On the use of graph neural networks and shape-function-based gradient computation in the deep energy method

Abstract

A graph neural network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in 3D for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. Its accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN-based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD-based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF-based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN-based DEM model and SF-based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities.

References in corpus (15)

• An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications
• Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
• Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
• Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks
• Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
• Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity
• Graph neural networks for laminar flow prediction around random 2D shapes
• Learning to fail: Predicting fracture evolution in brittle material models using recurrent graph convolutional neural networks
• FEA-Net: A Physics-guided Data-driven Model for Efficient Mechanical Response Prediction
• Graph Convolutional Neural Networks for Body Force Prediction
• Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D
• Automatic differentiation in ML: Where we are and where we should be going
• Flow Completion Network: Inferring the Fluid Dynamics from Incomplete Flow Information using Graph Neural Networks

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