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

• Published in: Advanced modeling and simulation in engineering sciences 2024-05, Vol.11 (1), p.11-30, Article 11
• Main Authors: Sahin, Tarik, von Danwitz, Max, Popp, Alexander
• Format: Article
• Language: English
• Subjects: Advances in Machine Learning and Computational Mechanics; Boundary conditions; Classical and Continuum Physics; Computational Science and Engineering; Contact mechanics; Dirichlet problem; Elastoplasticity; Enforcing inequalities; Engineering; Engineering Sciences; Fischer–Burmeister NCP-function; Inverse problems; Mechanics (physics); Mixed-variable formulation; Neural networks; Parameter identification; Partial differential equations; Physics-informed neural networks; Research Article; Solvers; Theoretical and Applied Mechanics
• ISSN: 2213-7467; 2213-7467
• DOI: 10.1186/s40323-024-00265-3

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.