Consistent Tangent Stiffness of a Three-Dimensional Wheel–Rail Interaction Element

Abstract

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.

An Effective Tangent Stiffness of Train–Track–Bridge Systems Based on Artificial Neural Network

“…where the subscript "i" can be w or r, denoting the force of wheel and rail, respectively. The derivation and implementation of Equation ( 5) require significant efforts, and the detailed derivation formula can be found in [13].…”

“…For the WRI element mentioned above, the computation of the CTS requires the differentiation of the internal forces with respect to the nodal displacement, following the complicated calculation process of the internal forces. Although the CTS is accurate and efficient and is able to guarantee the quadratic rate of convergence of Newton's method [13], its derivation and software implementation require significant efforts. Therefore, it is necessary to find an alternative method for computing the tangent stiffness that can reduce the efforts of derivation and software implementation while maintaining high computational efficiency.…”

“…The element tangent stiffness is obtained straightforwardly based on the output tangent stiffness. The trained BPTS model is implemented in a general finite element software, OpenSees [18], to substitute the calculation of the consistent tangent stiffness (CTS) [13] 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 the BPTS and the CTS.…”

“…The norm of the unbalance force in a few representative time steps (wheelset). 10 3 3.04 ´ 10 3 6.83 ´ 10 1 8.63 ´ 10 −1 2.38 ´ 10 −4 BPTS 8.37 ´ 10 3 6.58 ´ 10 3 4.65 ´ 10 2 0.36 ´ 10 1 9.07 ´ 10 −2 10 4 1 13. ´ 10 4 5.37 ´ 10 1 2.65 ´ 10 −1 7.54 ´ 10 −5 BPTS 1.25 ´ 10 4 8.45 ´ 10 3 8.51 ´ 10 2 1.88 ´ 10 1 0.83 ´ 10 −1 ´ 10 5 8.79 ´ 10 4 2.06 ´ 10 2 0.39 ´ 10 1 1.67 ´ 10 −3BPTS 5.45 ´ 10 5 8.64 ´ 10 3 9.64 ´ 10 2 2.60 ´ 10 1 1.26 ´ 10 −1 4.31 ´ 10 −2…”

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

“…Liu et al (2021) developed a three-dimensional (3D) WRI element based on the same idea wrapping the WRI into a 3D element, the WRI force of which is obtained by using Kalker's linear theory, nonlinear Hertz theory, and Polach formulation. The consistent tangent stiffness (Gu Q et al, 2021) of the 3D WRI element is derived, the accuracy and the efficiency of the algorithm are compared with the perturbation method (Liu et al, 2021) and the quadratic convergence rate is proved.…”