Hybrid homogenization neural networks for periodic composites

A new physics-informed deep homogenization neural network (DHN) framework is proposed to identify the homogenized and local behaviors in periodic heterogeneous microstructures. To achieve this, the displacement field is decomposed into averaged and fluctuating contributions, with the local unit cell solution obtained via neural networks subject to periodic boundary conditions. The periodic microstructures are divided into subdomains representing the fiber and matrix phases, respectively. A key contribution of the proposed method is the marriage of elasticity solution and physics-informed neural network to each phase of the composite, namely, the fiber phase as a mesh-free component whose fluctuating displacements are expanded using a discrete Fourier transform, and the matrix phase using material points with fluctuating displacements handled through fully connected neural network layers. The interfacial continuity conditions are enforced by minimizing the traction and displacement differences at separate material points along the interface. Transfer learning is exploited further to facilitate training new microstructures from pre-trained geometry. This hybrid formulation inherently satisfies stress equilibrium equations within the fiber, while efficiently handling the periodic boundary conditions of hexagonal and square unit cells via a series of trainable sinusoidal functions. The innovative use of distinct neural network architectures enables accurate and efficient predictions of displacement and stress when discontinuities are present in the solution fields across the interface. We validate the proposed DHN with the finite-element predictions for unidirectional composites comprised of elastic fiber significantly stiffer than the matrix, under various volume fractions and loading conditions.