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

2025-05-28

AI Summary

Low-order conforming finite elements suffer from volumetric locking in nearly incompressible linear elasticity, leading to non-robust behavior—i.e., severe accuracy degradation as the Lamé parameter λ → ∞ or Poisson’s ratio ν → 1/2. Method: This paper proposes a locking-free physics-informed neural network (PINN) framework. It reformulates the governing equations into a displacement–pressure coupled system and introduces a bidirectional joint optimization strategy with a soft-constraint loss function, thereby circumventing traditional mesh- and polynomial-degree-dependent discretization paradigms. Contribution/Results: The method demonstrates high accuracy and strong robustness across diverse numerical experiments—including constant-coefficient, variable-coefficient, and parametric Lamé coefficient settings—fully eliminating accuracy deterioration inherent to low-order elements. To the best of our knowledge, this is the first end-to-end PINN-based locking-free solver for incompressible and nearly incompressible elasticity.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $lambda oinfty$, or equivalently as the Poisson ratio $ u o1/2$. 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.

Problem

• Addressing divergence instability in nearly incompressible elasticity equations
• Overcoming locking in low-order finite element methods
• Robustly solving linear elasticity for nearly incompressible materials

Innovation

• Decompose equations to balance coefficients
• Solve forward and inverse problems simultaneously
• Use PINNs for nearly incompressible elasticity