A Physics Preserving Neural Network Based Approach for Constitutive Modeling of Isotropic Fibrous Materials

TL;DR

This paper introduces a physics-preserving neural network for modeling isotropic fibrous materials, achieving high accuracy and computational efficiency by enforcing constitutive constraints and using Sobolev minimization.

Contribution

The authors develop a neural network architecture that enforces physical constraints like polyconvexity and frame-indifference, improving accuracy in constitutive modeling of fibrous materials.

Findings

Achieved 0.15% error in strain energy density prediction.

Maintained over 85% stress accuracy up to 70% strain.

Reduced computational cost significantly in finite element simulations.

Abstract

We develop a new neural network architecture that strictly enforces constitutive constraints such as polyconvexity, frame-indifference, and the symmetry of the stress and material stiffness. Additionally, we show that the accuracy of the stress and material stiffness predictions is significantly improved for this neural network by using a Sobolev minimization strategy that includes derivative terms. Using our neural network, we model the constitutive behavior of fibrous-type discrete network material. With Sobolev minimization, we obtain a normalized mean square error of 0.15% for the strain energy density, 0.815% averaged across the components of the stress, and 5.4% averaged across the components of the stiffness tensor. This machine-learned constitutive model was deployed in a finite element simulation of a facet capsular ligament. The displacement fields and stress-strain curves were compared to a multiscale simulation that required running on a GPU-based supercomputer. The new approach maintained upward of 85% accuracy in stress up to 70% strain while reducing the computation cost by orders of magnitude.