Energy-based physics-informed neural network for frictionless contact problems under large deformation

TL;DR

The paper tackles frictionless contact under large deformation by formulating an energy-based PINN that minimizes the total potential energy $\Pi = E_{\text{in}} - E_{\text{ex}} + E_{\text{c}}$, where $E_{\text{in}} = \int_{\Omega} \Psi(F)\,d\Omega$, $E_{\text{ex}} = \int_{\Gamma^{\boldsymbol{t}}} \bar{\boldsymbol{t}} \cdot \boldsymbol{u}\,d\Gamma$, and $E_{\text{c}} = \iint_{\Gamma_1 \Gamma_2} \beta_1 \beta_2 \phi(r) \,d\Gamma_1 d\Gamma_2$ with an exponential contact potential $\phi(r) = \phi_0 \exp(-r/r_0)$. Boundary conditions are imposed via soft or hard schemes, and the training exhibits a pseudo-dynamic interpretation where energy dissipation drives the solution; ADAM enhances robustness over standard gradient descent. The framework is demonstrated on Hertz contact and three nonlinear rubber contact problems, showing accurate displacements and stress fields, robust convergence, and competitive computational efficiency relative to commercial FEM. Overall, the method provides a flexible, data-friendly approach to nonlinear contact mechanics, with clear pathways to incorporate friction and more advanced contact detections in future work.

Abstract

Numerical methods for contact mechanics are of great importance in engineering applications, enabling the prediction and analysis of complex surface interactions under various conditions. In this work, we propose an energy-based physics-informed neural network (PINNs) framework for solving frictionless contact problems under large deformation. Inspired by microscopic Lennard-Jones potential, a surface contact energy is used to describe the contact phenomena. To ensure the robustness of the proposed PINN framework, relaxation, gradual loading and output scaling techniques are introduced. In the numerical examples, the well-known Hertz contact benchmark problem is conducted, demonstrating the effectiveness and robustness of the proposed PINNs framework. Moreover, challenging contact problems with the consideration of geometrical and material nonlinearities are tested. It has been shown that the proposed PINNs framework provides a reliable and powerful tool for nonlinear contact mechanics. More importantly, the proposed PINNs framework exhibits competitive computational efficiency to the commercial FEM software when dealing with those complex contact problems. The codes used in this manuscript are available at https://github.com/JinshuaiBai/energy_PINN_Contact.(The code will be available after acceptance)

Figures (10)

• Figure 1: (a) The training dynamics of energy-based PINNs for solving a cantilever beam ($0.25$ m $\times1$ m) problem with a linear elastic material. The Young's modulus and the Poisson's ratio are $1\times10^4$ Pa and $0.3$, respectively. The left boundary of the beam is fixed. A parabolic distributed force of $10$ N is downwardly applied on the right boundary. Two feedforward neural networks with $3$ hidden layers and $5$ neurons per layer are used for predicting displacements. When using gradient descendant algorithms, the overall potential energy decreases during the training of energy-based PINNs. The intermediate absolute displacement mappings are shown. Along with the decreasing energy functional, the beam seems to be dynamically bent to the equilibrium state. Note that the VGD in the figure refers to the vanilla gradient descendant algorithm. A learning rate of $1\times10^{-4}$ is applied. (b) Comparisons between $\eta\frac{\partial \boldsymbol{u}}{\partial t}$ and actual displacement increment when using the VGD at the $20000^{\text{th}}$ epochs. (c) Comparisons between $\eta\frac{\partial \boldsymbol{u}}{\partial t}$ and actual displacement increment when using the ADAM optimiser at the $20000^{\text{th}}$ epochs.
• Figure 1: (a) The training dynamics of energy-based PINNs for solving a cantilever beam problem with neo-Hookean material and large deformation. A parabolic distributed force of $30$ N is downwardly applied on the right boundary. (b) Absolute displacement contours from the FEM, the ADAM optimiser and the VGD algorithm and the corresponding absolute error contours.
• Figure 2: Schematics of contact model discretisation wriggers_Contact_textbook. (a) Point-to-point contact model. (b) Point-to-surface model.
• Figure 3: (a) The Lennard-Jones potential plot; (b) The exponential form surface contact potential used in this work. Note that the $\phi_0$ and $r_0$ are the pre-defined potential constant and the effective radius of the discrete points, respectively.
• Figure 4: The schematic of the proposed energy-based PINN framework for contact problems.