an Initial Stage to pre-train the state and adjoint networks. This is followed by the main Alternating Optimization stage, where the neural networks for the state, adjoint and topology are refined by sequentially minimizing their respective loss functions.
• Figure 4: Problem setups for the three 2D compliance minimization benchmarks: (a) Cantilever beam, (b) Offset-loaded beam, and (c) MBB beam.
• Figure 5: Topological evolution for the 2D compliance minimization benchmarks. Each row illustrates the progression from a diffuse, ambiguous initial state (left) to a crisp, well-defined final topology (right) for the (a) Cantilever, (b) Offset-loaded and (c) MBB beam cases, respectively.