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

TL;DR

This paper explores the integration of graph neural networks into the deep energy method for solving 3D deformation problems, comparing gradient computation techniques and demonstrating the robustness of shape function-based gradients.

Contribution

It introduces a GCN-based DEM model and evaluates shape function gradients, showing improved robustness and efficiency over traditional MLP-based DEM.

Findings

GCN-based DEM achieves similar accuracy with shorter run times.

SF-based gradient computation is more robust than AD-based methods.

The combined GCN and SF approach effectively handles severe nonlinearities.

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.