Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks

TL;DR

This paper demonstrates that physics-informed neural networks can effectively solve forward and inverse contact mechanics problems, including boundary conditions and inequality constraints, with applications to Hertzian contact modeling.

Contribution

It introduces a mixed-variable PINN formulation with output transformation and KKT constraint enforcement for contact mechanics, advancing PINN capabilities in this domain.

Findings

PINNs can serve as PDE solvers, data-enhanced models, inverse solvers, and surrogate models.

Proper hyperparameter tuning improves accuracy and training efficiency.

Fischer-Burmeister NCP function aids in constraint enforcement.

Abstract

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush-Kuhn-Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer-Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.