[Paper Review] Comparison of two artificial neural networks trained for the surrogate modeling of stress in materially heterogeneous elastoplastic solids

Sarthak Kapoor, Jaber Rezaei Mianroodi+3 more|arXiv (Cornell University)|Oct 31, 2022

Microstructure and Mechanical Properties of Steels4 citations

TL;DR

This study compares a U-Net-based convolutional neural network (CNN) and a Fourier neural operator (FNO) for surrogate modeling of stress fields in heterogeneous elastoplastic polycrystalline microstructures. Trained on finite-element solutions of boundary-value problems, the FNO achieves 3.5–7.5× lower normalized mean absolute error (0.25–0.40%) than U-Net (1.41–2.15%) and exhibits superior robustness to spatial resolution and grain density variations, though both models fail to capture strong stress gradients near stress concentrators.

ABSTRACT

The purpose of this work is the systematic comparison of the application of two artificial neural networks (ANNs) to the surrogate modeling of the stress field in materially heterogeneous periodic polycrystalline microstructures. The first ANN is a UNet-based convolutional neural network (CNN) for periodic data, and the second is based on Fourier neural operators (FNO). Both of these were trained, validated, and tested with results from the numerical solution of the boundary-value problem (BVP) for quasi-static mechanical equilibrium in periodic grain microstructures with square domains. More specifically, these ANNs were trained to correlate the spatial distribution of material properties with the equilibrium stress field under uniaxial tensile loading. The resulting trained ANNs (tANNs) calculate the stress field for a given microstructure on the order of 1000 (UNet) to 2500 (FNO) times faster than the numerical solution of the corresponding BVP. For microstructures in the test dataset, the FNO-based tANN, or simply FNO, is more accurate than its UNet-based counterpart; the normalized mean absolute error of different stress components for the former is 0.25-0.40% as compared to 1.41-2.15% for the latter. Errors in FNO are restricted to grain boundary regions, whereas the error in U-Net also comes from within the grain. In comparison to U-Net, errors in FNO are more robust to large variations in spatial resolution as well as small variations in grain density. On other hand, errors in U-Net are robust to variations in boundary box aspect ratio, whereas errors in FNO increase as the domain becomes rectangular. Both tANNs are however unable to reproduce strong stress gradients, especially around regions of stress concentration.

Motivation & Objective

• To systematically compare the performance of a U-Net-based CNN and an FNO-based neural operator in surrogate modeling of stress fields in heterogeneous elastoplastic microstructures.
• To evaluate the accuracy, generalization, and robustness of both models across varying spatial resolution, grain density, and domain aspect ratios.
• To assess the models' ability to reproduce stress fields under uniaxial tensile loading with minimal error and high computational speed.
• To investigate limitations in capturing stress concentrations and violations of mechanical equilibrium in learned models.

Proposed method

• The U-Net is a fully convolutional encoder-decoder network with skip connections, trained to map spatial distributions of material properties to stress tensor fields in 2D periodic microstructures.
• The FNO uses Fourier-based spectral operations to learn mappings between function spaces, enabling resolution-invariant inference and single-shot superresolution.
• Synthetic 2D grain microstructures were generated via Voronoi tessellation with isotropic, perfect elasto-plastic grains and subjected to uniaxial tensile loading via spectral solvers.
• Both models were trained on 20-grain microstructures at 256×256 resolution, with validation and testing on unseen microstructures across diverse configurations.
• The models were evaluated using normalized mean absolute error (NMAE) across all 9 components of the first Piola-Kirchhoff stress tensor.
• Generalization was tested under variations in spatial resolution, grain density, and boundary box aspect ratio, with stress concentration regions analyzed for error localization.

Experimental results

Research questions

• RQ1 How do the U-Net and FNO perform in terms of accuracy when predicting stress fields in heterogeneous elastoplastic microstructures?
• RQ2 How does each model generalize to changes in spatial resolution and grain density?
• RQ3 How do the models respond to variations in the aspect ratio of the simulation domain?
• RQ4 Where are the primary sources of error localized—within grains or at grain boundaries?
• RQ5 To what extent do both models fail to capture strong stress gradients near stress concentrators?

Key findings

• The FNO-based surrogate model achieves a normalized mean absolute error (NMAE) of 0.25–0.40% across stress components, which is 3.5 to 7.5 times lower than the U-Net’s 1.41–2.15% error.
• FNO errors are localized primarily at grain boundaries, whereas U-Net errors originate both within grains and at boundaries, indicating a fundamental difference in error propagation.
• FNO generalizes robustly to higher spatial resolutions and maintains consistent error scaling with grain boundary fraction, while U-Net errors increase significantly with resolution changes.
• FNO is more robust to small variations in grain density, though large density variations cause a steep rise in error; U-Net shows higher sensitivity even to small density changes.
• U-Net is more robust to aspect ratio variations in the boundary box, while FNO errors increase significantly as the domain becomes rectangular, due to altered frequency content in the input.
• Both models fail to accurately capture strong stress gradients near geometric stress concentrators, such as sharp corners, indicating a shared limitation in resolving high-gradient features.