[Paper Review] Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

Panos Pantidis, Mostafa E. Mobasher|arXiv (Cornell University)|Jul 14, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper introduces the Integrated Finite Element Neural Network (I-FENN), a physics-informed framework that embeds a trained neural network within the finite element method to enable efficient, mesh-independent non-local continuum damage analysis. By using a physics-informed neural network (PINN) to compute non-local equivalent strain and its derivatives, I-FENN achieves the accuracy of non-local models at the computational cost of local approaches, eliminating mesh dependency without added computational overhead.

ABSTRACT

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost.

Motivation & Objective

• Address the mesh-dependence issue in local damage models while avoiding the high computational cost of traditional non-local methods.
• Develop a framework that integrates neural networks into the finite element method to accelerate nonlinear computational mechanics simulations.
• Enable accurate, diffused damage profiles in quasi-brittle materials using a non-local gradient-based damage model with minimal computational overhead.
• Ensure physical consistency and generalization by training a physics-informed neural network (PINN) on non-local strain relationships.
• Provide a reproducible, open-source solution by releasing training code and data for the PINN component.

Proposed method

• Train a physics-informed neural network (PINN) offline to predict non-local equivalent strain and its derivative with respect to local strain, based on a non-local gradient model.
• Embed the trained PINN into the finite element element stiffness matrix to perform local-to-non-local strain transformation during Newton-Raphson iterations.
• Use the PINN's output to compute the element residual vector and Jacobian matrix directly within the nonlinear solver, replacing conventional non-local integration schemes.
• Construct the global system of equations using the modified element-level Jacobian and residual, maintaining the structure of standard FEM while embedding neural network predictions.
• Integrate the PINN predictions into the weak form of the variational principle, ensuring consistency with the underlying mechanics and energy-based formulations.
• Leverage shape functions and isoparametric elements to map nodal displacements to strain fields, with the PINN handling the non-local transformation at integration points.

Experimental results

Research questions

• RQ1 Can a neural network be embedded within the finite element method to achieve the accuracy of non-local damage models while preserving the computational efficiency of local models?
• RQ2 How can a physics-informed neural network be trained to accurately approximate non-local equivalent strain and its derivatives in a way that ensures mesh independence?
• RQ3 To what extent does the I-FENN framework maintain convergence and stability in nonlinear iterative solvers compared to conventional non-local FEM?
• RQ4 Can the proposed method generalize across different mesh densities and loading conditions without retraining?
• RQ5 Does the integration of a PINN into the finite element stiffness matrix preserve the physical consistency of the solution while reducing computational cost?

Key findings

• The I-FENN framework achieves mesh-independent results in quasi-brittle damage simulations, eliminating the spurious mesh sensitivity common in local damage models.
• The computational cost of I-FENN matches that of the conventional local damage approach, despite incorporating non-local effects.
• The PINN successfully learns the non-local strain transformation, producing diffused damage profiles consistent with gradient-enhanced models.
• The method maintains numerical convergence in Newton-Raphson iterations, with convergence tolerances set in the range of $10^{-4}$ to $10^{-6}$.
• The framework demonstrates strong generalization capability across different mesh densities and loading conditions, as validated through multiple numerical examples.
• The authors released the PINN training code and associated data, enabling full reproducibility and future extension of the method.