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

arXiv: https://arxiv.org/abs/2412.09022 · HTML: https://mlsanity.com/paper/2412.09022/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2024.

## Built on by (most cited first, 2 of 2)

- [Graph Neural Network Surrogates for Contacting Deformable Bodies with Necessary and Sufficient Contact Detection](https://mlsanity.com/paper/2507.13459/) (2025)
- [Deep learning-based prediction of time-resolved adhesive forces in viscoelastic Hertzian contacts](https://mlsanity.com/paper/2607.19060/) (2026)

## Abstract (arXiv)

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.