Micrometer: Micromechanics Transformer for Predicting Mechanical Responses of Heterogeneous Materials

TL;DR

The Micromechanics Transformer (Micrometer), an artificial intelligence (AI) framework for predicting the mechanical response of heterogeneous materials, bridging the gap between advanced data-driven methods and complex solid mechanics problems is introduced.

Abstract

Heterogeneous materials, crucial in various engineering applications, exhibit complex multiscale behavior, which challenges the effectiveness of traditional computational methods. In this work, we introduce the Micromechanics Transformer ({\em Micrometer}), an artificial intelligence (AI) framework for predicting the mechanical response of heterogeneous materials, bridging the gap between advanced data-driven methods and complex solid mechanics problems. Trained on a large-scale high-resolution dataset of 2D fiber-reinforced composites, Micrometer can achieve state-of-the-art performance in predicting microscale strain fields across a wide range of microstructures, material properties under any loading conditions and We demonstrate the accuracy and computational efficiency of Micrometer through applications in computational homogenization and multiscale modeling, where Micrometer achieves 1\% error in predicting macroscale stress fields while reducing computational time by up to two orders of magnitude compared to conventional numerical solvers. We further showcase the adaptability of the proposed model through transfer learning experiments on new materials with limited data, highlighting its potential to tackle diverse scenarios in mechanical analysis of solid materials. Our work represents a significant step towards AI-driven innovation in computational solid mechanics, addressing the limitations of traditional numerical methods and paving the way for more efficient simulations of heterogeneous materials across various industrial applications.

Figures (12)

• Figure 1: Pipeline of Micrometer: Micrometer is a transformer-based deep learning model used to predict the mechanical responses of fiber-reinforced composite materials. It takes representative volume elements (RVEs) and material properties at the microscale (e.g., Poisson's ratio, Young's modulus) as inputs, and outputs the corresponding point-wise strain concentration tensor governed by Lippmann-Schwinger equation. Micrometer employs an encoder-decoder architecture. The input is first embedded using a resolution-invariant Fourier neural operator encoder. The resulting latent outputs are then patchified into a sequence of tokens, added with positional embeddings, and processed through a standard transformer backbone. A key feature of Micrometer is its ability to continuously evaluate the outputs at any query coordinate. This is achieved by interpolating the FNO encoder outputs using Nadaraya-Watson kernel interpolation to obtain latent query-specific features. These features are then decoded by a standard transformer decoder with cross-attention between the query features and the outputs of the transformer encoder. Finally, a multilayer perceptron (MLP) is used to decode the output into the physical space. The pre-trained Micrometer model can be easily integrated with computational micromechanics frameworks, facilitating fast and accurate homogenization and multiscale modeling of composite materials. Furthermore, Micrometer can be further fine-tuned for a variety of downstream tasks with limited data.
• Figure 2: Scaling Analysis of Micrometer: Left: Relative $L^2$ errors for various model configurations as a function of training sample size. Right: Relative $L^2$ errors for different model configurations as a function of GPU hours. The results demonstrate the favorable scaling properties of Micrometer, with performance improvements observed for larger models, increased training samples, and additional computational resources.
• Figure 3: Performance analysis of Micrometer across volume fractions and material properties: Left: Distribution of relative $L^2$ errors with respect to volume fraction across the test dataset. Right: Distribution of relative $L^2$ errors with respect to Young's modulus ratio between fiber and matrix, evaluated over all test samples.
• Figure 4: Representative predictions by Micrometer for microstructures with varying volume fractions and Young's modulus ratios between fiber and matrix. (a) Low volume fraction and low Young's modulus ratio. (b) Medium volume fraction and medium Young's modulus ratio. (c) High volume fraction and high Young's modulus ratio.
• Figure 5: Computational cost regarding FFT based homogenization vs. Micrometer: Wall-clock time of using the conventional FFT based homogenization versus the Micrometer to predict the full field strain concentration tensor for 3,000 microstructures with volume fractions uniformly distributed between 40% and 60%. The FFT based homogenization was implemented by MATLAB$^\circledR$ R2024a and tested on an HP precision Z2G5 mini working station (CPU: Intel$^\circledR$ Xeon$^\circledR$ W1290 3.2 GHz 10-cores), while the Micrometer model was evaluated on NVIDIA$^\circledR$ A100 GPUs. We acknowledge that this may not represent a unique and entirely fair comparison in terms of computational costs, as the hardware and software environments differ. This result is intended to provide a general sense of the relative runtime performance using common computational resources.