面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

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. 

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

A neural network method is developed to solve the bending problems of functionally graded thin plates with in-plane stiffness gradient in this paper. The partial differential equation (PDE) of the bending of thin plates with in-plane stiffness gradient is a complex fourth-order PDE. The conventional neural network solution based on strong form, may face the problem of slow convergence and the boundary conditions are difficult to handle when solving the PDE. According to the Kirchhoff thin plate bending theory, a neural network solution to the bending problem of thin plates with any in-plane stiffness gradient in rectangular coordinate system is proposed in this paper. The neural network model includes deflection network and bending moment network, which are used to predict the deflection and bending moment of the thin plate respectively. Thus, the solution of the fourth-order PDE is transformed into a series of second-order PDEs. By constructing trial function of the deflection and bending moment, the results of neural network calculation can be strictly satisfied the boundary conditions. In the back propagation process, training error is calculated according to the error function formula proposed in this paper and combining Adam optimization algorithm to update the internal parameters of the neural networks. In this paper, the bending problems of thin plate with in-plane stiffness gradient with different boundary conditions and shapes are solved, and the calculated results are compared with theoretical solutions or those of finite element solutions. It shows that the proposed method is suitable for solving the bending problem of thin plate with in-plane stiffness gradient. And the convergence of bending moment network is slower than the deflection network. However, it is robust and easier in dealing with boundary conditions.