Geometric Non-linear Beam Analysis Using Physics-informed Neural Networks

Abstract

The finite element method has proven itself to be an effective method of performing both linear and non-linear structural analyses. This numerical method, however, has several limitations, of one which is that the discretization of complex geometry requires human effort. The rise of the application of machine learning models has opened new possibilities of approaching challenges within scientific fields. The neural network, and more specifically, the physics informed neural network, is a promising method and allows for performing analyses without any discretization that is necessary. This type of neural network, which is less of a black box than the standard neural network, utilizes the concept of partial differential equations to accurately predict solutions.

The objective of this thesis is to perform a structural analysis of a geometrically non-linear Timoshenko beam using a physics informed neural network. The network is built using a variational principle (the principle of virtual work) and a force residual. Furthermore, two optimization algorithms, the adaptive weight loss algorithm and the adaptive activation function are separately used in conjunction with the model to examine the potential improvements on the convergence rate.

It is found that the geometrically non-linear Timoshenko beam can be accurately modeled (relative error of below 2% with respect to the finite element output) with a physics informed neural network. This accuracy can be achieved with a model possessing a relatively shallow size of four hidden layers containing eight neurons each. The adaptive weight loss algorithm and the adaptive activation algorithm both improve the convergence rate of the model, though they are not necessary to maintain the practicality of the model, as the convergence rate is adequate without these. It is recommended that the hyperbolic tangent function is utilized in conjunction with the Adam optimizer. The adaptive activation function can be incorporated into the model to improve the convergence rate significantly without substantially increasing the computational cost of the model.