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| NEURAL NETWORK METHOD FOR THIN PLATE BENDING PROBLEM |
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Abstract Recently, deep learning has made good progress in various disciplines. In order to develop the application of deep learning technology in solid mechanics, a neural network method with fully connected layers is proposed to solve the Kirchhoff thin plate bending problems which is governed by the fourth-order partial differential equation (PDE). Firstly, the training points from domain and boundary are randomly generated and feed into the forward propagation system of neural network to obtain the prediction solution. Then the errors are calculated by the loss function proposed in this paper. The parameters inside neural network are then optimized by the back propagation system. Finally, the neural network is trained continuously to make the errors converge, and the deflection solution of thin plate bending is then obtained. Taking triangle, ellipse and rectangular thin plates with different boundary and load conditions as examples, the partial differential equation is solved by the method proposed, and the results are compared with the theoretical solution or finite element method solution. In the end, the factors affecting the convergence of the neural network method are studied. It is found that the method is capable of solving the fourth order partial differential equations of thin plate bending problems. The convergence of this method is affected by the boundary conditions, optimization algorithms, numbers of hidden layers and neurons, and the chosen of learning rate. Compared to finite element method, the neural network method faces the problem of slow convergence speed. However, it is not based on the variational principle. It can obtain high accuracy without the calculation of stiffness matrix. The solution domain is discretized by the randomly generated points. The neural network method is flexible and can also be treated as meshless method. It can provide new ideas in the research of large deformation and nonlinear problems in the future.
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Received: 18 March 2021
Published: 14 December 2021
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2021, Vol. 42 
