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## 留言板 引用本文: 黄钟民, 谢臻, 张易申, 彭林欣. 面内变刚度薄板弯曲问题的挠度−弯矩耦合神经网络方法. 力学学报, 2021, 53(9): 2541-2553 Huang Zhongmin, Xie Zhen, Zhang Yishen, Peng Linxin. Deflection-bending moment coupling neural network method for the bending problem of thin plates with in-plane stiffness gradient. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2541-2553 doi: 10.6052/0459-1879-21-273
 Citation: Huang Zhongmin, Xie Zhen, Zhang Yishen, Peng Linxin. Deflection-bending moment coupling neural network method for the bending problem of thin plates with in-plane stiffness gradient. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2541-2553 • 中图分类号: TU339

## DEFLECTION-BENDING MOMENT COUPLING NEURAL NETWORK METHOD FOR THE BENDING PROBLEM OF THIN PLATES WITH IN-PLANE STIFFNESS GRADIENT

• 摘要: 发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.

• 图  1  本文神经网络模型示意图

Figure  1.  The schematic diagram of neural network model in this paper

图  2  圆形面内变刚度薄板

Figure  2.  Circular thin plate with in-plane stiffness gradient

图  3  神经网络训练误差曲线

Figure  3.  The convergence curve of neural network training error

图  4  PINN求解圆形面内变刚度薄板弯曲问题的训练误差收敛曲线(m = 2)

Figure  4.  Training error convergence curve of PINN (m = 2)

图  5  方形面内变刚度薄板

Figure  5.  Thin square plates with in-plane stiffness gradient

图  6  神经网络训练误差曲线

Figure  6.  Neural network training error-curve

图  7  不同梯度参数下本文挠度计算结果与有限元对比(y = 0.5 m)

Figure  7.  Comparison of dimensionless deflection calculation results in this paper with FEM when different gradient parameters (y = 0.5 m)

图  8  不同梯度参数下本文弯矩${\overline{M}}_{x1}$ 计算结果与有限元对比(y = 0.5 m)

Figure  8.  Comparison of dimensionless bending moment ${\overline{M}}_{x1}$ calculation results of this paper with FEM when different gradient parameters (y = 0.5 m)

图  9  三角形面内变刚度薄板

Figure  9.  Thin triangular plate with in-plane stiffness gradient

图  10  三角形面内变刚度薄板沿轴线x = 0上的挠度$\bar {w}$ 分布

Figure  10.  Dimensionless deflection variation of thin triangular plate with in-plane stiffness gradient along axis x = 0

图  11  m = 0.2时三角形面内变刚度薄板弯矩的有限元计算结果

Figure  11.  Finite element calculation of bending moment of thin plate with triangular in-plane variable stiffness (m = 0.2)

图  12  m = 0.2时三角形面内变刚度薄板弯矩的本文计算结果

Figure  12.  Neural network method calculation of bending moment of thin plate with triangular in-plane variable stiffness (m = 0.2)

图  13  有限元计算所需内存与节点数的关系(薄板单元)

Figure  13.  The relationship between the number of nodes and the memory needed in finite element calculation using thin plate bending element

图  14  神经网络训练时不同${k_{\rm{p}}}$ 下薄板变形能与外力功的变化情况

Figure  14.  Changes in the deformation energy and the external force work during the training process with different ${k_{\rm{p}}}$ 图  15  隐藏层数、每层神经元个数对计算误差的收敛影响对比

Figure  15.  Comparison of the effects of different number of hidden layers and neurons on the convergence of loss function

图  16  不同的激活函数对计算误差的收敛影响对比

Figure  16.  Comparison of the effects of different activation functions on the convergence of loss function

表  1  本文方法计算$\bar w$ 与理论解对比(无量纲)

Table  1.   Comparison of dimensionless $\bar w$ calculated by neural network method and the theoretical solution

 m (x, y) $\bar {w}$ Theory Relative error/% 0 (0.0, 0) −1.5625 −1.5625 0 (0.4, 0) −1.1024 −1.1025 −0.009 (0.8, 0) −0.2025 −0.2025 0 1 (0.0, 0) −0.8420 −0.8418 0.024 (0.4, 0) −0.5517 −0.5525 −0.145 (0.8, 0) −0.0877 −0.0877 0 2 (0.0, 0) −0.4520 −0.4522 −0.044 (0.4, 0) −0.2725 −0.2725 0 (0.8, 0) −0.0374 −0.0373 0.267 注: 本文误差计算公式为$e = \dfrac{{u - {u^*}}}{u} \times 100\%$, $u$为本文方法计算结果, ${u^*}$为理论解 (Note: The relative error in this paper is calculated by $e = \dfrac{ {u - {u^*} } }{u} \times$$100\%$, where $u$ is the calculation results of this paper, ${u^*}$ is the theoretical solution)

表  2  本文方法计算${\bar M_x}$ 与理论解对比(无量纲)

Table  2.   Comparison of dimensionless ${\bar M_x}$ calculated by neural network method and the theoretical solution

 m (x, y) ${\bar M_{x1}}$ ${\bar M_{x2}}$ Theory Relative error 1/% Relative error 2/% 0 (0.02, 0) −0.8122 −0.8118 −0.8117 0.062 0.020 (0.4, 0) −0.4828 −0.4824 −0.4825 0.071 −0.027 (0.8, 0) 0.5077 0.5075 0.5075 0.037 −0.004 (1.0, 0) 1.2512 1.2499 1.2500 0.094 −0.008 1 (0.02, 0) −0.6027 −0.5910 −0.5853 2.885 0.956 (0.4, 0) −0.3064 −0.3076 −0.3103 −1.278 −0.886 (0.8, 0) 0.6264 0.6249 0.6282 −0.287 −0.524 (1.0, 0) 1.3536 1.3634 1.3525 0.077 0.797 2 (0.02, 0) −0.4335 −0.4150 −0.4158 4.086 −0.192 (0.4, 0) −0.1652 −0.1651 −0.1695 −2.599 −2.684 (0.8, 0) 0.7273 0.7268 0.7319 −0.637 −0.712 (1.0, 0) 1.4344 1.4543 1.4409 −0.458 0.921

表  3  本文各算例求解所需的数据点数、内存、时间

Table  3.   The number of training data points, computational memory and computing time of numerical examples in this paper

 Numerical example Number of data points Computational memory/MiB $T(s)$ ${T^\prime }(s)$ ${T^\prime }(s)/T(s)$/% 1 560 121.3 551.99 374.66 67.8 2 450 139.4 673.81 475.22 70.5 3 610 169.3 646.14 479.81 74.3

表  4  算例2、算例3的有限元求解收敛所需节点数目

Table  4.   The number of nodes needed for the convergence of the finite element solution of numerical example 2 and 3

 Numerical example m Number of nodes 2 0 441 1 841 2 1681 3 0 305 0.2 1331 0.5 1604
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##### 出版历程
• 收稿日期:  2021-06-16
• 录用日期:  2021-08-17
• 网络出版日期:  2021-08-18
• 刊出日期:  2021-09-18

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