Physics-Informed Neural Networks for Solving Contact Problems in Three Dimensions

TL;DR

This paper applies physics-informed neural networks to 3D contact mechanics problems, incorporating boundary and inequality constraints through advanced formulations and benchmark testing.

Contribution

It introduces a novel PINN framework that enforces boundary conditions as hard constraints and KKT conditions as soft constraints using NCP and Fischer-Burmeister functions.

Findings

Successfully applied PINNs to 3D contact problems

Demonstrated effectiveness on benchmark contact tests

Enhanced constraint enforcement in neural network solutions

Abstract

This paper explores the application of physics-informed neural networks (PINNs) to tackle forward problems in 3D contact mechanics, focusing on small deformation elasticity. We utilize a mixed-variable formulation, enhanced with output transformations, to enforce Dirichlet and Neumann boundary conditions as hard constraints. The inherent inequality constraints in contact mechanics, particularly the Karush-Kuhn-Tucker (KKT) conditions, are addressed as soft constraints by integrating them into the network's loss function. To enforce the KKT conditions, we leverage the nonlinear complementarity problem (NCP) approach, specifically using the Fischer-Burmeister function, which is known for its advantageous properties in optimization. We investigate two benchmark examples of PINNs in 3D contact mechanics: a single contact patch test and the Hertzian contact problem.