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

TL;DR

This work presents a physics-preserving ICNN for constitutive modeling of isotropic fibrous materials, enforcing polyconvexity and frame-indifference to enable stable FE simulations. By combining a multiscale FE framework with a Sobolev-trained ICNN surrogate, the approach achieves accurate predictions of energy, stress, and stiffness while dramatically reducing computational cost compared to full microscale MuMFiM simulations. The method is demonstrated on a facet capsular ligament in a 3D FE setting, showing close agreement with reference multiscale results and robustness across large deformations. Key contributions include the architecture that guarantees constitutive constraints, the Sobolev training protocol that enhances derivative predictions, and a public dataset and code to promote broader adoption in biomechanical multiscale modeling.

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.

Figures (13)

• Figure 1: (a) Schematic of a typical workflow of MuMFiM multiscale procedure that involves two finite element solution modules that handle the macro and microscale part of the problem, respectively. The deformation gradient, as informed by the macroscale module, is used to simulate RVEs to estimate the material response. For fibrous materials, the RVE is a discrete fiber network model (e.g., 3D Delaunay network shown in (b)). The machine learning model presented in this study is intended to be an accurate model of the RVE and aims to replace the 'microscale' module in panel (a).
• Figure 2: Schematic of the input convex neural network we implement. The input vector consists of invariants of the right Cauchy-Green deformation tensor ($\vb{C}$), and the output is the predicted strain energy density ($\mathcal{W}$). The components of the PK2 stress ($\vb{\Pi}$) and tangent stiffness tensor ($\mathcal{\mathbb{C}}=\pdv{\vb{\Pi}}{\vb{E}}$) are computed by back-propagation through the ICNN neural network. Cauchy stress $\sigma$ and the updated-Lagrangian material stiffness are needed in the finite element implementation and are computed through the appropriate push forward operations.
• Figure 3: Mean squared error (loss) of ICNN during training. The average errors associated with strain energy density, stress, and stiffness tensors are shown in the first, second, and third columns, respectively. The top row shows the loss for the training dataset, and the bottom row represents the corresponding results for the test dataset. The legend $\mathcal{L}^{k} := \sum_{i=0}^{i=k} H_i$ indicates the variant of the loss function used to train ICNN, with $k$ being the highest order of derivative included in the composite loss function. The input array consists of all isotropic invariants.
• Figure 4: True (MuMFiM) vs ICNN-predicted strain energy density for various deformation gradients ($F$) in test dataset using ICNN trained with $\mathcal{L}^{2}$ loss function. The dashed red line indicates the ideal fit, and scatters indicate predictions.
• Figure 5: True (MuMFiM) vs ICNN-predicted values for unique components of Cauchy stress tensor for various deformation gradients ($F$) in the test dataset. The variant of the loss function used and invariants of $\vb{C}$ included in the input array are indicated in sub-figure titles.