ViT-Transformer: Self-attention mechanism based constitutive modeling for nonlinear heterogeneous materials

Multi-scale simulations of nonlinear heterogeneous materials and composites are challenging due to the prohibitive computational costs of high-fidelity simulations. Recently, machine learning (ML) based approaches have emerged as promising alternatives to traditional multiscale methods. However, existing ML surrogate constitutive models struggle in capturing long-range dependencies and generalization across microstructures. The recent advancements in attention-based Transformer architectures open the door to a more powerful class of surrogate models. Attention mechanism has demonstrated remarkable capabilities in natural language processing and computer vision. In this work, we introduce a surrogate (meta) model, namely ViT-Transformer, using a Vision Transformer (ViT) encoder and a Transformer-based decoder which are both driven by the self-attention mechanism. The ViT encoder extracts microstructural features from material images, while the decoder is a masked Transformer encoder that combines the latent geometrical features with the macroscopic strain input sequence to predict the corresponding stress response. To enhance training, we propose a random extract training algorithm that improves robustness to sequences of variable length. We design and construct a compact yet diverse dataset via data augmentation, and validate the surrogate model using various composite material images and loading scenarios. Several numerical examples are provided to show the effectiveness and accuracy of the ViT-Transformer model and the training algorithm.