用神经网络求解材料局部化变形的位移跳跃位置与大小
Exploring energy minimization to model strain localization as a strong discontinuity using Physics Informed Neural Networks
- 通过能量最小化构建变分框架，用神经网络直接预测位移间断
- 一维和二维案例验证方法可行，能同时求解平衡与断裂带位置
- 适合研究材料塑性失效、裂纹扩展等强间断问题的科研人员
本文探索利用能量最小化方法对弹塑性固体中的应变局部化进行数值建模，将其视为位移场中的强间断。考虑正则化的强间断运动学，采用人工神经网络（ANN）对相应数学模型进行离散化，目标是通过能量最小化从训练参数中同时预测位移跳跃的大小和位置。网络结构处理运动学约束，损失函数实现边界值问题的变分形式。核心思想是将平衡问题与局部化带位置的求解统一为可训练参数的优化过程。作为概念验证，通过一维和二维数值算例表明，基于能量最小化的弹塑性固体应变局部化计算建模是可行的。
原文摘要 · Abstract (English)
We explore the possibilities of using energy minimization for the numerical modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this purpose, we consider (regularized) strong discontinuity kinematics in elastoplastic solids. The corresponding mathematical model is discretized using Artificial Neural Networks (ANNs), aiming to predict both the magnitude and location of the displacement jump from energy minimization, $\textit{i.e.}$, within a variational setting. The architecture takes care of the kinematics, while the loss function takes care of the variational statement of the boundary value problem. The main idea behind this approach is to solve both the equilibrium problem and the location of the localization band by means of trainable parameters in the ANN. As a proof of concept, we show through both 1D and 2D numerical examples that the computational modeling of strain localization for elastoplastic solids using energy minimization is feasible.
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