Consistent tangent stiffness plays a crucial role in delivering a quadratic rate of convergence when using Newton’s method in solving nonlinear equations of motion. In this paper, consistent tangent stiffness is derived for a three-dimensional (3D) wheel–rail interaction element (WRI element for short) originally developed by the authors and co-workers. The algorithm has been implemented in finite element (FE) software framework (OpenSees in this paper) and proven to be effective. Application examples of wheelset and light rail vehicle are provided to validate the consistent tangent stiffness. The quadratic convergence rate is verified. The speeds of calculation are compared between the use of consistent tangent stiffness and the tangent by perturbation method. The results demonstrate the improved computational efficiency of WRI element when consistent tangent stiffness is used.

Dynamic response analysis of a train–track–bridge (TTB) system is a challenging task for researchers and engineers, partially due to the complicated nature of the wheel–rail interaction (WRI). When Newton’s method is used to solve implicit nonlinear finite element equations of a TTB system, consistent tangent stiffness (CTS) is essential to guarantee the quadratic convergence rate. However, the derivation and software implementation of CTS for the WRI element require significant efforts. Artificial neural network (ANN) can directly obtain a potentially good tangent stiffness by a trained relationship between input nodal displacement/velocity and output tangent stiffness. In this paper, the backpropagation neural-network-based tangent stiffness (BPTS) of the WRI element is derived and implemented into a general finite element software, OpenSees, and verified by dynamic response analysis of a high-speed train running on a seven span simply supported beam bridge. The accuracy and efficiency are compared between the use of BPTS and CTS. The results demonstrate that BPTS can not only save the significant efforts of deriving and software implementing CTS but also improve computational efficiency while ensuring good accuracy.

State-based peridynamics (SPD) is effective for simulating fracture and damage in different materials. However, the solutions may suffer from numerical instabilities, particularly for strong nonlinearity cases or large displacement cases, leading to inaccurate predictions or oscillation in responses. This paper proposed a novel practical method to solve the instability problem of SPD based on stress equilibrium equation, referred to herein as the stress correction method (SCM). A correction force is applied on each SPD point surrounding the loading points. The correction force is defined as the difference between an internal force obtained by the stress equilibrium equation and that obtained by the force states of the SPD. Four examples are presented herein to verify the accuracy and stability of the proposed method in various conditions, e.g. static and dynamic analyses of elastic and plastic models subjected to force and displacement boundary conditions.

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.