[论文解读] 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 Steels被引用 4

一句话总结

本研究比较了基于U-Net的卷积神经网络（CNN）与基于傅里叶神经算子（FNO）在异质弹塑性多晶微观结构中应力场代理建模中的表现。模型基于边界值问题的有限元解进行训练，FNO的归一化平均绝对误差（NMAE）为0.25–0.40%，比U-Net的1.41–2.15%低3.5–7.5倍，且在空间分辨率和晶粒密度变化下表现出更优的鲁棒性，但两者均未能准确捕捉应力集中区附近的强应力梯度。

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.

研究动机与目标

• 系统比较基于U-Net的卷积神经网络与基于FNO的神经算子在异质弹塑性微观结构中应力场代理建模中的性能。
• 评估两种模型在不同空间分辨率、晶粒密度和域纵横比下的精度、泛化能力与鲁棒性。
• 评估模型在单轴拉伸载荷下以最小误差和高计算速度重现应力场的能力。
• 探究模型在捕捉应力集中现象及违反力学平衡方面存在的局限性。

提出的方法

• U-Net是一种带有跳跃连接的全卷积编码器-解码器网络，用于将材料属性的空间分布映射到2D周期性微观结构中的应力张量场。
• FNO利用基于傅里叶的谱运算，学习函数空间之间的映射，实现分辨率不变的推理与单次超分辨率。
• 通过Voronoi镶嵌生成了合成的2D晶粒微观结构，晶粒为各向同性、理想弹塑性，利用谱求解器施加单轴拉伸载荷。
• 两种模型均在256×256分辨率的20晶粒微观结构上进行训练，并在不同配置的未见微观结构上进行验证与测试。
• 通过第一Piola-Kirchhoff应力张量的全部9个分量的归一化平均绝对误差（NMAE）对模型进行评估。
• 通过改变空间分辨率、晶粒密度和边界框纵横比，测试泛化能力，并对应力集中区域进行误差定位分析。

实验结果

研究问题

• RQ1 在预测异质弹塑性微观结构中的应力场时，U-Net与FNO的精度表现如何？
• RQ2 每种模型对空间分辨率和晶粒密度变化的泛化能力如何？
• RQ3 模型对模拟域纵横比变化的响应如何？
• RQ4 主要误差来源位于晶粒内部还是晶界处？
• RQ5 两种模型在多大程度上无法准确捕捉应力集中区附近的强应力梯度？

主要发现

• 基于FNO的代理模型在应力分量上的归一化平均绝对误差（NMAE）为0.25–0.40%，比U-Net的1.41–2.15%低3.5至7.5倍。
• FNO的误差主要集中在晶界处，而U-Net的误差则同时产生于晶粒内部与晶界，表明误差传播机制存在根本性差异。
• FNO对更高空间分辨率具有强泛化能力，并在晶界比例变化时保持一致的误差尺度；而U-Net在分辨率变化时误差显著增加。
• FNO对晶粒密度的小幅变化更具鲁棒性，但密度大幅变化时误差急剧上升；U-Net即使在小密度变化下也表现出更高敏感性。
• U-Net对边界框纵横比变化更具鲁棒性，而FNO在域变为矩形时误差显著增加，原因在于输入频率成分发生变化。
• 两种模型均无法准确捕捉几何应力集中区（如尖锐拐角）附近的强应力梯度，表明两者在解析高梯度特征方面存在共同局限性。