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面内变刚度薄板弯曲问题的挠度−弯矩耦合神经网络方法

黄钟民 谢臻 张易申 彭林欣

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 doi: 10.6052/0459-1879-21-273
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 doi: 10.6052/0459-1879-21-273
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 doi: 10.6052/0459-1879-21-273

面内变刚度薄板弯曲问题的挠度−弯矩耦合神经网络方法

doi: 10.6052/0459-1879-21-273
基金项目: 国家自然科学基金资助项目(12162004)
详细信息
    作者简介:

    彭林欣, 教授, 主要研究方向: 无网格方法. E-mail: penglx@gxu.edu.cn

  • 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} $TheoryRelative error/%
    0(0.0, 0)−1.5625−1.56250
    (0.4, 0)−1.1024−1.1025−0.009
    (0.8, 0)−0.2025−0.20250
    1(0.0, 0)−0.8420−0.84180.024
    (0.4, 0)−0.5517−0.5525−0.145
    (0.8, 0)−0.0877−0.08770
    2(0.0, 0)−0.4520−0.4522−0.044
    (0.4, 0)−0.2725−0.27250
    (0.8, 0)−0.0374−0.03730.267
    注: 本文误差计算公式为$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)
    下载: 导出CSV

    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}}$TheoryRelative error 1/%Relative error 2/%
    0(0.02, 0)−0.8122−0.8118−0.81170.0620.020
    (0.4, 0)−0.4828−0.4824−0.48250.071−0.027
    (0.8, 0)0.50770.50750.50750.037−0.004
    (1.0, 0)1.25121.24991.25000.094−0.008
    1(0.02, 0)−0.6027−0.5910−0.58532.8850.956
    (0.4, 0)−0.3064−0.3076−0.3103−1.278−0.886
    (0.8, 0)0.62640.62490.6282−0.287−0.524
    (1.0, 0)1.35361.36341.35250.0770.797
    2(0.02, 0)−0.4335−0.4150−0.41584.086−0.192
    (0.4, 0)−0.1652−0.1651−0.1695−2.599−2.684
    (0.8, 0)0.72730.72680.7319−0.637−0.712
    (1.0, 0)1.43441.45431.4409−0.4580.921
    下载: 导出CSV

    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 exampleNumber of data pointsComputational memory/MiB$T(s)$${T^\prime }(s)$${T^\prime }(s)/T(s)$/%
    1560121.3551.99374.6667.8
    2450139.4673.81475.2270.5
    3610169.3646.14479.8174.3
    下载: 导出CSV

    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 examplemNumber of nodes
    20441
    1841
    21681
    30305
    0.21331
    0.51604
    下载: 导出CSV
  • [1] 徐芝纶. 弹性力学简明教程. 北京: 高等教育出版社, 2013

    (Xu Zhilun. A Concise Course in Elasticity. Beijing: Higher Education Press, 2013 (in Chinese))
    [2] 于天崇, 聂国隽, 仲政. 变刚度矩形板弯曲问题的Levy解. 力学季刊, 2012, 33(1): 53-59 (Yu Tianchong, Nie Guojun, Zhong Zheng. Levy-type solution for the bending of rectangular plates with variable stiffness. Chinese Quarterly of Mechanics, 2012, 33(1): 53-59 (in Chinese) doi: 10.3969/j.issn.0254-0053.2012.01.008
    [3] 朱竑祯, 王纬波, 高存法等. 面内变刚度功能梯度圆形薄板的轴对称弯曲. 船舶力学, 2018, 22(11): 1364-1375 (Zhu Hongzhen, Wang Weibo, Gao Cunfa, et al. Axisymmetric bending of functionally graded thin circular plates with in-plane stiffness gradient. Journal of Ship Mechanics, 2018, 22(11): 1364-1375 (in Chinese) doi: 10.3969/j.issn.1007-7294.2018.11.006
    [4] 何建璋, 褚福运, 仲政等. 面内变刚度矩形薄板自由振动问题的辛弹性分析. 同济大学学报(自然科学版), 2013, 41(9): 1310-1317 (He Jianzhang, Chu Fuyun, Zhong Zheng, et al. Symplectic elasticity approach for free vibration of a rectangular plate with in-plane material inhomogeneity. Journal of Tongji University (Natural Science) , 2013, 41(9): 1310-1317 (in Chinese) doi: 10.3969/j.issn.0253-374x.2013.09.006
    [5] Santare MH, Lambros J. Use of graded finite elements to model the behavior of nonhomogeneous materials. Journal of Applied Mechanics, 2000, 67(4): 819-822 doi: 10.1115/1.1328089
    [6] Kim JH, Paulino GH. Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. Journal of Applied Mechanics, 2002, 69(4): 502-514
    [7] 黄立新, 姚祺, 张晓磊等. 基于分层法的功能梯度材料有限元分析. 玻璃钢/复合材料, 2013, 2: 43-48 (Huang Lixin, Yao Qi, Zhang Xiaolei, et al. Finite element analysis of functionally graded materials based on layering method. Composites Science and Engineering, 2013, 2: 43-48 (in Chinese)
    [8] 田云德, 秦世伦. 梯度厚板热应力分层计算方法的改进. 计算力学学报, 2009, 26(6): 961-965 (Tian Yunde, Qin Shilun. The improvement of stratified method about thermal stress of functionally gradient material thick plates. Chinese Journal of Computational Mechanics, 2009, 26(6): 961-965 (in Chinese)
    [9] Lee H, Kang IS. Neural algorithm for solving differential equations. Journal of Computational Physics, 1990, 91(1): 110-131 doi: 10.1016/0021-9991(90)90007-N
    [10] Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 1998, 9(5): 987-1000
    [11] Eukinan W, Yu B. The deep ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 2018, 6(1): 1-12 doi: 10.1007/s40304-018-0127-z
    [12] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 2019, 378: 686-707 doi: 10.1016/j.jcp.2018.10.045
    [13] Sirignano JA, Spiliopoulos K. DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 2018, 375: 1339-1364 doi: 10.1016/j.jcp.2018.08.029
    [14] Samaniego E, Anitescu C, Goswami S, et al. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020: 362
    [15] 瞿同明, 冯云田, 王孟琦等. 基于深度学习和细观力学的颗粒材料本构关系研究. 力学学报, 2021, 53(7): 1-12 (Qu Tongming, Feng Yuntian, Wang Mengqi, et al. Constitutive relations of granular materials by integrating micromechanical knowledge with deep learning. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 1-12 (in Chinese)
    [16] 谢晨月, 袁泽龙, 王建春等. 基于人工神经网络的湍流大涡模拟方法. 力学学报, 2021, 53(1): 1-16 (Xie Chenyue, Yuan Zelong, Wang Jianchun, et al. Artificial neural network-based subgrid-scale models for large-eddy simulation of turbulence. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1): 1-16 (in Chinese)
    [17] 刘宇翔, 王东东, 樊礼恒等. 基于卷积神经网络的无网格形函数影响域优化研究. 固体力学学报, 2021, 42(3): 302-319 (Liu Yuxiang, Wang Dongdong, Fan Liheng, et al. A CNN-based approach for optimizing support selection of meshfree methods. Chinese Journal of Solid Mechanics, 2021, 42(3): 302-319 (in Chinese)
    [18] 郭宏伟, 庄晓莹. 采用两步优化器的深度配点法与深度能量法求解薄板弯曲问题. 固体力学学报, 2021, 42(3): 249-266 (Guo Hongwei, Zhuang Xiaoying. The application of deep collocation method and deep energy method with a two-step optimizer in the bending analysis of kirchhoff thin plate. Chinese Journal of Solid Mechanics, 2021, 42(3): 249-266 (in Chinese)
    [19] 陈豪龙, 柳占立. 基于数据驱动模型求解热传导反问题. 计算力学学报, 2021, 38(3): 272-279 (Chen Haolong, Liu Zhanli. Solving the inverse heat conduction problem based on data driven model. Chinese Journal of Computational Mechanics, 2021, 38(3): 272-279 (in Chinese)
    [20] Cai Z, Chen J, Liu M, et al. Deep least-squares methods:an unsupervised learning-based numerical method for solving elliptic PDEs. Journal of Computational Physics, 2020: 420
    [21] Tang SQ, Yang Y. Why neural networks apply to scientific computing? Theoretical and Applied Mechanics Letters, 2021 (in press)
    [22] Glorot X, Bengio Y. Understanding the difficulty of training deep feedforward neural networks. Journal of Machine Learning Research, 2010, 9: 249-256
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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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