Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum

TL;DR

This paper introduces a neural network-based method for solving parametrized elasticity problems that explicitly enforces conservation of linear and angular momentum, improving computational efficiency and accuracy.

Contribution

It presents a novel decomposition strategy combined with neural networks to efficiently solve saddle-point elasticity systems while maintaining physical conservation laws.

Findings

Neural network correction improves solution accuracy.

Decomposition reduces computational complexity.

Method successfully applied to nonlinear elasticity test cases.

Abstract

We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce conservation of linear and angular momentum. Due to the saddle-point structure, the resulting system is computationally demanding to solve directly, and we therefore propose an efficient solution strategy based on a decomposition of the stress variable. First, a triangular system is solved to obtain a stress field that balances the body and boundary forces. Second, a trained neural network is employed to provide a correction without affecting the conservation equations. The displacement and rotation can be obtained by post-processing, if necessary. The potential of the approach is highlighted by three numerical test cases, including a non-linear model.