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.

References in corpus (13)

• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications
• Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• Physics-informed neural networks for inverse problems in nano-optics and metamaterials
• Physics-informed deep learning for incompressible laminar flows
• Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networks
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics
• Physics-informed neural networks for solving parametric magnetostatic problems
• Optimization with Equality and Inequality Constraints Using Parameter Continuation

Cited by in corpus (4)

• k-space Physics-informed Neural Network (k-PINN) for Compressed Spectral Mapping and Efficient Inversion of Vibrations in Thin Composite Laminates
• Towards a Hybrid Digital Twin: Physics-Informed Neural Networks as Surrogate Model of a Reinforced Concrete Beam
• Graph Neural Network Surrogates for Contacting Deformable Bodies with Necessary and Sufficient Contact Detection
• Physics-Informed Neural Networks for Solving Contact Problems in Three Dimensions