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 引用本文: 冯唐思捷, 梁伟. 基于物理信息神经网络的薄壁结构屈曲分析. 力学学报, 2023, 55(11): 2539-2553.
Feng Tangsijie, Liang Wei. The buckling analysis of thin-walled structures based on physics-informed neural networks. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(11): 2539-2553.
 Citation: Feng Tangsijie, Liang Wei. The buckling analysis of thin-walled structures based on physics-informed neural networks. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(11): 2539-2553.

## THE BUCKLING ANALYSIS OF THIN-WALLED STRUCTURES BASED ON PHYSICS-INFORMED NEURAL NETWORKS

• 摘要: 基于物理信息神经网络(PINN)建立了一种求解薄壁结构屈曲非线性控制方程组的方法. 薄壁结构的非线性控制方程可由挠度和应力函数表示成复杂的4阶非线性偏微分方程组, 使用物理信息神经网络(PINN)解法可以克服传统数值方法对求解域网格的依赖性. 文中建立的神经网络模型根据基于加权的均方误差的损失函数更新网络参数, 并用弧长法迭代的思想进行外层迭代控制以应对屈曲问题的迭代特性. 将弧长法, 硬边界条件, 基于预训练的权重调整策略, 以及自适应激活函数策略融合进网络优化的过程中使得PINN能够更为高效地求解线性与非线性屈曲问题. 文章对两种典型的薄壁结构进行了屈曲模态和带有缺陷的非线性后屈曲问题求解, 并将神经网络获得的解和有限元结果进行了对比. 结果分析表明, 物理信息神经网络方法能够在不需要标签数据的前提下对薄壁结构的屈曲问题进行有效分析, 并且给予的额外标签数据能够提高此方法的求解效率. 该方法虽较成熟的有限元解法收敛速度较慢, 但不需要对求解域进行人为的前处理, 有一定工程应用可行性.

Abstract: This paper proposes a method based on the physics-informed neural networks (PINN) for solving the thin-walled structure buckling problem. The governing equations of thin-walled buckling can be expressed as a fourth-order nonlinear partial differential equation system with the in-plane displacement and stress functions as variables. The PINN solution can overcome the dependence of traditional numerical methods on the mesh partition of the computational domain and conduct mesh-free calculations on the entire computational domain. The neural network model presented in the paper utilizes a weighted mean square error loss function composition for updating network parameters and employs the arc-length method for the outer-loop iteration control to deal with the iteration characteristic of buckling problems. The incorporation of the arc-length method, hard constraints, weight adjustment strategy based on trial-and-error pre-training, and self-adaptive activation function strategy enables PINN to solve linear and nonlinear buckling problems effectively. Two types of problems are investigated in the study, including buckling mode analysis and nonlinear post-buckling problems with geometry deficiencies. A comparison is made between the solutions obtained from the neural network and finite element results. The results demonstrate the efficacy of the proposed method in accurately solving both linear and nonlinear buckling problems in thin-walled structures, highlighting its potential applications in structural engineering and design optimization. The research results show that the physics-informed neural network can effectively analyze the buckling problem of thin-walled structures without requiring artificial preprocessing of the computational domain. Additionally, PINN retains the traditional characteristic of normal DNNs and can accept labeled data for faster calculations. The paper shows that the labeled buckling mode data can accelerate the convergence of the network. The drawback of PINN is that it converges slower than the mature finite element method, but the feature of requiring no artificial preprocessing of the solution domain before the training process makes PINN feasible in engineering.

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