Energy-based methods for solving forward and inverse linear elasticity problems in 2D structures

Physics Informed Neural Networks (PINNs) and the Deep Energy Method (DEM) are two recently developed approaches for solving partial differential equations (PDEs) using deep neural networks. While PINNs aim to minimize the residual of the strong form of PDEs, DEM solvers work by minimizing the total potential energy. However, these methods have limitations in capturing the complex characteristics of displacement and stress fields. To overcome these limitations, we propose two new extensions of DEM: the deep energy method with traction-free boundary loss term (t-DEM) and the energy minimization method (EMM) with finite element method (FEM) basis. The t-DEM includes an additional loss term to enforce traction-free boundary conditions, and the EMM combines the FEM basis with the DEM to efficiently minimize the total potential energy of the system.

In our paper, we apply these techniques to solve linear elasticity forward and inverse problems and compare their performance. We demonstrate the effectiveness of these methods through various elasticity standard problems and compare the results with true solutions. In addition, we conducted a parametric study to analyze the impact of different parameter variations on the proposed frameworks. The numerical results highlight the accuracy and reliability of the proposed approaches for the forward and inverse linear elasticity problems, underlining their potential, particularly for the solution of stress concentration problems.