DeepFEM: A Novel Element-Based Deep Learning Approach for Solving Nonlinear Partial Differential Equations in Computational Solid Mechanics

Abstract

In this paper, an element-based deep learning approach named DeepFEM for solving nonlinear partial differential equations (PDEs) in solid mechanics is developed to reduce the number of sampling points required for training the deep neural network. Shape functions are introduced into deep learning to approximate the displacement field within the element. A general scheme for training the deep neural network based on derivatives computed from the shape functions is proposed. For the sake of demonstrations, the nonlinear vibration, nonlinear bending, and cohesive fracture problems are solved, and the results are compared with those from the existing methods to evaluate the performance of the present method. The results demonstrate that DeepFEM can effectively approximate the solution of the nonlinear mechanics problems with high accuracy, while the shape functions can significantly improve the computational efficiency. Moreover, with the trained DeepFEM model, the solutions of nonlinear problems with different geometric or material properties can be obtained instantly without retraining. Finally, the proposed DeepFEM is employed in the identification of material parameters of composite plate. The results show that the longitudinal and transverse elastic moduli of the ply in the composite plates can be accurately predicted based on the nonlinear mechanical response of plates.