Locking-Free Training of Physics-Informed Neural Network for Solving Nearly Incompressible Elasticity Equations

Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible homogeneous elasticity equations deteriorates as the Lam\'e coefficient $\lambda\to\infty$, or equivalently as the Poisson ratio $\nu\to1/2$. This phenomenon, known as locking or non-robustness, remains not fully understood despite extensive investigation. In this paper, we propose a robust method based on a fundamentally different, machine-learning-driven approach. Leveraging recently developed Physics-Informed Neural Networks (PINNs), we address the numerical solution of linear elasticity equations governing nearly incompressible materials. The core idea of our method is to appropriately decompose the given equations to alleviate the extreme imbalance in the coefficients, while simultaneously solving both the forward and inverse problems to recover the solutions of the decomposed systems as well as the associated external conditions. Through various numerical experiments, including constant, variable and parametric Lam\'e coefficients, we illustrate the efficiency of the proposed methodology.