Supervised Learning for Finite Element Analysis of Holes Under Tensile Load

Abstract

As the use of machine learning becomes more common, there are many algorithms that are readily available to perform supervised learning. This paper is to evaluate the feasibility of supervised learning in simplifying the finite element analysis of holes under tensile load. The objective of this approach is to determine if the mesh size can be significantly reduced with supervised learning. The neural network training is performed with just a small set of 55 course mesh with 2-D linear element against the analytical solution of a hole under tensile load in an infinite width plate. The coarse mesh only has 2 elements along the quarter hole perimeter. The training would be done using the displacement nodal solution of the nearest 6 nodes to the hole’s edge. Three common back propagation network algorithms are evaluated; Conjugate Gradient, Bayesian and Levenberg-Marquart methods. These algorithms are used along with the tangent sigmoid and pure linear transfer functions. In the infinite width problem, the Bayesian algorithm with the tangent sigmoid function offers the highest accuracy in the testing of the network model. However, this model performs poorly when it is applied to the finite width problem. To reduce the prediction error, the training would be done solely based on the displacement \\( u_{y} \\) component. The displacement \\( u_{x} \\) component is removed from the network training since the displacement field in the x direction is quite different between the infinite hole solution and finite hole solution. Further testing shows the prediction can be improved by using the Levenberg-Marquart method with pure linear function. With these options, the prediction error is just 3% even though the mesh size is relatively coarse with only 2 elements along the perimeter. In contrast, the normal finite element method with this coarse mesh has an error of 26%. To achieve similar accuracy, the standard FEM would require 3 times the number elements along the perimeter to achieve similar accuracy. This initial result shows there is synergy between machine learning and finite element method in reducing the mesh size requirement and yet achieve good accuracy.