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

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.

References in corpus (13)

• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches
• Physics informed neural networks for continuum micromechanics
• Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)
• Data-driven fracture mechanics
• Hybrid FEM-NN models: Combining artificial neural networks with the finite element method
• Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems
• Impact of Data Normalization on Deep Neural Network for Time Series Forecasting
• Uncovering turbulent plasma dynamics via deep learning from partial observations
• Finite Element Network Analysis: A Machine Learning based Computational Framework for the Simulation of Physical Systems
• What machine learning can do for computational solid mechanics
• Lipschitz regularization for softening material models: the Lip-field approach
• Lipschitz regularization for fracture: the Lip-field approach

Cited by in corpus (9)

• Deep Learning in Deterministic Computational Mechanics
• Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks
• Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance
• A machine learning-based viscoelastic-viscoplastic model for epoxy nanocomposites with moisture content
• I-FENN for thermoelasticity based on physics-informed temporal convolutional network (PI-TCN)
• I-FENN with Temporal Convolutional Networks: expediting the load-history analysis of non-local gradient damage propagation
• Variational PINNs with tree-based integration and boundary element data in the modeling of multi-phase architected materials
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems