Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity

TL;DR

This work addresses forward and inverse modeling of finite-strain hyperelasticity using a differentiable meshfree solver (NIM). It integrates hyperelastic constitutive laws into a local variational loss via NeuroPU, a hybrid neural–PU approximation, and enforces essential boundaries with a boundary singular kernel, all implemented in JAX for GPU acceleration. Forward tests yield relative $L_2$ errors in displacement on the order of $10^{-3}$ to $10^{-5}$, competitive with finite-element methods, while inverse experiments recover heterogeneous elastic moduli from strain data with errors at a few percent. The results demonstrate a meshfree, differentiable, data–physics–driven framework for nonlinear elasticity that generalizes to heterogeneous materials and large deformations.

Abstract

The present study aims to extend the novel physics-informed machine learning approach, specifically the neural-integrated meshfree (NIM) method, to model finite-strain problems characterized by nonlinear elasticity and large deformations. To this end, the hyperelastic material models are integrated into the loss function of the NIM method by employing a consistent local variational formulation. Thanks to the inherent differentiable programming capabilities, NIM can circumvent the need for derivation of Newton-Raphson linearization of the variational form and the resulting tangent stiffness matrix, typically required in traditional numerical methods. Additionally, NIM utilizes a hybrid neural-numerical approximation encoded with partition-of-unity basis functions, coined NeuroPU, to effectively represent the displacement and streamline the training process. NeuroPU can also be used for approximating the unknown material fields, enabling NIM a unified framework for both forward and inverse modeling. For the imposition of displacement boundary conditions, this study introduces a new approach based on singular kernel functions into the NeuroPU approximation, leveraging its unique feature that allows for customized basis functions. Numerical experiments demonstrate the NIM method's capability in forward hyperelasticity modeling, achieving desirable accuracy, with errors among $10^{-3} \sim 10^{-5}$ in the relative $L_2$ norm, comparable to the well-established finite element solvers. Furthermore, NIM is applied to address the complex task of identifying heterogeneous mechanical properties of hyperelastic materials from strain data, validating its effectiveness in the inverse modeling of nonlinear materials. To leverage GPU acceleration, NIM is fully implemented on the JAX deep learning framework in this study, utilizing the accelerator-oriented array computation capabilities offered by JAX.

Figures (22)

• Figure 1: Schematic of neuro-partition of unity (NeuroPU) approximation for the solution $u(\bm x, \bm \mu)$, which is constructed based on the inner product of two function blocks: a neural network block and a block of symbolic PU functions. The neural network block takes the system parameters $\bm \mu$ as inputs, and outputs a set of nodal coefficient functions $\{\hat{d}_I\}_{I=1}^{N_h}$. The symbolic PU block produces a set of PU shape functions $\{\Psi_I\}_{I=1}^{N_h}$ that possess the PU condition and local compactness.
• Figure 2: The workflow of the neural-integrated meshfree (NIM) nonlinear modeling framework for hyperelastic materials, where the strain energy density $\hat{W}^h$ is modeled as a function of the displacement $\hat{\bm u}^h$ and its derivatives, approximated using the NeuroPU approach. The PK stress $\hat{\bm P}^h = \frac{\partial \hat{W}^h}{\partial \hat{\bm F}^h}$ is computed through automatic differentiation provided by JAX. Within the meshfree discretization, the red box indicates one of the subdomains $\Omega_s$, which is associated with a local variational residual term $\bm {\mathcal{R}}_s$ (Eq. \ref{['eq:local_inte']}). The summation of $\bm{\mathcal{R}}_s$ over the subdomains $\{\Omega_s\}_{s=1}^{N_{\mathcal{T}}}$ constitutes the total loss function $\mathcal{L}(\boldsymbol{\theta})$.
• Figure 3: Illustration of the 1D bar problem subjected to an uniaxial tension.
• Figure 4: Comparison of the displacement solutions (Left) and the point-wise absolute errors (Right) of NIM/c and NIM/h for the 1D bar problem.
• Figure 5: Comparison of the displacement gradient solutions (Left) and the point-wise absolute errors (Right) of NIM/c and NIM/h for the 1D bar problem.