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

T. Sahin, M. von Danwitz, A. Popp

TL;DR

It is shown 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, and the importance of choosing proper hyperparameters is demonstrated.

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.

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

TL;DR

It is shown 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, and the importance of choosing proper hyperparameters is demonstrated.

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.
Paper Structure (26 sections, 38 equations, 19 figures, 9 tables)

This paper contains 26 sections, 38 equations, 19 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Contact mechanics
  4. Mixed-variable formulation: the Hellinger–Reissner principle
  5. Physics-Informed Neural Networks for Solid and Contact Mechanics
  6. Generic PINNs with output transformation
  7. Application of PINNs to solid and contact mechanics
  8. Enforcing the Karush-Kuhn-Tucker inequality constraints
  9. Sign-based method
  10. Sigmoid-based method
  11. A nonlinear complementarity problem function: Fischer-Burmeister
  12. Algorithmic challenges
  13. Domination of $p_n$ over $g_n$
  14. Importance of output scaling
  15. Numerical examples
  16. ...and 11 more sections

Figures (19)

  • Figure 1: Contact problem between an elastic body and a rigid obstacle. (a) Reference configuration, (b) current configuration, (c) accompanying boundary conditions, illustration of the gap $g_n$, tangential traction $t_{\boldsymbol{\tau}}$ and contact pressure $p_n$.
  • Figure 2: Tonti's diagram of Hellinger–Reissner (HR) principle for contact problems with small strain theory and frictionless sliding condition.
  • Figure 3: The general representation of a physics-informed neural network for a BVP.
  • Figure 4: Physics-informed neural networks in the mixed-variable form to solve quasi-static solid and contact mechanics problems without additional network parameters.
  • Figure 5: An illustration of the sign-based function depending on gap $g_n$ and contact pressure $p_n$
  • ...and 14 more figures