A novel machine learning based tangent stiffness calculation method for 3D wheel-rail interaction element

A machine learning (ML) based method is presented in this paper for obtaining tangent stiffness of a complicated three-dimensional wheel-rail interaction element that is able to practically and effectively simulate the complicated dynamic responses of vehicle-track problems. The element tangent stiffness, defined as differentiation of element insisting force to nodal displacement, is important in improving efficiency when Newton's method is used to solve implicit nonlinear finite element equations. However, deriving and software implementing the tangent stiffness require significant efforts, and calculating the tangent stiffness in each iteration of the Newton method is usually time consuming. On the other hand, ML can directly obtain the implicit mapping between inputs and outputs of complex calculation process with limited programming effort and high computational efficiency, and is potentially a good alternative way to calculate the tangent stiffness of complicated element. In this paper, a feedforward artificial neural network is trained for obtaining the tangent stiffness, while inputs are the displacement and velocity of the element and outputs are the entries of the tangent stiffness matrix. The ML based tangent stiffness are implemented in an open source finite element software framework, OpenSees, and verified by application examples of a wheelset and a light rail vehicle running on straight rigid rail. The accuracy and efficiency are compared between the use of ML based tangent stiffness (MLTS) and the consistent tangent stiffness obtained at different speeds. The results demonstrate the MLTS can ensure the calculation accuracy and significantly improve the calculation efficiency.