Solving forward and inverse problems of contact mechanics using physics-informed neural networks

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.

A model hierarchy for predicting the flow in stirred tanks with physics-informed neural networks

“…Physics-Informed Neural Networks (PINNs) integrate the governing equations of the problem into the loss function of the Neural Network (NN), making them a natural fit for our objectives. Since the seminal paper of Raissi et al (2019) [31], PINNs have been successfully applied to problems across diverse domains, including solid mechanics [24,13,34], molecular dynamics [15], chemical reaction kinetics [2], the wave equation [28], hemodynamics [21,38,32], cardiac activation mapping [35], and various other fields [7]. Furthermore, PINNs have been applied to address numerous problems in fluid mechanics [19,6,18,40,12,47].…”