Multi-scale Topology Optimization using Neural Networks

TL;DR

The paper tackles the challenge of designing multi-scale structures with strong inter-cell connectivity by introducing a direct multi-scale topology optimization method that uses a neural network to represent a continuous, rotation-aware density field across both micro- and macro-scales. It casts the problem as a $2n$-dimensional inverse homogenization, employing an extended boundary strategy and a boundary loss to enforce compatibility between neighboring cells, and optimizes each microstructure via an energy-based homogenization of the effective stiffness tensor $E_{ijkl}^H$. A topology network with local $(u,w)$ and global $(x,y)$ coordinates, $T(oldsymbol{X})=oldsymbol{\\sigma}(oldsymbol{W}\\sin(oldsymbol{K}oldsymbol{X}+1))$, encodes the microstructure and scales to graded designs, while training jointly minimizes a loss combining homogenized stiffness, volume constraints, and boundary consistency. Demonstrations on graded multi-scale structures and metamaterial systems show good cell connectivity, rotation-aware microstructures, and competitive performance relative to Hashin–Shtrikman bounds, with efficiency gains from mini-batch and mini-epoch strategies and the ability to upsample density fields for higher-resolution results.

Abstract

A long-standing challenge is designing multi-scale structures with good connectivity between cells while optimizing each cell to reach close to the theoretical performance limit. We propose a new method for direct multi-scale topology optimization using neural networks. Our approach focuses on inverse homogenization that seamlessly maintains compatibility across neighboring microstructure cells. Our approach consists of a topology neural network that optimizes the microstructure shape and distribution across the design domain as a continuous field. Each microstructure cell is optimized based on a specified elasticity tensor that also accommodates in-plane rotations. The neural network takes as input the local coordinates within a cell to represent the density distribution within a cell, as well as the global coordinates of each cell to design spatially varying microstructure cells. As such, our approach models an n-dimensional multi-scale optimization problem as a 2n-dimensional inverse homogenization problem using neural networks. During the inverse homogenization of each unit cell, we extend the boundary of each cell by scaling the input coordinates such that the boundaries of neighboring cells are combined. Inverse homogenization on the combined cell improves connectivity. We demonstrate our method through the design and optimization of graded multi-scale structures.

Figures (13)

• Figure 1: The neural network outputs density $\rho$ at each coordinate point. By sampling coordinate points across the design domain, we obtain the density field. From the density field, we calculate the current volume fraction and the homogenized stiffness tensor from an FEA solver. The homogenized stiffness tensor and volume fraction are then formulated as a loss function which is used in backpropagation of the training process until convergence.
• Figure 2: Sampling inside the continuous topology domain for finite element analysis. When the rotation is at $0^\circ$, the boundary is extended by 1.2 times to ensure all 8 neighboring cells are covered. With a $45^\circ$ rotation, a 1.6 times increase is applied.
• Figure 3: The boundary loss is defined as the L1 loss between the red solid line region and the blue dashed line region in (a). When we zoom into the right bottom corner of the multi-scale structure shown in (b), we can observe that with boundary loss (c), the connection between cells is more continuous when upsampled compared to without boundary loss (d).
• Figure 4: Given a changing stiffness tensor weights, we can observe in (a) that, as the combined weights increase to the top right corner, the objective function value for each cell is also smaller. From the converged multi-scale structure, we can observe good connectivity between the cells.
• Figure 5: The convergence history of the multi-scale structure from the neural network. Microstructures emerge as clusters and from these clusters gradually propagate through the design domain.