THE BUCKLING ANALYSIS OF THIN-WALLED STRUCTURES BASED ON PHYSICS-INFORMED NEURAL NETWORKS
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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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图 2以弧长法为外层控制的物理信息神经网络屈曲/后屈曲求解框图
Figure 2.The framework of PINN as PDE sovler towards buckling/postbuckling problems governed by arcs method
图 3带有缺陷的均质薄板的非线性屈曲位移云图对比 (上: PINN结果, 下: FEM结果)
Figure 3.The comparison of nonlinear buckling with deficiency (up: PINN results, down: FEM results)
图 5带有缺陷的均质圆筒壳在最大$L P F_{\operatorname{max}}$处非线性屈曲位移云图对比 (续)
Figure 5.The comparison of nonlinear buckling displacement at$ LP{F_{\max }} $ (continued)
图 8不同权重策略对损失函数收敛特性的影响
Figure 8.The influence of different weighting scheme to loss functions
图 9不同网络参数对损失函数收敛特性的影响
Figure 9.The influence of different network parameters to loss functions
图 10各优化器/求解器迭代到收敛的计算时间对比, 以Adam优化器的PINN算法为基准(3种边界条件算例综合的平均值)
Figure 10.The influence of optimizers and solvers on computation time, PINN with Adam optimizer as standard (the average of three different boundary conditions)
图 13均质薄板算例中不同激活函数对损失函数的影响
Figure 13.The influence of the activation function on the loss function
图 14relu激活函数导致的求出不符合物理规律的解
Figure 14.A result obtained by relu activation function that does not meet the governing equation
图 16本文算例所需时间及其组成
Figure 16.The computation time of the numerical examples and its composition
算法1: 基于预训练的损失函数权重的PINN训练过程 Input: 预训练任务最大迭代数$ite{r_{{\rm{pre}}}}$, 主要训练任务最大迭代数$ ite{r_{\max }} $, 数量级超参$ {\boldsymbol{\alpha }} $ 1: 初始化神经网络权重和偏置向量$\left[{{\boldsymbol{W}}}^{i},{{\boldsymbol{b}}}^{i}\right], i\in La$ 2: ${\left( {\tilde W,\tilde F} \right)_{ite{r_{{\rm{pre}}}}}} = \Im (x,y,\varTheta )$(神经网络预训练${N_{{\rm{pre}}}}$次) 3: $\begin{gathered}Los{s_{ite{r_{{\rm{pre}}}}}} = {\left\| {{\boldsymbol{MS}}{{\boldsymbol{E}}_P}} \right\|_1} + {\left\| {{\boldsymbol{MS}}{{\boldsymbol{E}}_B}} \right\|_1} + {\left\| {{\boldsymbol{MS}}{{\boldsymbol{E}}_I}} \right\|_1} \\ \lg \left( {{{\boldsymbol{w}}_P}} \right) = \left[ {\boldsymbol{\alpha }} \right] - \left[ {{\text{lg}}{{\left( {{\boldsymbol{MS}}{{\boldsymbol{E}}_P}} \right)}_{{N_{{\rm{pre}}}}}}} \right] \\ \lg \left( {{{\boldsymbol{w}}_B}} \right) = \left[ {\boldsymbol{\alpha }} \right] - \left[ {{\text{lg}}{{\left( {{\boldsymbol{MS}}{{\boldsymbol{E}}_B}} \right)}_{{N_{{\rm{pre}}}}}}} \right] \\ \lg \left( {{{\boldsymbol{w}}_I}} \right) = \left[ {\boldsymbol{\alpha }} \right] - \left[ {{\text{lg}}{{\left( {{\boldsymbol{MS}}{{\boldsymbol{E}}_I}} \right)}_{{N_{{\rm{pre}}}}}}} \right] \\ {\text{ }} \\ \end{gathered}$ 4: for $ epoch = 0 $ to $ epoch = ite{r_{\max }} $ do:(主训练任务) { ${\left( {\tilde W,\tilde F} \right)_{{\rm{epoch}}}} = \Im (x,y,\varTheta )$(神经网络前向传播) $Los{s_{{\rm{epoch}}}} = {{\boldsymbol{w}}_{{P}}}{\boldsymbol{MS}}{{\boldsymbol{E}}_P} + {{\boldsymbol{w}}_{{B}}}{\boldsymbol{MS}}{{\boldsymbol{E}}_B} + {{\boldsymbol{w}}_{{I}}}{\boldsymbol{MS}}{{\boldsymbol{E}}_I}$ Update $ \varTheta $(神经网络反向传播) } End for 
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算法2: 基于弧长法的PINN近似解增量求解算法 Input: 利用抽样得到求解板壳上的点集$ \left( {{x_i},{y_i}} \right) $, 初始载荷系数$ LP{F_0} $, 误差容限$ \varepsilon $, 迭代次数上限$ ite{r_{\max }} $ Output: 屈曲偏微分控制方程近似挠度$ \tilde W\left( {x,y} \right) $及其最大值输出$ {W_{\max }} $, 对应的载荷因子$ LP{F_{\max }} $ 1: 构造一个前馈神经网络 $ \tilde W = \Im (x,y,LPF,\varTheta ) $, 其拥有$ La $个隐藏层, 每层有$ {N_u} $个神经元, 训练批次数为$ ite{r_{\max }} $ 2: 进行PDE求解域内和求解边界上的撒点, 域内为$ {N_P} $个, 域边界上为$ {N_B} $个, 采样点总数为$ {N_S} = {N_P} + {N_B} $个 3: $ Loss = {{\boldsymbol{w}}_P}{\boldsymbol{MS}}{{\boldsymbol{E}}_P} + {{\boldsymbol{w}}_B}{\boldsymbol{MS}}{{\boldsymbol{E}}_B} + {{\boldsymbol{w}}_I}{\boldsymbol{MS}}{{\boldsymbol{E}}_I} $ 4: $ \tilde {{W_0}}\left( {{x_i},{y_i}} \right) = \hat u\left( {{x_i},{y_i},LP{F_0},\varTheta } \right) $ 5: while $ \left( {\left| {LP{F_j} - LP{F_{j - 1}} > \varepsilon } \right|} \right)\& j \leqslant ite{r_{\max }} $ do: { $\Delta LP{F_{j + 1}} = \dfrac{{ - \displaystyle\sum\limits_{s = 1}^{{N_S}} {\left[ {\tilde {{W_j}}\left( {LP{F_j},{x_s},{y_s}} \right)\Delta \tilde {{W_j}}\left( {LP{F_{j + 1}},{x_s},{y_s}} \right)} \right]} }}{{{\omega ^2}LP{F_j} - \displaystyle\sum\limits_{s = 1}^{{N_S}} {\left[ {\tilde {{W_j}}\left( {LP{F_j},{x_s},{y_s}} \right)} \right]\frac{{\partial F}}{{\partial w}}\Delta LP{F_j}} }}$ $ LP{F_{j + 1}} = LP{F_j} + \Delta LP{F_{j + 1}} $ $ {\tilde W_{j + 1}}\left( {{x_i},{y_i}} \right) = \Im \left( {{x_i},{y_i},LP{F_{j + 1}},\varTheta } \right) $ } 6: return最佳网络参数 $ {\varTheta ^*} $ 下载:
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表 1第一阶屈曲模态对应的外载对比
Table 1.The buckling load of the first buckling mode
Boundary conditions PINN results/
(N·m−1)FEM results/
(N·m−1)Relative error SSSS 1466.2 1537.6 4.86% SCSC 4092.1 4048.5 1.07% SCSF 893.2 868.7 2.82% 下载:
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表 2带缺陷的圆筒壳屈曲行为最大比例载荷因子$ \left( {LP{F_{\max }}} \right) $与无量纲挠度$ \left( {{w_{\max }}} \right) $误差对比
Table 2.The $ LP{F_{\max }} $and $ {w_{\max }} $of shell buckling with initial deficiency and their relative error
Network parameters $ LP{F_{\max }} $ $ {w_{\max }} $ Relative
error$ \left( {LP{F_{\max }}} \right) $Relative
error$ \left( {{w_{\max }}} \right) $L= 3,Nu= 40,
Adam0.522 3.3 5.66% 4.76% L= 3,Nu= 40,
L-BFGS0.455 3.64 8.57% 15.5% L= 4,Nu= 40,
Adam0.480 2.97 2.92% 6.06% L= 4,Nu= 40,
L-BFGS0.432 2.88 14.35% 9.38% L= 4,Nu= 50,
Adam0.531 2.77 7.48% 13.7% L= 4,Nu= 50,
L-BFGS0.399 2.99 23.8% 5.35% 下载:
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表 3拥有标签数据训练得到的第一阶屈曲载荷
Table 3.The buckling load of the first buckling mode trained with labeled data
Boundary conditions PINN results/(N·m−1) FEM results/×103% Relative error/% SSSS 1721.8 1583.1 8.76 SCSC 4388.2 4169.4 5.24 SCSF 1009.5 894.7 12.77 下载:
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Variable name Definition $ a $ shell structure dimension $ h $ shell structure thickness $ E $ modulus of elasticity $ \nu $ Poisson's ratio $ P $ shell edge load $ U,V,W $ in-plane displacement $ \tilde W $ fit displacement generated by NN $ Q $ middle surface stress $ F $ airy stress function $ D $ bending stiffness $ {\varepsilon ^0} $ middle surface strain $ La $ NN hidden layer number $ {N_u} $ neuron number for hidden layer $ {\boldsymbol{W}} $ weight matrix for NN $ {\boldsymbol{b}} $ bias for NN $ {\sigma _i} $ activation function for NN $ {\boldsymbol{MSE}} $ mean square error vector $ \varTheta $ hyperparameters for NN ${ {\boldsymbol{w} }_{{P} } }{{,\;} }{ {\boldsymbol{w} }_{{B} } }{{,\;} }{ {\boldsymbol{w} }_{{I} } }$ weights vector for loss function $ {N_P} $ sample points inside the PINN domain $ {N_B} $ sample points on the PINN domain boundary $ {N_S} $ total sample points of PINN $ {N_T} $ test points of PINN $ LPF $ load proportional factor $ lr $ PINN learning rate $ite{r_{{\rm{pre}}} }$ maximum iteration for PINN pre-training $ ite{r_{\max }} $ maximum iteration for PINN training $ {a_{mn}} $ deficiency coefficient $ R $ cylindrical shell radius $ L $ cylindrical shell height $ {\bar Z _B} $ batdorf parameter 下载:
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Numerical example $ \alpha $ ${ {\boldsymbol{w} }_{{P} } }$ ${ {\boldsymbol{w} }_{{B} } }$ plate, SSSS −4 [1,1] [104,104,104,104] plate, SCSC −4 [1,1] [104,103,104,103] plate, SCSF −4 [1,1] [104,103,104,102] shell −4 [104, 10−2] [104,102,102,102,105] 下载:
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Hyperparameter setup Number of random seeds Variance of lg (Loss) $L = 3, {N}_{u} = 60\;\; {\rm{Adam}}\;\; \left[{w}_{P},{w}_{B}\right] = \left[1,1\right]$ 10 1.900 6 $L = 3, {N}_{u} = 60\;\; {\rm{Adam}}\;\; \left[{w}_{P},{w}_{B}\right] = {\left[{w}_{P},{w}_{B}\right]}_{{\rm{epoch}}}$ 10 0.880 6 $L = 3, {N}_{u} = 40\;\;{\rm{Adam}}$ 10 0.931 0 $L = 3, {N}_{u} = 50\;\; {\rm{Adam}}$ 10 0.726 7 $L = 3, {N}_{u} = 70\;\; {\rm{Adam}}$ 10 0.372 5 $L = 3, {N}_{u} = 80\;\; {\rm{Adam}}$ 10 0.883 0 $L = 4, {N}_{u} = 60\;\; {\rm{Adam}}$ 10 0.590 7 $L = 5, {N}_{u} = 60\;\; {\rm{Adam}}$ 10 0.974 5 $L = 3, {N}_{u} = 60\;\; {\rm{Adagrad}}$ 10 0.933 3 $L = 3, {N}_{u} = 60\;\;{\rm{Adadelta}}$ 10 0.794 1 $L = 3, {N}_{u} = 60\;\; {\rm{rmsprop}}$ 10 0.490 9 $L = 3, {N}_{u} = 60\;\; {\rm{L-BFGS}}$ 10 1.041 9 $L = 3, {N}_{u} = 60\;\; {\rm{LAAF} } = \text{relu}$ 10 1.451 8 $L = 3, {N}_{u} = 60\;\; {\rm{LAAF}} = \mathrm{tanh}$ 10 0.652 9 下载:
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